ASSESSMENT OF THE THERMOELASTIC RESPONSE OF A DIFFUSION
BOND USING TELEVISION LASER HOLOGRAPHIC INTERFEROMETRY
by
John A. Hargreaves
B. Mus., University of Colorado, 1973
B. S. M. E., University of Colorado, 1988
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Mechanical Engineering
1997
This thesis for the Master of Science
degree by
John A. Hargreaves
has been approved
by
Date
James C. Gerdeen
Hargreaves, John A. (M. S., Mechanical Engineering)
Assessment of the Thermoelastic Response of a Diffusion Bond Using Television
Laser Holographic Interferometry
Thesis directed by Professor William H. Clohessy
ABSTRACT
Previously, no nondestructive technique has existed allowing conclusive
characterization of the strength and structure of a diffusion bond between different
metals. Television Laser Holographic Interferometry is used to assess the transient
thermoelastic response of wellbonded and poorlybonded metal composites. A
coupled static finite element model is used to predict this transient thermoelastic
response allowing comparison with measured data. An attempt is made to detect a
void at the interface of one bonded composite when compared to another composite
without a void. This attempt is unsuccessful though the transient thermoelastic
response of both composites is successfully modeled when compared with measured
deformations on the submicron level.
This abstract accurately represents the contents or the candidates thesis. I
recommend its publication.
Signed
ACKNOWLEDGMENT
The author would like to thank many people who contributed much time, advice, and
effort assisting in the completion of this thesis: my advisor, Dr. William H. Clohessy
who gave many helpful suggestions; Dr. Keith Axler for his assistance with materials
science questions, helpful insights, and for his critical reading of the text; Dr. P.
Shivakumar of the MARC corporation for his assistance with the FEA model; Dr.
John A. Jack Hanlon for giving so unselfishly of his time and talents and, by his
example, teaching me to think as an experimentalist; my wife, Elizabeth, whose help
and encouragement through difficult times never failed.
CONTENTS
Chapter
1. Introduction ................................................................... 1
1.1 Problem Statement ......................................................... 1
1.2 Structure of Thesis ....................................................... 2
1.3 The Basis of Holograph Interferometry ..................................... 3
1.4 The Design and Operation of TVLaser Holography............................. 3
1.5 The Diffusion Bond ........................................................ 5
1.51 Introduction ............................................................. 5
1.52 Theoretical Basis of Diffusion Bonding ................................... 5
1.53 Diffusion Bonding Process Variables .......................................... 7
1.54 Metallurgical Factors ........................................................ 8
1.6 Design and Manufacture of CuAl, CuMgOAl, and CuSb Bonded
Bonded Composites ..................................................... 9
1.7 Properties of Experimental Materials ......................................... 11
1.71 Chemical, Thermal, and Physical Properties of Copper ....................... 12
1.72 Chemical, Thermal, and Physical Properties of Aluminum ..................... 14
1.73 Chemical, Thermal, and Physical Properties of Antimony ..................... 16
1.74 Properties of Macor Ceramic and RTV Silicon Potting ......................... 19
2. The Linear, Coupled, QuasiStatic Theory of Thermoelasticity ................ 21
2.1 Introduction ................................................................. 21
2.2 The Basic Laws of Mechanics and Thermodynamics ............................ 22
2.3 Elastic Materials and the Consequences of the Second Law .................. 24
2.4 Material FrameIndifference................................................... 27
2.5 Consequences of the Heat Conduction Inequality ............................ 30
2.6 Derivation of the Linear Thermoelastic Theory ............................. 30
2.7 OneDimensional Linear Thermoelasticity ...................................... 37
3. Analytical Modeling of Heat Flow through Test Cylinders .................... 42
3.1 Introduction ................................................................. 42
3.2 Formula for T(x,t) ........................................................... 44
3.3 Modeling T(x,t) in a SemiInfinite Slab ................................... 46
3.4 Formula for T(r,z,t) in a Finite Cylinder ................................. 50
3.5 Modeling T(r,z,t) in a Finite Cylinder ....................................... 54
3.6 Characterization of Temperature Difference (AT) on the Vertical Face ...... 56
3.7 Comparison of Calculated and Measured T(x,t) for t < 120 Seconds .......... 59
4. Analytical Method for Characterizing Heat Conduction across a Bonded Interface ... 60
4.1 Introduction and Statement of problem .................................... 60
4.2 Derivation of the Governing Equations .................................... 62
4.3 Conversion of the Governing Equations to Dimensionless Form .............. 64
4.4 Thermoelastic Coupling ................................................... 65
4.5 Initial and Boundary Conditions .......................................... 67
4.6 Transform of the Governing Equations into Laplace Space .................. 68
4.7 Formulation of a Laplace Transform Definition of ................... 70
4.8 Inversion of (*,/?) ...................................................... 74
4.9 Calculation of the SteadyState Form of 6 (jc, t) ........................ 77
5. Development of Finite element Models for a Bonded Metal Composite ...... 85
5.1 Introduction ............................................................. 85
5.2 FEA Modeling of the Copper and Aluminum Test Cylinders ................... 85
5.3 Modeling Heat Transfer in the Experimental Configuration .............. 87
5.4 Coupled Thermoelastic Analysis of the CuAl Bonded Composite ............. 90
5.5 Coupled Thermoelastic Model of the AlMgOCu Bonded Composite ............ 94
6. Assessment by TVH Interferometry of Diffusion Bonded Composites ............ 98
6.1 Introduction ............................................................. 98
6.2 Preliminary Tests ........................................................ 99
6.3 Results of Experiment No. 1 ............................................... 101
6.4 Results of Experiment No. 2 ............................................... 105
6.41 Comparison of Modeling and Experimental Data for Cylinder A ............... 105
6.42 Comparison of Modeling and Experimental Data for Cylinder B ............... 109
6.5 Conclusions and Areas for Future Research ................................ 115
Appendix
A. Derivation of Formula for T(r,z,t) ........................................... 116
Bibliography .................................................................... 128
FIGURES
Figures
1.1 Schematic of Los Alamos National Laboratory TVH ........................ 5
1.2 ALMgOCu Assembly in PreBonded Configuration ......................... 9
1.3 CuSb Assembly in PreBonded Configuration ............................... 10
1.4 Temperature and Pressure used for Bonding Specimens ...................... 10
1.5 Phase Diagram for the CopperAntimony System ............................. 11
1.6 Photograph of Metallurgical Structure of Copper, 25X ..................... 13
1.7 Thermal Conductivity Data for Copper ..................................... 14
1.8 Photograph of Metallurgical Structure of Aluminum, 100X .................. 15
1.9 Thermal Conductivity Data of Aluminum .................................... 16
1.10 Thermal Conductivity Data for Antimony ................................... 18
1.11 Photograph of Metallurgical Structure of Antimony, 100X .................. 18
3.1 Schematic of SemiInfinite Experiment .................................... 43
3.2 Configuration of Potted AlCu Composite Plate ............................ 44
3.3 Listing of Mathematica Program JCUM7.DAT ................................. 47
3.4 T(x,t) for Measurement of Copper cylinder, L = .00635 m .................. 47
3.5 Listing of Mathematica Program CPPR1 ..................................... 48
3.6 Comparison of T(x,t) with varying q; hi = h2 = 12 W/m2 C................. 49
3.7 Comparison of T(x,t) with Measured Data for Aluminum ..................... 50
3.8 Boundary Conditions and Spatial Coordinates for Finite Cylinder .......... 51
3.9 Mathematica Program CPPR ................................................. 53
3.10 Comparison of Calculations of T(r,z,t) with Measured Data ................ 54
3.11 T(r,z,t) as a Function of Different Vertical Convection Rates ............ 55
3.12 T(r,z,t) as a Function of Different Convection Coefficients .............. 56
3.13 Experimental Configuration for Determination of AT on z = .0165 m Face ... 57
3.14 Comparison of Calculated and measured T(r,z,t) for Antimony Cylinder ..... 58
3.15 Calculated T(r,z,t) for Sb cylinder at t = 2290 s......................... 59
3.16 Comparison of Calculated and Measured T(x,t) for 0 < t Â£ 120 Seconds ..... 60
4.1 nLayered Bonded Plate ................................................... 61
4.2 Bromwich Contour for Inversion of Oj' (x, p) ............................. 75
4.3 Configuration for First Solution for 0(x,t) .............................. 78
4.4 Listing of Program ARARP1 ................................................ 80
4.5 6 U,) for a 1Material Sb Composite Plate............................... 81
4.6 Convergence of Solution for 9 (*,~) as the Laplace Variable Approaches 0 ... 81
4.7 Configuration and Boundary Conditions for
Calculation of 9j (*, CuSb Plate ................................ 82
4.8 9j for SbCu Composite Plate; mi = 100 W/m2 C ......................... 83
4.9 Thermal Interference at Interface between WellBonded CuSb Plate .. 84
4.10 Plate Thermal interface between CuSb Plate with MgO Barrier .......84
5.1 Axisymmetric Model of Temperature in Cu Test Cylinder ................ 86
5.2 Axisymmetric Model of Temperature in A1 Test Cylinder ................ 87
5.3 Schematic of 4Thermocouple Experiment ............................... 88
5.4 Temperature Data for 4Thermocouple Experiment ....................... 89
5.5 FEA Temperature Curves for Thermocouples 2 and 4 ..................... 89
5.6 FEA Temperatures for 4Thermocouple Experiment, t = 1200 s............ 90
5.7 Mesh for Coupled Analysis of Axisymmetric AlCu Plate ................ 91
5.8 Boundary Condition for AlCu Bonded Plate ............................ 92
5.9 Deformed Shape of AlCu Bonded Composite, t = 300 s .................. 92
5.10 Magnitude of xDisplacement in AlCu Bonded Composite, t = 100 s.... 93
5.11 AlSSTCu Development Model .......................................... 94
5.12 Mesh, xDisplacement Field, and Deformation for AlSSTCu Model, t = 300s 95
5.13 Successful Axisymmetric Model for Composite with Void ................ 96
5.14 Temperature Contours for AlMgOCu Model, t = 300 s .................. 97
6.1 A and B Bonded Composites as used for Experiment No. 2 ................ 98
6.2 SteadyState Deformation of Homogenous Cu Test Cylinder ............... 99
6.3 OffCenter Fringe Pattern, AlCu Bonded Composite....................... 100
6.4 Fringe Pattern Centered by TVH Compensation ............................ 101
6.5 Temperature Data for Experiment No. 1 .................................. 102
6.6 Thermoelastic Response of Cylinder A; t = 155 s, qC0Upied = 2100 W/m2 .. 103
6.7 Thermoelastic Response of Cylinder A; t = 300s, qcoupied = 2100 W/m2 .. 103
6.8 Thermoelastic Response of Cylinder B; t=150s, qcoupIed = 2100 W/m2 .. 104
6.9 Thermoelastic Response of Cylinder B; t = 300s, qcoupled = 2100 W/m2 .. 104
6.10 T(r=0, z = .00635,t) Calculated for Cylinder A, 0 51 > 300 s, qCOUpied = 1200 s 106
6.11 T(r=0, z = .00635,t) Measured for Cylinder A: qcoupied = 1200 W/m2 ..... 106
6.12 xDisplacement as a Function of Time for Cylinder A; 0 < t ^ 300 s ..... 107
6.13 Measured Au for Cylinder A; t= 155 s, qC0Upii = 1200 W/m2.............. 108
6.14 Measured Au for Cylinder A; t= 300 s, qcoupied = 1200 W/m2.............. 108
6.15 xDisplacement for cylinder A; t = 300s, qC0Upied = 1200 W/m2 .......... 109
6.16 Temperature Data for Cylinder B, Experiment 2;, qC0Upied = 1200 W/m2 ... 110
6.17 MARC Temperature Curve for Cylinder B, Experiment No. 2 ................ 110
6.18 xDisplacement as a Function of Time for Cylinder B; 0 < t > 300 s ..... Ill
6.19 Measured Au for Cylinder B, Experiment No. 2; t = 300 s ................ 112
6.20 xDisplacement for Cylinder A; t = 300s, qC0Upied = 1200 W/m2 .......... 113
6.21 Calculation of Au, Cylinder B, Experiment No. 2; 0 < t> 167 s .......... 114
6.22 Interferometric Measurement of Au, Experiment No. 2, Cylinder B; t = 167 s 114
Al.l
A1.2
A1.3
Cylinder with Initial and Boundary Conditions ..
Positive Roots of pm J0 (Pm b) + H3 J0 (Pm b) = 0
Positive Roots for tan
% {Hi + H2)
n2P (H1 + H2)
108
115
116
TABLES
Table
1.1 Trace Elements in Copper ................................ 12
1.2 Thermal and Physical Properties of Copper ............... 13
1.3 Trace Elements in Aluminum .............................. 15
1.4 Thermal and Physical Properties of Aluminum ............. 15
1.5 Major Contaminants of Antimony .......................... 16
1.6 Trace Contaminants of Antimony .......................... 17
1.7 Thermal and Physical Properties of Antimony ............. 17
1.8 Properties of Macor Ceramic ............................. 19
1.9 Properties of RTV Potting ............................... 19
3.1 Comparison of Calculated and Measured Initial
Temperatures ................................. 54
4.1 Numerical Results for 0(x,oo) at Various Values for p ... 82
1. Introduction
1.1 Problem Statement
This work examines the use of television laser holographic interferometry (TVH) to assess
nondestructive^ the existence and nature of a diffusion bond between dissimilar metals. This work
attempts to examine the thermoelastic response of actual diffusion bonds to a known heat flux
passing through the parent metals and the bond between them. The thermoelastic response of two
bonded composites of identical metals and dimensions are compared using TVH, one composite
having an insulated void at the interface. This A/B comparison is intended as a proofofprinciple.
The diffusion bond represents a discontinuity which acts as a thermal resistance to any heat flux.
This thermal resistance is modeled analytically and numerically; the resulting response is modeled
using a finite elementbased coupled thermoelastic analysis. The predicted thermoelastic response
is then compared to the response observed by the TVH.
The different metals in a bonded composite have differing thermal conductivites, thermal
diffiisivities, and coefficients of expansion. It is possible these inherent differences provide the
basis for a specific and identifiable thermoelastic response in a bonded metallic composite. Such a
thermoelastic response might be characterized in terms of displacement, strain, or even strain rate.
Given the realtime capability provided by TVH to detect small changes on the submicron level,
could thermoelastic displacements, strains, or strain rates created by introducing a heat flux to the
bonded metal composite be detectable and quantifiable? If so, could these results be the basis for
the nondestructive evaluation of a metallic diffusion bond?
Xray radiography and ultrasonic testing are two techniques by which such a nondestructive
assessment can be made currently. These techniques, however, have limitations and usually cannot
provide conclusive information as to how well two metals are diffusion bonded or even if they are
bonded at all. Recent developments in the field of TV laser holographic interferometry offer a
possible alternative as a tool for the nondestructive characterization of diffusion bonded metals.
Holographic interferometry has become a wellestablished technique used for the measurement of
physical quantities such as displacements, strains, and densities. Initially used for the inspection of
aircraft components, holographic interferometry has been used in recent years to measure biological
changes in bone tissue and the time dependent shrinkage of concrete structures. In the field of
experimental mechanics, holographic interferometry has been used to detect flaws such as
microcracks, voids, delaminations, and material inhomogeneities. TVlaser holographic
interferometry makes use of recent advances in digital image acquisition and processing allowing
automatic (and near realtime) reduction and analysis of fringe data. The Los Alamos TVlaser
interferometer can measure displacements as small as two tenths of one micron (10 pinch) at a
speed of 30 Hz. This thesis seeks to address if the ability to detect submicron displacements in
realtime can be used to characterize the existence and extent of a diffusion bond.
Assuming homogeneity and isotropy, any metal will conduct heat according to the Fourier equation
q(r, t) = k VT(r, t). (1.1)
The temperature gradient is a vector normal to the isothermal surface; the heat flux vector, q(r,t),
represents heat flow per unit time per unit area of the isothermal surface in the direction of
decreasing temperature. Thermal conductivity, k, is a specific property of each metal. In a bonded
composite comprised of two metals having different thermal conductivities, i.e., k\ and fc2 the
1
temperature profile through the metals experiences a discontinuity through the bond that joins them.
The boundary condition at the bonded interface can be expressed as an energy balance as

dx
(1.2)
where i denotes the interface and h (in units W/m2 C) is the contact conductance, or film
coefficient, across the bonded interface. It is only in the ideal case of perfect thermal contact that h
eo, and equation 1.2 reduces to
_k^u
and Ti = T2.
(13), (1.4)
In most cases the film coefficient at the interface is finite. Analysis of the heat flow across the
interface can be done using a semiinfinite model, and axisymmetric model, or a full 3D solid
model. In this work, semiinfinite and axisymmetric models are used separately and in combination
to assess the affect of convection boundary conditions and material properties on experimental data.
Mathematica version 2.2.3 is used to support the analytical analyses, the finite elementbased heat
transfer and static coupled thermoelastic analyses are done using MARC version K6.
1.2 Structure of Thesis
Chapter 1 is an introduction giving the problem statement and the engineering basis on which the
problem is investigated. Chapter 1 also discusses the design, operation, and capabilities of the Los
Alamos National Laboratory (LANL) TV Laser Holographic Interferometer used for experimental
measurements. Chapter 1 also presents an explanation of the bonding process by which the copper
aluminum and copperantimony composites used for testing were manufactured as well as a
description of the materials used in various experiments.
Chapter 2 is a derivation of the equations of quasistatic coupled thermoelasticity from first
principles. Also included are the assumptions and restrictions underlying the historical
development of this theory.
Chapter 3 presents a discussion of modeling heat transfer through semiinfinite and finite cylinders.
The goal of this chapter is to examine to what degree analyticallyderived formulae for T(x,t) and
T(r,z,t) agree with each other and with measured experimental data for heated singlemetal test
samples.
Chapter 4 extends the analyses of Chapter 3 to the case of an nlayered bonded metal composite. A
set of governing thermoelastic equations are given, solved by use of the Laplace transform and the
complex inversion theorem, and a formula for T(x,t) derived for the steady state condition; a
program in Mathematica is given for computation of T(x,t). Results for ideal and experimental
bonded configurations in thermal equilibrium are presented.
Chapter 5 explores numerical analyses for T(x,t) singlemetal and bonded composites. These are
compared to some of the analytical results presented in chapters 3 and 4. Finite element models of
the two principal test configurations are developed.
2
Chapter 6 describes the experimental measurements of the bonded and unbonded composites using
TV laser interferometric holography. Comparisons are made between results predicted by the FEA
model developed in chapter 5 and data measured in the experiments. Chapter 6 concludes with a
discussion of the success of the model in predicting measured results and areas for future research.
1.3 The Basis of Holograph Interferometry
Holography is an interferometric technique that generally uses a laser as a coherent source of
radiation. Optical components such as mirrors and beamsplitters are used to combine laser light
reflected from the subject with a second reference beam produced from the same laser. The
combined beams produce an interference pattern which is recorded on an appropriate photographic
medium. After photographic processing, the hologram can be used to reconstruct a three
dimensional image of the original subject, again usually employing a laser.
An important advantage in this process is the ability to make engineering measurements based on
the holograms ability to store threedimensional information. Using with a doubleexposure or
timeaveraging technique, a variety of parameters can be measured: displacement, distortion,
vibration, shape, density, and others. This information is displayed as fringes striating the
holographic reconstruction of the subject. These fringes are contours of equal optical path length
can represent outofplane distortion, inplane displacement, and other quantities.
Conventional holography is based on the phenomenon of interference: phase differences across the
two wave fields, one from the subject under study and the other directly from the laser. These are
converted into density distributions across the emulsion of the photographic recording medium. A
typical holographic plate has a resolution greater than 3000 line pairs per millimeter and contains
approximately 1010 bits of information. For all this, conventional holography has a number of
limitations:
Slow
Photography based
Requires highpowered laser
Qualitative analogue output
To some extent, TVLaser Holographic Interferometry (TVH) mitigates these limitations.
1.4 The Design and Operation of TVLaser Holography (TVH)
As explained by J. C. Davies and C. H. Buckberry,1 TVH is a technique using a laser, CCD camera
and digital processing to create holograms at TV frame rate. This technique was first known as
electronic speckle pattern interferometry in the early 1970s. Recently other names have been used
such as electronic holography and electrooptic holography. The advent of smaller computers and
a technique for phase shifting in the 1980s have made TVH more robust and versatile.
A TV camera has resolution capability several orders of magnitude less than a photographic plate
(1010 bits compared to 106 bits for a camera). However, an optical system using a TV camera can
be configured so that it can capture holographic data and enable all measurements available to
1 Optical Methods in Engineering Metrology, ed. D. C. Williams, Chapman & Hall, London, pg.
277.
3
conventional holography to be made at TV frame rates. This, in effect, means TVH can furnish
holographic information in real time, though with some technologyrelated limitations. One of
these limitations is the resolution constraint introduced by the camera: some information contained
in a conventional hologram must be sacrificed.
The reference beam in a conventional hologram typically illuminates the photographic plate at an
angle of 45 degrees. This allows sufficient space for the subject to be separated from the reference
source. As the subject gets larger, the reference beam has to be moved further off axis during
formation of the hologram. If the reference beam is not sufficiently offaxis during formation of the
hologram, the image will produced will be partially overlapped and swamped by the reconstruction
beam. If the reference beam is moved farther off axis, the structure in the interference pattern
generated becomes more finely spaced and the resolution of the recording medium has to be
increased.
If the reference beam could be positioned in the center of the subject at the reconstruction stage,
illuminating along the axis of the holographic system, the resolution requirement for the recording
medium would be reduced to a minimum. This arrangement would be of little practical use in
conventional holography, however, since the resulting image would not be discernible in the
presence of the highintensity reconstruction beam. However, recording holographic data with a
TV camera allows use of a lowresolution operating mode obtained from an onaxis reference beam
geometry in conjunction with an electronic filter or subtraction stage to eliminate the reconstruction
signal. This optical subtraction process can not be realized physically in ordinary holography.
Resolution requirements would still be too high for a system based on a TV sensor requiring further
limitation of the holograms spatial resolution. Resolution is achieved by using a lens to form an
image of the subject on the sensor. The bandwidth is reduced by use of a small aperture on this
lens; the lateral resolution of the system is reduced proportionately.
A schematic of the Los Alamos TVH system is shown on Figure 1.1. A TVH reference beam may
have about 5 per cent of the intensity of the subject illumination beam. After detecting the
holographic interference pattern, the lowfrequency signal due to the onaxis reference beam
geometry can be eliminated electronically by highpass filtering. The modulation that remains can
then be viewed by feeding the filtered signal to a TV monitor. This signal is the reconstructed TV
hologram. This signal is usually squared or rectified to avoid losing negative components. The
hologram is generated at the TV frame rate; each new image seen on the monitor being the result of
filtering a new interferogram collected by the camera. Change in the optical path length between
subject and camera changes the interferogram and thereby modifies the image. Phase shifting is
accomplished using a piezoelectric crystal, denoted in Figure 1.1 as PZT.
4
MONITOR
Figure 1.1
Schematic of Los Alamos National Laboratory TVH
TVH normally operates in two modes: static and dynamic. Shape changes that are slow in
comparison to the 30 Hz TV frame rate are measured in the static mode. Shape changes are
displayed by fringes, or contour lines, with 7J2 spacing; X being the wavelength of light. A typical
value for X is 0.532 pm; this value for X/2 provides the 10 pinch limit of resolution of the Los
Alamos TVH. All TVH measurements presented in this thesis are taken in the static mode.
1.5 The Diffusion Bond
1.51 Introduction
Diffusion bonding, a.k.a. diffusion welding, is a metal joining technique used in a number of fields,
most notably in the nuclear power and aerospace industries. In nuclear power generation, a fuel
element must be constructed able to operate at elevated temperatures, possess hightemperature
strength, and be highly corrosion resistant. These characteristics can be achieved by diffusion
bonding a thin corrosionresistant cladding to a fuelbearing core which provides the needed
strength at high temperatures.
1.52 Theoretical Basis of Diffusion Bonding
The formation of a diffusion weld can be separated into three distinct stages. The first stage
involves the initial contact of the interfaces. This initial contact may include deformation of surface
asperities or roughness if that is needed to establish initial mechanical contact. These surface
asperities are deformed plastically accompanied by the simultaneous rupturing and/or displacement
of surface films or oxides. This initial plastic deformation is caused by a compressive load applied
5
to the interface of the metals to be joined. In the second stage there is a timedependent diffusion
activated deformation (creep) of the original interface which establishes more intimate interfacial
contact. In the third stage, diffusioncontrolled processes eliminate the original interface. The third
stage may occur through one or more mechanisms: grain growth across the interface, the solution
or dispersion of an interfacial contaminant, or by simple diffusion of atoms along or across the
original interface.
The first and second stages set up the third stage: diffusiondriven elimination of the original
interface. Four mechanisms have been proposed
1. Atom transport occurring across the original interface creating bonds across the parts.2
This is a volume diffusion process. This explanation appears somewhat inadequate.
Atomic interchange in volume (or bulk) is not possible until interatomic distances across
the interface are approximately equal to the lattice parameter. If the original interfaces
reach this state of proximity they become a grain boundary and (except for contaminant
atoms) this state is as strong as the joined parent metals which contain many grain
boundaries.
2. Recyrstallization and/or grain growth occurring at the interface. This growth results in
formation of a new grain structure which sweeps across the original interface boundary. It
is possible recyrstallization causes the yield strength of a metal to approach zero. In this
case, total accommodation of the interface can occur with little of no applied pressure.
This brings surface atoms into sufficiently close proximity to permit metallic bonding at
the interface.
3. Surface diffusion and sintering action cause the interfaces to grow together rapidly.3
4. Surface films or oxides are dissolved into the base metal and in so doing eliminate these
barriers resisting the formation of normal metallic bonds.4
The single model for diffusion bonding is probably not possible to postulate. Such a model may
include one or many mechanical or metallurgical processes. The specific conditions under which a
diffusion bond is made, as well as the results desired from the bond, determine which processes
contribute or predominate. However, Schwartz5 proposed an elementary model as a description of
the mechanism of diffusion bonding.
Two surfaces are initially brought together under a load. If high pressure is exerted on the interface
or the temperature is high, the surface asperities will experience considerable plastic flow until the
interfaces achieve a high degree of conformity. At this point in the process, the joint has
considerable strength due to metallic bonds forming at various regions of the interface. If the initial
pressure is lower, the same surface conformity may be achieved at longer times due to creep and/or
surface diffusion of atoms. During this deformation thin surface films are disrupted and some
plastic work may be put into the surfaces.
2 Feduska, W., and Horigan, W. L., Weld. Met. Fabrication, 35(12),, December 1967, pgs. 483
489.
3 Vaidyanath, L. R., Nicholas M. G., and Milner, D. R., British Welding Journal, 6(1), January 13,
1959.
4 Wiliford, C. F., ;and Tylecote, R. F., British Welding Journal, 7(12), December 1960, pgs. 708
712.
5 Schwartz, M. M., Metals Joining Manual, McGrawHill, 1979, pg. 105.
6
If the joint is held at higher temperature for an extended period of time, a large degree of atom
mobility may be realized at the joint. Recyrstallization or grainboundary motion may occur to
extend and further strengthen atomic bonds and further disrupt any surface film present at the joint.
These processes occur to varying degrees depending on temperature, time, interfacial deformation,
metal properties, and other factors. In the last stages of joint formation, some additional exchange
of atoms occurs across the initial interface with the affect of furthering structural or chemical
homogenization of the joint area.
As atoms redistribute in the interface region, phase changes take place in the metals at the interface.
These changes occur at rates that are dependent on the speed of migrating atoms. Diffusion in
metal systems may be categorized into three different processes depending on the path of the
diffusing element. These three processes are 1) volume diffusion, 2) grainboundary diffusion, and
3) surface diffusion. Each of these processes has a different diffusivity constant. The specific rates
for grainboundary diffusion and surface diffusion are higher than the rate for volume diffusion.
The governing equation for diffusion in metals is Ficks first law.
1.53 Diffusion Bonding Process Variables
For this work, hot isostatic pressing is the process by which diffusion bonding is accomplished.
Also known as hot (or cold) isostatic bonding, the metals to be joined are placed in a welded thin
wall metal container which is subsequently evacuated to a high vacuum. The container is then
placed in the isostatic chamber (or autoclave) and subjected to a combination of high pressure and
high temperature. After the bonding cycle is complete, the container with the nowbonded metals is
removed from the autoclave and the bonded metals are removed from the thinwalled container.
Several parameters are key to effective diffusion bonding. These parameters must be controlled in
theory. In practice, these factors are in fact highly controlled. These critical parameters are
temperature, time, pressure, and surface cleanliness; each of these is discussed in turn.
All mechanisms involved in diffusion bonding are temperature dependent. Since diffusion bonding
is a thermallyactivated process, incremental changes in temperature affect the greatest changes in
process kinetics compared with other parameters. Temperature is a readily controlled and
measurable process variable. Temperature is an effective crossreference with physical and
mechanical properties as well as metallurgical phase transformations. A systematic parametric
evaluation of the diffusion bonding process was done by Kazakov.6 He showed a continuous
temperaturetimepressure interdependence in which increased temperature resulted in increased
bond strength. Increased pressure and time also resulted in increased bond strength. Temperatures
used in diffusion bonding vary from 500 F (260 C) for joining some aluminum alloys to Ti5A1
2.5Sn to silver as a diffusion aid, to as high as 2800 F (1538 C) for joining pieces of tungsten.
Typically, the temperature at which diffusion bonding takes place is greater than 0.5 Tm where Tm
is the melting point of the material. Temperatures between 0.6 Tm and 0.8 Tm are commonly used
in diffusion bonding many metals.
6 Kazakov, N. F., Diffusion Welding in a Vacuum, Moska, Izdvo, Mashionostroyniye, pp. 1332,
1968.
7
Most diffusioncontrolled reactions vary with time. Data presented by Kazakov indicate that
increasing time at temperature and pressure increases joint strength up to a point. However, beyond
this time no further increase is achieved. This indicates that although atom motion continues
indefinitely, structural changes in diffusion bonding tend to approach an equilibrium. This behavior
is illustrated in recrystallization. A deformed sample first undergoes recovery when first heated; it
then recrystallizes. Initially, the formation and growth of new grains are rapid, but as time
increases, the rate of grainboundary motion and physical change diminishes. The decrease is due
to stabilization of the microstructure through reduction of internal energy. Thus the driving force
for continued structural change is also reduced. The rate of atom motion, however, does not
decrease significantly throughout this process.
Pressure is an important process parameter in diffusion bonding. As a quantitative variable, it is
less applicable than either temperature or time. The initial phase of bond formation is most
certainly affected by the amount of deformation induced by the pressure applied. Increased
pressure invariably results in higherstrength joints for any given timetemperature combination.
This is probably due to the greater interface deformation and asperity breakdown resulting from
higher pressures. Also, increased pressure (and deformation) leads to a lower recrystallization
temperature. This is to say increased deformation accelerates the process of recrystallization at a
given temperature.
Experience in practice regarding surface preparation can be summed up by the rule: maximum
attainable cleanliness. Surface preparation includes cleanliness, finish, removal of chemically
combined films, and removal of gaseous, aqueous, or organic surface films. Surface finish is
ordinarily achieved by machining, grinding, or polishing. Surface flatness is another necessary
component of surface preparation. A minimum degree of flatness and smoothness is required to
assure interfaces can achieve necessary compliance without excessive deformation. Machine
finishes, grinding, of polishing are usually adequate to obtain the needed degree of surface flatness
and smoothness. Machining also introduces a measure of plastic flow into the surface. Such cold
worked surfaces have a lower recrystallization temperature than unworked bulk metal. Processes
such as vapor degrease, vacuum bakeout, and chemical etching have also been used to achieve
needed surface cleanliness.
1.54 Metallurgical Factors
Two factors of particular importance in diffusion bonding different metals are allotropic
transformations and microstructural factors which result in modification of diffusion rates.
Allotropic or phase transformations occur in many metals and alloys. They are important in that
metal is very plastic when undergoing a transformation. This permits more rapid interfacial
accommodation at lower pressures in much the same manner as recrystallization. Diffusion rates
are typically higher in plastically deformed metals.
Diffusion can also be enhanced by alloying, i.e., introducing elements with high diffusivity into the
system. The highdiffusivity element accelerate the process of atomic motion at or across the
interface. Elements selected as diffusionaccelerating are usually those which have reasonable
solubility in the metal to be joined, do not form stable compounds, and depress the melting point
locally. Melting point depression by alloying must be controlled due to the possibility of
liquefaction at the joint interface.
8
1.6 Design and Manufacture of CuAl, CuMgOAl, and CuSb Bonded Composites
Three bonded composites were manufactured for use in this work. The first specimen was made of
copper and aluminum; the second was identical except for a cavity machined at the interface which
was filled with MgO powder before bonding. The third composite was made of copper and
antimony. In the first two cases, the bonded composites were made of pure metals of relatively
similar thermal conductivity. MgO powder was added to specimen number 2 to create a void at the
interface insuring a boundary of comparatively poor thermal conductivity. In the third case, the Cu
Sb composite was made at a temperature and pressure designed to maximize the formation of
discrete intermetallic layers between the two parent metals. Figure 1.2 shows a schematic of the pre
bonded configuration for the AlMgOCu composite. The AlCu composite was similar with the
exception of the MgO inclusion at the interface. Figure 1.3 shows a schematic of the prebonded
configuration of the SbCu composite. All three samples were prepared and processed by Messrs.
Patrick Rodriguez, Peter Lopez, and David Huerta of LANLs NMT5 group
Each prebonded assembly was encapsulated in an evacuated tantalum container. Closure was
achieved by electronbeam welding top and bottom lids to the main body of the container. Each
assembly was evacuated to approximately Iff4 torr prior to welding. Once welded, each
encapsulated assembly was placed in the hot isostatic press and bonded. The CuAl, CuMgOAl
and SbCu composites were bonded in the same run of the hot isostatic press (HIP).
fit Melting Point 660.5 C
Prebrushed wt. 128.76 g
 Posebrushed wt. 128.71 g
1.50 Ld X jDBQ'T cavity
MgO Powder wt. 1.05 g
 Cu Melting Point 1084.9 C
Prebrushed wt. 328.SB g
Postbrushed wt. 328.55 g
_Q.U2trCtaaiimt&
ftWZ" CUnnnp* . Kdo
X______________ *Th CBn EB welded at both ends
(Drawing not to scale) *2.625 ojd. X .020" well canister
Resistance press fit lid 0.020" T
*Ta net wt. 130.21 g
Grass wt. 588.53 g
Prist melding dimensions:
2.625* o,cLX 1.455 H
AlMgOCu Assembly in PreBonded Configuration
Figure 1.2
Each sample was wire brushed under freon prior to insertion into the tantalum cladding to insure
surfaces as clean as reasonably possible.
Typically HIP bonding is done at a temperature approximately 50 percent to 75 percent of the
melting temperature, Tm, of the metal possessing the lowest melting temperature. Bonding
temperature was raised to approximately 80 to 85 percent of Tm for all three bonded composites
used in this study. This higher HIP bonding temperature was employed to 1) insure a good bond
9
and 2) in the case of the SbCu composite, insure growth of a discrete
rrmTTTTrrmp
::*:::xKPPPsr*:.......
0.042? Gearance ea. side
(Drawing not to scale)
Sb Melting Point 630.5 C
 Preb rushed wt. 350.19 g
Postbrushed wt. 350.10 g
Cu Molting Point 1084.9 C
Prebrushed wt. 296.65 g
Postbrushed wt. 296.62 g
*Ta can EB welded at both ends
*2.625" o.d. X .020" wall canister
"Resistance pres9 fit lid 0.020" T
*Ta net wt. 130.05 g
"Gross wt. 776.74 g
"Post welding dimensions:
2.626" o.d X 1.456" H
CuSb Assembly in PreBonded Configuration
Figure 1.3
intermetallic layer between the two parent metals. Since there is very little difference between Tm
for aluminum and Tra for antimony, all three specimens were bonded in the same HIP run. Figure
1.4 shows a graph of the temperature and pressures used for bonding the three composite
specimens.
HIP RUN Â£96.01 A
530 C / 10 KSI /60 MIN SOAK
SEPTEMBER 26, 1996
ELAPSED TIMS
Temperature and Pressure Used for Bonding Specimens
Figure 1.4
10
1.7 Properties of Experimental Materials
The chemical, thermal, and metallurgical properties of the copper, aluminum, and antimony used in
this work are taken from standard reference works or, where these values seemed possibly incorrect,
determined experimentally. Properties determined experimentally are so identified. The copper
and aluminum used in manufacturing the diffusionbonded specimens were purchased from
JohnsonMatthey as 99.999 per cent pure.
The antimony (Sb) used for this study was contributed by the LANL Material Science and
Technology MST) division. TTiis antimony, used to create antimonal lead alloy at Los Alamos, is
itself comparatively less pure being about 98 per cent antimony. This compares with 99 per cent
pure antimony, i.e., the most pure metal available commercially and then only in pellet form. The
antimony contributed by MST division was taken from the sprue of a large casting, the form in
which it is stored. Subsequent shaping of the antimony was done by Electric Discharge Machining
(EDM) due to its extreme brittleness.
Antimony was chosen for use in this study for two reasons. First, it offers an advantageous thermal
impedance (15.2 W/m C) when compared to copper (384 W/m C) and aluminum (202 W/m
C). Second, as the phase diagram for the copperantimony system indicates in figure 1.5, copper
and antimony readily form intermetallic compounds when diffusion bonded. Radiography of the
copper/antimony composite made after bonding showed five discrete density zones. Two were the
parent metals and three were identifiable intermetallic layers the largest of which was
approximately 2.3 mm thick.
W Â£ I (5 H r %
2D VO AC TO 60 90 3k
1l l .1_____11______ ' *________t_________
Phase Diagram for the CopperAntimony System
Figure 1.5
11
1.71 Chemical, Thermal, and Mechanical Properties of Copper
The trace elements in the copper used in this study are shown in Table 1.1. This chemical analysis
was provided by David Wayne of LANLs group CST8. Two samples of the JohnsonMatthey
copper were analyzed using Glow Discharge Mass Spectroscopy (GDMS). The GDMS analysis
indicated the copper was probably between 99.900 and 99.990 percent pure, the largest trace
constituent being oxygen. The thermal and mechanical properties used in the various analyses
appearing in chapters 3,4, and 5, are shown in Table 1.2.7
Element JohnsonMatthey Comparison Sample Parts per Million (ppm) Experimental Sample Parts per Million (ppm)
C12 32 8 319 28
016 <1000 <1500
Mg24 0.22 0.04 0.019 0.009
Al27 <0.05 <0.08
Si28 0.49 0.04 <2
P31 3.2 0.4 0.013 0.004
S32 39 6 9.7 0.4
Cr52 <1.5 <4.0
Mn55 0.30 0.04 <0.03
Fe56 1.1 0.2 12.8 0.2
Co59 <0.01 0.07 0.01
Ni60 0.68 0.06 3.2 0.1
Zn68 0.20 0.04 4.1 0.2
As75 0.98 0.03 2.3 0.2
Se82 1.2 0.2 <1.0
Mo95 5.1 1.0 1.1 0.1
Ag107 19 2 11.2 0.4
Sn118 <0.1 <0.1
Sb121 0.50 0.05 0.33 0.02
Te125 <0.5 1.3 0.3
Pb208 1.7 0.1 0.50 0.02
Bi209 0.32 0.02 0.19 0.03
Trace Elements in Copper
Table 1.1
7 ASM Handbook, Vol. 2, ASM International, pgs. 11101113.
12
Thermal Conductivity (k) 384 W/m C
Thermal Diffusivity (a) 11.234 10'W/s
Heat Capacity (Cv) 383 J/kg C
Density (p) 8954 kg/m3
Poissons ratio 0.343
Youngs Modulus 12.98 1010 Pa
Coefficient of Thermal Expansion (3) 20 106 C
Thermal and Physical Properties of Copper
Table 1.2
Figure 1.6 shows a photograph of the metallurgical structure of the copper used in this work. The
photograph is at 25X. The dark spots are 1) artifacts or the etching process and 2) indication of
preferential attack by the etchant at the grain boundaries, probably due to the presence of oxygen at
those locations. The metallography in Figures 1.6, 1.7 and 1.9 was performed by Mr. Ramiro
Pereyra of Los Alamos CST15 group.
Photograph of Metallurgical Structure of Copper, 25X
Figure 1.6
Dr. Michael Hundley of LANLs MST10 group measured thermal conductivity for each metal used
in this work. Figure 1.7 gives his measurement of 1^^ as a function of temperature. As may be
seen, the thermal conductivity shows marked temperature dependence. Figure 1.7 represents
13
kcopper as measured on an axis normal to the vertical or diametrical surfaces of the test cylinder. The
dark line in figure 1.7 is Holmans value for thermal conductivity of copper at 0 C.8
Thermal
Conductivity
(W/mC)
30 40 50
60
70
Temp (C)
Thermal Conductivity Data for Copper
Figure 1.7
1.72 Chemical, Thermal, and Mechanical Properties of Aluminum
The aluminum used for this work was purchased to a specification of 99.999 per cent pure. The
analysis of race elements present in the aluminum is shown in Table 1.3. As with the copper, the
chemical analysis of pure aluminum was provided by David Wayne of Los Alamos National
Laboratorys CST8 group. Samples were analyzed using Glow Discharge Mass Spectroscopy
(GDMS). The thermal properties of aluminum are given in Table 1.4.9 Figure 1.8 shows 100X
metallography of the aluminum used in this work. Again, the dark areas are artifacts of the etching
process used to prepare this sample.
Element Parts per Million (ppm) Sigma1 Parts per Million (ppm)
Li < 0.004
Be <0.01
B <0.05
C 111 5
N <10
Na <0.5
Mr 0.52 0.06
Si 24 1
P 0.40 0.07
S 0.43 0.09
K <2.0
Mn 0.5 0.1
Fe 19 4
8 Heat Transfer, Holman, J. P., McGrawHill, New York, 1986, pgs. 635636.
9 ASM Handbook, Vol. 2, ASM International, pgs. 10991100.
14
Co <0.1
Ni 2.8 0.5
Cu 0.84 0.03
Zn <0.1
Ga <0.01
Ge <0.04
As <0.01
Zr <0.05
Nb 0.069 0.005
Ag <0.02
Sn < 0.05
Pb 0.32 0.07
Trace Elements in Aluminum
Table 1.3
Thermal Conductivity (k) 205 W/m C
Thermal Diffusivity (a) 8.418 10'5 mz/s
Heat Capacity (Cv) 896 J/kg C
Density (p) 2707 kg/m3
Youngs Modulus 62 GPa
Coefficient of Thermal Expansion (3) 23.6 10'6 C1
Thermal and Physical Properties of Aluminum
Table 1.4
Photograph of Metallurgical Structure of Aluminum, 100X
Figure 1.8
Figure 1.9 shows Dr. Hundleys measurement of k^ for the aluminum used in this work. Two
functions of kai are shown; the first is k^ as measured in a direction normal to the diametrical
surfaces of the test cylinder (transverse), the second kai was measured in a direction from the center
of the test cylinder to the circumferential edge (longitudinal). Figure 1.9 indicates considerable
15
anisotropy in k^. The dashed line indicates Holmans value for the thermal conductivity of pure
aluminum at 0C.
Thermal Conductivity
(W/mC)
Figure 1.9
1.73 Chemical, Thermal, and Mechanical Properties of Antimony
Two samples of antimony were also analyzed by David Wayne using GDMS. These samples were
dried in a vacuum at 200 C prior to analysis. Results for C, Fe, Ni, Si, Co, and Zn are regarded as
semiquantitative, i.e., off of real values by as much as 30 per cent. The large standard deviation
for sulfur resulted from a peak intensity that decreased over time during the GDMS analysis. As
can be seen in Table 1.5, the predominant contaminant of the antimony is iron with smaller but
significant amounts of nickel, copper, and lead. Trace constituents are shown in Table 1.6.
Element Weight Percent Sample 1 1sigma Sample 2 Weight Percent Sample 2 1sigma Sample 2
Sb121 97.94 97.94
Fe56 1.02 0.04 1.22 0.06
Ni60 0.50 0.05 0.46 0.01
Cu63 0.12 0.01 0.114 0.005
Pb208 0.21 0.01 0.25 0.02
Major Contaminants of Antimony
Table 1.5
Element Parts per Million (ppm) Sample 1 1sigma Sample 2 Parts per Million (ppm) Sample 2 1sigma Sample 2
C12 43.6 4.9 62.7 4.6
Al27 <0.03 <0.03
Si28 7.23 1.44 4.65 0.31
16
P31 <0.03 <0.05
S32 523 149 584 66
Cr52 <1.2 <1.2
Mn55 <0.05 <0.05
Co59 382 23 349 10
Zn68 1.99 0.52 2.98 0.17
As75 938 50 908 38
Se82 129 14 124 9
Mo92 1.03 0.09 1.02 0.13
Ag107 8.23 1.38 8.65 0.17
Cd111 <0.02 <0.15
In115 <0.2 <0.2
Sn120 22.0 5.5 24.6 0.6
Trace Constituents of Antimony
Table 1.6
The thermal and physical properties for antimony are given in Table 1.7. All of these values, with
the exception of thermal conductivity, are taken from the ASM Hanbook.10 Reference values of this
type are typically derived from very pure examples of the metal of interest. It appeared likely,
given the impurities known to present in the antimony used for this work, that actual thermal
properties would vary from values found in standard references.
Thermal Conductivity (k) 15.2 W/m C
Thermal Diffusivity (a) 1.33 Iff5 m2/s
Heat Capacity (Cv) 205 J/kg C
Density (p) 6650 kg/m3
Poissons Ratio Not Available
Youngs Modulus 77.759 GPa
Coefficient of Thermal Expansion (0) 9.2 106 C1
Thermal and Physical Properties of Antimony
Table 1.7
In addition, reference values for the thermal conductivity of antimony (kSb) varied widely
throughout the literature; they range from 19 W/m C to 26 W/m C. Initial modeling of heat
transfer through the Sb test cylinder assumed kSb = 26 W/m C. Initial analyses provided
inconsistent and often unsatisfactory results. Discrepancies between the several analyses and
experimental data suggested the possibility that the reference value of kSb= 26 W/m C was
incorrect for the particular antimony used for this study.
Figure 1.10 shows measurements of thermal conductivity taken over a range of approximately 20
C to 47 C. One measurement was taken of antimony machined radially from a cylindrical
shape, i.e., with z coordinates held constant. A second measurement was taken using a specimen of
antimony machined in the transverse direction from the same sample cylinder, i.e., with values of r
held constant. This was done to determine if thermal conductivity was anisotropic in the antimony
used for this work. As figure 1.10 indicates, no appreciable anisotropy was found.
10 ASM Handbook, Vol. 2, ASM International, pgs. 11001101.
17
Thermal Conductivity
(W/m C)
Temperature (C)
Thermal Conductivity of Antimony
Figure 1.10
Metallography of the antimony used in this work is shown in figure 1.11.
Photograph of Metallurgical Structure of Antimony, 100X
Figure 1.11
18
1.74 Properties of Macor Ceramic and RTV Silicon Potting
Figure 3.2 shows a bonded AlCu cylinder mounted in a ceramic ring. This was the configuration
used for temperature and dimensional measurements in the majority of experiments described in
this work. Test cylinders were potted into the ceramic collars using a commercial hightemperature
siliconbased RTV gasket material. This configuration was designed to 1) support the test cylinders
while imposing no significant constraint, 2) insulate the circumference of the test cylinders, and 3)
damp incipient vibration, and 4) provide the ability to conduct a number of experiments in as
repeatable a fashion as possible. The Macor ceramic collar was machined to the same thickness as
the test cylinders, i.e., .00635 m. The inside diameter of the collar was made 24 mm larger than
the outside diameter of the test cylinders. The Macor collar was 12.7 mm wide between the inside
and outside diameters. The mechanical and thermal properties of Macor are well documented and
appear in table 1.8.
Thermal Conductivity (k) 1.46 W/m C
Thermal Diffiisivity (a) 7.33 Iff7 m2/s
Heat Capacity (Cv) 790 J/kg C
Density (p) 2520 kg/m3
Youngs Modulus 66.9 GPa
Poissons Ratio .29
Coefficient of Thermal Expansion (3) 9.3 10'6 Cl
Properties of Macor Ceramic
Table 1.8
The properties of the RTV potting were difficult to obtain. Properties appearing in table 1.9 were
provided by the manufacturers technical support group. A value for specific heat was unavailable;
Cv for silicon (the main component of RTV potting) was used instead.
Thermal Conductivity (k) 0.19 W/m C
Density (p) 895.9 kg/m3
Heat Capacity (Cv) 790 J/kg C
Properties of RTV Potting
Table 1.9
19
NOMENCLATURE USED IN CHAPTER 2
Symbol
P
A
B
3B
C
D
E
F
G
H
K
K
M
0
P
S
S
a
b
b
c
e
f
g
k
m
n
P
3P
Q
q
q
q
A
<1
R
r
s
S
t
U
u
A
u
w
x
P
Definition
Unit Area
Thermal expansion tensor
Body
Boundary of B
Elasticity tensor
Finite strain tensor
Infinitesimal strain tensor
Deformation gradient
Any symmetric tensor
Any symmetric tensor
Conductivity tensor
Compliance tensor
. Stresstemperature tensor
Origin, zero vector, zero tensor
Part of B
Stress tensor
Prescribed stress on boundary
Thermoelastic coupling constant
Noninertial body force
Inertial constant
Specific heat
Internal energy
Inertial body force
Temperature gradient
Thermal conductivity
Stress temperature modulus
Unit outward normal to 3B
Position vector
Boundary of P
Orthogonal tensor corresponding to a change of observer
Heat flux vector
Heat flux
Prescribed heat flux vector on boundary
Prescribed heat flux on boundary
Orthogonal tensor such that Q = RT
Heat supply
Surface traction
Prescribed traction on boundary
Time
Positive square root of FTF
Displacement vector
Prescribed displacement on boundary
Rate of working
Point in B
Coefficient of thermal expansion
20
8
T\
fj
e
0O
x
v
V
V?
1
C)
()
(*)
U)
(*)
()r
curl
det
div
sym
tr
Kronecker delta, arbitrarily small quantity
Entropy
Prescribed entropy on boundary
Absolute temperature
Reference temperature
Lame constant
Lame constant, shear modulus
Free energy
Prescribed free energy on boundary
Density
Unit tensor
First derivative with respect to time
Second derivative with respect to time
Response form of quantity x, e.g., q = q(F, 0, g, x)
Reduced form of quantity x, e.g., q = q(D, 0, g)
Description of quantity x when T is taken as an independent variable
Transpose of a tensor
Curl
Determinant
Divergence
Symmetric part of tensor
Trace
2. The Linear, Coupled, QuasiStatic Theory of Thermoelasticity
2.1 Introduction
The following discussion attempts to provide a development of the general theory of thermoelasticity as
it proceeds from first principles of mechanics and thermodynamics. Following this, the governing
equations of thermoelasticity are derived within the context of various assumptions, explicit and
implicit, which historically have applied to the development of this discipline. This discussion then
provides the mathematical basis for giving the general theory a linear form. Assumptions for isotropy
and homogeneity are introduced. Finally, a discussion of the basis for the socalled coupled and
uncoupled, quasistatic and general theories within thermoelasticity is given. Important terms and
concepts are italicized.
Much of the following material is taken from Handbuch der Physik, and Heat Conduction Within
Linear Thermoelasticity as noted below. The explanation presented in chapter 2 seeks to provide the
motivation for pursuing the coupled quasistatic analysis presented in chapters 4, 5, and 6
21
2.2 The Basic Laws of Mechanics and Thermodynamics
The motion of a body B bounded by a regular region 3P is described by a vector field u on B x (0, to)
where (0, to) is a fixed, open interval of time. The vector field u represents the displacement of a
material point x (contained within B) at time t. A spatial gradient F of function x > x + u(x, t) is the
deformation gradient defined as
It is assumed for each t that the mapping x  x + u(x, t) is onetoone on B and that its inverse is
smooth, i.e., det F & 0.
If S(x, t) denotes the first PiolaKirchoff stress tensor measured per unit surface area in a reference
configuration, and if f(x, t) denotes a body force per unit volume in the reference configuration, then
the laws of balance of forces and moments can be expressed as equations (2.2) and (2.3) respectively:
Where P is a part of B, 3P is the boundary of P, P = x 0 is the position vector of x relative to the
origin 0, and n is the outward unit normal of 3P. Equations (2.2) and (2.3) are good for all times t. If f
is continuous on B x (0,/o), then equations (2.2) and (2.3) are equivalent to
F = 1 + Vu.
(2.1)
p
(2.2)
(2.3)
VS + f = 0
(221)
and
sft=fst.
(2.31)
The rate of work of part P at time t is given by equation (2.4)
dv
(2.4)
dp
p
(2.5)
3P
P
P
Equations (2.21) and (2.4) imply that
22
(2.6)
W(P) = J S F dv
p
where the quantity S F is called the stress power.
The first law of thermodynamics is given in equation (2.7)
e dv = W(P) J q
3p
n da +
r dv
(2.7)
for every part P at time t and where e(x,t) is the internal energy per unit volume of the reference
configuration; q (x, t) is the heat flux vector per unit volume in the reference configuration and r(x, t)
is the heat supply per unit volume in the reference configuration external to the configuration.
Equation (2.7) has the local equivalent
e = S FVq + r
(2.8)
when combined with equation (2.6). The second law of thermodynamics is given as equation (2.9)
d_
dt
/* J V1 ^+J Â£ *
P 3P P
(2.9)
for every part P and every time t; ti(x, t) is entropy per unit volume in the reference configuration and
0 (x, t) is the absolute temperature. Therefore J 77 dv is the entropy of P, J da is the total
entropy flux across 9P due to conduction and J dv represents the total entropy flux into the
interior of P from outside the reference configuration. Given previous assumptions, the second law is
therefore equivalent to
7?
(2.10)
where q is based on the unit surface area of the reference configuration.
Free energy, \(f, is defined as
yr = e 7)6 , (2.11)
which, when combined with equations (2.8) and (2.10), gives the local dissipation inequality in
equation (2.12)
yf + rjdSF + ^g < 0,
(2.12)
23
where g is the temperature gradient, g = V0. Given equations (2.8) and (2.11), equations (2.10) and
(2.12) are equivalent.
2.3 Elastic Materials and the Consequences of the Second Law
An elastic material is defined by constitutive equations giving the stress S, free energy \jr, entropy r),
and the heat flux q at each point x whenever the deformation gradient F, temperature 6, and
temperature gradient g are known at point x:
iff = v?(F, 0, g, x) (2.131)
S = S(F, 0, g, x) (2.132)
T] = fj(F, 0, g, x) (2.133)
q = q(F, 0, g, x) (2.134)
where \jr, S fj, and q are prescribed by the boundary of P and are called response functions; \jf, S,
fj, and q also must hold for all points in B. It is assumed these response functions are smooth on
their domain which is the set of all (F, 0, g, x) where F is a tensor with det F 0, 0 is a positive scalar,
g is a vector, and all x are contained within B. It is also assumed that S is subject to equation (2.14):
S(F, 0, g,x)Fr=FS (F, 0, g, x)T (2.14)
It is necessary to introduce the concept of an admissible thermodynamic process. Such a process
operates within an ordered array [u, 0, vf, S, T, q] with the following properties
1. All functions have a common domain in the form of P x T where P is a part of B and T is
an open time interval of (0, to);
2. u is a motion on P x T and 0 is a positive scalar;
3. y, S, t, and q are defined on P x T through the constitutive relations with (as before)
F = 1 + Vu and g = V0.
P x T is called the domain of the process and may vary from process to process. Given an admissible
thermodynamic process, equations (2.21) and (2.8) yield the inertial body force f and heat supply r
necessary to support the process
f = VS and r = e S F + Vq (2.151), (2.152)
The local dissipation inequality, equation (2.12), implies certain restrictions on the response functions.
A necessary and sufficient condition that every admissible thermodynamic process obey the local
dissipation inequality is that the following three statements hold: 1
1. The response functions iff, S, and fj are independent of the temperature gradient g:
24
y/ = y/(F, 0); S = S(F, 0); and 7J = tj(F, 6);
2. yr determines S through the stress relation
S(F, 0) =
dy/{F,0)
dF 1
and fj through the entropy relation
7?(F, 0) =
dij{F, 0)
90 :
3. q obeys the heat inequality
q(F, 0, g) g < 0.
A proof of the necessity and sufficiency of this condition is given by D. E. Carlson.11
In an admissible thermodynamic process, the energy equation, equation (2.8), takes the form
0TJ  Vq + r . (2.16)
Thus an admissible thermodynamic process is adiabatic (Vq + r = 0) if and only if it is isentropic
(dt)/dt = 0). Also, in an admissible thermodynamic process, the response function for stress (S ) and
entropy (fj ) satisfy the Maxwell relation
olS(F, 0) 9fj(F, 0)
90 9F
(2.17)
By the relationships yr = e t] 0 yr = y?(F, 0), S = S(F, 0), rf = fj(F, 0) and equations (2.131)
through (2.134), internal energy obeys a constitutive relation of the form
e(F, 0) = yr(F, 0) + 0fj(F, 0) (2.18)
Specific heat, c, is defined by
c(F, 0) =
9y{F, 0)
90
+ tj(F, 0) + 0
9ij(F, 0)
90
The relation rj(F, 0)
9yr(F, 0)
90
implies
11 Handbuch derPhysik, D. E. Carlson, Volume Via/2, springerVerlag, 1972, pgs. 297312.
25
c(F, 9) = e ^ ^ (2.19)
c (F, 0) is assumed to be positive; this combined with 0 > 0 and equation (2.19) implies rj(F, 6) is
smoothly invertible in 0 for each choice of F. The constitutive relations (2.131) through (2.134) can
be rewritten
e = e(F, Tj), S = S(F, Tj)
(2.201), (2.202)
6 = 0(F, tj), q = q(F, 77, g)
(2.203), (2.204)
For a given F, the function 77  9 (F, 77) is the inverse of the function 9 fj(F, 0). The function
e for example is given by
e(F, tj) = ?(F, 9, (F, 77))
(2.21)
It follows from equations (2.18) and (2.21) that
e(F, 77) = v?(F, 9, (F, 77)) + 770 (F, tj) ,
which gives
9e(F, tj) 9y 9\jr 99 99
9F = ~9F + ~99 ~9F + 71 ~9F
(2.22)
and
9e(F, 77) 9\j/ 99 99
=^r~r + 9 +77
077 00 077 07J
(2.23)
From the relations s(f, 0) = and tj(F, 0) =  ^
it can be shown that
and
S(F, 77) =
0e(F, 7?)
0F
0(F, 77) =
9e(F, 77)
077
(2.24)
(2.25)
Equations (2.24) and (2.25) are the stress and temperature relations when entropy is taken as an
independent thermodynamic variable.
26
2.4 Material FrameIndifference
The principle of material frameindifference holds that the constitutive relationships assumed in (2.13
1) through (2.134) are independent of any frame of reference (or the observer). Changing the frame of
reference transforms the relevant mechanical and thermodynamic quantities as
FQF
SQS
0 0
gg
T1 r
q > q
where Q is the orthogonal tensor corresponding to the change.12 Since equations in (2.131) through
(2.134) are to be invariant under all such changes, the following relationships must hold
12 The transformation rule for F is a consequence of Truesdell and Noll, Encyclopedia of Physics, the NonLinear
Field Theories of Mechanics, Volume III/3, section 5, pages 44. The rule for the first PiolaKirchoff stress tensor,
S (Truesdell and Nolls TR), follows from the transformation law for the Cauchy stress, T. Constitutive equations
must be invariant under changes of frame of reference. If a constitutive equation is satisfied for a process with a
motion and a symmetric stress tensor given by
x = X(X,t), T = T(X,t)
then it must be satisfied also for any equivalent process {%*,T*}. This is to say the constitutive equation must be
satisfied also for the motion and stress tensor given by
* = X*(X,t*) = c(t) + Q(t)x(X,t),
T* = T*(X,t) = Q(t)T(X,t)Q(t)T,
t* = t a
where c(t) is an arbitrary point function, Q(t) an arbitrary orthogonal tensor function of the time t, and an arbitrary
number. Further, let dP be the boundary of a part P of a body in some configuration % defined by its deformation x
= x(X) from a fixed global reference configuration. The contact force acting on P is
fc = jTn<Â£s,
dP
where n is the outward unit vector normal to 3P in the configuration %. Let dsR be the surface element and nR the
outward unit normal to 3P in the reference configuration. By the laws of the transformation of surface integrals,
fc = jTnrfa= Jt* nR dsR
3P dP
where TR is defined by the relation
T = J~l To Fr in which J sldet f = .
* 1 ' Pr
27
F, 9) = V/(QF, 9) (2.261)
S(F, 9) = QrS{QF, 9) (2.262)
fj(F, 9) = fj(QF, 9) (2.263)
q(F, 9, g) = q(QF, 9, g) (2.264)
for every orthogonal tensor Q and for all (F, 9, g) in the domain.
Using the polar decomposition theorem,
F = RU (2.27)
where R is orthogonal and U is positive definite square root of FTF.* 13 Choosing
Q = RT in equation (2.261) through (2.264) and using equation (2.27), \jr, S, T, and q have the
relationships
V = lP(U, 9)
S = FU"S(U, 9)
V = ri{U> 0)
q = q(U, 9, g).
Defining the finite strain tensor, D, as
D = ^ (u2 l) = (FTF l) (2.28)
leads to the consequences of material frameindifference.
The constitutive equations in (2.131) through (2.134) satisfy the principal of material frame
indifference if and only if they can be written in reduced form:
V = D, 9)
S = FS(D, 9)
7] = tj(D, 9)
q = q(D, 9, g).
The temperature gradient g is invariant since it is the gradient relative to the reference configuration p. The heat
flux, q, is invariant since it is measured per unit area p.
13 H. Richter first employed the polar decomposition theorem to explore the implications of material frame
indifference in Sur Elastizitatstheorie endlicher Verformungen, Mathematische Nachrichtem, Vol. 8, pp. 6573;
translated into English in Foundations of Elastic Theory, ed. C. Truesdell, Gordon and Breach. The polar
decomposition theorem states that any invertible tensor F can be expressed as F = VQ = QU where Q is an
orthogonal tensor and U and V are symmetric tensors.
28
In addition, the stress and entropy relations (2.131) through (2.134) reduce to equations (2.29) and
(2.30):
s(d, 0) dy{D, 0) dD (2.291)
rj(D, 9) dy{D, 0) de (2.292)
The derivation of equations (2.291) and (2.292) proceeds in terms of components:
dy dy dDtj
dFa dDu dFki
where, from equation (2.28),
1 /
Ay = 2 \FmlFmj dFkl 2(F^ + W (2.30)
Since cT ii ^ <7S
it can be shown d\j/ df dFkl ta dD
or equivalently dy F dy (2.31)
dF dD
Equation (2.31), the stress relation of (2.132), and S = FU 1s(u, d) imply equation (2.291). It is
interesting to note that
dy{D, 0)
S = F
<9D
automatically satisfies
S F = F S1
The implication here is that in thermoelasticity the balance of moments is a consequence of balance of
forces, the two laws of thermodynamics, and material frameindifference.
29
2.5 Consequences of the Heat Conduction Inequality
From equation (2.133), the reduced response function q must obey the inequality
q(D, 9, g) g < 0. (2.32)
The conductivity tensor is defined as
K(D,e) = >&**>
dg
(2.33)
g = o
Holding D and 0 fixed and expanding q(g) s q(D, 9, g) in a Taylor series about g = 0 gives
q(g) = 5(0) Kg + o(  g ) as g > 0. (2.34)
Combining equations (2.32) and (2.34) gives
5(0) g g K + o(g2) < 0 as g 0.
The inequality holds for all g if and only if q(0) = 0 and g K g ^ 0 for all g. This has the
consequence that the heat flux vanishes whenever the temperature gradient vanishes, i.e.,
q(D, 9,0) = 0.
(2.35)
From equation (2.35)
^q(D, 6,0)
99
= 0
= 0.
(2.36)
(2.37)
2.6 Derivation of the Linear Thermoelastic Theory
The complete system of field equations for nonlinear thermoelastic theory has been derived consisting
of equations (2.21) and (2.16):
VS + f = 0 (2.21)
Ofj = Vq + r (2.16)
30
combined with the constitutive equations
V' = ^(D. 0)
S = FS(D, 9)
n = fj(D, 0)
q = q(D, 9, g)
where
F = 1 + Vu ;
D =I(FrFl);
and
g = V0.
(2.381)
(2.382)
(2.383)
(2.384)
(2.391)
(2.392)
(2.393)
These constitutive relations are subject to the thermodynamic restrictions
S(D, 9) =
dijr[D, 6)
dD
jj(D, e) = 
df(D, 6)
36
q(D, 9, g) g < 0.
A linear approximation of equations (2.21), (2.16), (2.38), (2.1), (2.28), and (2.39) proceeds under the
assumptions:
1. the displacement gradient and its time rate of change are small
2. the temperature field is nearly equal to a given uniform field 0O called a reference temperature
3. the temperature rate and the temperature gradient are small.
Thus it is assumed that
Vu[ < 8 ; (2.401)
Vu < 8 ; (2.402)
\e e0 < 8 ; (2.403)
31
\9  < 5; (2.404)
g ^ 5; (2.405)
where 5 is arbitrarily small. Due to inequalities (2.401) through (2.405), the deformation gradient F becomes F = 1 + 0(5) as 5 > 0. The infinitesimal strain tensor E is (2.41)
E = 1 (Vu + VuT); by (2.40), E becomes (2.42)
o II (2.431)
E = 0(5), (2.432)
From equations (2.391), (2.392), (2.40), (2.42), and (2.43), the finite strain tensor, D, is
D = E + ^(VurVu); (2.44)
D = E + 0(5 2 ); (2.45)
D = 0(5). Equations (2.44) and (2.401) through (2.405 imply (2.46)
D = E +  (VuTu + VurVu); or (2.47)
D = E + 0(5 2 ). (2.48)
When F = 1, 6 = 90, D = 0, and S = S(0, 90), the quantity S(0, 9 0) represents the residual stress
at the reference temperature, i.e., the stress that the body would experience if it were held in the
reference configuration at the uniform temperature 9 0 It is assumed
S(0, 0o)=O. (2.49)
32
The assumptions of zero residual stress and uniform reference temperature are foundational to the
classic theory of thermoelasticity. A Taylor series expansion of S(Â£), 0), with S(0, 0O) =0, around
D = 0 and 6 = 6 0 gives
S(D, 0) = S(O,0o)
3s(d, 6)
3D
D=0
e=ea
(do)
5s(d, 0)
do
D=0
e=en
(ee0) + 0(5)
(2.50)
Equation (2.50) can be rewritten
S(D, 6) = C[E] + (0 0O)M + 0(5)
where
C =
<9s(d, e)
3D
D=0
6=60
and
M =
Ye
D=0
8 = 6o
(2.51)
(2.52), (2.53)
With equations (2.40), (2.41), and (2.431), equation (2.51) can be written as
S(D, 6) = C[E] + (6 60)M + 0(5).
(2.54)
C is a fourthorder tensor called the elasticity tensor. With equation S(D, d) =
dy/(D, 6)
3D
and equation (2.52), equation (2.54) can be written
3D d=o 3D 0 = 0 3D D=0 0=0O 3D2
(2.55)
For any pair of symmetric tensors G and H,
32
G C[H] = ijf(aG + /JH, 0o)=/=o = C[G]
which shows the elasticity tensor is symmetric. Tensor M is called the stresstemperature tensor; since
S = ST, the stresstemperature tensor is also symmetric.
Expanding q = q(D, 0, g) in a Taylor series about (0, 0o,O) yields with equations (2.35), (2.36),
(2.40), and (2.44) through (2.46), q can be written
33
( \ ( \
q(D, 0, g) = q(o, 0,0) + (D 0) + \0 ! (0 0O) + (g) + h. o.
de
or
q(D, 6, g)p=o = Kg + h.o. = K V0. (2.56)
000
g=0
where K is the conductivity tensor. At (0, 00,0), K is
K = 
. 0q, g)
dg
g=0
(2.57)
This result requires that within an error of 0(5) the heat flux depends linearly on the temperature
gradient and is independent of the strain and the temperature. This result follows from the heat
conduction inequality; note that no argument concerning symmetry is involved.
The asymptotic form of the law of forces is equation (2.2.1)
VS + f = 0. (2.21)
If the noninertial body force, b, is introduced, then f becomes
f = b p ii. (2.56)
where p is density. Equation (2.21) becomes
VS + b = pii. (2.57)
The last equation needed to complete the classical linearized theory of thermoelasticity is a linear form
of the energy equation (2.16), 6l) = Vq + r. Given the relationship Tf = d), the time
derivative of T is
n =
dH(D, 0) t  dfj(D, 0).
dD
90
(2.58)
Since ff is smooth,
dD
dD
D = 0
9=0O
+ 0(1)
(2.59)
34
+ 0(1).
(2.60)
dfj{D, 0)
de
dfj{v, e)
de
D=0
e=e.
Utilizing equations (2.29),
d2y{D, 0)
du de
dw{ d, e)
de
(2.61)
d_
de
[s(d, 0)]
d_ \dy/{p, e)
de dn
a2y(p, 0)
dDdO
(2.62)
Therefore
07jf(D, 0) 0S(D, 0)
de
de
(2.63)
d\j/(D, 0) . dw(D, 0) dn d S
From equations S(D, 0) =  and tj(D, 0) =, it follows that = .
o D dO aD o 0
This is a reduced form of the Maxwell relation equation (2.17) which, when combined with equation
(2.53), gives
5tj(d, 0)
de
= M.
D=0
e=e0
By equations (2.19) and (2.381) through (2.384), the number
dq(D, 0)
C 0n
de
D=0
6=S0
(2.64)
(2.65)
is the specific heat corresponding to D = 0 and 8 = 60. Equations (2.58), (2.60), (2.64), (2.65),
(2.40), (2.432), and (2.48) give
077 = 0OM E + c0 + 0(5) (2.66)
This combined with equation (2.16) gives the asymptotic form of the energy equation
 Vq + 0OM E + r + 0(5) = c6 (2.67)
The basic equations for linear thermoelasticity theory are then
35
E = I (Vu + Vur) (2.42)
VS + f = p ii (2.57)
Vq + 0OM E + r + 0(<5) = cQ (2.67)
s = c[e] + (e e0)M (2.51)
q = K V0 . (2.56)
M and the values of C are symmetric. From equation (2.54)) it can be seen S = ST. strain vanishes, equation (2.54) reduces to When E = 0 and
S = (00o) M. (2.68)
Since the elasticity tensor, C, is invertible (Cs domain being restricted to the space of all symmetric
tensors), equation (2.54) can be solved for E:
E = K[S] + (0 0O) A (2.69)
where
K = and A = K[M]. (2.70), (2.71)
K and A are called the compliance tensor and the thermal expansion tensor, respectively. From
equation (2.69), A gives the strain resulting from a given temperature distribution when S vanishes:
e = (e e0)A = (e o0) k [m] . (2.72)
Material functions C, K, M, c, and p generally depend on the reference temperature G0, and do not
depend on temperature 6 Dependence on 6 is inconsistent with the assumptions leading to equation
(2.67). Unless the body B is homogenous, the material functions depend on the position of x in B.
Material symmetry effects the form of the elasticity tensor C, the stresstemperature tensor M, and the
conductivity tensor K. When a material is isotropic
and
2p E + A (ir E) 1, (2.731)
M = m 1, (2.732)
iH II (2.733)
36
The scalars p and X are the Lame moduli; p is the shear modulus, m is the stress temperature modulus,
and k is thermal conductivity. Combining the relations (2.73) with equations (2.42), (2.57), (2.66),
(2.54), and (2.56) gives the basic equations of linear thermoelasticity for an isotropic body:
E = (Vu +Vur) (2.74)
VS + b = pii (2.75)
 Vq + md0 tr E + r c8 (2.76)
S = 2 /i E + tr e) 1 + m(d 80) 1 (2.77)
q = kVG. (2.78)
It follows from equations (2.78) that in an isotropic body, the stress is equal to vanishes: a pressure when strain
S = m (d Q0) 1 when E = 0.
If p 5* 0 and 3X + 2p ^ 0, equation (2.77) can be inverted to give
1 X E = S ;  (tr S) 1 + a(0 0) 1 2 In (3A + 2 ft) V V 01 (2.79)
where P is the coefficient of thermal expansion
p= m 3A + 2 n (2.80)
Under isotropy assumptions, the thermal expansion tensor, A, becomes A = P 1. (2.81)
According to (2.79), the strain is equal to a dilatation when the stress vanishes:
E = P [d 60) 1 when S = 0.
2.7 OneDimensional Linear Thermoelasticity
Assume a homogenous and isotropic body bounded by a pair of parallel planes at x = 0 and x = 1.
Assume the body has a uniform stressfree reference state at absolute temperature 0O throughout.
Further assume no external supply of heat or external body force is available. All displacement is
measured from the reference state configuration and remains parallel to the xaxis for all times t. Under
37
these restrictions, equations for onedimensional linear thermoelasticity can be derived, or reduced,
from the more general forms given as equations (2.75) and (2.76). The equation of energy (2.76) is
 Vq + md0 tr E + r = c9.
By the assumptions, r = 0 and the trace of E becomes the strain rate. Recognizing from
dxdt
equation (2.78) that q= k VO and from equation (2.80) that the stresstemperature modulus is
m = p( 3A + 2p), the equation of energy becomes
k j = c + ft (3X + 2fi) 90
OX Ot
Similarly, the equation of motion (2.75) is given as
VS + b = pii.
Under the same restrictions and assumptions as before, b = 0. Also, the infinitesimal strain tensor E
becomes
E = I (Vu + Vur) = i [2 Vu] = V u = (2.83)
Equation (2.77) combined with equation (2.83) becomes
S = (2 // + A) ~ + m{0
The divergence of S for the onedimensional case becomes
VS (2 n + X) + m
dx2
where 0* denotes the temperature change from the reference state, (0 0O) The divergence of S in
equation (2.75), combined with the definition of the stresstemperature modulus,
m = P (3/1 + 2/t), becomes the equation 0/motion stated as
(2p+A) Jpf = [/3(3p+2A)] ^ + pu (2.84)
Introducing equations (2.85) and (2.86) for entropy, 77, and stress, a, respectively,
7] = (0 0O) + P{3A + 2p) ,
VQ V X
o)
99*
d2u
dxdt
(2.82)
38
a = (A + 2p) ^ /3(3A + 2/z)(0 60),
d x
combined with q = k VO, allow equations (2.82) and (2.84) to be written as
dq_
dx
do d2u
dx P dt2
The partial derivatives appearing in equations (2.82)
du du d2u
dx' dt' dxdt'
and (2.84)
d2u
dt2
(2.85)
(2.86)
are the strain, velocity, strain rate, and acceleration, respectively.
Day14 discusses using these relationships to transform the equations of energy and motion into the two
governing thermoelastic equations for a homogenous, isotropic body in 1space. The number of
constants in the equations is reduced by introducing a change of variables
x 
x
7
kt
t > 5,
cl2
9 >
u
u >
l
2. + Ifi
e0c
This change results in replacing the spatial interval [0,1] with a unit interval [0,1]; the reference
temperature B0 by 6 = 0. Use of the changed variables in equations (2.82) and (2.84) give equations
(2.87) and (2.88). These equations are a version of the two governing equations for linear, isotropic,
homogenous thermoelasticity for the half space x > 0:
and
d2d d9 I d2 u
= =  + Va 
dx2 dt dxdt
d2u 1 d9 d2u
T = Jd + b 7T
dx2 V dx dt2
(2.87)
(2.88)
The constants a and b are the coupling constant and the inertial constant, respectively. In the
formulation of equations (2.87) and (2.88), constants a and b are defined as
14 Heat Conduction Within Linear Thermoelasticity, William Alan Day, Springer Tracts in Natural Philosophy,
vol. 30, SpringerVerlag, 1985, pg 15.
39
60p2{3X + 2m)2
c (A + 2fij
b =
k2 p
c2 (A + 2p) l2
The coupling constant, a, is usually small compared to unity and is independent of the thickness of the
halfspace. The coupling constant is a measure of the coupling between thermal and mechanical
effects. It can also be divided into two factors, one the inverse of the other; one factor describes the
effect of the thermal field on mechanical response, the other factor describes the effect of the
mechanical response on the thermal field. As will be discussed later, the inertial constant, b,
approaches zero for metals and is usually disregarded. Given the specific change in variables which
result in equations (2.87) and (2.88), the definitions of heat flux, q, entropy, t, and stress, a, become
d6 I du
, du I
and a = \ad.
dx
If constants a and b are set equal to zero, and if displacement at the boundaries x = 0 and x = 1 is
constrained such that
the equations of the onedimensional version of Fouriers theory are obtained. In this theory,
temperature is a solution of the heat equation
d20 d6
dx2 dt
(2.89)
and displacement vanishes. As a comparison, if a and b are considered positive and displacement
between the thermoelastic equations is eliminated, temperature becomes a solution of the fourthorder
equation
d40
dx4
+ b
d30
dt3
(l + a)
d3e
dx2dt
+ b
d40
dx2dt2
(2.90)
Equation (2.90) is the onedimensional, homogenous, and isotropic version of Carlsons temperature
equation as given in his explanation of dynamic thermoelasticity.15
Two simplifying assumptions can be made to facilitate the solution of actual problems. First, the term
(o M e) in the equation (2.67) can be neglected; the resulting theory is referred to as uncoupled.
Second, the term (pii) can be neglected in equation (2.57) resulting in the quasistatic theory of
15 Handbuch derPhysik, D. E. Carlson, Vol. VIa/2, SpringerVerlag, 1972, pg. 328.
40
thermoelasticity; in practice, the quasistatic assumption applies best when changes of temperature
proceed comparatively slowly. Even though the term (pii) is canceled, the functions involving u and 0
remain functions of time, t. This distinguishes the quasistatic approach from the stationary approach.
Typically, both of these approximations are made together giving the quasistatic uncoupled theory of
thermoelasticity. For reasons discussed in Chapter 4, the analysis performed here adopts the coupled
quasistatic approach to thermoelastic theory.
41
NOMENCLATURE USED IN CHAPTER 3
Symbol Definition
hi
x
L
T
I
R
r
t
P
A
^calculated
^coupled
T(x,t)
T(r,z,t)
cp
k
P
a
P
Â¥
Heat convection coefficient for the ith surface
Spatial coordinate for semiinfinite analysis
Thickness of semiinfinite plate or slab
Ambient air temperature
Electrical current
Electrical resistance, region of space
Radius of experimental cylinder, radial coordinate
Time
Power, calculated from i2R
Area of experimental cylinders vertical surface,
Heat flux calculated from P/A where P = i2R
Heat flux coupled to test cylinder
Temperature as a function of time and onedimensional space
Temperature as a function of time and cylindrical coordinates r and z
Specific heat at constant pressure
Thermal conductivity
Density
k
Thermal diffusivity, a = 
cvP
Coefficient of thermal expansion
Any eigenfunction generally
Pm Eigenvalue(s) summed over m
Tp Eigenvalue(s) summed over n
Si ith boundary surface
Sb Chemical symbol for antimony
X(pm,x) Eigenfunction related to a specific boundary condition
H, Parameter, H = h/k
J0 Bessel function of the first kind, order 0
AT Difference in temperature, specifically temperatures measured by different
thermocouples
ATcu Difference in temperature between the center and edge of a face of the copper
cylinder
ATSb Difference in temperature between the center and edge of a face of the antimony
cylinder
3. Analytical Modeling of Heat Flow through Test Cylinders
3.1 Introduction
Modeling heat flow across a diffusion bond between dissimilar metals requires an understanding of
how heat flows through each separate metal separately. Such modeling can be done either analytically
or numerically. The accuracy of various modeling techniques can be assessed by comparing analytical
42
and numerical models with experimental data and with each other. Experimental data developed using
simple, carefully controlled geometry and pure materials provide the best basis for these comparisons.
Boundary conditions can be ascertained and the correctness of material properties of the different test
metals determined. Two analytical methods are used to model heat conduction through a solid
cylinder. These are semiinfinite and finite formulations. In the former, temperature is calculated as a
function of distance and time, T(x,t); the latter calculates temperature as a function of radius (r),
thickness (z), and time, i.e., T(r,z,t). The results of the semiinfinite and finite models are compared
with experimental data.
Experimental data for cylinders of copper, aluminum, and antimony are presented. Copper and
aluminum are of primary interest since the bonded composites used in the experiments and analyses
presented in chapters 5 and 6 are made of these metals. Also, it was possible to obtain very pure
examples of both these metals as described in chapter 1. The data obtained from the antimony is of
secondary value and not presented in detail. It is useful, however, to examine how successfully the
models used for copper and aluminum (both of which have high thermal conductivity) could predict
heat transfer through antimony possessing low thermal conductivity.
Schematic of SemiInfinite Experiment
Figure 3.1
Figure 3.1 illustrates the configuration of the heat conduction experiments involving cylinders of single
metals. The heat flux, q, is supplied by an adhesive resistance heater placed on the x = 0 surface. The
relative orientation of the Macor collar and RTV potting is shown. An adhesive thermocouple by
which measurements of temperature are taken is placed on the x = L surface of the test cylinder.
Boundary conditions and an initial condition typical of a semiinfinite analysis are also shown. T_
denotes the temperature of the surrounding atmosphere, k is the thermal conductivity of the test metal,
hi and h2 are the convection coefficients corresponding to x = 0 and x = L surfaces of the test cylinder
respectively. These coefficients important in approximating two of the boundary conditions in a semi
infinite analysis (see equations 3.2and 3.3).
Cylinders were machined of each of the three test metals. The test cylinders of copper and aluminum
each had a diameter of .03175 .00005 m and a thickness of .00635 .00005 m. The cylinder of
43
antimony (Sb) was machined using an electronic discharge machining method (EDM) due to its
brittleness. The diameter and thickness of the antimony was .03048 .00005 m and .0165 .00005 m,
respectively. Siliconbased RTV potting was used to 1) support the samples without constraining the
samples dimensionally, 2) provide consistent insulation on the radial surface of all samples tested, and
3) provide some damping of incipient vibration. Section 3.2 discusses the first model is based on a
semiinfinite formulation. Figure 3.2 shows the configuration of a AlCu composite plate potted into its
Macor collar; an example of the adhesivebacked resistance heater used to supply q is also shown.
RTV Potting
Resistance
Heater
Macor
Collar
AlCu Bonded
Composite
(Cu Face
Showing)
Los Alamos
Configuration of Potted AlCu Composite Plate
Figure 3.2
3.2 Derivation of Formula for T(x ,t)
The first modeling of heat transfer through a test cylinder was done using a formulation for a semi
infinite slab. The coordinate system has one spatial variable, x. Since the analysis is done on the half
space, the problem is for a slab with a thickness of L. The following are the governing equation,
applicable boundary conditions, and initial condition:
d2T 1 dT(x, t)
dx2 a dt
in 0L, t>0 (3.1)
44
hxT = {q h{T) = f:{t)
at x = 0, t > 0
(3.2)
+ hj = h,T = f2{t)
at x = L, t > 0
(3.3)
T(x, t) = F(x)
for
0^x>L att = 0.
(3.4)
Since the boundary conditions are not equal to zero, the problem is nonhomogeneous. The solution
employs an integraltransform approach based on the following eigenvalue problem:
V2 v^(r) + A.2v^(r) = 0 in a region R, and
t M*i)
1 dr];
+ hjifffc) = 0 on the boundary of Sj.
d .
R is the region of interest, i]/m (rj) are eigenfunctions, is the normal derivative in the outward
dni
direction at the ith surface, i is the number of continuous boundary surfaces, S,, of region R (the x = 0
and x = L surfaces in the semiinfinite formulation). An integraltransform pair is constructed allowing
reduction of equation (3.1) to an ordinary differential equation. Similarly, the initial condition is
reduced from a function of x and t to a function of one eigenvalue alone. A full explanation of this
procedure is given in appendix 1 for the case of a finite cylinder. The solution for the semiinfinite
problem is stated directly as
tU. >) = 1 %j <*[%,)+Â£ * Mi. <)
(3.5)
where
A{Pm, t') = a
X{Pm,x)
AM +
x=0
x= L
and
p(pm) = r x(f}m,x')F(x) dx'.
X(J5m, x) is an eigenfunction, N(Pm) is a normalization integral, and pm are eigenvalues. Particular
values for X(/)m, x) and N(Pm) are taken from Ozisik16. The final form of the functional
representation for T(x,t) for the specific boundary conditions of the experiment is shown as equation
(3.6):
16 Heat Conduction, M. Nectar Ozisik, 2nd Edition, Wiley & Sons, New York, 1993, pp. 4849.
45
T(x, t) = 2^
PmCos(pmx) + H{Sin(pmx)
m 1
te + ?)
L +
H,
+
r0c_^'pmCos(pnx) + HlSnfaKx')dx'
(3.6)
+ ^(i e0*) + + //,5rn(i3mL
1 e
1
In equation (3.6), Hi and H2 are constants given by the ratio of thermal conductivity to the heat transfer
convection coefficient for the ith surface of the region, 0<*;>L. The eigenvalues 3m are the positive
roots of equation (3.7)
tan Pm L =
+ ^2)
 hxh2
(3.7)
3.3 Modeling T(x,t) in a SemiInfinite Slab
The first measurements of heat transfer and temperature through a copper slab were made using a
resistance heater rated at 264 ohms. Tests were conducted using varying power levels including 2 W, 5
W, and 8 W. The 8 W test is discussed here to allow a basis for comparison since an 8 W measurement
were also made using the aluminum and antimony test pieces. A .188 ampere current at 44.3 V
produced an 8.0 W load. This, when divided by the 0.00317 m2 surface area of the copper cylinder,
produced a qcaicuiated of 2519 W/ m2. Two type K thermocouples were used to experimentally determine
temperature; the center thermocouple was placed at coordinates x = L in the center of the cylinder. The
second thermocouple was placed at the outside diameter of the cylinder on the x = L surface.
Temperature measurements indicated the copper cylinder had a uniform initial temperature of 25.1 C.
The data from the thermocouples was edited in an Excel file and this file was in turn collated and
graphed using Mathematica program JCUM7.DAT. Figure 3.3 gives a listing of Mathematica program
JCUM7.DAT; the output of JCUM7.DAT is shown in figure 3.4.
(*** Program JCUM7 ***)
(*** Records and Plots Thermocouple Data ***)
!!a:\jalm7.csv
a = Flatten[ReadList["a:\jcum7.csv",
Number,RecordLists>True,RecordSeparatois >( "\n",'',"}]];
b = Partition[a,2];
c = Transpose^];
dl=c[[l]];
d2 = c[[2]];
Lengthfdl]
46
x = Range[236] 10/60//N;
y = Tcanspose[{x,dl}];
yl = Transpose[{x,d2)];
picl = ListPlot[y, PlotJoined>True,PlotRange>{20,70},GridLines>Automatic,
DisplayFunction>Identity]
pic2 = ListPlot[yl, PlotJoined>True,PlotRange>{20,70},GridLines>Automatic,
DisplayFunction>Identity]
Show[picl,pic2,DisplayFunction>$Display Function]
Listing of Mathematica Program JCUM7.DAT
Figure 3.3
At approximately t = 1700 seconds the exterior thermocouple detached from and was restored to the
surface of the copper cylinder, as can be seen in figure 3.4. Figure 3.4 shows T(x=.00635,t) as
measured using 8W of power to heat the copper test cylinder.
0
500
1000
1500
2000
Time (s) 
T(x,t) for Measurement for Copper Cylinder, L = .00635 m
Figure 3.4
T(x,t) calculated using Mathematica program CPPR1, the listing of which is shown as Figure 3.5. The
first attempt at modeling T(x,t) employed convective heat transfer coefficients hi = h2= 10 W/m2 C, a
standard value for h for a vertical plate under free convection in atmospheric air. On this basis,
comparison of T(x,t) as calculated and T(x,t) as measured indicated that either 1) values used for hi
and/or h2 were incorrect, 2) the heat flux through the cylinder was incorrect, i.e., the 8 W created by the
resistance heater was not in fact entirely transferred to the test cylinder, 3) the thermal properties used
for pure copper were incorrect, or 4) some combination of all of these. That the published values for
thermal conductivity and thermal diffusivity of pure copper could be in error to the degree necessary to
create so significant a discrepancy was considered unlikely. The error seemed likely to be a
combination of 1 and 2.
(*** SemiInfinite Plate Problem ***)
(*** 21596 ***)
(*** Copper ***)
(*** Comparison Run, Semi to Measurement, 31697 ***)
(*** Physical Constants ***)
Alphal = .00011234
47
L = 0.00635
To = 25
Tinf = 22
k = 384
hi = 12
h2= 12
g = 2519
HI =hl/k
H2 = h2/k
fl = g (Tinf hi)
f2 = h2*Tinf
lint = 3
(*** Initialize Eigenvalues ***)
Array [Betal,lmt]
(Betal [1 ] .Betal [2],Beta 1 [3], Betal [4],Betal [5],
Betal [6],Betal [7],Betal [8],Betal [9],Betal [10] }=
{3.73401,494.76718, 989.49209, 2968.44008221, 3957.91656364,
4947.39365295, 5936.8710462, 6926.34861313,7915.82628861,
8905.30403645)
(** Formulation of the Temperature Function **)
Num = (Betal[m] Cos[Betal[m] x]) + (HI Sin[Betal[m] x])
Deni = (Betal [m]A2 + H1A2)
Den2 = L + (H2/(Betal[m]A2 + H2A2))
Den = (Deni Den2) + HI
Expo = Exp[Alphal Betal [m]A2 t]
Intgmd=(Betal[m] Cos[Betal[m]*xp]) + (HI Sin[Betal[m] xp])
Int 1 =Integrate [Intgmd, {xp,0,L) ]
Mull = To Expo Inti
Mul2=(fl/(k Betal [m])) (1 Expo)
Mul3=(f2/(k Betal [m])A2) ((Betal [m] Cos[Betal[m] L]) + (HI Sin[Betal[m] L])) (1 
Expo)
(** Construct T(x) and Sum **)
Tfunc = (Num/Den) (Mull + Mul2 + Mul3)
Tsum = Sum[Tfunc,{m,l,lmt)]
Txt = 2 Tsum
(** Post Process T(x,t) **)
N[Txt/.{ t > 0, x > L},12]
plal2d=Plot[Txt/.{x > L), {t,0,2400), Frame > False,
GridLines > Automatic,
AxesLabel > {"time (s)", "temp (C)"},
PlotRange > {20,70),
PlotLabel > "h = 20,14, z = .006; q=2519"]
Listing of Mathematica Program CPPR1
Figure 3.5
A number of calculations of T(x,t) were done using varying values of qi...u,.^. hj and h2. The first
calculation employed a nominal qcaiCuiated for the 8W experiment of 2519 W/m2. This nominal qCaicuiated
when combined with hi = h2 = 12 W/m2 C, approximated the measured data for T(x,t) poorly. Many
more calculations were performed using various combinations of qcaicuiated> hi, and h2. The calculations
best approximating T(x,t) as measured are shown in figure 3.6. These approximations all hold ht and
h2 constant at 12 W/m2 C and are compared to measured data in the inset of figure 3.6.
48
Temp (C) 
Comparison ofT(x,t) with varying q. hj= h2= 12 W/m2 C
Figure 3.6
Calculations based on hi = h2 = 12 W/m2 C and 1800 W/m2 < qcaicuiated 2200 W/m2 provided
results most closely approximating measured data. The approximations appearing in figure 3.6
provided the first evidence not all of the heat flux produced by the resistance heater was coupling with
the copper test cylinder. This assumption was equivalent to increasing the convection from the
resistance heater to 20 W/m2 C <; ht > 26W/m2oC. For analyses of T(x,t) throughout this work,
it was necessary to use a reduced figure for q or a large figure for hi if calculations were to agree with
measurement. An arbitrarily reduced q was chosen and is designated as qCOupied throughout this work.
Subsequent calculations of T(x,t), and later T(r,z,t), for aluminum and antimony consistently yielded
good approximations when the same reduced figure for qcupied was used for them all. These consistent
results were considered some validation for the assumption of resistance heater coupling loss.
The first measurements of heat transfer through the aluminum test cylinder employed the same
resistance heater as was in the copper experiment. As before, the heater was found to have a resistance
of 264 ohms A .188 ampere current at 44.3 V produced an 8.0 W load. This, when divided by the
0.00317 m2 surface area of the cylinder, produced a qCakuiated of 2519 W/ m2. Two type K
thermocouples were used to experimentally determine temperature; the center thermocouple was placed
at coordinates x = L in the center of the cylinder. The second thermocouple was placed near to the
outside diameter of the cylinder on the x = L = .03170 m surface. The aluminum cylinder was assumed
to have a uniform initial temperature of 21.2 C, this being the initial temperature measured at three
locations on the test cylinder prior to the experiment. Mathematica program CPPR1 (with material
properties and eigenvalues appropriate for aluminum) was used to compute T(x,t). The measured data
for temperature given by two thermocouples located at (r = 0, z = .00635 m) and (r = .03170 m, z =
.00635 m) are shown in figure 3.7.
49
Comparison ofT(x,t) with Measured Data for Aluminum
Figure 3.7
Proceeding from the assumption that approximately 12 to 17 per cent of qcaiCuiated is lost, or uncoupled,
from the aluminum test cylinder, figure 3.7 shows two plots of T(x = .00635 m ,t). Since the
calculations of T(x,t) closely approximate the measured data, the plots are given as dashed lines for
clarity. One plot is based on convection coefficients of hi = h2 = 15 W/m2 C and a qCOuPied of 2200
W/m2; the second plot is based on hi = h2 = 14 W/m2 C and a qC0Upied of 2100 W/m2. The semiinfinite
numerical approximations of T(x,t) are quite accurate, given the chosen boundary conditions and the
assumption of some heat loss, i.e., the difference between qcaicuiated and qCOupied
3.4 Formula for T(r,z,t) in a Finite Cylinder
Modeling heat transfer through a finite cylinder provides a useful comparison to and check of the
results of the preceding analysis of a semiinfinite slab. The boundary conditions, position of
thermocouples, and spatial coordinates used for the formulation of T = T(r,z,t) which follows are
shown in figure 3.8. A coefficient for heat transfer, h3, is added corresponding to the radial surface at r
= b.
50
Boundary Conditions and Spatial Coordinates for Finite Cylinder
Figure 3.8
The solution for temperature in a finite cylinder as a function of r, z, and t is given as equation (A 1.43).
The coordinate system is cylindrical, i.e., T = T(r,
, T(r,(p,z,t) reduces to T(r,z,t). Equation (A1.43) is stated directly since its derivation is shown fully in
appendix 1.
T{r, z, t) = 4 X
Jo{Pmr)(ilp cos T]pz + Hx sin z)$
m1p = 1
[j0{Pmbf\(Hl + /?2) (172 + H*)
Ho
c +
% + H.
+ Hy
c b
z' = 0 r' = 0
(A 1.43)
+ 0})
c
hMh J nP cos 7]pz'+H1 sin z' dz'
z=0
fnP (% cos Tjpz + Hx Sin z)f2
+ ~~a J\ W>) + B J1
Hn rra
The specific boundary conditions and initial condition appearing in equation (A 1.43) are given in
appendix 1 in equations (A1.2) through (A1.6). As discussed in appendix 1, the eigenvalues associated
with hi and I12 are the roots of the transcendental equation
51
(3.8)
single metal, k3 equals k2. With k\ = k2, eigenvalues rip resulting from equation (3.8) relate directly to
the average of hx + h2. For example, eigenvalues tip depending on h3 = 20 W/m2 C and h2 12 W/m2
C are identical to eigenvalues tip depending on h\ =16 W/m2 C and h2 = 16 W/m2 C. Eigenvalues
Pm are the positive roots of equation (3.9)
H3 is the ratio of 113/k. As may be inferred by equations (3.8) and (3.9), eigenvalues pm and Tprelate to
the z and r coordinates respectively. A program performing a numerical calculation of T(r,z,t) for a
finite cylinder of copper is done by Mathematica program CPPR, the listing for which appears in figure
A check of numerical accuracy was made for three points on the test cylinder to determine how well
the CPPR program could calculate the initial temperature of 25.0 C at T(r=0, z=0, t=0) T(r = 0, z = c,
t =0), and T(r = b, z = c, t =0). It is an assumption of the analysis that the initial temperature of the test
cylinder is consistent and uniform throughout. The location of thermocouple 1 (see figure 3.7) was set
at (r = 0 m, z = c = .003175 m); the location of thermocouple 2 was set at (r = b = .03170 m, z = c =
.003175 m). Measured data for the 8 W experiment showed the initial temperature at these two
locations to be 25.0 C and 24.9 C, respectively. Table 3.1 shows the calculated values for T(0,c,0)
and T(b,c,0); the first
(*** Finite Cylinder Problem, T = f(z,r,t) ***)
(*** Cu ***)
(*** Physical Constants ***)
Alphal = .00011234
c = 0.00635
b = 0.03175
To = 25
Tinf = 22
k = 384
hi = 15
h2= 15
h3 = 15
Hll = hl/k
H22 = h2/k
H33 = h3/k
q = 2100
fl = q (hi Tinf)
f2 = Tinf h2
f3 = To h3
lmt = 3
tlim = 20
templim = 50
PmJ0{Pmb) + H3J0{Pmb) = 0.
(3.9)
3.9.
52
(*** Initialize Eigenvalues ***)
Array[Betal,lmt]
{Betal [1 ],Betal [2],Betal [3], Betal [4],Betal [3],
Betal [6],Betal [7],Betal [8], Betal [9],Betal [10] }=
{1.568,120.70404,220.97449,320.42675298,
419.64579504,518.76000318,617.82231662,716.853052501,
815.86368880,914.86074215}
Array[Etal,lmt]
(Etal [1 ],Etal [2],Etal [3],Etal [4],Etal [5],
Etal[6],Etal [7],Etal [8],Etal [9],Etal [10]} =
[3.507,494.75889,989.48795,1484.222528,1978.960147,
2473.69831855,2968.4367665,3463.17537233,3957.91407685,
4452.65284716}
(** Formulation of the Temperature Function **)
Numeta = (Etalfp] Cos[Etal[p] z]) + (Hll Sin[Etal[p] z])
Numbeta = Betal [m]A2 BesselJ[0,Betal[m] r]
Denleta = Etal [p]A2 + HI 1A2
Den2eta = (c + (H22/(Etal [p]A2 + H22A2)))
Deneta = (Denleta Den2eta) + HI 1
Deni beta = (Bessel J[0,Betal [m] b])A2
Den2beta = (H33A2 + Betal [m] A2)
Denbeta = Denlbeta Den2beta
Den = Deneta Denbeta
Num = Numeta Numbeta
Expo=Exp[Alpha 1 (Etal[p]A2 + Betal [m]A2) t]
Inteta=(Etal[p] Cos[Etal[p]*zp]) + (Hll Sin[Etal[p] zp])
Intbeta = rp BesseU[0,Betal[m] rp]
Intgmd = Inteta Intbeta
lntl=lntegrate[lntgmd,[rp,0,b},{zp,0,c}]
N[Intl,8]
IntCond = To Expo Inti
(** Add Boundary Conditions **)
Expnt = 1 Expo
Eigen = (Betal [m]A2 + Etal [p]A2)
TranB = Integrate[lnteta,(zp,0,c}]
TranE = Integrate[Intbeta,(rp,0,b}]
Mull = (((b*BesselJ[0,Betal[m]*b])/(k*Eigen)) f3) Expnt TranB
Mul2 = ((Etal [p]/(k*Eigen)> fl) Expnt TranE
Mul3num = ((Etal[p]*Cos[Etal[p]*c]) + (Hll Sin[Etal[p]*c])) f2
Mul3den = k Eigen
Mul3 = (Mul3num/Mul3den) Expnt TranE
(** Construct T(r,z,t) and Sum **)
Tfunc = (Num/Den) (IntCond + Mull + Mul2 + Mul3)
Tsum = Sum[Tfunc,(m,l,lmt},{p,l,lmt}]
N[Tsum,8]
Trzt = (4/bA2) Tsum
N[Trzt/.(r>0, t> 0, z > 0},8]
N[Trzt/.[r>0,t> 0, z > c},8]
N[Trzt/.{r>b,t> 0, z > c},8]
(** Post Process Trzt *)
pi 11 =PIot[Trzt/. [ r >0,z > 0.00635), {t.0,2400],
PlotRange> {20,70} ,GridLines> Automatic,
PlotLabel > "k=384;hs=12,12,h3=30,q =2200,c=.006"]
Mathematica Program CPPR
Figure 3.9
53
calculation was based on a summation over one eigenvalue for both Pm and Tp; the second calculation
was based on a summation of three eigenvalues for both Pm and T]p. Measured initial temperatures for
each location appear in parentheses beside the calculated initial temperatures. As can be seen from
table 3.1, a summation over even one eigenvalue yields calculated results well within the accuracy of
measured data. A summation over three eigenvalues permits the calculated value to approach the
measured value even more closely. Throughout this work, temperatures calculated from equations
(3.6) or (A1.43) result from summations over three eigenvalues unless otherwise noted.
Number of Eigenvalues Calculated Values for T(0,0,0) (C) Calculated Values for T(0,c,0) (C) Calculated Values for T(b,c,0) (C)
1 25.0198 (25.0) 25.0198 (25.0) 24.9785 (25.0)
3 25.0034 (25.0) 25.0034 (25.0) 24.9935 (25.0)
Comparison of Calculated and Measured Initial Temperatures
Table 3.1
3.5 Modeling T(r,z,t) in a Finite Cylinder
A number of attempts were made to model the observed change in temperature using equation (A 1.43)
and the CPPR program shown in figure 3.9. Values for ht and h2 were chosen arbitrarily as 15 W/m2 C
or 16 W/m2 C based on the previous success in the semiinfinite slab formulation, figure 3.7. Similarly, qcoupied
was chosen as 2100 W/m2 or 2200 W/m2. Given that hi, h2, and qcoupied were essentially fixed, h3 was the
only remaining variable that could be changed. The CPPR algorithm was run many times using values
for h3 ranging from 5 to 100 W/m2 C. The most successful combinations of hi, h2 qcoupied> and h3 are
shown in figure 3.10. As can be seen in this figure, approximations where hi = h2= 15 W/m2 C, h3 
30 or 35 W/m2 C, and qcalculated = 2100 W/m2 were generally good, especially for values of t ranging
from 0 1500 s.
Comparison of Calculations ofT(r,z,t) with Measured Data
Figure 3.10
54
It is useful when modeling heat flow through a finite cylinder to investigate the influence of
convection from the vertical surfaces as compared to conduction from the radial or radial surface.
Figure 3.11 shows T(r,z,t) for a qcoupied of 2200 W/m2 for three values of hi and h2 while h3 is held
constant at 30 W/m2 C.; hi and h2 are as shown. The temperature curves shown in figure 3.11 are
each comprised of a solid line and a dashed line. The solid line is T(r,z,t) = T(0,.00635,t), i.e.,
calculates the temperature in the center of the z = .00635 m face. The dashed line represents T(r,z,t)
= T(.03175,.00635,t), i.e., calculates the temperature at the position of the exterior thermocouple. As
a consequence of the high thermal conductivity of copper, the solid and dashed lines nearly coincide
for all times t.
As can be seen from figures 3.11 and 3.12, changes in convection from the vertical surfaces tend to
manifest themselves early in the time history of T(r,z,t). Conduction from the circumferential surface
tends to determine the time needed to achieve equilibrium. Convection from all surfaces evidently
contribute to the magnitude of T(r,z,t) in the steady state. As can be seen in modeling of T(r,z,t) in
succeeding sections, convection from the vertical surfaces effects the slope of the temperature curve
primarily in the first 300 500 seconds. The speed with which equilibrium is achieved and the
ultimate magnitude of the steady state temperature is more a function of conduction from the
cicumferential surface of the finite cylinder. Figure 3.12 shows T(r,z,t) as a function of different
convection coefficients applied to the r = b surface.
0 500 1000 1500 2000
Time (s)
T(r,z,t) as a Function of Differing Vertical Convection Rates
Figure 3.11
55
0 500 1000 1500 2000
Time (s)
T(r,z,t) as a Function of Different Convection Coefficients Applied to the r = b Surface
Figure 3.12
3.6 Characterization of the Temperature Difference (AT) on the Vertical Face
Figure 3.7 shows the temperature curve measured for an aluminum cylinder given an 8W power load.
Figure 3.4 shows a temperature curve for a copper cylinder for the same 8W power load. These curves
result from simultaneous measurements from two thermocouples: one placed at the center of the
vertical face each test cylinder and one placed near the edge of the vertical face. For early times t the
temperatures measured by the thermocouples nearly coincide. Though less obvious in figure 3.4, in
both cases a separation in the curve is apparent for later times t. A measurable difference in
temperature was observed between the center and edge of both cylinders as they approached thermal
equilibrium.
Formulation of T(r,z,t) for the antimony test cylinder was done using the experimentally determined
value for thermal conductivity reported in chapter 1, i.e., ksb = 15.2 W/m C. The measured data in
figure 3.14 show two distinct values for T(r,z,t) related to the placement of the two thermocouples.
Thermocouple one was placed at coordinates r = 0 m, z = .0165 m; thermocouple 2 was placed at r =
.03048 m and z = .0165 m. It is evident that the magnitude of the measured (and calculated) difference
between T(0, .0165, t) and T(.03048, .0165, t) is a strong function of thermal conductivity. Data for
temperature from thermocouples placed similarly on the aluminum test cylinder are shown in the inset
of figure 3.14.
As can be seen in the inset of figure 3.14, thermocouple 2 (placed at r = .03040 m, z = .0165 m)
records cooler temperatures than the thermocouple 1 (placed at r = 0 m) for both aluminum (at t > 200
s) and antimony (at t > 750 s). The AT across the z = c face is smaller for aluminum than for
antimony. A similar AT exists for copper though smaller in magnitude. The AT for copper can be seen
in figure 3.4; it is so small, however, that the plots for T(0, .003175, t) and T(0.03175, .003175, t)
56
almost coincide. Thus empirical evidence indicates the observed AT between the center and edge of
the test cylinders increases as the thermal conductivity of each test metal decreases.
Experimental Configuration for Determination of AT onz = .0165 m Face
Figure 3.13
57
0 500 1000 1500 2000
Time (s)
Comparison of Calculated and Measured T(r,z,t) for Antimony Cylinder
Figure 3.14
The measured temperature data from thermocouples 1 and 2 indicate AT approaches a constant as heat
transfer through the test cylinders approaches steady state, as should be expected. In the case of the
antimony cylinder, the measured ATsb approached a constant value beginning at approximately t =
2000 seconds. From t = 2000 s and later, measured ATSb equals 1.0 C varying from this by .1 C at
most. Experimentally, temperature measured at location (r = 0, z = .0165, t = 2290 s) was 50.4 C;
temperature measured at location (r = .03048 m, z = .0165 m, t = 2290 s) was 49.5 C. The measured
ATSb was therefore .9 C.
Figure 3.15 is the graph of T(r,z,t) at t = 2290 s with hi = h2 = 15 W/m2 C and h3 = 25 W/m2 C.
Comparing this with figure 3.14, the discrepancy between the calculated and measured temperatures at
the locations of the thermocouples is apparent, being on the order of 4.0 C. This is an indication that
the model represented by equation (A1.43) is somewhat inaccurate in calculating T(r,z,t), especially for
later times t. Figure 3.15 calculates the temperature profile over the entire antimony test cylinder based
on equation (A 1.43). Time at t = 2290 s is chosen because ATSb is greatest for late times t and
therefore more easily graphed. ATsb at t = 2290 s was calculated as approximately .77 C compared
with a measured ATSb of .9 C at that time. Though the prediction of T(r,z,t) is not completely
successful for the antimony cylinder, calculation of ATSb compares well with measured values. ATCu as
calculated for the copper test cylinder is shown in the inset of figure 3.15 for comparison. The smaller
magnitude of ATCu is attributed to the higher thermal conductivity of copper.
58
0.015
Calculated T(r,z,t) for Sb Cylinder at t = 2290 s
Figure 3.15
3.7 Comparison of Calculated and Measured T(x,t) for t Â£ 120 Seconds
The various calculations of T(x,t) and T(r,z,t) presented in preceding sections have been carried out
for all times through thermal equilibrium. Since a characterization of a transient thermoelastic
response was considered as potentially necessary, an assessment of the accuracy of the calculation of
T(x,t) given by equation (3.5) for shorter times t was done. Figure 3.16 compares a measurement of
T(x = .003175, t) for times 0 < t ^ 120 seconds with a calculation of T(x = .003175, t) using
Mathematica program CPPR1. The calculation of T(x = .003175, t) in figure 3.16 is based on a
copper cylinder with a qCoupied = 2100 W/m2, h, = h2= 12 W/ m2 C. The measurement of T(x,t) was
done at 10second intervals. Prediction of T(x,t) based on equation (3.5) is reasonably good.
59
Temp (C)
Comparison of Calculated and Measured T(x,t) for 0 < t > 120 Seconds
Figure 3.16
4 Analytical Method for Characterization of Heat Conduction Across a Bonded Interface
4.1 Introduction and Statement of the Problem
If a plate is initially at rest and at a uniform constant initial temperature, introduction of heat to the
plate will create a flow of heat accompanied by fields of stress, strain, and temperature. A description
of this heat flow, as well as the fields of temperature, stress, strain, displacement, etc., that it creates is
available through the theory of linear thermoelasticity set forth in Chapter 2. As in that chapter,
assumptions of isotropy and homogeneity apply. This analysis is done for the halfspace.
The method developed in this chapter calculates the transient and steadystate temperature profiles
across a bonded metal interface. Having given the derivation for the formula to calculate 6(x,t), the
author was successful in writing a Mathematica program (figure 4.4) to calculate the steadystate
temperature profile given any arbitrary film coefficient at the bond interface. Work was done provide
the means to calculate the transient changes in temperature as well. Successful calculation of the
transient temperature profile proved more difficult and could not be obtained. This was probably due
to the inability to find the roots of G in equation (4.78) possessing sufficiently large imaginary
components to allow computation of . The method developed in this chapter has utility and
oz
60
generality and (in the authors opinion) possesses a certain mathematical elegance. For these reasons it
is included here.
As discussed in Chapter 2, the treatment of thermoelastic problems fall into several categories
depending on which different assumptions and restrictions are adopted. The coupled thermoelastic
theory is used for this analysis; details of this theory are given later in this chapter. Much of the
method used in this chapter is taken from work done by Atarashi and Minagawa.17 This method
incorporates film coefficients between the layers of a composite plate permitting modeling of the
thermal resistance usually encountered between bonded plates. Much of the analysis that follows is
given for bonded, composite semiinfinite plate having two layers.
Figure 4.1 shows a plate consisting of n parallel layers, labeled 1 to n from left to right. The spatial
coordinate is x in a direction perpendicular to the layers of the plate. Each interface between layers is
designated jcj for the jth layer (j = 1,2,..., Nl). The atmospheric temperature to the left of the
composite plate is TA atmospheric temperature to the right of the composite is TB. T0 is the initial
temperature of the plate and is uniform throughout. Theta, 9, is temperature deviation from To; Uj is the
displacement of the jth layer. This analysis is onedimensional and all field variables are functions of x
and t alone.
nlayered Bonded Plate
Figure 4.1
17 Transient CoupledThermoelastic Problem of Heat Conduction in a MultiLayered Composite Plate, T.
Atarashi and S. Minagawa, International Journal of Engineering Science, Vol. 30, No. 10,1992, pgs. 1543
1550,
61
4.2 Derivation of the Governing Equations
For the jth layer of the bonded metallic composite plate, the governing equations are:
da j d2Uj
 = p (equation of motion)
dx dt2
dx
ds i
 + T j
O dt
ddj
dx + qj =
du:
Oj c, y jd (Constitutive equation)
dx
du j
Sj = Yj (Constitutive equation)
OX
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
where CTj is the normal component of stress in the direction perpendicular to the jth layer, qj the heat
flux, Sj the entropy density, pj the mass density of the jth material, kj the thermal conductivity and
Cj = I, + 2fij; yj = (3A} + 2p; )/3>; and y/ =
where X,j and ij the Lame constants, (ij the coefficient of thermal expansion, and Cj the heat capacity of
the jth material under constant volume.
Taking the derivative of equation (4.4) with respect to the space variable x gives
 r  V
dx dx2 Yj dx
Combining equations (4.6) and (4.1) gives
d2Uj ddj d2u
ci TT = y> T + Pj TT
(4.6)
(4.7)
Taking derivatives of equation (4.5) with respect to time and equations (4.2) and (4.3) with respect to
the space variable
ds: d2Uj dd i
~dT = Yj dxdt ~ Â¥j ~dT
(4.8)
62
(4.9)
dsj _ 1 dqj
dt T0 dx
d_i
dx
From equations (4.9) and (4.10)
dt
1 dq} 1 d20
T dx 1 Ta dx2
Combining equations (4.8) and (4.11) and dividing by pjCj gives
kj d20
p^7 *T
f Y> '
PJCJ
i7
r ^ 2 >\
d_Uj_
dxdt
dO
dt
Equation (4.12) can then be written as
a,
Pe
dx2
dO: d2u,
L + T E 
dt 01 dxdt
where
a, = 
' PJCJ
and Â£,
_ ' i
PjCj
The two governing equations are (4.7) and (4.13):
d2Uj dOj d2u
= Y^ + P^
J dx2
#0
j dx2
a
dO: d2Uj
= L + TnÂ£i L
dt
0 J
dxdt
where
ai =
Pjcj
and
Â£J =
Yj
PjCj
(4.10)
(4.11)
(412)
(4.13)
(4.7)
(4.13)
63
4.3 Conversion of Governing Equations to Dimensionless Form
It is necessary to convert the governing equations (4.7) and (4.13) into dimensionless form. Units of
time, displacement, and temperature are made dimensionless using the following relationships:
x (*i x0)
x = = , (unit space coordinate)
t* t , (unit of dimensionless time)
l2 /2p,C,
[T T0)
9 = , (unit of dimensionless temperature)
Equation (4.7) is nondimensionalized as follows:
9 Uj ddj d2u
Cj IhS ~ rj~&+ Pi ~d?
T
\ci J
99,
99j_
9x
92uj
~9xF
92Uj
Pj
vcj 
.2A
d2u
9t2
Pu
9t2
where
ToY j , Pj
Pj = , and
a,
Equation (4.13) is nondimensionalized as
929 99j
a
1 9x2
ctj 929 99
a, 9x2
92uj
+ Te 
9t 1 9x9t
92u
+ Â£i 
9t 1 9x9t
c 929 99 92u
^~9S = Ik + Ej ~9x9t
(4.7)
(4.14)
(4.15)
64
U j
where Â£ = is the ratio of thermal diffusivities of the first and jth layers.
a,
Therefore equations (4.14) and (4.15) are the nondimensionalized form of the governing equations:
d2u
+ gj dt2
(4.14)
d20 d6 d2u
7 dx2 dt 7 dxdt
(4.15)
The coefficient is simply the ratio of thermal diffusivity of the jth layer to the thermal diffusivity of
the first layer. Epsilon, Ej, is a coefficient showing the effect of the mechanical field on temperature: Tj
is the coefficient for effect of the thermal field on the mechanical; gy is a coefficient denoting the
effect of inertia of the materials. These coefficients can be expressed in terms of lame constants (^ and
Xj) and the coefficient of linear thermal expansion ((3j):
(3lj + 2j)P,
*1
Si
K + ^i)t>i
Xj + 2Hj x0
f \2
Pi Kj 1
(Xj + 2 Iij) [Pfi u
(4.16)
(4.17)
(4.18)
In a coupled thermoelastic analysis, Â£; may be neglected since for most metals this coefficient is on the
order of 10'17; gy for copper is approximately 8.0 E17, Â£y for aluminum is approximately is 4.5 E17.
The last term of equation (4.14) is therefore neglected.
4.4 Thermoelastic Coupling
The coefficient for thermoelastic coupling for the jth layer, 8y, is given by the relationship
Sj = VjÂ£j (4.19)
65
The coefficient Sj is on the order of 10'2 to 10'1 for metals; 5j for copper is approximately 0.057, 8j for
aluminum is approximately 0.116. An alternative formula for the thermoelastic coupling coefficient,
5j, is given by Boley:18
B (3A + 2P2 T0
* _2 ,.2
(4.191)
j(A + 2 fi)
ve, the velocity of propagation of dilatational waves in an elastic medium, is v =  and
V P
cE is specific heat at constant deformation. Specific heat at constant deformation is defined as
d2q>
C F
d2T
T, where (p is free energy. Specific heat at constant volume and specific heat at
constant deformation can be used interchangeably in linear thermoelastic theory. If an external
mechanical agency produces variations of strain within a body, these variations are in general
accompanied by variations in temperature and consequently by a flow of heat. This process leads to an
increase in entropy and an increase in energy stored in a mechanically irrecoverable manner. This is
thermoelastic dissipation, the study of which requires the coupled heat equation.
If deformations due to external loads are accompanied by only small changes in temperature, it is
possible to neglect thermal expansion when calculating these deformations. Similarly, if strains
produced in a body by a nonuniform distribution of temperature are small, their influence on strain
should also be small. It then follows that the coupling term can be disregarded except when
thermoelastic dissipation is of particular interest. Omitting the thermoelastic coupling term depends on
the inequality S 1 must hold (it does for most metals) and that strain rates must be on the same
order of magnitude as temperature rates. This last condition implies that the time history of
displacements closely follows that of temperature.
Equations (4.13) and (4.7) can be restated19 as
,d2T dT d2u
k17pc^iu + 2^nMi
 0
(4.192)
and
(A + 2p) JÂ£ (3A + 2fi)P ^ = 0, (4.193)
respectively. Integrating this last equation with respect to x gives the equation
18 Theory of Thermal Stresses, B. A. Boley and J. H. Weiner, John Wiley & Sons, New York, 1970, pp. 4244.
19 Survey of Recent Developments in the Fields of Heat Conduction in Solids and ThermoElastcity, B.
A. Boley, Nuclear Engineering and Design, v. 18 (1972), pp. 379380.
66
(4.194)
(A + 2fi) (3A + 2fi)p T = ox= /(f)
The arbitrary function f(t) must vanish identically if the surface x = 0 is traction free. If this is the case,
equation (4.191) reduces to equation (4.195)
rji
k = pcE {1 + 6). (4.195)
dx at
Equation (4.195) is identical in form to the ordinary heat conduction equation. The solution of a
coupled problem is derived from the corresponding uncoupled problem by replacing thermal
diffusivity, a, by the quantity .
P CE
4.5 Initial and Boundary Conditions
At the left boundary of the composite (x = xo), the boundary condition is one of known convection:
^ + HA{9A 0,) = 0 (4.20)
At the interface of the two layers of the composite (x = xt) two boundary conditions are
~lt= Ml(0 ^
and
d9i
dx
k2 dd2
kY dx
(4.21) and (4.22)
At the right boundary of the composite (x = xN), the boundary condition is based solely on convection:
where
and
at), / \
+ hb{q2 eB) = 0
(4.23)
Ha
IK
Hb =
lh.o
and
M,
lmy
K
9b
67
and
Ta = absolute temperature of the atmosphere on the A side
Tb = absolute temperature of the atmosphere on the B side
hA = coefficient of heat transfer from outer surface A
hB = coefficient of heat transfer from outer surface B
mi = the film coefficient at the interface between the two layers of the composite.
4.6 Transformation of the Governing Equations in Laplace Space
The quantities 6j and Uj are transformed into Laplace space according the definition of the Laplace
transform, where p is the Laplace variable:
df(x, p) = jQ e pldj(x, t)dt
Equation (4.14), the governing equation of motion, becomes
d2uf(x,p) dOf(x, p)
dx2 ^ dx
d2Uf(x, p) ddf(x, p)
dx2 1,1 dx
(4.24)
Equation (4.15), the governing equation of energy, becomes
(4.25)
Equation (4.25) is differentiated partially with respect to the space variable:
68
which gives
d_
dx
'
q dx2
' r ^1 Pfi
dx
d2J
dx2
= 0.
(4.26)
Substituting equation (4.24) for the last term on the left of (4.26) gives
which becomes
d_
dx
dW
E.J
dxL
B' Vi V
dx
def
dx
= 0
(4.27)
d I d2ef . , ,
wa =
This yields
= 
(4.28)
Similarly, the boundary conditions represented by equations (4.20) through (4.23) become
2 aL
dLe[
dx
(4.29)
+ Ht
eel
 0 at X = X0;
2 aL
dAe{
dx
= Mx {Q\ $2 )
dej k2 de2
dx ki dx
at X = X,;
d2e2l
dx
+ H,
ef
= 0 at X = x,.
The quantity dj1 may be expressed in the following form
0f(x, p) = Aje ,x +
where
(4.301)
(4.302)
(4.31)
(4.32)
69
S; sjjp
and
SJ
1 + VjÂ£i
1 + g;
(4.33)
(4.34)
and A,, B, and S, are functions of the Laplace variable p alone. Variables shown in bold face are
functions of p alone.
4.7 Formulation of a Laplace Transform Definition of df(x,pj
Equations (4.29) and (4.32) are combined to given equation (4.35):
[Aies,j:+B,e"SlJtl + Ha
dx 1 J p '
= 0.
(435)
Separating Bi in equation (4.35) gives
Bj = Aj e 1
2S,x0
H, S,
Ha
e
Ml
a+ S,
which can alternatively be stated in an abbreviated form
Bx = A0b0 e2S'x + B0
where
ha+ S, p
Ai A0
b0
Ha "Si
Ha+ St
B0
S1*0
H.
Ha+S1
and
do
has1
ha+ Sx
and where A0 is an unknown function of p to be determined later. Combining the other boundary
conditions, i.e., equations (4.301) and (4.302), and equation (4.32) gives equations defining A2 and
B2 in terms of Aj, Bi, and by implication, A0. Equation (4.32) is substituted into
70
de[_
dx
at x = Xj = xi. Separating A2 gives
= A, Si 1 + 7 (s, s2) ^ B fiA1
1 M\ 1 Ml\
e(s,s2) B2e_2S2j:'. (4.36)
Similarly, B2 can be expressed as
B2 = A2 e2S^> NtAt J?1 +S2)" + JV, B2 jS2 "Sl)Jl
(4.37)
where
N. =
isA
k2 S2
Substituting equation (4.37) into equation (4.36) gives A2 in terms of Aj, Bj, Mi, and Ni:
A* A,
'r
,2,
Si
B(s. + Bl
rv
<2/
In terms of Ab B), Mi, and Nt, B2 is expressed as
1
B2 = A, 
1 + N1
Mi
(Sl + S2) +
Si
1 17" Ni
M1
ih + Ni
M,
e(s, +sa) Xl (43g)
e(s.s2)x, (439)
Equations (4.38) and (4.39) are written for a 2layer composite. These equations can be expressed in
general terms for a composite of j + 1 layers:
A;+' A; I 2
B7+l A7
s. 1+ L + N, +B, [ 1]
Mj 1 1 V V
\ SJ 1+ IV, (s,+ S ] x, ev > j*) j + B t
Mj 1 j
S;
1 N>
Mj
SJ
iTr+Nj
Mj 1
(s;
(4.40)
(4.41)
Equations (4.40) and (4.41) are then rewritten again in terms of specific coefficients as
A;+i = A; GAj + Bj FAj and B;+1 = A} GBj + B, FBj (4.42), (4.43)
71
where
GM ~ T
SJ
1 + i N:
M: 1
JS1 + SJ +1 )XJ
(4.44)
GBj 2
1 + ^ ~ N>
(Sj + SJ+1)jy
(4.45)
FAl =
S,
1 N:
Mj 1
"(Sj + Sj +l)Xj
(4.46)
F 2
(si sj
(4.47)
where
Substituting equations (4.352) and (4.353) in the righthand side of equations (4.40) and (4.41) gives
Aj + j and Bj +1 in terms of A0 and B0:
A j + \ = A0 L Aj + B0 (4.48)
By + j = A0 Ligj + B0 MBj (4.49)
where the coefficients LAj, LBj, MAj, MBj are given by the recurrence formulae
LA; GAyj LAy jj + FAj LB[j _ (4.50)
LOj = Gfl/J LA(j1) + FBj LB(j1) (4.51)
^A/' GAyj jj + FAj _jj (4.52)
Mfl/ = GBjl UA(jl) + FBj MB(j 1) (4.53)
where
72
(4.54)
Lao 1
Lfl0 = b e2S'*
Mao = 0
M flo = 1.
(4.55)
(4.56)
(4.57)
Equations (4.32), (4.42), and (4.43) are combined to give the Laplace form of temperature in the last,
j = n, layer of the composite:
eji{x, p) = Ajy A* + Bw e~s"x ,
(4.58)
where
and
Aw A0 L^Ar_1j + B0
Bw A0 Lfl^JV_1j + B0 .
Finally, the quantity A0 can be defined as
An
D
P G
where
D iN 0B Q eSlX d0 dA
(1 1 , , J?nxn) j. h T , ,
^ ~ LiA(N 1) e + N ^BiN1) e
Q M / , z>(s"x") + h M z ~(snxn)
~ 1V1a0V1) e + N mB(N1) e
(4.59)
(4.60)
(4.61)
(4.62)
(4.63)
(4.64)
and
bv
Hb ~ Sy
hb+sn'
dv
UB
Hg + SN
(4.65), (4.66)
73
The Laplace form of temperature can then be written
Rj {x, p) D + [e, (jc, p) d0eA es'x G
P G
(4.67)
or
, . R,(x.p) D [fi, (x, p) d0e, es'M
') = ~ p~G + 1, <4
where
= Lx(,o (4.69)
eS,X + MbOi) (4.70)
The term Oj" (x, p) in equation (4.68), the deviation from the initial temperature T0, is expressed as a
function of position and the Laplace variable p for the jth layer of the composite plate. It is necessary
to gain an expression for dj (x, t) expressed as a dimensionless number, and as a function of position
and time. The complex inversion formula is used to invert the Laplace form of dj1 (x, p) to 0j (x,t).
4.8 Inversion of (^(x, p)
If f(p) = eÂ£ {F(t)}, then eÂ£_1 {f(p)} is given by the complex inversion formula:
1 fY+i"
eP,f(p)dP (471)
2ki Jyi
and F(t) = 0 for all t < 0. The integration is performed along a line p = y in the complex plane where p
 x + iy. The real number y is chosen so that s = y lies to the right of all singularities (or poles) but is
otherwise arbitrary. The integral in equation (4.71) is evaluated by considering the contour integral
^iep,f(p)dp (4.72)
2ni Jc
where C is the Bromwich contour shown in figure 4.2.
74
Bromwich Contour for Inversion of 9j (*, p)
Figure 4.2
The Bromwich contour is composed of line AB and the arc BJKLA. T represents the arc of the
contour; R represents the radius of the circular part of the contour. It then follows that T is defined as
T Jr2 y2 F(t) becomes equation (4.73):
1 ry+iT
F(t) = lim I ept f (p) dp (4.73)
2Kl Jy+iT
or
F(t) = lim  I ept f(p)dp  [ ep,f(p)dp\.
*>[ 2m Jc 2m Jr J
Supposing 1) the only singularities of f(p) are poles all of which lie to the left of the line p = y for some
real constant y and 2) that the integral around T in equation (4.73 ) approaches zero as R >. Then
Equation (4.73) can then be written using the residue theorem as F(t) = Z residues of ept f(p), at poles of
(P)
Let
g(p) = (P~a) (P~ b) (p c)
(4.74)
then
75
p=a
p=a
dg{p)
dp
dg(s) d(g a)
d(g a) dp
dg(p)
d(p a)
= (pb) {pc) =
g(p)
(pa)
dg(p)
dp
p = p
(pa) (pc) =
dg{p)
ds
p=e
= {pa) (pb)
g(.P)
[P~c)
therefore
(p ~ ri) = 1
g{p) dg(p)
This leads to the conclusion that
eT
q(p) .. q(p) ! y q(p)
_P k(p) Kp) po
(4.75)
?(/>)
k(p)
p = o
+
I
q(p) (prj)
P k(p)
p = r,
(4.76)
where ^L
P k(p)
are the residue of k(p). The result of equation (4.76) can be restated as
tf1
q(p) q(p)
_p k(p)_ k(p)
q(p)
' dk(p)
dp
p = n
(4.77)
76
Equation (4.77) permits equation (4.68)
to be written as equation (4.78)
(4.78)
where
(4.79)
and Zqs are the roots, other than zero, of the transcendental equation G(z) = 0. G(z) is given by the
substitution of the complex variable z for the Laplace variable p in G.
Equation (4.79) represents 6J in equilibrium or steady state; it derives from the first term on the right
hand side of equation (4.77). As can be seen in equation (4.79), 6J approaches equilibrium in the
limit as the Laplace variable, p, approaches zero. The second term on the righthand side of equation
(4.78) supplies the transient component of 0(x,t). The next section examines a Mathematica algorithm
which calculates 0(x,t) in the steady state for a composite nlayered plate.
4.9 Calculation of the SteadyState Form of 0(x,t)
Mathematica program ARARP1 calculates the steady state solution for 0(x,t) in equation (4.79). The
first test of ARAP1 was done for the configuration shown in figure 4.3. This theoretical composite is
made by joining two pieces of antimony; thermal resistance at the interface is effectively eliminated by
setting the film coefficient, mu equal to 10000 W/m2C. The goal of this calculation was to test if
ARARP1 could produce an uninterrupted temperature profile across a perfect thermal interface
between the same metals. In addition, it was expected (under these idealized conditions) that the
temperatures at either side of the bonded composite should match the ambient temperatures existing on
either side of the 2layered plate. A listing of the program ARARP1 appears as figure 4.4
77
Configuration for First Solution of 6(x,t)
Figure 4.3
Since this composite plate of antimony is, for this analysis, essentially one uniform piece of
homogenous metal with no thermal resistance at the interface, the steadystate temperature should have
a constant slope through the thickness of the plate. Figure 4.5 shows the plot for 6(x,t) under the
thermal, geometric, and material constraints shown in figure 4.3 with mi = 10000 W/m2C. Setting mi
= 10000 W/m2 oC essentially forces thermal resistance at the interface to zero. If the algorithm is
consistent and accurate in its treatment of the interface film coefficient, lower values for mi should
introduce a discontinuity in 6(x,t) at Xi = 0. The insets of figure 4.5 show plots for 6(x,t) with mi = 100
W/m2 oC and with mi = 10 W/m2 oC. These smaller values for the film coefficient do in fact introduce
finite discontinuities at the interface in inverse proportion to the magnitude of m],
(*** Algorithm for calculating T(x,t) for an Atarashi Plate ***)
(*** Experimental Values for Um p > 0 ***)
(*** yy ***^
deltal = .031
kl = 202
Cvl= 896
rhol = 2707
alphal = kl/(Cvl rhol)
(** Cu **)
delta2 = .057
k2 = 384
Cv2 = 398
rho2 = 8930
alpha2 = k2/(Cv2 rho2)
(** Temperature **)
To = 300
Ta = 900
Tb = 600
ThetaA = (Ta To)/To
ThetaB = (Tb To)/To
78
(** Dimension **)
1 = .00635
xO = .00635/1
xl = 0/1
x2 = (.00635)/!
(** Convection Parameters **)
ha =15
hb= 15
Ha=(l*ha)/kl
Hb = (l*hb)/kl
(** Laplace Variable **)
p = l 10M0
(** Interface Film Coefficient **)
ml = 10
(** S and Sp **)
51 =Sqrt[(l + deltal)/(alphal/alphal)]
52 = Sqrt[(l + delta2)/(alpha2/alphal)]
Spl = SI Sqrt[p]
Sp2 = S2 Sqrt[p]
(** Laplace Variables **)
bO = ((Ha Spl)/(Ha + Spl))
dO = Ha/(Ha + Spl)
b2 = ((Hb Sp2)/(Hb + Sp2))
d2 = Hb/(Hb + Sp2)
Ml = (1 ml)/kl
N1 = (kl Sl)/(k2 S2)
Gal = .5 (1 + (Spl/Ml) + Nl) Exp[(Spl Sp2) (xl)]
Gbl = .5 (1 + (Spl/Ml) Nl) Exp[(Spl + Sp2) (xl)]
Fal = .5 (1 (Spl/Ml) Nl) Exp[(Spl + Sp2) (xl)]
Fbl = .5 (1 (Spl/Ml) + Nl) Exp[(Spl Sp2) (xl)]
LaO = 1
LbO = bO Exp[2 Spl xO]
Ma0 = 0
MbO = 1
U1 = (Gal LaO) + (Fal LbO)
Lbl = (Gbl LaO) + (Fbl LbO)
Mai = (Gal MaO) + (Fal MbO)
Mbl = (Gbl MaO) + (Fbl MbO)
Qp = (Mai Exp[Sp2 x2]) + (b2* Mbl Exp[Sp2 x2])
Dp = (d2 ThetaB) (Qp Exp[Spl xO] dO ThetaA)
Gp = (Lai Exp[Sp2 x2]) + (b2 Lbl Exp[Sp2 x2])
R1 = LaO Exp[Spl (x/1)] + (LbO Exp[Spl (x/1)])
R2 = Lai Exp[Sp2 (x/1)] + (Lbl Exp[Sp2 (x/1)])
Q1 = MaO Exp[Spl (x/1)] + (MbO Exp[Spl (x/1)])
Q2 = Mai Exp[Sp2 (x/1)] + (Mbl Exp[Sp2 (x/1)])
SST1 = (R1 (Dp/Gp)) + (Q1 dO ThetaA Exp[Spl xO])
SST2 = (R2 (Dp/Gp)) + (Q2 dO ThetaA Exp[Spl xO])
(*** Post Processing ***)
pi =Plot[SSTl,{x,xO,xl],
79
PlotRange > {0,2.0}, GridLines > Automatic,
PlotLabel > "SST, steady state"]
p2 = Plot[SST2,(x,xl,x2},
PlotRange > {0,2.0}, GridLines > Automatic,
PlotLabel > "Cu, steady state"]
p3 = Showfpl, p2, PlotLabel > AlCu Composite, m = 10 ]
Listing of Program ARARP1
Figure 4.4
In terms of dimensionless variables, 0A and 0B are given by
and
Ta T0 f 900 K 300 AT^
T0 ~ { 300 K >
Tg T0 f 600 K 300 K'
T0 ~ { 300 K
which gives the temperatures at the left and righthand surfaces of the composite antimony plate. In
terms of dimensionless temperature, the initial temperature of the composite plate is by definition equal
to 0.
As discussed earlier, 0(x,t) as given by equation (4.79) approaches equilibrium as the Laplace variable,
p, approaches zero in the limit. Values for 6(x,) as given in figure 4.5 are predicated on/? = 1 10'10.
It was necessary to explore how the algorithm ARARP1 performed when the Laplace variable p was
greater than 0, and when p was much greater than 0. It was discovered that equation (4.79) does indeed
converge to a linear, steady state solution for 6(x,t) as p approaches 0. The speed with which equation
(4.79) converges to a linear form for 6(x,t) is shown in figure 4.6.
80
6(x, )for the Idealized 1Material Sb Composite Plate
Three Values for mi
Figure 4.5
1 0.5 0 0.5 1
Dimensionless Length
Convergence of Solution for 8(x,) as the Laplace Variable approaches 0
Figure 4.6
81
Table 4.1 gives the numerical values for Q(x,o) in four locations of the composite plate for four
increasingly smaller values of p. 8(l,) denotes 6(x,) as the lefthand side of the composite plate;
Q(+l,o),) denotes 6(x,) as the righthand side of the composite plate. 6(0,) and 6(+0,) denote
6(x,) an infinitesimal distance to the left and right of the interface, respectively. As table 4.1 shows,
6(x,oo) is acceptably accurate when p = 10'5 and for all smaller values of p. All calculations of 6(x,)
shown in this chapter are based on p = 10'10 unless stated otherwise.
Laplace Variable P (0,oo) 6(+0,oo) 6(+l,)
10" 4.5281 1.4760 1.4753 3.0719
10" 2.2046 1.4979 1.4972 1.1815
10" 2.0114 1.5001 1.4994 1.0265
10" 1.9986 1.5003 1.4996 1.0098
Numerical Results for 6(x,) at Various Values for p
Table 4.1
In developing the Mathematica programming to solve equation (4.79), a model was chosen next of a
bonded composite of antimony and copper. Convection from both exterior surfaces was equal. The
ambient temperature on the left and righthand sides of the composite were again 900 K, or 0A = 2 and
600 K, or 0B = 1. The initial temperature of the composite was chosen to be 300 K and was assumed to
be uniform throughout the composite.
Thickness of the 316 stainless steel and copper plates were chosen as .00635 m each. In terms of
dimensionless length, each plate of the composite has a length of 1. The film coefficient was
arbitrarily chosen to be 100 W/m2 oC. With this geometry, the coordinate of the left surface becomes 
1 and the coordinate of the right face becomes +1. Figure 4.7 summarizes this configuration.
Configuration and Boundary Conditions for Calculation of Qj (.x, >); CuSb Plate
Figure 4.7
82
6j {x, oo) for the SbCu composite plate is shown in figure 4.8. The affect of the higher thermal
conductivity of copper as compared to that of antimony is apparent. The slope of 0] (x,) for Sb is
much more pronounced than that for 02 (x,) for Cu.
Dimensionless Temperature
1 0.5
0
0 5
1
Dimensionless Length
6j (x, o)for an SbCu Composite Plate; m; = 100 W/m2C
Figure 4.8
Figures 4.9 and 4.10 show similar temperature profiles for bonded composites of copper and
aluminum. The film coefficients between the two composites are m = 200 W/m2 and m = 10 W/m2.
The difference in temperatures across the two interfaces are some indication of the relative thermal
interference to be expected between the Cu/Al wellbonded composite and the Cu/Al composite
possessing a MgOfilled void at the interface.
83
Dimensionless
Temperature
zr 1 n c
1 E
1 ^ c.
1
n tc
a e
a a e
1 0.5 0 0.5 1
Dimensionless Length
Thermal Interference at Interface between WellBonded CuAl Composite Plate
Figure 4.9
Dimensionless Temperature o
1 75
1 5
1 n e
1
A 7 E
A E
A A E
1 0.5 0 0.5 1
Dimensionless Length
Thermal Interference at Interface between CuAl Composite Plate with MgO Barrier
Figure 4.10
84
5.0 Development of Finite Element Models for a Bonded Metal Composite
5.1 Introduction
This chapter 1) compares the results of numerical calculations with experimental data taken from
thermal tests performed on homogenous copper and aluminum test cylinders, 2) develops an finite
element analysis (FEA) model for the MacorRTVAluminum configuration used to accrue
temperature and displacement data for this work, and 3) creates and tests FEA models of the AlCu
and AlMgOCu test specimens. The FEA models of the AlCu and AlMgOCu specimens are
designed to characterize temperature and displacement on the vertical face observed by the TVH.
The FEA models are then used to compare with displacements measured by the TVH as a function of
temperature and time. The FEA software used in the analyses presented in chapters 5 and 6 is the K6
version developed by the MARC corporation.
The type of analysis used in this chapter for the bonded composite plate is described by MARC as
static coupled. This is in contrast to a dynamic transient coupled analysis which is also available.
Both types of coupled analyses effectively reconcile the effect of the thermal and mechanical fields on
each other. The dynamic transient coupled analysis incorporates the inertial term, q, as given in
equation 4.7:
MARCs static coupled analysis neglects q. This is acceptable for the AlCu and AlMgOCu plates,
however, since q is on the order of 1017 for copper and for aluminum. T(r,z,t) for singlemetal shapes
as calculated in section 5.2 are calculated as a heat transfer problem rather than as a static coupled
problem.
It is the transient thermoelastic response of the test specimens that is of greatest interest. Thermal
equilibrium will tend to minimize (or eliminate) differences in temperature profiles between bonded
composites. This is particularly true of composites made of metals with high thermal conductivity.
For this reason, any difference created by the thermal resistance at the interface of a bonded
composite is likely to be most obvious during the transient phase at early times t.
S.2 FEA Modeling of the Copper and Aluminum Test Cylinders
FEA modeling was first done on the .00635 mthick homogenous test cylinders modeled analytically
in chapter 3. The FEA analyses of T(r,z,t) for Cu and A1 test cylinders were compared to
experimental data and the corresponding analytical results developed using equation (A1.43). This
was done to confirm that the boundary conditions used in previous analytical models yielded results
comparable to those produced by FEA techniques. If T(r,z,t) as calculated by very different analytical
and numerical methods compared well with experimental data, this was considered evidence material
properties and boundary conditions had been assessed correctly.
T(r,z,t) for the copper test cylinder is calculated using the material properties, convection boundary
conditions, and qe0upii cited below. An axisymmetric analysis is employed and MARC element type
85
10 is used in the mesh, i.e., a fournode, isoparametric quadrilateral. MARC specifically
recommends use of element type 10 rather than higherorder elements when performing a contact
analysis. The material properties shown in table 1.2 are used for the MARC FEA analysis of T(r,z,t).
The initial conditions and boundary conditions used are hi = h2 = 12 W/m2 C for the right and left
vertical faces, h3 = 5 W/m2 C for the transverse face, and an initial temperature (25 C) is uniform
throughout the Cu test cylinder (see figure 3.10). A qCOupied = 2100 W/m2 is used for the heat flux 
this being qcaicuiated discussed in chapter 3. The convection coefficients, initial temperature, and
qcoupied used in the FEA analysis shown in figure 5.2 are similar to those used for the aluminum test
cylinder in section 3.3.
T(r,z,t) was calculated for 0 < t > 300 seconds since this is the time period of greatest change in the
temperature and displacement fields. Figure 5.1 shows the temperature resulting from the 8W
experiment discussed in section 3.3. Measured data for the first 300 seconds of the 8W experiment,
and a leastsquares fit of that data, are shown in the inset of figure 5.1. T(r,z,t) calculated by the
FEA model can be compared to similar calculations based on equation A1.43 shown in figure 3.10.
As indicated in figure 5.1, the temperature curve is associated with node 13. This node is located at
the lower right comer of the mesh and therefore represents temperature at the center of the x =
.00635 m face of the test cylinder. Analysis of displacement in the x direction (u) resulting from
T(r,z,t) was calculated for nodes 13 and 793, located at (0.00635,0,0) and (0.00635, .03175,0)
respectively.
The difference in displacement, Am, across the x = .00635 m face of the cylinder was calculated as .3
im for all t, 0 < t > 300 seconds. Figure 5.2 shows the FEA approximation of the .00635 mthick
homogenous aluminum cylinder. Superimposed in the inset is the data measured during the 8W
experiment characterizing the aluminum. In this case, the approximation is extended to 0 < t >
2400 seconds. The quality of the approximation can be judged against the measured data appearing
in the inset of the figure. The results appearing in figure 5.2 proceed
Axisymmetric Model of Temperature in Cu Test Cylinder, 0 < t >300 Seconds
Figure 5.1
86
from the same model producing the results appearing in figure 5.1, except for the substitution of
material properties for aluminum and a different initial temperature. Thus the relative accuracy of
the FEA model, based on the boundary conditions developed through the analytical analysis in
chapter 3, is compared to measured data for two metals for short and long t.
The calculations of axisymmetric T(r,z,t) shown in figures 5.1 and 5.2 represent good agreement with
experimental data and the analytical calculation of T(r,z,t). In keeping with these results, the values
for qCOupied (2100 W/m2), Iq and h2 (12 W/m2 C), and h3 (5 W/m2 C ) used in the analyses
appearing in figures 5.1 and 5.2. These values are hereafter referred to as the guideline values and
are used throughout the remainder of chapters 5 and 6 unless otherwise noted.
Axisymmetric Model of Temperature in Al Test Cylinder, 0 < t >2400 Seconds
Figure 5.2
5.3 Modeling Heat Transfer in the Experimental Configuration
As a final (and most exacting) test of the accuracy of the material properties and boundary conditions
used for the remaining analyses of the bonded composites, heat transfer in the entire experimental
configuration was modeled. The experimental configuration referred to is that discussed in section
87
1.74 and shown in figure 3.2, except that an aluminum test cylinder is employed in place of the AlCu
composite shown in figure 3.2. The A1 cylinder was potted into its Macor collar and instrumented
with four thermocouples as shown schematically in figure S.3. The test used an 8 W load and
measured temperatures from all four thermocouples simultaneously. Ambient air temperature (T_)
was ~ 22.8 C; initial temperature of the assembly was 23.7 C. Figure 5.4 gives the temperature
data from the test through a time of 2400 seconds.
Thermocouples 2 and 4 recorded the temperatures at the center of the aluminum cylinder and at top
dead center of the Macor collar. These plots are of primary interest and are modeled in the FEA
analysis. As can be seen in figure 5.3, the temperature measured at the exterior of the Macor collar is
markedly less. This is due to the comparatively small thermal conductivities of the RTV potting and
the Macor. The FEA model employed guideline boundary conditions and the material properties
shown in chapter 1. The only exceptions to this was 1) the thermal conductivity of the RTV potting
was increased to 0.4 W/m C (from 0.19 W/m C supplied from the manufacturer) and 2) qCOupied
was increased to 2200 W/m2 from the guideline value of 2100 W/m2.
Given kRTv = 0.4 W/m C, q~TM = 2200 W/m2, and hi = h2 = h3 = 12 W/ m2 oC, and material
properties taken from tables 1,4,1.8, and 1.9, figure 5.5 gives T(r,z,t) calculated axisymmetrically for
the entire experimental configuration. These data may be compared to the measured data shown in
figure 5.4. Agreement between the FEA model and measured data is good. The greatest divergence
occurs at t Â£ 1800 seconds. The FEA model fails to achieve equilibrium as quickly as the measured
data in figure 5.3 indicate.
Schematic of 4Thermocouple Experiment
Figure 5.3
Figure 5.6 shows T(r,z,t = 1200 s) over part of the cross section of the experimental configuration.
This figure indicates how effectively the RTV potting insulates the test cylinder on its circumference.
Given the agreement between the measured data of figure 5.4 and the results heat transfer calculation
shown in figure 5.5, it is possible to conclude that the boundary conditions and material properties
needed for the composite plate analysis are well understood.
88
The analysis which generated the results in figure 5.5 was based on MARC element type 40. This
element is defined by MARC as a fournode, isoparametric, arbitrary quadrilateral used for
axisymmetric heat transfer analysis. Higher order elements were used during development of the
program used in this section but without significant improvement of the results.
Temperature Data for 4Thermocouple Experiment
Figure 5.4
FEA Temperature Curves for Thermocouples 2 and 4
Figure 5.5
89
FEA Temperatures for 4Thermocouple Experiment, t = 1200 s
Figure 5.6
5.4 Coupled Thermoelastic Model of the AlCu Plate
A simple model of an aluminumcopper plate was developed. This model was axisymmetric,
possessed an infinite film coefficient at the interface, and used MARCs static coupled analysis
capability to determine temperature and displacement simultaneously. The mesh used for the model
is shown in figure 5.7. MARC element 10 is used as discussed in section 5.2. The glue all command
is used enforcing contact of the interface between the two metals at all times. Two initial conditions
are invoked: a uniform temperature prescribed for all nodes and fixed x displacement for two nodes at
the lower left of the model. This latter initial condition gives translational and rotational stability to
the model. The element mesh for copper is slightly offset with respect to the mesh for aluminum.
This in no way reflects the actual geometry of the AlCu test piece. The offset is present because the
coupled static MARC analysis is optimized when node communicates with element face at the
interface of the two metals. The mesh shown in figure 5.7 represents the solid AlCu bonded
composite with a diameter of 61.72 mm; the two metals in the composite are 6.35 mm thick. This is
the configuration pictured in figure 6.1 and analyzed in test number 2 in chapter 6.
90
MARC
Mesh for Coupled Analysis of Axisymmetric AlCu Plate
Figure 5.7
The boundary conditions in this model are shown in figure 5.8. The convection coefficients are hi =
h2 = 12 W/m2oC on the vertical surfaces. The film coefficient on the top of the composite
(corresponding to the cylindrical or transverse surface) is = h3 = 5 W/m2C. The heat fluxes qcaicuiated
and qcoupled vary for each experimental configuration. Material properties are those shown in tables
1.2 and 1.4. The initial temperature was assigned in each metals as experimental measurements
required.
The model shown in figure 5.7 was analyzed by MARC as a contact problem. This type of problem
required the aluminum and copper pieces be defined as contact bodies. It also required the metal with
the lowest elastic modulus, i.e., aluminum, by defined as contact body 1; copper was defined as
contact body 2. The contact table was used to prescribe a separation force of lei2 forcing the two
metals to remain in contact. The entire model was defined as a single geometric entity with
axisymmetric solid elements; constant dilatation was prescribed for all elements.
Figure 5.9 shows the original and deformed shapes of the AlCu bonded composite in response to a
qcoupied of 1200 W/m2. The deformation is magnified 500 times for clarity. The concave bowing
of the cylinder was typically observed in every calculation involving a bonded composite.
91

PAGE 1
ASSESSMENT OF THE THERMOELASTIC RESPONSE OF A DIFFUSION BOND USING TELEVISION LASER HOLOGRAPHIC INTERFEROMETRY by John A. Hargreaves B. Mus., University of Colorado, 1973 B.S. M. E., University of Colorado, 1988 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering 1997
PAGE 2
This thesis for the Master of Science degree by John A. Hargreaves has been approved by I Date
PAGE 3
Hargreaves, John A. (M.S., Mechanical Engineering) Assessment of the Thermoelastic Response of a Diffusion Bond Using Television Laser Holographic Interferometry Thesis directed by Professor William H. Clohessy ABSTRACT Previously, no nondestructive technique has existed allowing conclusive characterization of the strength and structure of a diffusion bond between different metals. Television Laser Holographic Interferometry is used to assess the transient thermoelastic response of wellbonded and poorlybonded metal composites. A coupled static finite element model is used to predict this transient thermoelastic response allowing comparison with measured data. An attempt is made to detect a void at the interface of one bonded composite when compared to another composite without a void. This attempt is unsuccessful though the transient thermoelastic response of both composites is successfully modeled when compared with measured deformations on the submicron level. This abstract accurately represents the contents or the candidate's thesis. I recommend its publication. S d 1gne __ William H. Clohessi.f'
PAGE 4
ACKNOWLEDGMENT The author would like to thank many people who contributed much time, advice, and effort assisting in the completion of this thesis: my advisor, Dr. William H. Clohessy who gave many helpful suggestions; Dr. Keith Axler for his assistance with materials science questions, helpful insights, and for his critical reading of the text; Dr. P. Shivakumar of the MARC corporation for his assistance with the FEA model; Dr. John A. "Jack" Hanlon for giving so unselfishly of his time and talents and, by his example, teaching me to think as an experimentalist; my wife, Elizabeth, whose help and encouragement through difficult times never failed.
PAGE 5
CONTENTS Chapter 1. Introduction ......... ............... ......... . ... ..................................... . .......... ........... . 1 1.1 Problem Statement . ................................ ........................................ .. .... . ....... 1 1.2 Structure of Thesis . . .. . . . . ...... . .. .. . . . . . . . . .. . . . . . . . . .. ... . . . . . . . . . . . . 2 1.3 The Basis of Holograph Interferometry ... .... ............ . ......... ........... . ........... ... 3 1.4 The Design and Operation of TVLaser Holography.... ................... .... ......... 3 1.5 The Diffusion Bond ..................... ... ..... . .............. ............... . ......... ...... ...... .. 5 1.51 Introduction . ... . . ....... ......... ... . . ....................................................... ...... .. 5 1.52 Theoretical Basis of Diffusion Bonding .................... .................................. 5 1.53 Diffusion Bonding Process Variables .. ...... ..... .... .. .. . ..... ... ..... . . . ..... . 7 1.54 Metallurgical Factors . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . 8 1.6 Design and Manufacture of CuAI, CuMgOAl, and CuSb Bonded Bonded Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.7 Properties of Experimental Materials . . . . . ................................... .. .... 11 1.71 Chemical, Thermal, and Physical Properties of Copper ........ ..... ... . . .......... 12 1.72 Chemical, Thermal, and Physical Properties of Aluminum .. ........................ 14 1. 73 Chemical Thermal, and Physical Properties of Antimony . . . . . . . . . . . . . 16 1.74 Properties ofMacor Ceramic and RTV Silicon Potting .............................. 19 2 The Linear, Coupled, QuasiStatic Theory of Thermoelasticity .......................... 21 2 1 Introduction ... ... .. ...... ... .......... ........................... .. ...... ........ ......... ... . ........... 21 2.2 The Basic Laws of Mechanics and Thermodynamics ..... ............................... 22 2.3 E lastic Materials and the Consequences of the Second Law ....................... .. .... 24 2.4 Material FrameIndifference................ ............................................. .............. 27 2.5 Consequences of the Heat Conduction Inequality ..... ... . . .... . . ......... .............. .. 30 2 6 Derivation of the Linear Thermoelastic Theory .................... .............. ............. 30 2.7 OneDimensional Linear Thermoelasticity ..................................... ... . . .... ... ... 37 3 Analytical Modeling of Heat Flow through Test Cylinders ............... : . .. .. . .. . .. 42 3 1 Introduction ..................................................................................... 42 3 2 Formula for T(x,t) . . .. . .. . . . . . . . . .. .. .. .. . .. .. .. . . .. .. .. .. .. .. .. .. . . .. .. 44 3.3 Modeling T(x,t) in a SemiInfinite Slab .. . . .. .. .. ... . . . .. . . .. .. ... .. . .. .. .. 46 3.4 Formula for T(r,z,t) in a Finite Cylinder .. .. .. .. .. .. . . . .. . .. .. . .. .. . ... .. . .. .. 50 3.5 Modeling T(r,z,t) in a Finite Cylinder . ...... ...... ..... ...... .. .. ....... .... .. .. .. . . 54 3.6 Characterization of Temperature Difference (LlT) on the Vertical Face ............ . 56 3.7 Comparison of Calculated and Measur ed T(x,t) fortS 120 Seconds .... ... . . ....... 59
PAGE 6
4 Analytical Method for Characterizing Heat Conduction across a Bonded Interface . 60 4.1 Introduction and Statement of problem .. ........... ....... .. ...................... .. ....... 60 4.2 Derivation of the Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3 Conversion of the Governing Equations to Dimensionless Form . . . . . . . . . . . 64 4.4 Thermoelastic Coupling . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.5 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... 67 4.6 Transform of the Governing Equations into Laplace Space . . . . . . . . . . . . . . 68 4.7 Formulation of a Laplace Transform Definition of 6f (x,p) ........................... 70 4 8 Inversion of ef(x,p) ...................... ................................. .................. 74 4.9 Calculation of the SteadyState Form of 6 (x, t) .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 77 5. Development of Finite element Models for a Bonded Metal Composite ....... 85 5 1 Introduction .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. . .. .. .. .. .. .. . . .. . 85 5.2 FEA Modeling of the Copper and Aluminum Test Cylinders . . . . . . . . . . . ... 85 5 3 Modeling Heat Transfer in the Experimental Configuration . . . . . . . . . . . . 87 5.4 Coupled Thermoelastic Analysis of the CuAl Bonded Composite .. .. .. .. .. .. .. .. .. .. 90 5.5 Coupled Thermoelastic Model of the AIMgOCu Bonded Composite .. .. .. .. .. .. .. .. 94 6. Assessment by TVH Interferometry of Diffusion Bond e d Composites 98 6.1 Introduction .. . . .. . . . . .. . . . . . . .. . . .. . . . . .. . . . .. . .. . . .. . . . .. ... 98 6.2 Preliminary Tests . . .. . .. . . . . . . . . . .. .. . . . . . . . .. .. . . . . . . . . . . . .. 99 6.3 Results ofExperimentNo. 1 .. .. ........................................................... 101 6.4 Results of Experiment No.2 .... . .... ... ..... ..... ...... ........... ... . . ... ...... .. . 105 6.41 Comparison of Modeling and Experimental Data for Cylinder A .......... ....... 105 6.42 Comparison of Modeling and Experimental Data for Cylinder B . . . . . . . . . 109 6.5 Conclusions and Areas for Future Research . . . . . .. . . . . .. .. .. . . . .. . . .. . 115 Appendix A. Derivation of Formula for T(r,z,t) . . . .. .. . . . . .. . . . .. . .. .. . . . . . .. . . .. . .. 116 Bibliography ............................. .... ....................... .................... .. ......... 128
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FIGURES 1.1 Schematic of Los Alamos National Laboratory TVH . . .. .. .. . .. . . .. . .. . 5 1 2 ALMgOCu Assembly in PreBonded Configuration .............................. 9 1.3 CuSb Assembly in PreBonded Configuration ...................................... 10 1.4 Temperature and Pressure used for Bonding Specimens ................ ........... 10 1.5 Phase Diagram for the CopperAntimony System .. .. .. .. . .. .. .. .. .. . .. .. .. ... 11 1.6 Photograph of Metallurgical Structure of Copper, 25X ............................ 13 1.7 Thermal Conductivity Data for Copper . ..... ... .. . . .. .. ..... .. ... ... ........ 14 1.8 Photograph of Metallurgical Structure of Aluminum, lOOX ....................... 15 1.9 Thermal Conductivity Data of Aluminum ....................................... .. .. 16 1.10 Thermal Conductivity Data for Antimony .......... ......... ........... .............. 18 1.11 Photograph of Metallurgical Structure of Antimony, 1 OOX . . . . . . . . . . . . 18 3.1 Schematic of SemiInfinite Experiment ... .......... .. ................ ............... 43 3.2 Configuration of Potted AlCu Composite Plate .. .. .... .. .. .. . .. .. .. .. .. . .. .. .... 44 3.3 Listing ofMathematica Program JCUM7.DAT ...................................... 47 3.4 T(x,t) for Measurement of Copper cylinder, L = 00635 m .. .. .. ... .. .. .. .. .. .. .. 47 3.5 Listing of Mathematica Program CPPRl . . . . . . . . . . . . . . .. . . . . . . . 48 3.6 Comparison ofT(x,t) with varying q; h1 = h2 = 12 W/m2 C ...................... 49 3.7 Comparison ofT(x,t) with Measured Data for Aluminum .......... ............... 50 3.8 Boundary Conditions and Spatial Coordinates for Finite Cylinder ................ 51 3.9 Mathematica Program CPPR . .. .. .. .. . ... . .. . . .. . .. .. . . . .. . . .. .. . .. 53 3.10 Comparison of Calculations of T(r,z,t) with Measured Data . . .. .. .. . . . . . .. 54 3.11 T(r,z,t) as a Function of Different Vertical Convection Rates ........ . .. .......... 55 3.12 T(r,z,t) as a Function of Different Convection Coefficients . ... ...... .. .......... 56 3.13 Experimental Configuration for Determination of LiT on z = .0165 m Face... 57 3.14 Comparison of Calculated and measured T(r,z,t) for Antimony Cylinder .. .. .. 58 3.15 Calculated T(r,z,t) for Sb cylinder at t = 2290 s....................................... 59 3.16 Comparison of Calculated and Measured T(x,t) for 0::;; t :2: 120 Seconds ...... 60 4.1 nLayered Bonded Plate ............................................................... 61 4.2 Bromwich Contour for Inversion of Of (x,p) ........................................ 75 4.3 Configuration for First Solution for 9(x,t) ............................................. 78 4.4 Listing of Program ARARPI .. .. .. .. . .. .. .... .. .. . .. .. . .. .. .. .. . .. .. .. .. .. 80 4 5 9 (x,oo) for a !Material Sb Composite Plate .......................................... 81 4.6 Convergence of Solution for 9 (x,oo) as the Laplace Variable Approaches 0 81 4.7 Configuration and Boundary Conditions for Calculation of Oi (x, oo) CuSb Plate ........... .............................. 82 4.8 (Ji (x,oo) for SbCu Composite Plate; m1 = 100 W/m2 C ........................... 83
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4.9 Thermal Interference at Interface between WellBonded CuSb Plate ........... . 84 4.10 Plate Thermal interface between CuSb Plate with MgO Barrier .............. .. 84 5.1 Axisymmetric Model of Temperature in CuTest Cylinder ......................... 86 5.2 Axisymmetric Model of Temperature in AI Test Cylinder ................. .... 87 5.3 Schematic of 4Thermocouple Experiment ........................................... 88 5.4 Temperature Data for 4Thermocouple Experiment . . . . . . . . . . . . . . . 89 5.5 FEA Temperature Curves for Thermocouples 2 and 4 ........ .. ............ ... 89 5.6 FEA Temperatures for 4Thermocouple Experiment, t = 1200 s ............... 90 5.7 Mesh for Coupled Analysis of Axisymmetric AlCu Plate ........................ 91 5.8 Boundary Condition for AlCu Bonded Plate .. .. .. .. .. . . . .. .. . .... ... . .. 92 5.9 Deformed Shape of AlCu Bonded Composite, t = 300 s . . . . . . . . . . . . 92 5.10 Magnitude ofxDisplacement in AlCu Bonded Composite, t = 100 s.. ....... 93 5.11 AlSST Cu Development Model . .. . . .. .. . . .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. 94 5.12 Mesh, xDisplacement Field, and Deformation for AlSSTCu Model, t = 300s 95 5.13 Successful Axisymmetric Model for Composite with Void . . . . . . . . . . 96 5.14 Temperature Contours for AlMgO Cu Model, t = 300 s . . . . . . . . . . . . 97 6.1 A and B Bonded Composites as used for Experiment No. 2 . .. .. .. .. .. .. .. .. 98 6.2 SteadyState Deformation of Homogenous CuTest Cylinder . .. .. .. .. .. .. .. .... 99 6.3 OffCenter Fringe Pattern, AlCu Bonded Composite ........ ...................... 100 6.4 Fringe Pattern Centered by TVH Compensation .................................... 101 6.5 Temperature Data for Experiment No. 1 .. .. .. .... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 102 6.6 Thermoelastic Response of Cylinder A; t = 155 s, Qcoupled = 2100 W/m2 ........ 103 6 7 Thermoelastic Response of Cylinder A; t = 300 s, Qcoupled = 2100 W/m2 ........ 103 6.8 Thermoelastic Response of Cylinder B; t = 150 s, Qcoupled = 2100 W /m2 .. .. .. .. 104 6.9 Thennoelastic Response of Cylinder B; t = 300 s, Qcoupled = 2100 W/m2 .. .. .. 104 6.10 T(r=O, z = .00635,t) Calculated for Cylinder A, 0 t 300 s, qcoupled = 1200 s 106 6.11 T(r=O, z = .00635,t) Measured for Cylinder A: Qcoupled = 1200 W/m2 ..... ... .. .. 106 6.12 xDisplacement as a Function of Time for Cylinder A; 0 t 300 s .. .. .. .. 107 6.13 Measured Au for Cylinder A; t = 155 s, Qcoupled = 1200 W/m2... ................... 108 6.14 Measured Au for Cylinder A; t = 300 s, Qcoupled = 1200 W/m2 ...................... 108 6.15 XDisplacement for cylinder A; t = 300 s, Qcoupled = 1200 W/m2 .. ........ .... 109 6.16 Temperature Data for Cylinder B, Experiment 2;, Qcoupled = 1200 W/m2 ...... 110 6.17 MARC Temperature Curve for Cylinder B, Experiment No.2 . . . . . . . . . 110 6.18 300s ......... 111 6.19 Measured Au for Cylinder B, Experiment No.2; t = 300 s ........................ 112 6.20 xDisplacement for Cylinder A; t = 300 s, Qcoupled = 1200 W/m2 ................. 113 6.21 Calculation of Au, Cylinder B, Experiment No.2; 0 t 167 s .... .. .. .. .. 114 6.22 Interferometric Measurement of Au, Experiment No.2, Cylinder B; t = 167 s 114 Al.l Al.2 Al.3 Cylinder with Initial and Boundary Conditions ..... ...... ......... ........ ... ..... 108 Positive Roots of Jo b)+ H3 Jo b)= 0 .................................... 115 . ( ) 71p (H1 + H2 ) PostUve Roots for tan 1Jp c = 2 ( ) ................................... 116 11pHI+ H2
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TABLES 1.1 Trace Elements in Copper . . . .. . . . . . . . . .. .. . .. . .. ... 12 1.2 Thermal and Physical Properties of Copper . . . . .. . .. . .. 13 1.3 Trace Elements in Aluminum . . . . . . . . .. .. . . . . .... ... 15 1.4 Thermal and Physical Properties of Aluminum . ....... ... ... 15 1.5 Major Contaminants of Antimony ................ .. ...... . ..... 16 1.6 Trace Contaminants of Antimony .. . .. .. .. .. .. . .. .. .. .. .... .. .. 17 1.7 Thermal and Physical Properties of Antimony ....... ...... 17 1.8 Properties ofMacor Ceramic ..................................... 19 1.9 Properties ofRTV Potting ......................................... 19 3.1 Comparison of Calculated and Measured Initial Temperatures . . . . . .. .. .. .. .. . . . . . . . . . 54 4.1 Numerical Results for 9(x,oo) at Various Values for p ......... 82
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1. Introduction 1.1 Problem Statement This work examines the use of television laser holographic interferometry (fVH) to assess nondestructively the existence and nature of a diffusion bond between dissimilar metals. This work attempts to examine the thermoelastic response of actual diffusion bonds to a known heat flux passing through the parent metals and the bond between them. The thermoelastic response of two bonded composites of identical metals and dimensions are compared using TVH, one composite having an insulated void at the interface This AlB comparison is intended as a proofofprinciple. The diffusion bond represents a discontinuity which acts as a thermal resistance to any heat flux. This thermal resistance is modeled analytically and numerically; the resulting response is modeled using a finite elementbased coupled thermoelastic analysis. The predicted thermoelastic response is then compared to the response observed by the TVH. The different metals in a bonded composite have differing thermal conductivites, thermal diffusivities, and coefficients of expansion. It is possible these inherent differences provide the basis for a specific and identifiable thermoelastic response in a bonded metallic composite. Such a thermoelastic response might be characterized in terms of displacement, strain, or even strain rate. Given the realtime capability provided by TVH to detect small changes on the submicron level, could thermoelastic displacements, strains, or strain rates created by introducing a heat flux to the bonded metal composite be detectable and quantifiable? If so, could these results be the basis for the nondestructive evaluation of a metallic diffusion bond? Xray radiography and ultrasonic testing are two techniques by which such a nondestructive assessment can be made currently. These techniques, however, have limitations and usually cannot provide conclusive information as to how well two metals are diffusion bonded or even if they are bonded at all. Recent developments in the field of TV laser holographic interferometry offer a possible alternative as a tool for the nondestructive characterization of diffusion bonded metals. Holographic interferometry has become a wellestablished technique used for the measurement of physical quantities such as displacements, strains, and densities. Initially used for the inspection of aircraft components, holographic interferometry has been used in recent years to measure biological changes in bone tissue and the time dependent shrinkage of concrete structures. In the field of experimental mechanics, holographic interferometry has been used to detect flaws such as microcracks, voids, delaminations, and material inhomogeneities. TV laser holographic interferometry makes use of recent advances in digital image acquisition and processing allowing automatic (and near realtime) reduction and analysis offringe data. The Los Alamos TVlaser interferometer can measure displacements as small as two tenths of one micron ( 10 at a speed of 30 Hz. This thesis seeks to address if the ability to detect submicron displacements in realtime can be used to characterize the existence and extent of a diffusion bond. Assuming homogeneity and isotropy, any metal will conduct heat according to the Fourier equation q(r, t) = k VT(r, t). (1.1) The temperature gradient is a vector normal to the isothermal surface; the heat flux vector, q(r,t), represents heat flow per unit time per unit area of the isothermal surface in the direction of decreasing temperature. Thermal conductivity, k, is a specific property of each metal. In a bonded composite comprised of two metals having different thermal conductivities, i.e., k1 and k2 the 1
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temperature profile through the metals experiences a discontinuity through the bond that joins them. The boundary condition at the bonded interface can be expressed as an energy balance as ( ) ki = h 1i 12 = j I dx j (1.2) where i denotes the interface and h (in units W/m2 0C) is the contact conductance, or film coefficient, across the bonded interface. It is only in the ideal case of perfect thermal contact that h oo, and equation 1.2 reduces to (1.3), (1.4) In most cases the film coefficient at the interface is finite. Analysis of the heat flow across the interface can be done using a semiinfinite model, and axisymmetric model, or a full 3D solid model. In this work, semiinfinite and axisymmetriC models are used separately and in combination to assess the affect of convection boundary conditions and material properties on experimental data. Mathematica version 2.2.3 is used to support the analytical analyses, the finite elementbased heat transfer and static coupled thermoelastic analyses are done using MARC version K6. 1.2 Structure of Thesis Chapter 1 is an introduction giving the problem statement and the engineering basis on which the problem is investigated. Chapter 1 also discusses the design, operation, and capabilities of the Los Alamos National Laboratory (LANL) TV Laser Holographic Interferometer used for experimental measurements. Chapter 1 also presents an explanation of the bonding process by which the copper aluminum and copperantimony composites used for testing were manufactured as well as a description of the materials used in various experiments. Chapter 2 is a derivation of the equations of quasistatic coupled thermoelasticity from frrst principles. Also included are the assumptions and restrictions underlying the historical development of this theory. Chapter 3 presents a discussion of modeling heat transfer through semiinfmite and finite cylinders. The goal of this chapter is to examine to what degree analyticallyderived formulae for T(x,t) and T(r,z,t) agree with each other and with measured experimental data for heated singlemetal test samples. Chapter 4 extends the analyses of Chapter 3 to the case of an nlayered bonded metal composite. A set of governing thermoelastic equations are given, solved by use of the Laplace transform and the complex inversion theorem, and a formula for T(x,t) derived for the steady state condition; a program in Mathematica is given for computation of T(x,t). Results for ideal and experimental bonded configurations in thermal equilibrium are presented. Chapter 5 explores numerical analyses for T(x,t) singlemetal and bonded composites. These are compared to some of the analytical results presented in chapters 3 and 4. Finite element models of the two principal test configurations are developed. 2
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Chapter 6 describes the experimental measurements of the bonded and unhanded composites using TV laser interferometric holography. Comparisons are made between results predicted by the FEA model developed in chapter 5 and data measured in the experiments. Chapter 6 concludes with a discussion of the success of the model in predicting measured results and areas for future research. 1.3 The Basis of Holograph Interferometry Holography is an interferometric technique that generally uses a laser as a coherent source of radiation. Optical components such as mirrors and bearnsplitters are used to combine laser light reflected from the subject with a second reference beam produced from the same laser. The combined beams produce an interference pattern which is recorded on an appropriate photographic medium. Mter photographic processing, the hologram can be used to reconstruct a three dimensional image of the original subject, again usually employing a laser. An important advantage in this process is the ability to make engineering measurements based on the holograms ability to store threedimensional information. Using with a doubleexposure or timeaveraging technique, a variety of parameters can be measured: displacement, distortion, vibration, shape, density, and others. This information is displayed as fringes striating the holographic reconstruction of the subject. These fringes are contours of equal optical path length can represent outofplane distortion, inplane displacement, and other quantities. Conventional holography is based on the phenomenon of interference: phase differences across the two wave fields, one from the subject under study and the other directly from the laser. These are converted into density distributions across the emulsion of the photographic recording medium. A typical holographic plate has a resolution greater than 3000 line pairs per millimeter and contains approximately 1010 bits of information. For all this, conventional holography has a number of limitations: Slow Photography based Requires highpowered laser Qualitative analogue output To some extent, TVLaser Holographic Interferometry (fVH) mitigates these limitations. 1.4 The Design and Operation of TV Laser Holography (TV .H) As explained by J. C Davies and C. H. Buckberry,l TVH is a technique using a laser, CCD camera and digital processing to create holograms at TV frame rate. This technique was first known as electronic speckle pattern interferometry in the early 1970s. Recently other names have been used such as electronic holography and electrooptic holography. The advent of smaller computers and a technique for phase shifting in the 1980s have made TVH more robust and versatile. A TV camera has resolution capability several orders of magnitude less than a photographic plate (1010 bits compared to 106 bits for a camera) However, an optical system using a TV camera can be configured so that it can capture holographic data and enable all measurements available to 1 Optical Methods in Engineering Metrology, ed. D. C. Williams, Chapman & Hall, London, pg. 277. 3
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conventional holography to be made at TV frame rates. This, in effect, means TVH can furnish holographic information in real time, though with some technologyrelated limitations One of these limitations is the resolution constraint introduced by the camera: some information contained in a conventional hologram must be sacrificed. The reference beam in a conventional hologram typically illuminates the photographic plate at an angle of 45 degrees. This allows sufficient space for the subject to be separated from the reference source. As the subject gets larger, the reference beam has to be moved further off axis during formation of the hologram. If the reference beam is not sufficiently offaxis during formation of the hologram, the image will produced will be partially overlapped and swamped by the reconstruction beam. If the reference beam is moved farther off axis, the structure in the interference pattern generated becomes more finely spaced and the resolution of the recording medium has to be increased. If the reference beam could be positioned in the center of the subject at the reconstruction stage, illuminating along the axis of the holographic system, the resolution requirement for the recording medium would be reduced to a minimum. This arrangement would be of little practical use in conventional holography, however since the resulting image would not be discernible i n the presence of the highintensity reconstruction beam. However, recording holographic data with a TV camera allows use of a lowresolution operating mode obtained from an onaxis reference beam geometry in conjunction with an electronic filter or subtraction stage to eliminate the reconstruction signal. This optical subtraction process can not be realized physically in ordinary holography Resolution requirements would still be too high for a system based on a TV sensor requiring further limitation of the hologram's spatial resolution Resolution is achieved by using a lens to form an image of the subject on the sensor. The bandwidth is reduced by use of a small aperture on this lens; the lateral resolution of the system is reduced proportionately. A schematic of the Los Alamos TVH system is shown on Figure 1.1. A TVH reference beam may have about 5 per cent of the intensity of the subject illumination beam. After detecting the holographic interference pattern, the lowfrequency signal due to the onaxis reference beam geometry can be eliminated electronically by highpass filtering. The modulation that remains can then be viewed by feeding the filtered signal to a TV monitor. This signal is the reconstructed TV hologram. This signal is usually squared or rectified to avoid losing negative components. The hologram is generated at the TV frame rate; each new image seen on the monitor being the result of filtering a new interferogram collected by the camera. Change in the optical path length between subject and camera changes the interferogram and thereby modifies the image. Phase shifting is ac c omplished using a p i ezoelectric crystal, denoted in Figure 1.1 as PZT. 4
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TEST PART MONITOR Figure 1.1 Schematic of Los Alamos National Laboratory TVH TVH normally operates in two modes: static and dynamic. Shape changes that are slow in comparison to the 30 Hz TV frame rate are measured in the static mode. Shape changes are displayed by fringes, or contour lines, with 'AJ2 spacing; A. being the wavelength of light. A typical value for A. is 0.532 J.1IIl; this value for "JJ2 provides the 10 limit of resolution of the Los Alamos TVH. All TVH measurements presented in this thesis are taken in the static mode. 1.5 The Diffusion Bond 1.51 Introduction Diffusion bonding, a.k.a. diffusion welding, is a metal joining technique used in a number of fields, most notably in the nuclear power and aerospace industries In nuclear power generation, a fuel element must be constructed able to operate at elevated temperatures, possess hightemperature strength, and be highly corrosion resistant. These characteristics can be achieved by diffusion bonding a thin corrosionresistant cladding to a fuelbearing core which provides the needed strength at high temperatures. 1.52 Theoretical Basis of Diffusion Bonding The formation of a diffusion weld can be separated into three distinct stages. The first stage involves the initial contact of the interfaces. This initial contact may include deformation of surface asperities or roughness if that is needed to establish initial mechanical contact. These surface asperities are deformed plastically accompanied by the simultaneous rupturing and/or displacement of surface films or oxides. This initial plastic deformation is caused by a compressive load applied 5
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to the interface of the metals to be joined In the second stage there is a timedependent diffusion activated deformation (creep) of the original interface which establishes more intimate interfacial contact. In the third stage, diffusioncontrolled processes eliminate the original interface. The third stage may occur through one or more mechanisms: grain growth across the interface, the solution or dispersion of an interfacial contaminant, or by simple diffusion of atoms along or across the original interface. The ftrst and second stages set up the third stage: diffusiondriven elimination of the original interface Four mechanisms have been proposed 1. Atom transport occurring across the original interface creating bonds across the parts.2 This is a volume diffusion process. This explanation appears somewhat inadequate Atomic interchange in volume (or bulk) is not possible until interatomic distances across the interface are approximately equal to the lattice parameter. If the original interfaces reach this state of proximity they become a grain boundary and (except for contaminant atoms) this state is as strong as the joined parent metals which contain many grain boundaries. 2. Recyrstallization and/or grain growth occurring at the interface. This growth results in formation of a new grain structure which sweeps across the original interface boundary. It is possible recyrstallization causes the yield strength of a metal to approach zero In this case total accommodation of the interface can occur with little of no applied pressure. This brings surface atoms into sufficiently close proximity to permit metallic bonding at the interface. 3. Surface diffusion and sintering action cause the interfaces to grow together rapidly.3 4 Surface ftlms or oxides are dissolved into the base metal and in so doing eliminate these barriers resisting the formation of normal metallic bonds.4 The single model for diffusion bonding is probably not possible to postulate. Such a model may include one or many mechanical or metallurgical processes The speciftc conditions under which a diffusion bond is made, as well as the results desired from the bond, determine which processes contribute or predominate. However, Schwartz5 proposed an elementary model as a description of the mechanism of diffusion bonding. Two surfaces are initially brought together under a load. If high pressure is exerted on the interface or the temperature is high, the surface asperities will experience considerable plastic flow until th e interfaces achieve a high degree of conformity. At this point in the process, the joint has considerable strength due to metallic bonds forming at various regions of the interface. If the initial pressure is lower, the same surface conformity may be achieved at longer times due to creep and/or surface diffusion of atoms. During this d e formation thin surface ftlms are disrupted and some plastic work may be put into the surfaces. 2 Feduska W., and Horigan, W. L. Weld. M et. Fabrication, 35(12),, December 1967, pgs 483 489. 3 Vaidyanath, L. R., Nicholas M G. and Milner D. R. British Welding Journal, 6(1), January 13, 1959. 4 Wiliford, C. F. ;and Tylecote, R F., British Welding Journal, 7(12), December 1960, pgs. 708712. 5 Schwartz, M. M., Metals Joining Manual McGraw Hill, 1979 pg. 105 6
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If the joint is held at higher temperature for an extended period of time, a large degree of atom mobility may be realized at the joint. Recyrstallization or grainboundary motion may occur to extend and further strengthen atomic bonds and further disrupt any surface film present at the joint. These processes occur to varying degrees depending on temperature, time, interfacial deformation, metal properties, and other factors. In the last stages of joint formation, some additional exchange of atoms occurs across the initial interface with the affect of furthering structural or chemical homogenization of the joint area. As atoms redistribute in the interface region, phase changes take place in the metals at the interface. These changes occur at rates that are dependent on the speed of migrating atoms. Diffusion in metal systems may be categorized into three different processes depending on the path of the diffusing element. These three processes are 1) volume diffusion, 2) grain boundary diffusion, and 3) surface diffusion. Each of these processes has a different diffusivity constant. The specific rates for grainboundary diffusion and surface diffusion are higher than the rate for volume diffusion. The governing equation for diffusion in metals is Pick's first law. 1.53 Diffusion Bonding Process Variables For this work, hot isostatic pressing is the process by which diffusion bonding is accomplished. Also known as hot (or cold) isostatic bonding, the metals to be joined are placed in a welded thin wall metal container which is subsequently evacuated to a high vacuum The container is then placed in the isostatic chamber (or autoclave) and subjected to a combination of high pressure and high temperature. After the bonding cycle is complete, the container with the nowbonded metals is removed from the autoclave and the bonded metals are removed from the thinwalled container. Several parameters are key to effective diffusion bonding. These parameters must be controlled in theory. In practice these factors are in fact highly controlled. These critical parameters are temperature, time, pressure, and surface cleanliness; each of these is discussed in tum. All mechanisms involved in diffusion bonding are temperature dependent. Since diffusion bonding is a thermallyactivated process, incremental changes in temperature affect the greatest changes in process kinetics compared with other parameters. Temperature is a readily controlled and measurable process variable. Temperature is an effective crossreference with physical and mechanical properties as well as metallurgical phase transformations. A systematic parametric evaluation of the diffusion bonding process was done by Kazakov .6 He showed a continuous temperaturetimepressure interdependence in which increased temperature resulted in increased bond strength. Increased pressure and time also resulted in increased bond strength Temperatures used in diffusion bonding vary from 500 op (260 C) for joining some aluminum alloys to Ti5Al2.5Sn to silver as a diffusion aid, to as high as 2800 op (1538 oq for joining pieces of tungsten. Typically, the temperature at which diffusion bonding takes place is greater than 0.5 Tm, where Tm is the melting point of the material. Temperatures between 0.6 Tm and 0.8 Tm are commonly used in diffusion bonding many metals. 6 Kazakov, N. F., "Diffusion Welding in a Vacuum," Moska, Izdvo, Mashionostroyniye, pp. 1332, 1968. 7
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Most diffusioncontrolled reactions vary with time. Data presented by Kazakov indicate that increasing time at temperature and pressure increases joint strength up to a point. However, beyond this time no further increase is achieved. This indicates that although atom motion continues indefinitely, structural changes in diffusion bonding tend to approach an equilibrium This behavior is illustrated in recrystallization. A deformed sample first undergoes recovery when first heated; it then recrystallizes. Initially, the formation and growth of new grains are rapid, but as time increases, the rate of grainboundary motion and physical change diminishes. The decrease is due to stabilization of the microstructure through reduction of internal energy. Thus the driving force for continued structural change is also reduced. The rate of atom motion, however, does not decrease significantly throughout this process. Pressure is an important process parameter in diffusion bonding. As a quantitative variable, it is less applicable than either temperature or time. The initial phase of bond formation is most certainly affected by the amount of deformation induced by the pressure applied. Increased pressure invariably results in higherstrength joints for any given timetemperature combination. This is probably due to the greater interface deformation and asperity breakdown resulting from higher pressures. Also, increased pressure (and deformation) leads to a lower recrystallization temperature. This is to say increased deformation accelerates the process of recrystallization at a given temperature. Experience in practice regarding surface preparation can be summed up by the rule: maximum attainable cleanliness. Surface preparation includes cleanliness, finish, removal of chemically combined films, and removal of gaseous, aqueous, or organic surface films. Surface finish is ordinarily achieved by machining, grinding, or polishing. Surface flatness is another necessary component of surface preparation A minimum degree of flatness and smoothness is required to assure interfaces can achieve necessary compliance without excessive deformation. Machine finishes, grinding, of polishing are usually adequate to obtain the needed degree of surface flatness and smoothness. Machining also introduces a measure of plastic flow into the surface Such cold worked surfaces have a lower recrystallization temperature than unworked bulk metal. Processes such as vapor degrease, vacuum bakeout, and chemical etching have also been used to achieve needed surface cleanliness 1.54 Metallurgical Factors Two factors of particular importance in diffusion bonding diffe rent metals are allotropic transformations and microstructural factors which result in modification of diffusion rates. Allotropic or phase transformations occur in many metals and alloys. They are important in that metal is very plastic when undergoing a transformation. This permits more rapid interfacial accommodation at lower pressures in much the same manner as recrystallization. Diffusion rates are typically higher in plastically deformed metals. Diffusion can also be enhanced by alloying, i.e., introducing elements with high diffusivity into the system. The highdiffusivity element accelerate the process of atomic motion at or across the interface. Elements selected as diffusionaccelerating are usually those which have reasonable solubility in the metal to be joined, do not form stable compounds, and depress the melting point locally. Melting point depression by alloying must be controlled due to the possibility of liquefaction at the joint interface. 8
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1.6 Design and Manufacture of CoAl, CuMgOAI, and CuSb Bonded Composites Three bonded composites were manufactured for use in this work. The first specimen was made of copper and aluminum; the second was identical except for a cavity machined at the interface which was filled with MgO powder before bonding. The third composite was made of copper and antimony. In the first two cases, the bonded composites were made of pure metals of relatively similar thermal conductivity. MgO powder was added to specimen number 2 to create a void at the interface insuring a boundary of comparatively poor thermal conductivity. In the third case, the CuSb composite was made at a temperature and pressure designed to maximize the formation of discrete intermetallic layers between the two parent metals. Figure 1.2 shows a schematic of the pre bonded configuration for the AlMgOCu composite. The AlCu composite was similar with the exception of the MgO inclusion at the interface. Figure 1.3 shows a schematic of the prebonded configuration of the SbCu composite. All three samples were prepared and processed by Messrs. Patrick Rodriguez, Peter Lopez, and David Huerta ofLANL's NMT5 group Each prebonded assembly was encapsulated in an evacuated tantalum container. Closure was achieved by electronbeam welding top and bottom lids to the main body of the container. Each assembly was evacuated to approximately 104 torr prior to welding. Once welded, each encapsulated assembly was placed in the hot isostatic press and bonded. The CuAl, CuMgOAl and SbCu composites were bonded in the same run of the hot isostatic press (HIP). (Drawing not to scare) 'Ta can EB welded at both encb "'2.625 .. o.d. X .O,ZO"' wall canister *Rest:stanc:e press fit lid T *Ta net wt. 130.21 g "Gross Vtot g "'Post v.'Eiding: dbnensipn5: 2.625" o.d. X 1.455" H AlMgO Cu Assembly in PreBonded Configuration Figure 1.2 Each sample was wire brushed under freon prior to insertion into the tantalum cladding to insure surfaces as clean as reasonably possible. Typically HIP bonding is done at a temperature approximately 50 percent to 75 percent of the melting temperature, T m of the metal possessing the lowest melting temperature. Bonding temperature was raised to approximately 80 to 85 percent of Tm for all three bonded composites used in this study. This higher HIP bonding temperature was employed to 1) insure a good bond 9
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and 2) in the case of the SbCu composite, insure growth of a discrete O.OZO Oearance (Dr.awlng not to scale) ..,.a can EB welded at both ends o.d. X .020" wall canister press fit l i d 0.020" T ..,.a net wt. 130.05 g *Gross wt. 776.74 g *Post welding dimensions: 2.626" o.d X 1.456" H CuSb Assembly in PreBonded Configuration Figure 1.3 intermetallic layer between the two parent metals. Since there is very little difference between T m for aluminum and Tm for antimony, all three specimens were bonded in the same HIP run Figure 1.4 shows a graph of the temperature and pressures used for bonding the three composite specimens HIP RUN .96014 530 C I 1 0 KSI /60 MIN SOM SEPTENBER Z6, [l.APS, 11WS: ( mm) !ll):xJ 6000 I) 41n i t I f wnE1f.'flll/'.TUE { Oil Temperature and Pressure Used for Bonding Specimens Figure 1.4 10
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1. 7 Properties of Experimental Materials The chemical, thermal, and metallurgical properties of the copper, aluminum, and antimony used in this work are taken from standard reference works or, where these values seemed possibly incorrect, determined experimentally. Properties determined experimentally are so identified. The copper and aluminum used in manufacturing the diffusionbonded specimens were purchased from JohnsonMatthey as 99.999 per cent pure The antimony (Sb) used for this study was contributed by the LANL Material Science and Technology MST) division. This antimony, used to create antimonallead alloy at Los Alamos, is itself comparatively less pure being about 98 per cent antimony. This compares with 99 per cent pure antimony, i.e., the most pure metal available commercially and then only in pellet form. The antimony contributed by MST division was taken from the sprue of a large casting, the form in which it is stored. Subsequent shaping of the antimony was done by Electric Discharge Machining (EDM) due to its extreme brittleness. Antimony was chosen for use in this study for two reasons. First it offers an advantageous thermal impedance (15.2 W/m 0C) when compared to copper (384 W/m 0C) and aluminum (202 W/m 0C). Second, as the phase diagram for the copperantimony system indicates in figure 1.5, copper and antimony readily form intermetallic compounds when diffusion bonded. Radiography of the copper/antimony composite made after bonding showed five discrete density zones Two were the parent metals and three were identifiable intermetallic layers the largest of which was approximately 2.3 mm thick. w lApproximate Temperature of the Hot Isostatic Bonding Cyle for Cu/Sb Specimen (Dashed Line) C14 .ID RlJ 3'0 qo .ro 40 'TO 8() 90 .Sb A r o .IV\ .z S Phase Diagram for the CopperAntimony System Figure 1.5 11
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1.71 Chemical, Thermal, and Mechanical Properties of Copper The trace elements in the copper used in this study are shown in Table 1.1. lbis chemical analysis was provided by David Wayne ofLANL's group CST 8. Two samples of the JohnsonMatthey copper were analyzed using Glow Discharge Mass Spectroscopy (GDMS) The GDMS analysis indicated the copper was probably between 99.900 and 99.990 percent pure, the largest trace constituent being oxygen. The thermal and mechanical properties used in the various analyses appearing in chapters 3,4, and 5, are shown in Table 1.2. 7 Element C12 016 Mg24 Al27 Si28 P31 S32 Cr52 Mn55 Fe56 Co59 Ni60 Zn68 As75 Se82 Mo95 Ag 107 Sn118 Sb121 Te125 Pb208 Bi209 Johnson Matthey Comparison Sample Parts per Million (ppm ) 32 <1000 0.22 .04 <0 05 0.49 .04 3.2 .4 39 <1.5 0.30. 04 1.1 .2 <0 .01 0.68.06 0.20.04 0.98 .03 1.2 .2 5.1 1.0 19 <0.1 0.50 .05 <0.5 1.7 .1 0.32 .02 Trace El e ments in Copper Table 1.1 7 ASM Handbook, Vol. 2, ASM Int e rnational, pgs. 11101113. 12 Experimental Sample Parts per Million (ppm) 319 <1500 0.019 .009 <0.08 <2 0.013 .004 9.7 0.4 <4.0 <0.03 12.8.2 0.07.01 3.2 .1 4.1 2 2.3 2 <1.0 1.1 .1 11. 2 .4 <0.1 0.33 .02 1.3 .3 0.50 .02 0 19 .03
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Thermal Conductivity (k) 384 W/m C Thermal Diffusivity (a) 11.234 wJ m;L Is Heat Capacity (Cv) 383 J/kg C Density (p) 8954kg/mj Poisson's ratio 0.343 Young's Modulus 12.98 lO'u Pa Coefficient of Thermal Expansion (j}) 20 w6 oct Thermal and Physical Properties of Copper Table 1.2 Figure 1.6 shows a photograph of the metallurgical structure of the copper used in this work. The photograph is at 25X. The dark spots are 1) artifacts or the etching process and 2) indication of preferential attack by the etchant at the grain boundaries probably due to the presence of oxygen at those locations The metallography in Figures 1.6, 1.7, and 1.9, was performed by Mr. Ramico Pereyra of Los Alamos CST15 group. Photograph of Metallurgical Structure of Copper, 25X Figure 1.6 Dr. Michael Hundley ofLANL's MST10 group measured thermal conductivity for each metal used in this work Figure 1 7 gives his measurement of k.:opper as a function of temperature As may be seen, the thermal conductivity shows marked temperature dependence. Figure 1.7 represents 13
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k.:opper as measured on an axis normal to the vertical or diametrical surfaces of the test cylinder The dark line in figure 1.7 is Holman's value for thermal conductivity of copper at 0 C.8 Thermal Conductivity (W/mC) 30 40 50 60 Thermal Conductivity Data for Copper Figure 1.7 70 1. 72 Chemical, Thermal, and Mechanical Properties of Aluminum The aluminum used for this work was purchased to a specification of 99.999 per cent pure. The analysis of race elements present in the aluminum is shown in Table 1.3. As with the copper, the chemical analysis of pure aluminum was provided by David Wayne of Los Alamos National Laboratory's CST8 group. Samples were analyzed using Glow Discharge Mass Spectroscopy (GDMS). The thermal properties of aluminum are given in Table 1.4.9 Figure 1.8 shows lOOX metallography of the aluminum used in this work. Again, the dark areas are artifacts of the etching process used to prepare this sample. Element Parts per Million (ppm) Sigma1 Parts per Million (ppm) Li < 0 004 Be < 0.01 B <0.05 c 111 5 N < 10 Na < 0.5 MK 0 52 0.06 Si 24 1 p 0.40 0.07 s 0.43 0.09 K < 2.0 Mn 0.5 0.1 Fe 19 4 8 Heat Transfer, Holman, J.P., McGrawHill, New York, 1986, pgs 635636. 9 ASM Handbook, Vol. 2, ASM International, pgs. 10991100. 14
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Co Ni Cu Zn Ga Ge As Zr Nb Sn Ph Thennal Conductivity (k) Thermal Diffusivity (a:) Heat Capacity (Cv) Density (p) Young's Modulus < 0.1 2.8 0.5 0.84 0.03 < 0.1 < 0.01 <0.04 < 0.01 <0. 05 0 069 0.005 <0.02 < 0.05 0.32 0.07 Trace Elements in Aluminum Table 1 3 205W/m C 8.418 w) m:.l /s C 2707 kg/mj 62GPa Coefficient of Thermal Expansion (j3) 23.6 10'6 0C'1 Thermal and Physical Properties of Aluminum Table 1.4 Photograph of Metallurgical Structure of Aluminum, 1 OOX Figure 1.8 Figure 1.9 shows Dr. Hundley's measurement of ka1 for the aluminum used in this work. Two functions of ka1 are shown; the first is ka1 as measured in a direction nonnal to the diametrical surfaces of the test cylinder (transverse), the second ka1 was measured in a direction from the center of the test cylinder to the circumferential edge (longitudinal). Figure 1.9 indicates considerable 15
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anisotropy in ka1. The dashed line indicates Holman's value for the thermal conductivity of pure aluminum at 0C. Thermal Conductivity (W/moC) 280 260 / Longitudinal ka1 I 240 7 Reference ka1 fTransverse ka1 I "...: v .o:.; ......... r.......... 220 200 20 30 40 50 60 Thermal Conductivity Data of Aluminum Figure 1.9 1.73 Chemical, Thermal, and Mechanical Properties of Antimony ........ 70 TempC I Two samples of antimony were also analyzed by David Wayne using GDMS. These samples were dried in a vacuum at 200 C prior to analysis. Results for C, Fe, Ni, Si, Co, and Zn are regarded as semiquantitative, i.e., off of real values by as much as 30 per cent. The large standard deviation for sulfur resulted from a peak intensity that decreased over time during the GDMS analysis. As can be seen in Table 1.5, the predominant contaminant of the antimony is iron with smaller but significant amounts of nickel, copper, and lead. Trace constituents are shown in Table 1.6. Element Weight Percent Sample 1 Sb121 97.94 Fe56 1.02 Ni60 0.50 Cu63 0.12 Pb208 0.21 Element Parts per Million (ppm) Sample 1 C12 43.6 Al27 <0.03 Si28 7 23 1sigma Weight Sample 2 Percent Sample 2 97.94 0.04 1.22 0.05 0.46 0.01 0.114 0.01 0.25 Major Contaminants of Antimony Table 1.5 1sigma Parts per Sample 2 Million (ppm) Sample 2 4.9 62.7 <0.03 1.44 4.65 16 1sigma Sample 2 0.06 0.01 0.005 0.02 1sigma Sample 2 4.6 0.31
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P31 <0.03 S32 523 Cr52 <1.2 Mn55 <0.05 Co59 382 Zn68 1.99 As75 938 Se82 129 Mo92 1.03 Ag107 8.23 Cd111 <0.02 ln115 <0.2 Sn120 22.0 <0.05 149 584 <1.2 <0.05 23 349 0.52 2.98 50 908 14 124 0.09 1.02 1.38 8.65 <0.15 <0.2 5.5 24.6 Trace Constituents of Antimony Table 1.6 66 10 0.17 38 9 0.13 0.17 0.6 The thermal and physical properties for antimony are given in Table 1.7. All of these values, with the exception of thermal conductivity, are taken from the ASM Han book. 10 Reference values of this type are typically derived from very pure examples of the metal of interest. It appeared likely, given the impurities known to present in the antimony used for this work, that actual thermal properties would vary from values found in standard references. Thermal Conductivity (k) 15.2 W/m C Thermal Diffusivity (a) 1.33 10' m2/s Heat Capacity (Cv) 205 J/kg C Density (p) 6650kg/mJ Poisson's Ratio Not Available Young's Modulus 77.759 GPa Coefficient of Thermal Expansion (j3) 9.2 10"6 C1 Thermal and Physical Properties of Antimony Table 1.7 In addition, reference values for the thermal conductivity of antimony (ksb) varied widely throughout the literature; they range from 19 W/m octo 26 W/m C. Initial modeling of heat transfer through the Sb test cylinder assumed ksb = 26 W/m C. Initial analyses provided inconsistent and often unsatisfactory results. Discrepancies between the several analyses and experimental data suggested the possibility that the reference value of ksb = 26 W /m oc was incorrect for the particular antimony used for this study. Figure 1.10 shows measurements of thermal conductivity taken over a range of approximately 20 C to 47 C. One measurement was taken of antimony machined radially from a cylindrical shape, i.e., with z coordinates held constant A second measurement was taken using a specimen of antimony machined in the transverse direction from the same sample cylinder, i.e., with values of r held constant. This was done to determine if thermal conductivity was anisotropic in the antimony used for this work As figure 1.10 indicates, no appreciable anisotropy was found 10 ASM Handbook, Vol. 2, ASM International, pgs. 11001101. 17
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Thermal Conductivity (W/m"C) 17 16.5 I l 16 15. 5 15 14.5 I 14 Thermal Con (Transv Thermal Cone CLom!itudi uctivity I se) rt:vity r25 30 35 40 45 Thermal Conductivity of Antimony Figure 1.10 Temperature ( "C) Metallography of the antimony used in this work is shown in figure 1.11. Photograph of Metallurgical Structure of Antimony, JOOX Figure 1.11 18
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1.74 Properties ofMacor Ceramic and RTV Silicon Potting Figure 3.2 shows a bonded AlCu cylinder mounted in a ceramic ring. This was the configuration used for temperature and dimensional measurements in the majority of experiments described in this work. Test cylinders were potted into the ceramic collars using a commercial hightemperature siliconbased RTV gasket material. This configuration was designed to 1) support the test cylinders while imposing no significant constraint, 2) insulate the circumference of the test cylinders, and 3) damp incipient vibration, and 4) provide the ability to conduct a number of experiments in as repeatable a fashion as possible. The Macor ceramic collar was machined to the same thickness as the test cylinders, i.e., .00635 m. The inside diameter of the collar was made24 mm larger than the outside diameter of the test cylinders. The Macor collar was 12 7 mm wide between the inside and outside diameters. The mechanical and thermal properties of Macor are well documented and appear in table 1.8. Thermal Conductivity (k) 1.46 W/m C Thermal Diffusivity (a) 7.33 107m2 Is Heat Capacity (Cv) 790J/kg C Density (p) 2520 kg/m3 Young's Modulus 66.9 GPa Poisson's Ratio .29 Coefficient of Thermal Expansion 9.3 106 C1 Properties of Macor Ceramic Table 1.8 The properties of the RTV potting were difficult to obtain. Properties appearing in table 1.9 were provided by the manufacturer's technical support group A value for specific heat was unavailable; Cv for silicon (the main component of RTV potting) was used instead. Thermal Conductivity (k) Density (p) Heat Capacity (Cv) Properties of RTV Potting Table 1.9 19 0.19 W/m C 895.9 kg/m3 790 Jlkg C
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Symbol f.J A B CJB c D E F G H K K M 0 p s a b b c e f g k m n p CJP Q q q ij A q R r s s t u u u w NOMENCLATURE USED IN CHAPTER 2 Definition Unit Area Thermal expansion tensor Body Boundary of B Elasticity tensor Finite strain tensor Infinitesimal strain tensor Deformation gradient Any symmetric tensor Any symmetric tensor Conductivity tensor Compliance tensor Stresstemperature tensor Origin, zero vector zero tensor Part ofB Stress tensor Prescribed stress on boundary Thermoelastic coupling constant Noninertial body force Inertial constant Specific heat Internal energy Inertial body force Temperature gradient Thermal conductivity Stress temperature modulus Unit outward normal to CJB Position vector Boundary of P Orthogonal tensor corresponding to a change of observer Heat flux vector Heat flux Prescribed heat flux vector on boundary Prescribed heat flux on boundary Orthogonal tensor such that Q = R T Heat supply Surface traction Prescribed traction on boundary Time Positive square root of FTF Displacement vector Prescribed displacement on boundary Rate of working Point in B Coefficient of thermal expansion 20
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0 ll 11 e Oo A. ll "' 1{J ll 1 ( .. ) c) (.X) (x) (.X) (f curl det div sym tr Kronecker delta, arbitrarily small quantity Entropy Prescribed entropy on boundary Absolute temperature Reference temperature Lame constant Lame constant, shear modulus Free energy Prescribed free energy on boundary Density Unit tensor First derivative with respect to time Second derivative with respect to time Response form of quantity x, e.g., q = q(F, 0, g, x) Reduced form of quantity x, e.g q = q{D, 0 g) Description of quantity x when ll is taken as an independent variable Transpose of a tensor Curl Detenninant Divergence Symmetric part of tensor Trace 2. The Linear, Coupled, QuasiStatic Theory ofThermoelasticity 2.1 Introduction The following discussion attempts to provide a development of the general theory of thennoelasticity as it proceeds from first principles of mechanics and thennodynarnics. Following this, the governing equations of thermoelasticity are derived within the context of various assumptions, explicit and implicit, which historically have applied to the development of this discipline This discussion then provides the mathematical basis for giving the general theory a linear fonn. Assumptions for isotropy and homogeneity are introduced. Finally, a discussion of the basis for the socalled coupled and uncoupled, quasistatic and general theories within thennoelasticity is given. Important tenns and concepts are italicized. Much of the following material is taken from Handbuch der Physik, and Heat Conduction Within Linear Thermoelasticity as noted below The explanation presented in chapter 2 seeks to provide the motivation for pursuing the coupled quasistatic analysis presented in chapters 4, 5, and 6 21
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2.2 The Basic Laws of Mechanics and Thermodynamics The motion of a body B bounded by a regular region dP is described by a vector field u on B x (0, to) where (0, to) is a fixed, open interval of time. The vector field u represents the displacement of a material point x (contained within B) at timet. A spatial gradient F of function x x + u(x, t) is the deformation gradient defined as F = 1 + Vu. (2.1) It is assumed for each t that the mapping x x + u(x, t) is onetoone on B and that its inverse is smooth, i.e., det F :1: 0. If S(x, t) denotes the first PiolaKirchoff stress tensor measured per unit surface area in a reference configuration, and if f(x, t) denotes a body force per unit volume in the reference configuration, then the laws of balance offorces and moments can be expressed as equations (2.2) and (2.3) respectively: J S n da + J f dv = 0 (2.2) i)p p J (p + u) x (S n)da + J (p + u) x fdv = 0. (2.3) p Where P is a part of B, CJP is the boundary of P, P = x 0 is the position vector of x relative to the origin 0, and n is the outward unit normal of dP. Equations (2.2) and (2.3) are good for all times t. Iff is continuous on B X ( O,t 0 ) then equations (2.2) and (2.3) are equivalent to VS + f = 0 and The rate of work of part P at time t is given by equation (2.4) da+ Jru dv p (2.21) (2.31) (2.4) Since F = V u it follows from the divergence theorem, J V n dS = J Vv dV that the first term S R on the right of equation (2.4) can be written as J (s n) u da = J S F dv + J u VS dv (2.5) i)p p p Equations (2.21) and (2.4) imply that 22
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W(P) == J S F dv p where the quantity S F is called the stress power. The first law of thermodynamics is given in equation (2.7) :C J e dv = W(P) J q n da + J r dv p p (2.6) (2.7) for every part P at time t and where e(x,t) is the internal energy per unit volume of the reference configuration; q (x, t) is the heat flux vector per unit volume in the reference configuration and r(x, t) is the heat supply per unit volume in the reference configuration external to the configuration. Equation (2 .7) has the local equivalent e == s F Vq + r (2 .8) when combined with equation (2.6). The second law of thermodynamics is given as equation (2.9) dJ Jqn Jr TJ dv da + dv dt (J (J p p (2.9) for every part P and every timet; 'fl(X, t) is entropy per unit volume in the reference configuration and a (x, t) is the absolute temperature. Therefore J TJ dv is the entropy ofP, J q n da is the total p Jp entropy flux across dP due to conduction and J dv represents the total entropy flux into the p interior of P from outside the reference configuration. Given previous assumptions, the second law is therefore equivalent to (2.10) where q is based on the unit surface area of the reference configuration. Free energy, 'If, is defined as 11' = eTJO, (2.11) which, when combined with equations (2.8) and (2.10), gives the local dissipation inequality in equation (2.12) (2.12) 23
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where g is the temperature gradient, g = Ve. Given equations (2.8) and (2.11 ), equations (2.10) and (2.12) are equivalent. 2.3 Elastic Materials and the Consequences of the Second Law An elastic material is defined by constitutive equations giving the stress S, free energy 'If, entropy "Jl, and the heat flux qat each point x whenever the deformation gradient F, temperature e, and temperature gradient g are known at point x: lfl = ljl(F, e, g, x) s = S(F, e, g, x) 11 = ij(F, e, g, x) q = q(F, e, g, x) (2.131) (2.132) (2.133) (2.134) where 1jl S ij and q are prescribed by the boundary of P and are called response functions; 1jl S ij, and q also must hold for all points in B. It is assumed these response functions are smooth on their domain which is the set of all (F, e, g, x) where F is a tensor with det F :F. 0, e is a positive scalar, g is a vector, and all x are contained within B. It is also assumed that S is subject to equation (2.14 ): S(F, e, g, x) FT = F s (F, e. g, x)T (2.14) It is necessary to introduce the concept of an admissible thermodynamic process Such a process operates within an ordered array [u, e, 'I' S, "Jl, q] with the following properties 1. All functions have a common domain in the form of P x T where P is a part of B and T is an open time interval of (0, to); 2. u is a motion on P x T and e is a positive scalar; 3. 'If, S, "Jl, and q are defined on P x T through the constitutive relations with (as before) F = 1 + Vu and g = VO. P x T is called the domain of the process and may vary from process to process. Given an admissible thermodynamic process, equations (2.21) and (2.8) yield the inertial body force f and heat supply r necessary to support the process f = VS and r = e S F + V q (2.151) (2.152) The local dissipation inequality, equation (2.12), implies certain restrictions on the response functions. A necessary and sufficient condition that every admissible thermodynamic process obey the local dissipation inequality is that the following three statements hold: I. The response functions 1jl, S, and ij are independent of the temperature gradient g: 24
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'I' = lji(F, 9); S = S(F, 9); and 1] = ij(F, 9); 2 1ji detennines S through the stress relation oo/(F. 9) S(F, 9} = .......;...,:_.......;...,:_ aF and i7 through the entropy relation 3. q obeys the heat inequality _( ) aij(F, 9). TJF,9o9 q(F, 9, g) g 0. A proof of the necessity and sufficiency of this condition is given by D. E. Carlson.11 In an admissible thermodynamic process, the energy equation, equation (2.8), takes the form 01] =Vq + r (2.16) Thus an admissible thermodynamic process is adiabatic (Vq + r = 0) if and only if it is isentropic (dT}/dt = 0). Also, in an admissible thermodynamic process, the response function for stress (S) and entropy ( fi ) satisfy the Maxwell relation dS(F, 9) = o9 aij(F, 9) (JF (2.17) By the relationships V' = e 1] e. V' = lji(F, e)' s = S(F, e). 1] = ij(F, e) and equations (2.131) through (2.134 ), internal energy obeys a constitutive relation of the form e(F, e) = lji(F, e) + Oij(F, 0) (2.18) Specific heat, c, is defined by c(F e) = atjf(F, e) + ii(F 9) + e di7(F, e) aB ., ae Th 1 (F 9) oo/(F. e) r e re au on 1J = ''0 9 __._ Imp 1es 11 Handbuch der Physik, D E Carlson, Volume Via/2, springerVerlag, 1972, pgs. 297312. 25
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c(F 9) = 9 itij(F, 9 ) d9 (2.19) c (F, 9) is assumed to be positive; this combined with 9 > 0 and equation (2.19) implies 9) is smoothly invertible in 9 for each choice ofF. The constitutive relations (2.13 1) through (2 134) can be rewritten e = e(F, 17), s = S(F, 77) (2.201), (2 202) 9 = O(F, 11)' q = q(F, 1], g) (2.203), (2.204) For a given F, the function 1] O(F, 17) is the inverse of the function 9 i](F, 9). The function e for example is given by e(F, 17) = e(F, ii, (F, 77)) (2.21) It follows from equations (2.18) and (2.21) that e(F, 17) = lji(F, ii, (F, 77)) + 17ii(F, 17), which gives (2.22) and (2 23) ( ) alji(F, o) ( ) dlji(F, 9) From the relations S F, 9 = dF and 1] F, 9 = ()9 it can be shown that ( ) ae(F, 11) SF, 1'/ = dF (2.24) and e (F, 11) = 1'/..:.:...) (2.25) Equations (2.24) and (2. 25) are the stress and temperature relations when entropy is taken as an independent thermodynamic variable. 26
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2.4 Material FrameIndifference The principle of material frameindifference holds that the constitutive relationships assumed in (2.131) through (2.134) are independent of any frame of reference (or the observer). Changing the frame of reference transforms the relevant mechanical and thermodynamic quantities as F7QF S 7 QS 9?9 g7 g 'lf?'lf Tt?Tt q?q where Q is the orthogonal tensor corresponding to the change.12 Since equations in (2.131) through (2. 134) are to be invariant under all such changes, the following relationships must hold 12 The transfonnation rule for F is a consequence of Truesdell and Noll, Encyclopedia of Physics, the NonLinear Field Theories of Mechanics, Volume 11113, section 5, pages 44. The rule for the first PiolaKirchoff stress tensor, S (Truesdell and Noll's TR ), follows from the transfonnation law for the Cauchy stress, T. Constitutive equations must be invariant under changes of frame of reference. If a constitutive equation is satisfied for a process with a motion and a symmetric stress tensor given by x =X (X,t), T = T(X,t) then it must be satisfied also for any equivalent process {X* ,T*} This is to say the constitutive equation must be satisfied also for the motion and stress tensor given by X = x* (X,t*) = c(t) + Q(t) X (X,t), T* = T*(X,t) = Q(t)T(X,t)Q(t)T, t* = ta where c(t) is an arbitrary point function, Q(t) an arbitrary orthogonal tensor function of the time t, and an arbitrary number. Further, let ()p be the boundary of a part P of a body in some configuration X defined by its defonnation x = x(X) from a fixed global reference configuration. The contact force acting on P is where n is the outward unit vector nonnal to ()p in the configuration X Let dsR be the surface element and nR the outward unit nonnal to ()p in the reference configuration. By the laws of the transfonnation of surface integrals, fc =I Tnds= I TR nR dsR ()p ()p where TR is defined by the relation T=T1TR FT inwhich J=ldetFI=_f!_. PR 27
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VI{F, e) = lji{QF, e) s(F, e)= QrS{QF, 6) ij(F, 6) = ij( QF, 6) ij{F, 6, g) = ij( QF, 6, g) for every orthogonal tensor Q and for all {F, e, g) in the domain Using the polar decomposition theorem, F=RU where R is orthogonal and U is positive definite square root of FTF. 13 Choosing (2.261) (2.262) (2 263) (2.264) (2.27) Q = RT in equation (2 261) through (2. 264) and using equation (2 27), 'I' S, 1'\. and q have the relationships 11' = VI{U, e) s = Fu's(u. e) 17 = ij(U, 6) q = ij(U, 6, g). Defining the finite strain tensor, D, as leads to the consequences of material frameindifference. (2.28) The constitutive equations in (2.131) through (2.134) satisfy the principal of material frame indifference if and only if they can be written in reduced form: 11' = Vi{D, 6) S = FS{D, 6) 11 = ff(D, e) q = ij{D, 6 g) The temperature gradient g is invariant since it is the gradient relative to the reference configuration p. The heat flux, q, is invariant since it is measured per unit area p. 13 H Richter first employed the polar decomposition theorem to explore the implications of material frame indifference in Sur Elastizitatstheorie endlicher Verformungen, Mathematische Nachrichtem, Vol. 8, pp. 6573; translated into English in Foundations of Elastic Theory, ed. C. Truesdell, Gordon and Breach. The polar decomposition theorem states that any invertible tensor F can be expressed as F = VQ = QU where Q is an orthogonal tensor and U and V are symmetric tensors. 28
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In addition, the stress and entropy relations (2.131) through (2.134) reduce to equations (2.29) and (2.30): s(n o) = av;(n. o) an ff(n. o) = av;(n. o) ao The derivation of equations (2.291) and (2 292) proceeds in terms of components: av; av; aDij =where, from equation (2 28), Since it can be shown or equivalently (2.291) (2.292) (2.30) aa_.!!_= F __.!!._. (2.31) aF an Equation (2.31), the stress relation of (2.132), and S = F ut S(U, 0) imply equation (2.291) It is interesting to note that automatically satisfies S = F _all'_('n_. 0....:....} an The implication here is that in thermoelasticity the balance of moments is a consequence of balance of forces, the two laws of thermodynamics, and material frameindifference. 29
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2.5 Consequences of the Heat Conduction Inequality From equation (2.133), the reduced response function q must obey the inequality ij(D, 9, g) g 0. (2.32) The conductivity tensor is defined as K (D, 9) = oq(D, 9, g) og g=o (2.33) Holding D and 9 fixed and expanding ij(g) = ij(D, 9, g) in a Taylor series about g = 0 gives (2.34) Combining equations (2.32) and (2.34) gives ij(O) gg K +o(lgl2 ) 0 The inequality holds for all g if and only if q(O) = 0 and g K g 0 for all g. This has the consequence that the heat flux vanishes whenever the temperature gradient vanishes, i.e., ij(D, 9,0) = 0. (2.35) From equation (2.35) oii(D, 9,o) _..;...__'= 0 an (2.36) oii(D, 9,o) _..:....a9'= 0 (2.37) 2.6 Derivation of the Linear Thermoelastic Theory The complete system of field equations for nonlinear thermoelastic theory has been derived consisting of equations (2.21) and (2.16): vs + f = 0 (2.21) 61j =Vq + r (2.16) 30
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combined with the constitutive equations where and lfl = Vi(D, o) s = FS(D, 8) 11 = ij(D, 8) q = ij(D, 8, g) F = 1 + Vu; D = (FTF t); g = vo. These constitutive relations are subject to the thermodynamic restrictions ( ) JVi(D, 8) S D, 8 = _....;______;_ JD _( ) JVi(D, 8) 11 D, 8 =_;,_JO'and ij(D, 8, g) g 0 (2.381) (2.382) (2.383) (2.384) (2.391) (2.392) (2 393) A linear approximation of equations (2.21 ), (2.16), (2.38), (2.1 ), (2.28), and (2.39) proceeds under the assumptions: 1. the displacement gradient and its time rate of change are small 2. the temperature field is nearly equal to a given uniform field 90 called a reference temperature 3. the temperature rate and the temperature gradient are small. Thus it is assumed that IVul IVul 10 801 31 (2.401) (2.402) (2.403)
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where 0 is arbitrarily small. Jol 8; I gl 8: Due to inequalities (2.401) through (2.405), the deformation gradient F becomes F = 1 + 0(8) as 0 0. The infinitesimal strain tensor E is by (2.40), E becomes E = 0(8), E = o(8), From equations (2.39 1), (2.392), (2.40), (2.42), and (2.43), the finite strain tensor, D, is D = o(8). Equations (2.44) and (2.401) through (2.405 imply or (2.404) (2.405) (2.41) (2.42) (2.431) (2.432) (2.44) (2.45) (2.46) (2.47) (2.48) WhenF=l, () =()0,D=O,and S = S(O, 90), thequantity S(O, 90)representstheresidualstress at the reference temperature, i.e., the stress that the body would experience if it were held in the reference configuration at the uniform temperature() 0 It is assumed s(o, 9o) = 0. (2.49) 32
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The assumptions of zero residual stress and uniform reference temperature are foundational to the classic theory ofthermoelasticity. A Taylor series expansion of S(D, 6), with S(O, 60 ) = 0, around D = 0 and (J = (J 0 gives Js(o, 6) JS(D, 6) S(D, 6) = s(o, 6o) + JD D=O (no)+ ()6 D=O (66o) + 0(8) (2.50) 6=80 6=8o Equation (2.50) can be rewritten S(D, 6) = c{E] + (6 60)M + o(i5) (2.51) where JS(D, 6) c = _..:...__'! JD D=O and JS(D, 6) M = J6 D=O (2.52), (2.53) 8=6o 8=80 With equations (2.40), (2.41), and (2.431), equation (2.51) can be written as S(D, 6) = C[E] + (6 60)M + 0(!5). (2.54) cis a fourthorder tensor called the elasticity tensor. With equation S(D, 6) = ay;: 6)' and equation (2.52), equation (2.54) can be written C = JS(D, 6) = _i_ rJVi(D, 6) ] = J2 Vi(D, 6) JD D=O JD JD D=O i)D2 D=O 6=60 6=60 8=60 (2.55) For any pair of symmetric tensors G and H, az G C[H] = JaJf3 Jii(aG + f3H, 60)la=JJ=O = H C[G] which shows the elasticity tensor is symmetric. Tensor M is called the stresstemperature tensor; since S = ST, the streSStemperature tensor is also symmetric. Expanding q = ij(D, 6, g) in a Taylor series about (0, 60,0) yields with equations (2.35), (2.36), (2.40), and (2.44) through (2.46), q can be written 33
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( ) ( ) c1q(0,00,0) c1q(0,00,0) ( ) c1q(0,00,0) ( ) q D, 0, g = q 0, 0,0 + c1D (D 0) + ao 0 00 + c1g g +h. o. or q(D, o, g)ID=o = Kg + h.o. = K ve. 9=90 g=O where K is the conductivity tensor. At (0, 90 ,0), K is c1q(o, 00 g) K=c1g g=O (2.56) (2.57) This result requires that within an error of O(o) the heat flux depends linearly on the temperature gradient and is independent of the strain and the temperature. This result follows from the heat conduction inequality; note that no argument concerning symmetry is involved The asymptotic form of the law of forces is equation (2.2.1) vs + f = 0. (2.21) If the noninertial body force, b, is introduced, then f becomes f=bpii. (2.56) where pis density. Equation (2.21) becomes VS + b = pii. (2.57) The last equation needed to complete the classical linearized theory of thermoelasticity is a linear form of the energy equation (2.16), 81] = V q + r Given the relationship 1J = ij ( D, 8 ) the time derivative of11 is Since if is smooth, c1ij(D, 0) c1ij(D, 0) 1}= D+ 0 GlD oO ar;(n. o) ar;(n. o) = c1D c1D D=O 9=90 34 + 0(1) (2.58) (2.59)
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afj(D, o) ao = afj(D, o) +0(1). Utilizing equations (2.29), Therefore r( )] = [ay;(n, o)] =a2y;(n, o) iJO S D, 6 ao an anao iJif(D, e) ae = as(D, 6) ao (2.60) (2.61) (2.62) (2.63) a(n o) a(n o) aas From equations S(D, o) = V' and ii(D, 6) = V' , it follows that ___!]__ = aD i}(J aD a 6 This is a reduced form of the Maxwell relation equation (2.17) which, when combined with equation (2.53), gives ae =M. (2.64) afj(D, o) By equations (2.19) and (2.381) through (2.384), the number aij(D, 6) c = 60 ''! ao 0=0 (2.65) 8=80 is the specific heat corresponding to D = 0 and (} = (} 0 Equations (2.58}, (2.60} (2.64 ), (2.65}, (2.40), (2.432), and (2.48) give or, = 60M E + cO + o(o) (2.66) This combined with equation (2.16) gives the asymptotic form of the energy equation V q + 00M E + r + o{ o) = cO (2.67) The basic equations for linear thermoelasticity theory are then 35
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(2.42) vs + f = p ii (2 57) Vq + 00M E + r + O(c5) = cO (2.67) S = C[E] + (0 00)M (2.51) q =K vo. (2.56) M and the values of Care symmetric From equation (2.54)) it can be seen S =sT. When E = 0 and strain vanishes, equation (2.54) reduces to S = (000 ) M. (2 68) Since the elasticity tensor, C, is invertible (C's domain being restricted to the space of all symmetric tensors), equation (2.54) can be solved forE: E = K[S] + (000 ) A (2 69) where 1 K = C and A = K[M]. (2.70), (2.71) KandA are called the compliance tensor and the thermal expansion tensor, respectively. From equation (2.69), A gives the strain resulting from a given temperature distribution when S vanishes: E = ( 000 )A = ( 000 ) K [M] (2.72) Material functions C, K, M, c, and p generally depend on the reference temperature ()0 and do not depend on temperature () Dependence on () is inconsistent with the assumptions leading to equation (2 67). Unless the body B is homogenous, the material functions depend on the position of x in B. Material symmetry effects the form of the elasticity tensor C, the stresstemperature tensor M, and the conductivity tensor K. When a material is isotropic C[E] = 2p E + It ( t r E) 1, (2.73 1) M=ml, (2 732) and K=kl. (2 .733) 36
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The scalars Jl and A. are the Lame moduli; Jl is the shear modulus, m is the stresstemperature modulus, and k is thermal conductivity. Combining the relations (2.73) with equations (2.42), (2.57), (2.66), (2.54), and (2.56) gives the basic equations of linear thermoelasticity for an isotropic body: E = (vu + Vu r) (2.74) VS + b = pii (2.75) Vq + m80 tr E + r = cO (2.76) S = 2 J.L E + A. ( tr E) 1 + m(e 80 ) 1 (2.77) q =k V9. (2.78) It follows from equations (2.78) that in an isotropic body, the stress is equal to a pressure when strain vanishes: S = m (990 ) 1 whenE=O. If Jl '# 0 and 3A. + 2Jl '# 0, equation (2.77) can be inverted to give E = ..!_ S 2 A. ( ) (tr s) 1 + a(e90 ) 1 2J.L 3A. + 2 J.L where is the coefficient of thermal expansion f3 = m 3A. + 2J.L Under isotropy assumptions, the thermal expansion tensor, A, becomes 1. According to (2.79), the strain is equal to a dilatation when the stress vanishes: E = f3 (980 ) 1 when S=O. 2.7 OneDimensional Linear Thermoelasticity (2.79) (2.80) (2.81) Assume a homogenous and isotropic body bounded by a pair of parallel planes at x = 0 and x = l. Assume the body has a uniform stressfree reference state at absolute temperature 90 throughout. Further assume no external supply of heat or external body force is available All displacement is measured from the reference state configuration and remains parallel to the xaxis for all times t. Under 37
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these restrictions, equations for onedimensional linear thermoelasticity can be derived, or reduced, from the more general forms given as equations (2.75) and (2 76). The equation of energy (2.76) is Vq + m80 tr E + r = cO. a2u By the assumptions, r = 0 and the trace of E becomes the strain rate. Recognizing from axat equation (2.78) that q=k V (} and from equation (2 80) that the stresstemperature modulus is m = f3 { 3). + 2f.L) the equation of energy becomes (2.82) Similarly, the equation of motion (2.75) is given as vs + b = pii. Under the same restrictions and assumptions as before, b = 0. Also, the infinitesimal strain tensor E becomes Equation (2.77) combined with equation (2.83) becomes au S = {2 f.L + A.) + m(O 90 ) ax The divergence of S for the onedimensional case becomes a2 u ao V S = (2 f.L + ).) 2 + max ax au ax (2.83) where a denotes the temperature change from the reference state, ( 0 00). The divergence of sin equation (2.75), combined with the definition of the stresstemperature modulus, m = f3 (3). + 2f.L), becomes the equation of motion stated as (2.84) Introducing equations (2 85) and (2.86) for entropy, Tf, and stress, a, respectively, 17 = (9 9o) + /3(3A. + 2p.) a u. dx 38
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) au CJ = (A. + 2J.J. .8(3A. + 2JJ.)(680), dx combined with q = k VO, allow equations (2.82) and (2.84) to be written as aa cPu = p2. dx dt The partial derivatives appearing in equations (2.82) and (2. 84) au au ax at. are the strain, velocity, strain rate, and acceleration, respectively. (2.85) (2.86) Day14 discusses using these relationships to transform the equations of energy and motion into the two governing thermoelastic equations for a homogenous, isotropic body in 1space. The number of constants in the equations is reduced by introducing a change of variables X 1 This change results in replacing the spatial interval [0,1) with a unit interval [0, 1]; the reference temperature 80 by 8 = 0. Use of the changed variables in equations (2.82) and (2. 84) give equations (2.87) and (2.88). These equations are a version of the two governing equations for linear, isotropic, homogenous thermoelasticity for the half space x > 0: = ao + .;;; cPu dt Jxat (2.87) and (2.88) The constants a and bare the coupling constant and the inertial constant, respectively. In the formulation of equations (2.87) and (2.88), constants a and bare defined as 14 H eat Conduction Within Linear Thermoelasticity William Alan Day, Springer Tracts in Natural Philosophy, vol. 30, Springer Verlag, 1985, pg 15 39
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a = 00,B2(3A. + 2pf c (A. + 2,u) b = k2 p c2 (A.+ 2,u) f The coupling constant, a, is usually small compared to unity and is independent of the thickness of the halfspace. The coupling constant is a measure of the coupling between thermal and mechanical effects. It can also be divided into two factors, one the inverse of the other; one factor describes the effect of the thermal field on mechanical response, the other factor describes the effect of the mechanical response on the thermal field. As will be discussed later, the inertial constant, b, approaches zero for metals and is usually disregarded. Given the specific change in variables which result in equations (2.87) and (2.88), the definitions of heat flux, q, entropy,n, and stress, a, become q = 1dU 7J = 0 + yu , dX Ju r and a = v a dX If constants a and b are set equal to zero, and if displacement at the boundaries x = 0 and x = 1 is constrained such that ul = ul = 0 x = O x=l the equations of the onedimensional version of Fourier's theory are obtained. In this theory, temperature is a solution of the heat equation = () x2 Jt (2.89) and displacement vanishes. As a comparison, if a and b are considered positive and displacement between the thermoelastic equations is eliminated, temperature becomes a solution of the fourthorder equation (2.90) Equation (2.90) is the onedimensional, homogenous and isotropic version of Carlson's temperature equation as given in his explanation of dynamic thermoelasticity.15 Two simplifying assumptions can be made to facilitate the solution of actual problems First, the term ( 00 M E) in the equation (2.67) can be neglected; the resulting theory is referred to as uncoupled. Second, the term (pii) can be neglected in equation (2.57) resulting in the quasistatic theory of 15 Handbuch der Physik, D. E. Carlson, Vol. Vla/2, SpringerVerlag, 1972, pg. 328. 40
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thermoelasticity; in practice, the quasistatic assumption applies best when changes of temperature proceed comparatively slowly. Even though the term (pu) is canceled, the functions involving u and a remain functions of time, t. This distinguishes the quasistatic approach from the stationary approach. Typically, both of these approximations are made together giving the quasistatic uncoupled theory of thermoelasticity. For reasons discussed in Chapter 4, the analysis performed here adopts the coupled quasistatic approach to thermoelastic theory. 41
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Symbol r t p A qcalculaled qcoupled T(x,t) T(r,z,t) Cp k p a Tip si Sb X(j}m,X) Hi Jo .'\T NOMENCLATURE USED IN CHAPTER 3 Definition Heat convection coefficient for the ith surface Spatial coordinate for semiinfinite analysis Thickness of semiinfinite plate or slab Ambient air temperature Electrical current Electrical resistance. region of space Radius of experimental cylinder, radial coordinate Time Power, calculated from i2R Area of experimental cylinder's vertical surface, Heat flux calculated from PIA where P = i2R Heat flux coupled to test cylinder Temperature as a function of time and onedimensional space Temperature as a function of time and cylindrical coordinates r and z Specific heat at constant pressure Thermal conductivity Density Thermal diffusivity, a = k CvP Coefficient of thermal expansion Any eigenfunction generally Eigenvalue(s) summed over m Eigenvalue(s) summed over n ith boundary surface Chemical symbol for antimony Eigenfunction related to a specific boundary condition Parameter, Hi= h/k Bessel function of the first kind, order 0 Difference in temperature, specifically temperatures measured by different thermocouples Difference in temperature between the center and edge of a face of the copper cylinder Difference in temperature between the center and edge of a face of the antimony cylinder 3. Analytical Modeling of Heat Flow through Test Cylinders 3.1 Introduction Modeling heat flow across a diffusion bond between dissimilar metals requires an understanding of how heat flows through each separate metal separately. Such modeling can be done either analytically or numerically. The accuracy of various modeling techniques can be assessed by comparing analytical 42
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and numerical models with experimental data and with each other. Experimental data developed using simple, carefully controlled geometry and pure materials provide the best basis for these comparisons. Boundary conditions can be ascertained and the correctness of material properties of the different test metals determined. Two analytical methods are used to model heat conduction through a solid cylinder. These are semiinfinite and finite formulations. In the former, temperature is calculated as a function of distance and time, T(x,t); the latter calculates temperature as a function of radius (r), thickness (z), and time, i.e., T(r,z,t). The results of the semiinfmite and finite models are compared with experimental data. Experimental data for cylinders of copper, aluminum, and antimony are presented. Copper and aluminum are of primary interest since the bonded composites used in the experiments and analyses presented in chapters 5 and 6 are made of these metals. Also, it was possible to obtain very pure examples of both these metals as described in chapter 1. The data obtained from the antimony is of secondary value and not presented in detail. It is useful, however, to examine how successfully the models used for copper and aluminum (both of which have high thermal conductivity) could predict heat transfer through antimony possessing low thermal conductivity. MacorCoUar 1+t Schematic of SemiInfinite Experiment Figure 3.1 Not to Scale Figure 3.1 illustrates the configuration of the heat conduction experiments involving cylinders of single metals. The heat flux, q, is supplied by an adhesive resistance heater placed on the x = 0 surface. The relative orientation of the Macor collar and RTV potting is shown. An adhesive thermocouple by which measurements of temperature are taken is placed on the x = L surface of the test cylinder. Boundary conditions and an initial condition typical of a semiinfinite analysis are also shown T_ denotes the temperature of the surrounding atmosphere, k is the thermal conductivity of the test metal, h 1 and h2 are the convection coefficients corresponding to x = 0 and x = L surfaces of the test cylinder respectively. These coefficients important in approximating two of the boundary conditions in a semi infinite analysis (see equations 3.2and 3.3). Cylinders were machined of each of the three test metals. The test cylinders of copper and aluminum each had a diameter of .03175 .00005 m and a thickness of .00635 .00005 m. The cylinder of 43
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antimony (Sb) was machined using an electronic discharge machining method (EDM) due to its brittleness. The diameter and thickness of the antimony was .03048 .00005 m and .0165 .00005 m, respectively. Siliconbased RTV potting was used to 1) support the samples without constraining the samples dimensionally, 2) provide consistent insulation on the radial surface of all samples tested, and 3) provide some damping of incipient vibration. Section 3.2 discusses the first model is based on a semiinfinite formulation. Figure 3.2 shows the configuration of a AlCu composite plate potted into its Macor collar; an example of the adhesivebacked resistance heater used to supply q is also shown. Configuration of Potted AlCu Composite Plate Figure 3.2 3.2 Derivation of Formula for T(x ,t) The first modeling of heat transfer through a test cylinder was done using a formulation for a semi infinite slab. The coordinate system has one spatial variable, x. Since the analysis is done on the half space, the problem is for a slab with a thickness of L. The following are the governing equation, applicable boundary conditions, and initial condition: = ..!.. ar(x, t) a at 44 in t>O (3.1)
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at X= 0, t> 0 (3.2) k + h,_T = = fl(t) at x=L, t>O (3.3) T(x, t) = F(x) for O:s;;x;:::L att=O. (3.4) Since the boundary conditions are not equal to zero, the problem is nonhomogeneous. The solution employs an integraltransform approach based on the following eigenvalue problem: V2VF(r) + A..2'1f(r) = 0 in a region R, and d'lf(r.) k; '+ h; VF( r ;) = 0 on the boundary of S;. d1J; R is the region of interest, 'I'm (r;) are eigenfunctions, / is the normal derivative in the outward on; direction at the ith surface, i is the number of continuous boundary surfaces, S;, of region R (the x = 0 and x = L surfaces in the semiinfinite formulation). An integraltransform pair is constructed allowing reduction of equation (3 1) to an ordinary differential equation. Similarly, the initial condition is reduced from a function of x and t to a function of one eigenvalue alone. A full explanation of this procedure is given in appendix 1 for the case of a finite cylinder. The solution for the semiinfinite problem is stated directly as ( ) L .. X(f3m, x) aAz,[( ) JL .Az ( ) J T t I'm F a + ea"m' A am t' dt' X, = m=l N(/3m) e I'm 0 M (3.5) where and F(.B"a) = r x(fim, x') F(x') dx'. X(/3m, x) is an eigenfunction, N(/3m) is a normalization integral, and eigenvalues. Particular values for X(/3m, x) and N(/3m) are taken from Ozisik16 The final form of the functional representation for T(x,t) for the specific boundary conditions of the experiment is shown as equation (3.6): 16 Heat Conduction, M Nectar Ozisik, 2"d Edition, Wiley & Sons, New York, 1993, pp. 4849. 45
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{ foL .BmCos(.Bmx') + HISin(.Bmx')dx' (3.6) In equation (3.6), H1 and H2 are constants given by the ratio of thennal conductivity to the heat transfer convection coefficient for the ith surface of the region, 0 S: x L. The eigenvalues are the positive roots of equation (3.7) tan,BmL = (3.7) 3.3 Modeling T(x,t) in a SemiInfinite Slab The first measurements of heat transfer and temperature through a copper slab were made using a resistance heater rated at 264 ohms. Tests were conducted using varying power levels including 2 W, 5 W, and 8 W. The 8 W test is discussed here to allow a basis for comparison since an 8 W measurement were also made using the aluminum and antimony test pieces. A .188 ampere current at 44.3 V produced an 8.0 W load. This, when divided by the 0.00317 m2 surface area of the copper cylinder, produced a qcaiculatcd of2519 W/ m2 Two type K thennocouples were used to experimentally determine temperature; the center thennocouple was placed at coordinates x =Lin the center of the cylinder. The second thennocouple was placed at the outside diameter of the cylinder on the x = L surface. Temperature measurements indicated the copper cylinder had a unifonn initial temperature of 25.1 C. The data from the thennocouples was edited in an Excel file and this file was in turn collated and graphed using Mathematica program JCUM7 .DAT. Figure 3.3 gives a listing of Mathematica program JCUM7.DAT; the output of JCUM7.DAT is shown in figure 3.4. (*** Program JCUM7 ***) (*** Records and Plots Thermocouple Data ***) l!a:\jalrn7 .csv a"' Flatten[ReadList["a:\jcum7 .csv", Number,RecordLists> True,RecordSeparators >{ "\n", "," }]]; b = Partition[a,2]; c = Transpose[b]; dl = c[[l]]; d2 = c[[2]]; Length[dl] 46
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x = Range[236]*10/60//N; y = Transpose[{ x,d 1 ) ] ; y 1 = Transpose[{ x,d2)]; picl = ListPlot[y, Plotioined> True,PlotRange>{20,70 },OridLines>Automatic, DisplayFunction>ldentity] pic2 = ListPlot[yl, Plotioined> True,PlotRange>( 20,70} ,GridLines>Automatic, DisplayFunction>ldentity] Show[picl,pic2,DisplayFunction>$DisplayFunction] Listing of Mathematica Program JCUM7.DAT Figure 3.3 At approximately t = 1700 seconds the exterior thermocouple detached from and was restored to the surface of the copper cylinder, as can be seen in figure 3.4. Figure 3.4 shows T(x=.00635,t) as measured using 8W of power to heat the copper test cylinder .,70 tfemp (C) I [Cenlf Thermocou le 60 ./' 7 v / / ouplel/ / IL ss of Thermo / I v I I Exterior I 50 40 30 0 500 1000 1500 2000 ITime (s) T(x,t)for Measurement for Copper Cylinder, L = .00635 m Figure 3.4 T(x,t) calculated using Mathematica program CPPRl, the listing of which is shown as Figure 3.5. The first attempt at modeling T(x,t) employed convective heat transfer coefficients h1 = h2 = 10 W/m2 C, a standard value for h for a vertical plate under free convection in atmospheric air. On this basis, comparison of T(x,t) as calculated and T(x,t) as measured indicated that either 1) values used for h1 and/or h2 were incorrect, 2) the heat flux through the cylinder was incorrect, i.e., the 8 W created by the resistance heater was not in fact entirely transferred to the test cylinder, 3) the thermal properties used for pure copper were incorrect, or 4) some combination of all of these. That the published values for thermal conductivity and thermal diffusivity of pure copper could be in error to the degree necessary to create so significant a discrepancy was considered unlikely. The error seemed likely to be a combination of 1 and 2. (* .. SemiInfinite Plate Problem ***) ( 21596 ***) (*** Copper ***) (*** Comparison Run, Semi to Measurement, 31697 ***) (** Physical Constants ***) Alpha! = .00011234 47
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L=0.00635 To=25 Tinf=22 k=384 h1 = 12 h2= 12 g= 2519 Hl =hl/k H2=h21k fl = g(Tinf h1) f2=h2 *Tinf lmt=3 (*** Initialize Eigenvalues ***) Array[Beta1,lmt] {Beta1[1],Beta1[2],Beta1[3], Beta1[4],Beta1[5], Beta1 [6],Betal [7],Beta1 [8],Beta1 [9],Betal [1 0] )= {3.73401, 494.76718, 989.49209, 2968.44008221, 3957.91656364, 4947.39365295, 5936.8710462, 6926.34861313,7915.82628861, 8905.30403645} (** Fonnulation of the Temperature Function **) Num = (Beta1[m] Cos[Beta1[m] x]) + (Hl Sin[Beta1 [m] x]) Dent = (Betal [m]112 + Hl112) Den2 = L + (H2/(Betal[m)112 + H2112)) Den= (Dent Den2) + Hl Expo= Exp[Alphal Betal[m)112 t] Intgmd=(Beta1[m] Cos[Beta1[m]*xp]) + (Hl Sin[Betal[m] xp]) lntl=lntegrate[Intgmd, { xp, O,L)] Mull = To Expo Inti Mul2=(fl/(k Betal[m])) *(IExpo) Mul3=(f2/(k Betal[m])112) ((Betal[m] Cos[Beta1[m] L]) +(HI Sin[Betal[m] L])) (1Expo) (** Construct T(x) and Sum **) Tfunc = (Num/Den) (Mull + Mul2 + Mul3) Tsum = Sum[Tfunc, { m,l,lmt)] Txt=2 Tsum (** Post Process T(x,t) **) N[Txtl.{ t > 0, x > L),12] pla12d=Plot[Txtl.{x > L), {t,0,2400}, Frame> False, GridLines >Automatic, AxesLabel> { time (s)", "temp (C)"), PlotRange > {20 70}, Plotl..abel> "h = 20,14, z = 006; q=2519"] Listing of Mathemntica Program CPPRJ Figure 3.5 A number of calculations of T(x,t) were done using varying values of qcalculated h1 and h2 The frrst calculation employed a nominal qcalculated for the 8W experiment of 2519 W /m2 This nominal qcalculated when combined with h1 = h2 = 12 W/m2 C, approximated the measured data for T(x,t) poorly. Many more calculations were performed using various combinations of qcalculale and h2 The calculations best approximating T(x,t) as measured are shown in figure 3.6. These approximations all hold h1 and h2constant at 12 W/m2 C and are compared to measured data in the inset of figure 3.6. 48
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0 500 1000 1500 2000 Comparison ofT(x,t) with varying q; hi= h2 = 12 W/m2 oc Figure 3.6 ITime (s) Calculations based on h1 = h2 = 12 W/m2 C and 1800 W/m2::;;; qcalculatcd 2200 W/m2 provided results most closely approximating measured data. The approximations appearing in figure 3 6 provided the first evidence not all of the heat flux produced by the resistance heater was coupling with the copper test cylinder. This asswnption was equivalent to increasing the convection from the resistance heater to 20 W/m2 oc::;;; hi 26 W/m2 C. For analyses ofT(x,t) throughout this work, it was necessary to use a reduced figure for q or a large figure for h1 if calculations were to agree with measurement. An arbitrarily reduced q was chosen and is designated as qcoupled throughout this work. Subsequent calculations of T(x,t), and later T(r,z,t), for aluminum and antimony consistently yielded good approximations when the same reduced figure for qcoupled was used for them all. These consistent results were considered some validation for the assumption of resistance heater coupling loss. The first measurements of heat transfer through the aluminum test cylinder employed the same resistance heater as was in the copper experiment. As before, the heater was found to have a resistance of 264 ohms A .188 ampere current at 44.3 V produced an 8.0 W load. This, when divided by the 0.00317 m2 surface area of the cylinder, produced a qcalculated of 2519 W/ m2 Two type K thermocouples were used to experimentally determine temperature; the center thermocouple was placed at coordinates x =Lin the center of the cylinder. The second thermocouple was placed near to the outside diameter of the cylinder on the x = L = .03170 m surface. The aluminum cylinder was assumed to have a uniform initial temperature of 21. 2 C, this being the initial temperature measured at three locations on the test cylinder prior to the experiment. Mathematica program CPPRl (with material properties and eigenvalues appropriate for aluminum) was used to compute T(x,t). The measured data for temperature given by two thermocouples located at (r = 0, z = .00635 m) and (r = .03170 m, z = .00635 m) are shown in figure 3.7. 49
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I Temp (C) I 70 I ,. m aod za.OO 3S m rr_ 60 / v 50 / /I Thermo ouple Posltlooe at r / r.0317 m aod z .OU 35m 40 7 h, = h2 = 15 W/m2 ococ /_ qcoupled = 2200 W /m 2 30 v o\.. 0 500 1000 1500 2000 6 n. 5 n 4 (). 3 n I Temp coc> I .. / ,=I I II' / llJ 112 1111 "" "" II qcoupled = 2100 w /ml I 0 500 1000 1500 2000 Time (s) Comparison ofT(x,t) with Measured Data for Aluminum Figure 3.7 Proceeding from the assumption that approximately 12 to 17 per cent of qcalculart.d is lost, or uncoupled, from the aluminum test cylinder, figure 3.7 shows two plots of T(x = .00635 m ,t). Since the calculations of T(x,t) closely approximate the measured data, the plots are given as dashed lines for clarity. One plot is based on convection coefficients of h1 = h2 = 15 W/m2 oc and a qcoupled of 2200 W/m2 ; the second plot is based on h1 = h2 = 14 W/m2 C and a qcoupted of 2100 W/m2 The semiinfinite numerical approximations of T(x,t) are quite accurate, given the chosen boundary conditions and the assumption of some heat loss, i.e., the difference between qcalcutart.d and qcoupted 3.4 Formula for T(r,z,t) in a Finite Cylinder Modeling heat transfer through a finite cylinder provides a useful comparison to and check of the results of the preceding analysis of a semiinfinite slab. The boundary conditions, position of thermocouples, and spatial coordinates used for the formulation ofT= T(r,z,t) which follows are shown in figure 3.8. A coefficient for heat transfer, h3 is added corresponding to the radial surface at r =b. 50
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Boundary Conditions and Spatial Coordinates for Finite Cylinder Figure 3.8 The solution for temperature in a finite cylinder as a function of r, z, and tis given as equation (Al.43). The coordinate system is cylindrical, i.e., T = T(r,q>,z,t). However, since symmetry about q> is assumed T(r,q>,z,t) reduces to T(r,z,t). Equation (Al.43) is stated directly since its derivation is shown fully in appendix 1. T(r, z, t) (Al.43) The specific boundary conditions and initial condition appearing in equation (Al.43) are given in appendix 1 in equations (Al.2) through (Al.6). As discussed in appendix 1, the eigenvalues associated with h1 and h2 are the roots of the transcendental equation 51
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(3.8) where H1 and H2 are the ratios of !'!1. and !!2:.._, respectively. In the case of a finite cylinder made of a kl k2 single metal, k1 equals k2 With k1 = k2, eigenvalues Tlp resulting from equation (3.8) relate directly to the average of h1 + h2 For example, eigenvalues Tlp depending on h1 = 20 W/m2 C and h2 = 12 W/m2 oc are identical to eigenvalues Tlp depending on h1 = 16 W/m2 C and h2 = 16 W/m2 C. Eigenvalues are the positive roots of equation (3.9) (3.9) H3 is the ratio of h:/k. As may be inferred by equations (3.8) and (3.9), eigenvalues Pm and Tlprelate to the z and r coordinates respectively. A program performing a numerical calculation of T(r,z,t) for a finite cylinder of copper is done by Mathematica program CPPR, the listing for which appears in figure 3.9. A check of numerical accuracy was made for three points on the test cylinder to determine how well the CPPR program could calculate the initial temperature of 25.0 C at T(r=O, z=O, t=O) T(r = 0, z = c, t =0), and T(r = b, z = c, t =0) It is an assumption of the analysis that the initial temperature of the test cylinder is consistent and uniform throughout. The location of thermocouple 1 (see figure 3.7) was set at (r = 0 m, z = c = .003175 m); the location of thermocouple 2 was set at (r = b = .03170 m, z = c = .003175 m). Measured data for the 8 W experiment showed the initial temperature at these two locations to be 25.0 oc and 24.9 C, respectively. Table 3.1 shows the calculated values for T(O,c,O) and T(b,c,O); the first (*** Finite Cylinder Problem, T = f(z,r,t) ***) (*** Cu ***) (*** Physical Constants ***) Alpha1 = .00011234 c=0.00635 b=0.03175 To=25 Tinf=22 k=384 h1 = 15 h2= 15 h3= 15 Hll =h1/k H22=h21k H33 =h3/k q = 2100 f1 =q(h1 *Tint) f2 = Tinf* h2 f3 =To* h3 lmt=3 tlim= 20 templim=50 52
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(*** Initialize Eigenvalues ***) Array[Betal,lmt] {Beta I [l],Betal [2],Betal [3], Beta I [4],Betal [5], Beta I [6],Betal [7],Betal [8], Beta1[9],Betal [10] }= ( 1.568,120.70404,220.97449,320.42675298, 419.64579504,518.76000318,617.82231662,716.853052501, 815.86368880,914.86074215} Array[Etal,lmt] (Eta I [ I],Etal [2],Etal [3],Etal [4],Etal [5], Eta1[6),Etal [7],Etal [8],Etal [9] Etal [10]} = { 3.507,494 75889,989 48795,1484 222528,1978 960147. 2473 69831855,2968.4367665,3463.17537233,3957.91407685, 4452. 65284716} (** Fonnulation of the Temperature Function **) Numeta = (Etal[p] Cos[Etal[p1 z]) +(Hit Sin[Etal [p1 z]) Numbeta= Betal[m1"2 Besse1J[O,Betal[m1 r1 Den leta= Etal [p1"2 + Hl1"2 Den2eta = (c + (H22/(Etal [p1"2 + H22"2))) Deneta =(Den leta* Den2eta) + HII Denlbeta = (BesseiJ[O,Betal[m1 b])"2 Den2beta = (H33"2 + Betal[m1"2) Denbeta =Dent beta* Den2beta Den = Deneta Denbeta Num = Numeta Numbeta Expo=Exp[Aiphal (Etal[p1"2 + Betal[m1"2) t1 lnteta =(Etal[p1 Cos[Etal[p]*zp]) +(HI I Sin[Etal[p] zp]) lntbeta = rp BesseU[O,Betal [m1 rp] lntgrnd = lnteta lnlbeta Inti =lntegrate[Intgrnd, ( rp,O,b}, { zp,O,c} 1 N[Int1,8] IntCond = To Expo Inti (** Add Boundary Conditions **) Expnt = I Expo Eigen = (Betal[m]"2 + Etal[p]"2) TranB = Integrate[lnteta, ( zp,O,c} 1 TranE = lntegrate[Intbeta,(rp,O,b}] Mull= (((b*Besseii[O,Betal[m]*b)}/(k*Eigen)) f3) Expnt TranB Mul2 = ((Etal[p]/(k*Eigen)) fl) Expnt TranE Mul3num = ((Etal[p]*Cos[Etal [p]*c]) + (Hll Sin[Etal[p]*c])) f2 Mul3den = k Eigen Mul3 = (Mu13num/Mul3den) Expnt TranE (** Construct T(r,z t) and Sum **) Tfunc = (Num/Den) (lntCond +Mull + Mul2 + Mul3) Tsum = Sum[Tfunc, ( m,l,lmt}. { p,l,lmt}] N[Tsum,8] Trzt = (4/b"2) Tsum N[Trzt/. { r>0, t> 0, z > 0} ,8] N[Trzt/.{r>O,t > 0, z > c},8] N[Trzt/.{r>b,t> 0, z > c},8] (** Post Process Trzt "'*) plll=Piot[Trzt/ {r >O,z > 0 00635),{t,0,2400), PlotRange>{20,70} ,GridLines>Automatic, PlotLabel> "k=384;hs=l2,12,h3=30,q =2200,c= 006"] Mathematica Program CPPR Figure 3.9 53
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calculation was based on a summation over one eigenvalue for both and llp; the second calculation was based on a summation of three eigenvalues for both and llp Measured initial temperatures for each location appear in parentheses beside the calculated initial temperatures As can be seen from table 3.1, a summation over even one eigenvalue yields calculated results well within the accuracy of measured data. A summation over three eigenvalues permits the calculated value to approach the measured value even more closely. Throughout this work, temperatures calculated from equations (3.6) or (Al.43) result from summations over three eigenvalues unless otherwise noted. Number of Eigenvalues Calculated Values for Calculated Values for T(O,O,o) (0C) T(O,c,O) (0C) 1 25.0198 (25.0) 25.0198 (25.0) 3 25.0034 (25.0) 25.0034 (25.0) Comparison of Calculated and Measured Initial Temperatures Table 3.1 3.5 Modeling T(r,z,t) in a Finite Cylinder Calculated Values for T(b,c,O) (0C) 24.9785 (25.0) 24.9935 (25.0) A number of attempts were made to model the observed change in temperature using equation (Al.43) and the CPPR program shown in figure 3 9. Values for h1 and h2were chosen arbitrarily as 15 W/m2 oc or 16 W/m2 C based on the previous success in the semiinfinite slab formulation, figure 3.7. Similarly, qcouplcd was chosen as 2100 W/m2 or 2200 W/m2 Given that h2 and qcouplcd were essentially fixed, h3 was the only remaining variable that could be changed. The CPPR algorithm was run many times using values for h3 ranging from 5 to 100 W/m2 C. The most successful combinations of h2 qcouplcd and h3 are shown in figure 3.10. As can be seen in this figure, approximations where h1 = h2 = 15 W/m2 oc, h3 = 30 or 35 W/m2 C, and qcalculated = 2100 W/m2 were generally good, especially for values oft ranging fromO 1500s. b1 = h1= IS Wlm"C; h,= IS W/m1"C 500 1000 1500 2000 Time(s) Comparison of Calculations ofT(r,z,t) with Measured Data Figure 3.10 54
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It is useful when modeling heat flow through a finite cylinder to investigate the influence of convection from the vertical surfaces as compared to conduction from the radial or radial surface. Figure 3.11 shows T(r,z,t) for a Qcouplect of 2200 W/m2 for three values of h1 and h2 while h3 is held constant at 30 W/m2 C.; h1 and h2 are as shown. The temperature curves shown in figure 3.11 are each comprised of a solid line and a dashed line. The solid line is T(r,z,t) = T(0,.00635,t), i.e., calculates the temperature in the center of the z = .00635 m face. The dashed line represents T(r,z,t) = T(.03175,.00635,t), i.e., calculates the temperature at the position of the exterior thermocouple. As a consequence of the high thennal conductivity of copper, the solid and dashed lines nearly coincide for all times t. As can be seen from figures 3.11 and 3 .12, changes in convection from the vertical surfaces tend to manifest themselves early in the time history of T(r,z,t). Conduction from the circumferential surface tends to determine the time needed to achieve equilibrium. Convection from all surfaces evidently contribute to the magnitude of T(r,z,t) in the steady state. As can be seen in modeling ofT(r,z,t) in succeeding sections, convection from the vertical surfaces effects the slope of the temperature curve primarily in the fust 300 500 seconds. The speed with which equilibrium is achieved and the ultimate magnitude of the steady state temperature is more a function of conduction from the cicumferential surface of the finite cylinder. Figure 3.12 shows T(r,z,t) as a function of different convection coefficients applied to the r = b surface. 0 500 1000 1500 2000 ITime (s) T(r,z.,t) as a Function of Differing Venical Convection Rates Figure 3.11 55
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!Temp (C) I 70,_.,,,r0 500 1000 1500 2000 !Time (s) T(r,z,t) as a Function of Different Convection Coefficients Applied to the r = b Surface Figure 3.12 3.6 Characterization of the Temperature Difference (AT) on the Vertical Face Figure 3.7 shows the temperature curve measured for an aluminum cylinder given an 8W power load. Figure 3.4 shows a temperature curve for a copper cylinder for the same 8W power load. These curves result from simultaneous measurements from two thermocouples: one placed at the center of the vertical face each test cylinder and one placed near the edge of the vertical face. For early times t the temperatures measured by the thermocouples nearly coincide. Though less obvious in figure 3.4, in both cases a separation in the curve is apparent for later times t. A measurable difference in temperature was observed between the center and edge of both cylinders as they approached thermal equilibrium. Formulation of T(r,z,t) for the antimony test cylinder was done using the experimentally determined value for thermal conductivity reported in chapter 1, i.e., ksb = 15.2 W/m C. The measured data in figure 3.14 show two distinct values for T(r,z,t) related to the placement of the two thermocouples. Thermocouple one was placed at coordinates r = 0 m, z = .0165 m; thermocouple 2 was placed at r = .03048 m and z = .0165 m. It is evident that the magnitude of the measured (and calculated) difference between T(O, .0165, t) and T(.03048, .0165, t) is a strong function of thermal conductivity. Data for temperature from thermocouples placed similarly on the aluminum test cylinder are shown in the inset of figure 3.14. As can be seen in the inset of figure 3.14, thermocouple 2 (placed at r = .03040 m, z = .0165 m) records cooler temperatures than the thermocouple 1 (placed at r = 0 m) for both aluminum (at t > 200 s) and antimony (at t > 750 s). The AT across the z = c face is smaller for aluminum than for antimony. A similar AT exists for copper though smaller in magnitude. The AT for copper can be seen in figure 3.4; it is so small, however, that the plots for T(O, .003175, t) and T(0.03175, .003175, t) 56
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almost coincide. Thus empirical evidence indicates the observed between the center and edge of the test cylinders increases as the thermal conductivity of each test metal decreases RTV Potting r = b = .03048 m k iJT/iJz + h1 = f1(r,t) = [q(h1T.)] T(r,z,t) = F(f,z) =To Uniform Initial Temperature Macor Collar Thermocouple 2 Thermocouple 1 z = c = .0165 m I x (semiinfinite coordinate) I Experimental Configuration for Determination of tlTon z = .0165 m Face Figure 3 13 57
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J TciDP (C) J Thermocouple Test Data .. Aluminum Test Cylinder / / 8WCase v II' Te mp(OC) I .. / / 60 55 50 45 40 35 30 25 )0 A Thermocouple l (Measured) .f 0 "' 1000 1500 HOD / Thermocouple 2 (Measured) I TICDt (1) I / i Thermocouple l (Predicted) / (Solid Line) \ .. ;::1 Thermocouple I (Predicted) (Dashed Line) h1 = 20 W/m2 oc, ""'.. ""' h2 = 15 W/m2 C, h3 =5 W/m2 oc f ....... Qsupplied = 2200 W /m2 ,. ksb = 15.5 W/m oc 0 500 1000 1500 2000 Time (s) Comparison of Calculated and Measured T( r,z,t) for Antimony Cylinder Figure 3.14 The measured temperature data from thermocouples 1 and 2 indicate aT approaches a constant as heat transfer through the test cylinders approaches steady state, as should be expected. In the case of the antimony cylinder, the measured aTsb approached a constant value beginning at approximately t = 2000 seconds. From t = 2000 s and later, measured aT sb equals 1.0 C varying from this by .1 oc at most. Experimentally, temperature measured at location (r = 0, z = .0165, t = 2290 s) was 50.4 C; temperature measured at location (r = .03048 m, z = .0165 m, t = 2290 s) was 49.5 C. The measured ATsb was therefore .9 C. Figure 3.15 is the graph ofT(r,z,t) at t = 2290 s with h1 = h2 = 15 W/m2 C and h3 = 25 W/m2 C. Comparing this with figure 3.14, the discrepancy between the calculated and measured temperatures at the locations of the thermocouples is apparent, being on the order of 4.0 C. This is an indication that the model represented by equation (A1.43) is somewhat inaccurate in calculating T(r,z t), especially for later times t. Figure 3.15 calculates the temperature profile over the entire antimony test cylinder based on equation (Al.43). Time at t = 2290 sis chosen because aTsb is greatest for late times t and therefore more easily graphed. aTsb at t = 2290 s was calculated as approximately .77 C compared with a measured aTsb of .9 oc at that time Though the prediction of T(r,z,t) is not completely successful for the antimony cylinder, calculation of aTsb compares well with measured values. aTcu as calculated for the copper test cylinder is shown in the inset of figure 3.15 for comparison. The smaller magnitude of aT cu is attributed to the higher thermal conductivity of copper. 58
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Temp (0C) 46.5 46 45.5 45 0.005 Thickness (m) 0 .01 46.68 46.66 46.64 46.62 46.6 0 Calculated T(r,z.t) for Sb Cylinder att = 2290 s Figure 3.15 3.7 Comparison of Calculated and Measured T(x,t) fort S 120 SKonds .003 T(r,z,t) at r = 03048 m z = .0165 m Radius (m ) The various calculations of T(x,t) and T(r,z,t) presented in preceding sections have been carried out for all times through thermal equilibrium Since a characterization of a transient thermoelastic response was considered as potentially necessary, an assessment of the accuracy of the calculation of T(x,t) given by equation (3.5) for shorter times t was done. Figure 3.16 compares a measurement of T(x = .003175 t) for times 0 s t 120 seconds with a calculation of T(x = .003175, t) using Mathematica program CPPR I. The calculation of T(x = .003175, t) in figure 3.16 is based on a copper cylinder with a qcoupled = 2100 W/m2 h1 = h2 = 12 W/ m2 C. The measurement ofT(x, t) was done at I 0second intervals. Prediction of T(x,t) based on equation (3.5) is reasonably good. 59
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30 28 26 24 0 20 40 Measurement of T(x,t) Calculation of T(x,t) 60 80 100 120 I Time (s) Comparison of Calculated and Measured T(x,t) for 0 :s; t 120 Seconds Figure 3.16 4 Analytical Method for Characterization of Heat Conduction Across a Bonded Interface 4.1 Introduction and Statement of the Problem If a plate is initially at rest and at a uniform constant initial temperature, introduction of heat to the plate will create a flow of heat accompanied by fields of stress, strain, and temperature. A description of this heat flow, as well as the fields of temperature, stress, strain, displacement, etc., that it creates is available through the theory of linear thermoelasticity set forth in Chapter 2 As in that chapter, assumptions of isotropy and homogeneity apply. This analysis is done for the halfspace. The method developed in this chapter calculates the transient and steadystate temperature profiles across a bonded metal interface. Having given the derivation for the formula to calculate 9(x,t), the author was successful in writing a Mathematica program (figure 4.4) to calculate the steadystate temperature profile given any arbitrary film coefficient at the bond interface. Work was done provide the means to calculate the transient changes in temperature as well. Successful calculation of the transient temperature profile proved more difficult and could not be obtained. This was probably due to the inability to find the roots of Gin equation (4.78) possessing sufficiently large imaginary components to allow computation of dG(z) The method developed in this chapter has utility and dz 60
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generality and (in the author's opinion) possesses a certain mathematical elegance. For these reasons it is included here. As discussed in Chapter 2, the treatment of thermoelastic problems fall into several categories depending on which different assumptions and restrictions are adopted. The coupled thermoelastic theory is used for this analysis; details of this theory are given later in this chapter. Much of the method used in this chapter is taken from work done by Atarashi and Minagawa.17 1bis method incorporates film coefficients between the layers of a composite plate permitting modeling of the thermal resistance usually encountered between bonded plates. Much of the analysis that follows is given for bonded, composite semiinfinite plate having two layers. Figure 4.1 shows a plate consisting of n parallel layers, labeled 1 to n from left to right. The spatial coordinate is x in a direction perpendicular to the layers of the plate. Each interface between layers is designated Xj for the jth layer (j = 1, 2, ... N1). The atmospheric temperature to the left of the composite plate is T A atmospheric temperature to the right of the composite is T B T 0 is the initial temperature of the plate and is uniform throughout. Theta, 9, is temperature deviation from T0 ; uj is the displacement of the jth layer. This analysis is onedimensional and all field variables are functions of x and t alone. nlayered Bonded Plate Figure 4.1 17 Transient CoupledThermoelastic Problem of Heat Conduction in a MultiLayered Composite Plate, T. Atarashi and S. Minagawa, lntematiotuJl Journal of Engineering Science, Vol. 30, No. 10, 1992, pgs. 15431550, 61
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4.2 Derivation of the Governing Equations For the jth layer of the bonded metallic composite plate, the governing equations are: (equation of motion) (4.1) dq ds __ + T ' = 0 (equation of energy balance) dx 0 dt (4.2) ao kj __ J + qj = 0 (Fourier equation) dx (4.3) y i(J i (Constitutive equation) (4.4) duj s i = Y i dx 'I' i(J i (Constitutive equation) (4.5) where cri is the normal component of stress in the direction perpendicular to the jth layer, 'li the heat flux, si the entropy density, Pi the mass density of the jth material, ki the thermal conductivity and p .c. ci = A.i + 2J.Li; yi = (31ti + 2J.Li)f3i; and 'I'= 1 1 T,, where Aj and fli the Lame constants, Pi the coefficient of thermal expansion, and Ci the heat capacity of the jth material under constant volume Taking the derivative of equation (4.4) with respect to the space variable x gives (4 .6) Combining equations ( 4.6) and ( 4.1) gives (4.7) Taking derivatives of equation (4 5) with respect to time and equations (4.2) and (4.3) with respect to the space variable (4.8) 62
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1 aqj = (4.9) (4.10) From equations (4.9) and (4.10) (4.11) Combining equations (4.8) and (4.11) and dividing by pjCj gives Equation (4.12) can then be written as (4.13) where The two governing equations are (4.7) and (4 13): (4.7) (4.13) where and 63
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4.3 Conversion of Governing Equations to Dimensionless Form It is necessary to convert the governing equations (4.7) and (4.13) into dimensionless form. Units of time, displacement, and temperature are made dimensionless using the following relationships : X = (J = (rTo) TO (unit space coordinate) (unit of dimensionless time) (unit of dimensionless temperature) Equation (4.7) is nondimensionalized as follows: where and Equation (4.13) is nondimensionalized as (}(}. (}2u = ' + T. __ J dt 0 J dXdt 64 (4.7) (4.14) (4.15)
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a. where ;1 = 1 is the ratio of thermal diffusivities of the first and jth layers. al Therefore equations ( 4 14 ) and ( 4 15) are the nondimensionalized form of the governing equations: (4.14) (4.15) The coefficient g is simply the ratio of thermal diffusivity of the jth layer to the thermal diffusivity of the first layer. Epsilon, E_;, is a coefficient showing the effect of the mechanical field on temperature; Tli is the coefficient for effect of the thermal field on the mechanical; 1 is a coefficient denoting the effect of inertia of the materials. These coefficients can be expressed in terms of lame constants and JL;) and the coefficient of linear thermal expansion ei = ( 3A.1 + 2J.L i )131 piCi (4.16) 1Jj = (3A.. + 2J.L. )13. J J J T. A.j + 2J.Lj 0 (4 17) = ( r p 1 ki 1 (.:t1 + 2J.L1 ) p 1c1 1 (4.18) In a coupled thermoelastic analysis, 1 may be neglected since for most metals this coefficient is on the order of 10"17; for copper is approximately 8 0 E17, for aluminum is approximately is 4.5 E17. The last term of equation (4.14) is therefore neglected. 4.4 Thermoelastic Coupling The coefficient for thermoelastic coupling for the jth layer Si, is given by the relationship (4.19) 65
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The coefficient is on the order of 102 to w t for metals; for copper is approximately 0.057, for aluminum is approximately 0.116. An alternative formula for the thermoelastic coupling coefficient, is given by Boley:18 (4.191) v. the velocity of propagation of dilatational waves in an el.,tic medium, is v, = ('. : and cE is specific heat at constant deformation. Specific heat at constant deformation is defined as ;:P
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du (A.+ 2J.L) dx(3A. + 2J.L)fJ T = C1x = f(t) (4.194) The arbitrary function f(t) must vanish identically if the surface x = 0 is traction free. If this is the case, equation (4.191) reduces to equation (4.195) Equation (4.195) is identical in form to the ordinary heat conduction equation. The solution of a coupled problem is derived from the corresponding uncoupled problem by replacing thermal diffusivity, a, by the quantity p CE 4.5 Initial and Boundary Conditions At the left boundary of the composite (x = Xo), the boundary condition is one of known convection: d8! ( ) iJx + H A ()A 8! = 0 (4.20) At the interface of the two layers of the composite (x = x1 ) two boundary conditions are (4.195) and (4 21) and (4.22) At the right boundary of the composite (x = xN), the boundary condition is based solely on convection: (4 23) where HA = lhA Ha = lh8 and Mt = lm1 k! k2 k! and 8A = TA To ()B = Ts To To To 67
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and T A = absolute temperature of the atmosphere on the A side T 8 = absolute temperature of the atmosphere on the B side hA = coefficient of heat transfer from outer surface A h8 = coefficient of heat transfer from outer surface B m1 = the film coefficient at the interface between the two layers of the composite. 4.6 Transformation of the Governing Equations in Laplace Space The quantities ei and ui are transformed into Laplace space according the definition of the Laplace transform, where p is the Laplace variable: Equation ( 4.14 ), the governing equation of motion, becomes aof(x, p) = 71j iJx (4.24) Equation ( 4.15), the governing equation of energy, becomes (4.25) Equation ( 4.25) is differentiated partially with respect to the space variable : 68
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which gives Substituting equation ( 4.24) for the last term on the left of ( 4.26) gives a [ azer:] a [ aol:] ;. 1 [poi:]PE 11. ' = o dx I dx2 dx I I I dx (4.27) which becomes This yields (4.28) Similarly, the boundary conditions represented by equations (4.20) through (4.23) become + HA[o; o.L] = 0 at x=Xo; (4.29) k aoL = __1_ 2 at X = X kl dx I' azof H [oL oPi J = 0 a;+ B 2 at X = X 2 The quantity ()iL may be expressed in the following form t:lL(x p) A + B .esJz Uj = } J where 69 (4.301) (4.302) (4 31) (4.32)
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(4.33) and (4.34) and A 1 B; and S1 are functions of the Laplace variable p alone Variables shown in bold face are functions of p alone. 4.7 Formulation of a Laplace Transform Definition of Of(x,p) Equations (4.29) and (4.32) are combined to given equation (4.35): :x[Ate81x+BteSx] + HA[O; (A1e81x0 +B1eSxo)] = 0. (4.35) Separating B1 in equation ( 4.35) gives which can alternatively be stated in an abbreviated form where bo = HAs. HA +St Bo = esxo( HA o;) = do o,t es.x o and do = HA +St p HA s1 HA +St and where A0 is an unknown function of p to be determined later. Combining the other boundary conditions, i.e equations (4.301) and (4. 302), and equation (4. 32) gives equations defining A2 and B2 in terms of and by implication, A0 Equation (4.32) is substituted into 70
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at x = Xj = x1 Separating A2 gives Similarly, B2 can be expressed as (4.37) where Substituting equation (4.37) into equation (4.36) gives A2 in terms of and N1 : In terms of Bit and N1 B2 is expressed as Equations (4.38) and (4.39) are written for a 2layer composite. These equations can be expressed in general terms for a composite of j + 1layers: (1) [ Si l (s+s )x (1) [ Si l (s s )x. B =A. 1+N. e' J+l 1 +B. 1+N e 1 J+l 1 (441) J + l 1 2 M. 1 1 2 M 1 J J Equations (4.40) and (4.41) are then rewritten again in terms of specific coefficients as 71
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where GAi I [ SJ Ni] ,(s, + s, .,)., (4.44) = 1 ++ 2 Mi GBi =HI sJ Ni] (s1 + s1 1).r, (4.45) +e Mi FAi = SJ Ni] (s1 + s1 +1).r1 (4.46) e Mi FBi = N ] (s, s,.,)., (4.47) 2 Mi i e where Ni = ki si k 1 S J + j +I Substituting equations (4.352) and (4.353) in the righthand side of equations (4.40) and (4.41) gives Ai + 1 and Bj + 1 in terms of A0 and B0 : (4.48) and (4.49) where the coefficients LAi L8j, MAi Maj are given by the recurrence formulae LAi = G Ajj L A(i 1) + FAi LB(i I) (4.50) LBj = G BiJ L A{i1) + FBi LB(i I) (4.51) MAi = G Aij M A(j 1) + FAj MB(i I) (4.52) MBi = GBjJ MA(i I) + FBj MB{i1) (4.53) where 72
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LAo = 1 (4.54) LBO = b e2 Slxo (4.55) MAo = 0 (4.56) MBo = 1. (4.57) Equations (4.32), (4.42), and (4.43) are combined to give the Laplace form of temperature in the last, j = n, layer of the composite: where and Finally, the quantity A0 can be defined as where and D Ao = pG (4.58) (4.59) (4 .60) (4.61) (4.62) (4.63) (4.64) (4.65), (4.66) 73
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The Laplace form of temperature can then be written or ( ) Rj(x,p)D (JJ!X, p = '''p G + where [ Qj (x, p) d09A e81x0 ] p (4.67) (4.68) (4.69) (4.70) The term of(x, p) in equation (4.68), the deviation from the initial temperature T0 is expressed as a function of position and the Laplace variable p for the jth layer of the composite plate. It is necessary to gain an expression for 9 j ( x, t) expressed as a dimensionless number, and as a function of position and time. The complex inversion formula is used to invert the Laplace form of (J f (X, p) to aj (x,t). 4.8 Inversion of 0 f ( x, p) lff(p) = <1, {F(t) }, then <1,1 {f(p)} is given by the complex inversion formula: 1 Iy+ioo F(t) = . eP' f(p) dp 2m r,.,.. (4.71) and F(t) = 0 for all t < 0. The integration is performed along a line p = r in the complex plane where p = x + iy. The real number y is chosen so that s = r lies to the right of all singularities (or poles) but is otherwise arbitrary. The integral in equation (4.71) is evaluated by considering the contour integral .( eP' f(p) dp 2m Yc where C is the Bromwich contour shown in figure 4.2. 74 (4.72)
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Bromwich Contour for Inversion of 8 f ( x, p) Figure4.2 The Bromwich contour is composed of line AB and the arc BJKLA. r represents the arc of the contour; R represents the radius of the circular part of the contour. It then follows that Tis defined as T = l F(t) becomes equation (4.73): 1 Jy+iT F(t) = lim 2 eP1 f(p) dp R71rl y+iT (4.73) or F(t) = lim { .( eP1 f(p)dp 1 rreP' f(p)dp}. R72m Yc 21ri J1 Supposing 1) the only singularities of f(p) are poles all of which lie to the left of the line p = 'Y for some real constant 'Y and 2) that the integral around r in equation ( 4.73 ) approaches zero as R 7 oo. Then Equation (4.73) can then be written using the residue theorem as F(t) = l: residues of eP' f(p), at poles of (p). Let g(p) = (pa) (pb) (pc) (4.74) then 75
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therefore dg(p)l dp p=a = dg(s) d(ga) np=a = dg(p) = (pb) (pc) = g(p) d(p a) (p a) = (pa) (pc) = g(p) (pb) This leads to the conclusion that c.t[ q(p) ] = q(p)[ + .L q(p) p k(p) k(p) p=O r; p k(p) (4.75) = q(p)l + .L q(p)
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Equation (4.77) permits equation (4.68) L( ) Ri (x, p) D ()j X, p p G to be written as equation (4.78) O(x, t) where = ()':" + J + [Qi (x, p) do()A eSixo] p (4.78) (4.79) and Zq's are the roots, other than zero, of the transcendental equation G(z) = 0. G(z) is given by the substitution of the complex variable z for the Laplace variable p in G. Equation (4.79) represents Oj in equilibrium or steady state; it derives from the first term on the right hand side of equation (4.77). As can be seen in equation (4.79), Oj approaches equilibrium in the limit as the Laplace variable, p, approaches zero. The second term on the righthand side of equation (4.78) supplies the transient component of 9(x,t). The next section examines a Mathematica algorithm which calculates 9(x,t) in the steady state for a composite nlayered plate. 4.9 Calculation of the SteadyState Form of 9(x,t) Mathematica program ARARPl calculates the steady state solution for 9(x,t) in equation (4.79). The first test of ARAPl was done for the configuration shown in figure 4.3. This theoretical composite is made by joining two pieces of antimony; thermal resistance at the interface is effectively eliminated by setting the film coefficient, mh equal to 10000 W/m2C. The goal of this calculation was to test if ARARPl could produce an uninterrupted temperature profile across a perfect thermal interface between the same metals. In addition, it was expected (under these idealized conditions) that the temperatures at either side of the bonded composite should match the ambient temperatures existing on either side of the 2layered plate. A listing of the program ARARPl appears as figure 4.4 77
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Antimony k 1 = 15.2 W/m C Cp"' 205 JlkgC, p = 6650 kg/m3 a"' .031,11 "'.00635 m m 1 = 10000 W/m2C m1 = 100 W/m2C m 1 "' 10 W/m2C Antimony k1 = 15.2 W/rn C Cp = 205 Jlkg C, p = 6650 kg/m3 a= .031 h"' .00635 m Xz = + x/1 Configuration for First Sol ution of 6( x,t) Figure4.3 Since this composite plate of antimony is, for this analysis, essentially one uniform piece of homogenous metal with no thermal resistance at the interface, the steadystate temperature should have a constant slope through the thickness of the plate Figure 4.5 shows the plot for (}(x, t) under the thermal, geometric, and material constraints shown in figure 4.3 with m1 = 10000 W/m2C. Setting m1 ""10000 W/m2C essentially forces thermal resistance at the interface to zero. If the algorithm is consistent and accurate in its treatment of the interface film coefficient, lower values for m1 should introduce a discontinuity in (}(x,t) at x1 ""0. The insets of figure 4.5 show plots for (}(x,t) with m1 = 100 W/m2 C and with m1"" 10 W/m2C. These smaller values for the film coefficient do in fact introduce finite discontinuities at the interface in inverse proportion to the magnitude of m1 (*** Algorithm for calculating T(x t) for an Atarashi Plate ***) (*** Experimental Values for Limp> 0 ***) (*** AI ***) delta1 = .031 k1 = 202 Cv1 = 896 rhol =2707 alpha1 = kl/(Cvl *rho I) (** Cu *) delta2 = .057 k2= 384 Cv2=398 rho2= 8930 alpha2 = k2/(Cv2 rho2) (** Temperature **) To=300 Ta = 900 lb=600 ThetaA = (TaTo)ffo ThetaB = (1bTo)ffo 78
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(** Dimension **) I= .00635 xO = .0063511 xl = 011 x2 = (.00635)11 (** Convection Parameters **) ha= 15 hb= 15 Ha = (l*ha)lkl Hb = (l*hb)lkl (** Laplace Variable **) p=ll0"10 (** Interface Film Coefficient **) ml=lO (** S and Sp **) Sl = Sqrt[(l + deltal)/(alphallalphal)] S2 = Sqrt[(l + delta2)/(alpha2falphal)] Spl =Sl Sqrt[p] Sp2 = S2 Sqrt[p] (** Laplace Variables **) bO = ((HaSpl)/(Ha + Spl)) dO= Haf(Ha + Spl) b2 = ((Hb Sp2)/(Hb + Sp2)) d2 = Hb/(Hb + Sp2) Ml =(I* ml)lkl Nl = (kl Sl)/(k2 S2) Gal = .5 (1 + (Spl/Ml) + Nl) Exp[(Spl Sp2) (xl)] Gbl = .S (1 + (Spl/Ml)Nl) Exp[(Spl + Sp2) (xl)] Fal = .5 (1 (Spl/Ml)Nl) Exp[(Spl + Sp2) (xi)] Fbi = .5 (1 (Spl/Ml) + Nl) Exp[(Spl Sp2) (xl)] LaO= 1 LbO = bO Exp[2 Spl xO] Ma0=0 MhO= 1 Lal = (Gal LaO) + (Fat LbO) Lbl = (Gbl LaO)+ (Fbi LbO) Mal = (Gal MaO) + (Fal MhO) Mbl = (Gbl *MaO)+ (Fbi MbO) Qp =(Mal Exp[Sp2 x2]) + (b2 Mbl Exp[Sp2 x2]) Dp = (d2 ThetaB) (Qp Exp[Spl xO] dO ThetaA) Gp = (Lal Exp[Sp2 x2]) + (b2 Lbl Exp[Sp2 x2]) Rl =LaO* Exp[Spl (x/1)] + (LbO Exp[Spl (x/1)]) R2 = Lal Exp[Sp2 (x/1)] + (Lbl Exp[Sp2 (x/1)]) Ql =MaO* Exp[Spl (x/1)] + (MbO Exp[Spl (x/1)]) Q2 =Mal Exp[Sp2 (x/1)] + (Mbl Exp[Sp2 (xll)]) SSTI = (Rl (Dp/Gp)) + (Ql *dO* ThetaA Exp[Spl xO]) SST2 = (R2 (Dp/Gp)) + (Q2 *dO* ThetaA Exp[Spl xO]) (*** Post Processing ***) pi = Plot[SSTl,[x,xO,xl }, 79
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PlotRange > {0, 2.0}, GridLines >Automatic, PlotLabel> "SST, steady state"] p2 = Plot[SST2,(x xl,x2}, PlotRange > {0, 2.0}, GridLines >Automatic, PlotLabcl > "Cu, steady stntc"] p3 = Show[pl, p2, PlotLabel>" AlCu Composite, m = 10 ] Listing of Program ARARP 1 Figure4. 4 In terms of dimensionless variables, eA and 98 are given by 6 A = TA T 0 = ( 900 K 300 K) = 2 1'o 300K and 6 = T8 1'o = ( 600 K 300 K) = 1 8 1'o 300K which gives the temperatures at the leftand righthand surfaces of the composite antimony plate In terms of dimensionless temperature, the initial temperature of the composite plate is by definition equal to 0 As discussed earlier, ()(x,t) as given by equation (4.79) approaches equilibrium as the Laplace variable, p, approaches zero in the limit. Values for ()(x,oo) as given in figure 4.5 are predicated on p = 1 10"10 It was necessary to explore how the algorithm ARARPl performed when the Laplace variable p was greater than 0, and when p was much greater than 0. It was discovered that equation ( 4. 79) does indeed converge to a linear, steady state solution for ()(x,t) asp approaches 0 The speed with which equation (4.79) converges to a linear form for (}(x,t) is shown in figure 4.6. 80
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... .............. Interface __....,... m1 = 100 W/m2C .... n n n .., I Dimensionless Temperature I I OA=2 I ') 1 0.5 0 0 5 1 1 7<; 'r' I I Os 1 I Interface 1 ")<; m1 = 10 W/m2C 1 r/ (1 7<; ./_ (1 <; Interface 1t(1 ')<; I m1 = 10000 W/m2C 1 0.5 0 0.5 1 1 0. 5 0 0.5 1 Dimensionless Length (](x, oo) for the Idealized 1Material Sb Composite Plate Three Values for m1 Figure 4.5 Dimensionless Temperature 1 0.5 0 0.5 1 Dimensionless Length I Convergence of Solution for (](X, oo) as the Laplace Variable approaches 0 Figure4.6 81
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Table 4.1 gives the numerical values for 8( x, oo) in four locations of the composite plate for four increasingly smaller values of p. 8(l,oo) denotes lJ(x,oo) as the lefthand side of the composite plate; (J(+l,oo) ,oo) denotes lJ(x,oo) as the righthand side of the composite plate 6(0,oo) and lJ(+O,oo) denote (J(x,oo) an infinitesimal distance to the left and right of the interface, respectively. As table 4.1 shows, (J(x,oo) is acceptably accurate when p = ws and for all smaller values of p. All calculations of 6(x,oo) shown in this chapter are based on p = 1010 unless stated otherwise. Laplace Variable p 104 w' w6 ws 6{J,oo) 8{0,oo) lJ{+O,oo) 4.5281 1.4760 1.4753 2.2046 1.4979 1.4972 2.0114 1.5001 1.4994 1.9986 1.5003 1.4996 Numerical Results for lJ(x,oo) at Various Values for p Table4.1 lJ{+l,oo) 3.0719 1.1815 1.0265 1.0098 In developing the Mathematica programming to solve equation (4.79), a model was chosen next of a bonded composite of antimony and copper. Convection from both exterior surfaces was equal The ambient temperature on the leftand righthand sides of the composite were again 900 K, or 9A = 2 and 600 K, or 98 = 1. The initial temperature of the composite was chosen to be 300 K and was assumed to be uniform throughout the composite. Thickness of the 316 stainless steel and copper plates were chosen as .00635 rn each. In terms of dimensionless length, each plate of the composite has a length of 1. The film coefficient was arbitrarily chosen to be 100 W/m2C. With this geometry, the coordinate of the left surface becomes1 and the coordinate of the right face becomes +I. Figure 4. 7 summarizes this configuration Antimony k1 = 15.2 W/m "C <; = 560 Ilkg c. p = 7970 kglm3 li = .031,11 = .00635 m h1 = 15W/m "C Copper k1 = 384 W/m"C <;. = 398 Jlkg c, p = 8930 kglm3 li = .057, 12 = .00635 m Configuration and Boundary Conditions for Calculation of 8j (x, oo); CuSb Plate Figure4.7 82
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8j (x, oo) for the SbCu composite plate is shown in figure 4.8 The affect of the higher thermal conductivity of copper as compared to that of antimony is apparent. The slope of 91 (x,oo) for Sb is much more pronounced than that for 92 (x,oo) for Cu. I Dimensionless Temperature ., 1 7 c; eA = 2 rsbl 1 1 ?.<; .1 ............... J Interface J 0 75 1 m1 = 100 W/m1C 0 5 u Ta 600 K I n ?c; 9a = I 0. 5 0 0.5 1 Dimensionless Length 8j(x, oo)foranSbCu Composite Plate; m1 = 100W/m2C Figure4.8 Figures 4.9 and 4.10 show similar temperature profiles for bonded composites of copper and aluminum. The film coefficients between the two composites are m = 200 W/m2 and m = 10 W/m2 The difference in temperatures across the two interfaces are some indication of the relative thermal interference to be expected between the Cu/Al wellbonded composite and the Cui AI composite possessing a MgOfalled void at the interface. 83
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Dimensionless Temperature 1 0.5 .., 1 '7C:: ' "' .... .,. 1 '71:" c:: vo" ..,, 0 0.5 1 Dimensionless Length Thermal Interference at Interface between WellBonded CuAl Composite Plate Figure4.9 I Dimens ionless .., Temperature 1 '7C:: 1 c:: 1 '"lC:: ....,, c:: . ")C:: 1 0. 5 0 0.5 1 Dimensionless Length Thermal Interference at Interface between CuAl Composite Plate with MgO Barrier Figure 4.10 84
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5.0 Development of Finite Element Models for a Bonded Metal Composite 5.1 Introduction 1bis chapter 1) compares the results of numerical calculations with experimental data taken from thermal tests performed on homogenous copper and aluminum test cylinders, 2) develops an finite element analysis (FEA) model for the MacorRTVAluminum configuration used to accrue temperature and displacement data for this work, and 3) creates and tests FEA models of the AlCu and AlMgOCu test specimens. The FEA models of the AlCu and AlMgOCu specimens are designed to characterize temperature and displacement on the vertical face observed by the TVH. The FEA models are then used to compare with displacements measured by the TVH as a function of temperature and time. The FEA software used in the analyses presented in chapters 5 and 6 is the K6 version developed by the MARC corporation. The type of analysis used in this chapter for the bonded composite plate is described by MARC as static coupled. This is in contrast to a dynamic transient coupled analysis which is also available. Both types of coupled analyses effectively reconcile the effect of the thermal and mechanical fields on each other. The dynamic transient coupled analysis incorporates the inertial as given in equation 4.7: iJlu. J = ax2 MARC's static coupled analysis This is acceptable for the AlCu and AlMgOCu plates, however, since is on the order of 10"17 for copper and for aluminum. T(r,z,t) for singlemetal shapes as calculated in section 5.2 are calculated as a heat transfer problem rather than as a static coupled problem. It is the transient thermoelastic response of the test specimens that is of greatest interest. Thermal equilibrium will tend to minimize (or eliminate) differences in temperature profiles between bonded composites. This is particularly true of composites made of metals with high thermal conductivity. For this reason, any difference created by the thermal resistance at the interface of a bonded composite is likely to be most obvious during the transient phase at early times t. 5.2 FEA Modeling of the Copper and Aluminum Test Cylinders FEA modeling was flfSt done on the .00635 mthick homogenous test cylinders modeled analytically in chapter 3. The FEA analyses of T(r,z,t) for Cu and AI test cylinders were compared to experimental data and the corresponding analytical results developed using equation (A1.43). This was done to confirm that the boundary conditions used in previous analytical models yielded results comparable to those produced by FEA techniques. If T(r,z,t) as calculated by very different analytical and numerical methods compared well with experimental data, this was considered evidence material properties and boundary conditions had been assessed correctly. T(r,z,t) for the copper test cylinder is calculated using the material properties, convection boundary conditions, and qcoupled cited below. An axisymmetric analysis is employed and MARC element type 85
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10 is used in the mesh, i.e a fournode, isoparametric quadrilateral. MARC specifically recommends use of element type 10 rather than higherorder elements when performing a contact analysis. The material properties shown in table 1.2 are used for the MARC FEA analysis ofT(r,z,t) The initial conditions and boundary conditions used are h1 = h2 = 12 W/m2 C for the right and left vertical faces, h3 = 5 W/m2 C for the transverse face, and an initial temperature (25 C) is uniform throughout the Cutest cylinder (see figure 3.10). A qcoupled = 2100 W/m2 is used for the heat fluxthis being qcaicwaled discussed in chapter 3. The convection coefficients, initial temperature, and qcoupled used in the FEA analysis shown in figure 5.2 are similar to those used for the aluminum test cylinder in section 3.3. T(r,z,t) was calculated for 0 t 300 seconds since this is the time period of greatest change in the temperature and displacement fields. Figure 5.1 shows the temperature resulting from the 8W experiment discussed in section 3.3. Measured data for the first 300 seconds of the 8W experiment, and a leastsquares fit of that data, are shown in the inset of figure 5.1. T(r,z,t) calculated by the FEA model can be compared to similar calculations based on equation Al.43 shown in figure 3.10. As indicated in figure 5.1, the temperature curve is associated with node 13. This node is located at the lower right comer of the mesh and therefore represents temperature at the center of the x = .00635 m face of the test cylinder. Analysis of displacement in the x direction (u) resulting from T(r,z,t) was calculated for nodes 13 and 793, located at (0. 00635,0,0) and (0. 00635, .03175,0) respectively. The difference in displacement, across the x = .00635 m face of the cylinder was calculated as .3 Jlill for alit, 0 1 300 seconds. Figure 5.2 shows the FEA approximation of the 00635 mthick homogenous aluminum cylinder. Superimposed in the inset is the data measured during the 8W experiment characterizing the aluminum. In this case, the approximation is extended to 0 1 2400 seconds. The quality of the approximation can be judged against the measured data appearing in the inset of the figure The results appearing in figure 5.2 proceed l&m=t ... 10 .loo .uo too tso ,oo Axisymmetric Model of Temperature in CuTest Cylinder, 0 t ;;? 300 Seconds Figure 5 1 86
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from the same model producing the results appearing in figure 5.1, except for the substitution of material propenies for aluminum and a different initial temperature. Thus the relative accurm:y of the FEA model, based on the boundary conditions developed through the analytical analysis in chapter 3, is compared to measured data for two metals for short and long t The calculations of axisynunetric T(r,z t) shown in figures 5.1 and 5.2 represent good agreement with experimental data and the analytical calculation of T(r,z,t). In keeping with these results, the values for qcoupled (21 00 W /m2), h1 and h2 (12 W /m2 0C), and h3 (5 W 1m2 C ) used in the analyses appearing in figures 5 1 and 5 2 These values are hereafter referred to as the guideline values and are used throughout the remainder of chapters 5 and 6 unless otherwise noted '1 / ...,..... I' .$ / I 4 II 0 500 1000 1500 2000 Axisymmetric Model of Temperature in AI Test C ylinder, 0 t 2400 Seconds Figure 5.2 5.3 Modeling Heat Transfer in the Experimental Configuration A s a final (and most exacting) test of the ac c uracy of the material properties and b o undary conditions used for the remaining analyses of the bonded composites, heat transfer in the entire experimental configuration was modeled. The experimental configuration referred to is that discussed in s ection 87
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1. 74 and shown in figure 3.2, except that an aluminum test cylinder is employed in place of the AlCu composite shown in figure 3.2. The AI cylinder was potted into its Macor collar and instrumented with four thermocouples as shown schematically in figure 5.3. The test used an 8 W load and measured temperatures from all four thermocouples simultaneously. Ambient air temperature (T_) was22.8 oc; initial temperature of the assembly was 23.7 C. Figure 5.4 gives the temperature data from the test through a time of 2400 seconds. Thermocouples 2 and 4 recorded the temperatures at the center of the aluminum cylinder and at top dead center of the Macor collar. These plots are of primary interest and are modeled in the FEA analysis. As can be seen in figure 5.3, the temperature measured at the exterior of the Macor collar is markedly less. 1bis is due to the comparatively small thermal conductivities of the R1V potting and the Macor. The FEA model employed guideline boundary conditions and the material properties shown in chapter 1. The only exceptions to this was 1) the thermal conductivity of the R1V potting was increased to 0.4 W/m C (from 0.19 W/m C supplied from the manufacturer) and 2) Qcoupled was increased to 2200 W/m2 from the guideline value of 2100 W/m2 Given ka'IV = 0.4 W/m C, Qcoupled = 2200 W/m2 and b1 = b2 = h3 = 12 W/ m2C, and material properties taken from tables 1,4, 1.8, and 1.9, figure 5.5 gives T(r,z,t) calculated axisymmetrically for the entire experilnental configuration. These data may be compared to the measured data shown in figure 5.4. Agreement between the FEA model and measured data is good. The greatest divergence occurs at t 1800 seconds. The FEA model fails to achieve equilibrium as quickly as the measured data in figure 5.3 indicate. Macor Collar Thermocouple 1 Thermocouple 2 Schematic of 4Thermocouple Experiment Figure 5.3 Figure 5.6 shows T(r,z,t = 1200 s) over part of the cross section of the experimental configuration. 1bis figure indicates how effectively the R1V potting insulates the test cylinder on its circumference. Given the agreement between the measured data of figure 5.4 and the results heat transfer calculation shown in figure 5.5, it is possible to conclude that the boundary conditions and material properties needed for the composite plate analysis are well understood. 88
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The analysis which generated the results in figure 5.5 was based on MARC element type 40 This element is defined by MARC as a fournode, isoparametric arbitrary quadrilateral used for axisymmetric heat transfer analysis. Higher order elements were used during development of the program used in this section but without significant improvement of the re s ults. 0 Thermocouple 2 600 1200 1800 Temperature Data for 4Thermocouple Experiment Figure 5.4 2000 Time (s) FEA Temperature Curves for Thermocouples 2 and 4 Figure 5.5 89
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FEA Temperaturesfor4Thermocouple Experiment, t = 1200 s Figure 5.6 5.4 Coupled Thermoelastic Model of the AICu Plate A simple model of an alurninwncopper plate was developed. This model was axisymmetric possessed an infinite film coefficient at the interface, and used MARC's static coupled analysis capability to detennine temperature and displacement simultaneously. The mesh used for the model is shown in figure 5.7. MARC element 10 is used as discussed in section 5.2. The glue all command is used enforcing contact of the interface between the two metals at all times. Two initial conditions are invoked: a uniform temperature prescribed for all nodes and fixed x displacement for two nodes at the lower left of the model. This latter initial condition gives translational and rotational stability to the model. The element mesh for copper is slightly offset with respect to the mesh for aluminum. This in no way reflects the actual geometry of the AlCu test piece. The offset is present because the coupled static MARC analysis is optimized when node communicates with element face at the interface of the two metals. The mesh shown in figure 5.7 represents the solid AICu bonded composite with a diameter of 61.72 mm; the two metals in the composite are 6.35 mm thick. This is the configuration pictured in figure 6.1 and analyzed in test number 2 in chapter 6. 90
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Mesh for Coupled Analysis of Axisymmetric AlCu Plate Figure 5.7 The boundary conditions in this model are shown in figure 5 8 The convection coefficients are h1 = h2 = 12 W /m2 oc on the vertical surfaces. The film coefficient on the top of the composite (corresponding to the cylindrical or transverse surface) is= h3 = 5 W/m2C. The heat fluxes qcalculated and q coupled vary for each experimental configuration. Material properties are those shown in tables 1.2 and 1.4. The initial temperature was assigned in each metals as experimental measurements required. The model shown in figure 5.7 was analyzed by MARC as a contact problem. This type of problem required the aluminum and copper pieces be defined as contact bodies. It also required the metal with the lowest elastic modulus, i.e., aluminum by defined as contact body I; copper was defined as contact body 2. The contact table was used to prescribe a separation force of 1e12 forcing the two metals to remain in contact. The entire model was defined as a single geometric entity with axisymmetric solid elements ; constant dilatation was prescribed for all elements. Figure 5.9 shows the original and deformed shapes of the AICu bonded composite in response to a q coupled of 1200 W/m2 The deformation is magnified 500 times for clarity. The concave bowing of the cylinder was typically observed in every calculation involving a bonded composite 91
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Boundary Conditions for AlCu Bonded Composite Figure 5 8 Deformed Shape of ALCu Bonded Composite, t = 300 s Figure 5.9 92
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Displacement resulting from the changing temperature field is of great interest. This displacement is calculated by the model in the x direction onlythis being the direction observed and quantified by the lVH. The LANL lVH arbitrarily eliminates all x displacements at any given time (I = 0 in the experiments) on the observed surface This creates a reference state for the test cylinder. Displacements observed by the lVH at times 1 > 0 are then the difference in magnitude of x displacement from the reference state. The displacements measured by the lVH at 1 > 0 are the difference between the magnitude of the x displacement of the center of the test cylinder compared with the magnitude of the x displacement at the cylinder's edge. This difference in displacement in the x direction over the observed surface of the bonded plate is denoted hereafter as tlu. As will be seen in chapter 6, displacements normal to the surface observed by the lVH appear as fringe bands. A tlu of even .5 micron is therefore easily detectable by the lVH and, since the lVH has a .25 !liD resolution, would be seen as 2 fringes Figure 5.10 shows a typical x displacement field for bonded composite A pictured in figure 6.l.where qcoupled == 1200 W/m2 t = 100 s. Magnitude ofxDisplacement in AlCu Bonded Composite, t = 100 s Figure 5 10 tlu on the face observed by the lVH (on the right side of the composite in figure 5.10) equals approximately I !lffi. and thus would be seen as four fringes. Figure 5.10 also indicates the fringes should not be equidistant across the face. Since the displacement gradient is greatest at the edge of the composite and smallest at the center, the fringes should reflect this by being somewhat more closely grouped at the edge and somewhat more widely spaced towards the center. 93
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5.5 Coupled Thermoelastic Model of the AlMgOCu Plate The A to B comparison used in this work compared the transient thermoelastic response of an AICu bonded composite cylinder (A) to the transient thermoelastic response of an AICu composite containing a void (B). The void in B was filled with magnesia (MgO) to create a thermal barrier at the interface of the two metals in the composite. Creating a model for the static coupled analysis of B proved a challenge. The AIMgOCu composite was first modeled as a threebody contact problem. In this model's first iteration, a stainless steel piece of identical size was substituted for the MgO filled void. 111is substitution was intended to make development and testing of the model easier and more predictable. Stainless steel was used because it offered low thermal conductance in comparison with AI and Cu (as the MgO would). Also, the significantly higher elastic modulus of stainless steel made it possible to designate it as the third contact body in the MARC analysis; aluminum and copper were first and second, respectively. Boundary conditions, the initial temperature condition, and material properties were otherwise identical with the model shown in figure 5.8 The mesh for the AISSTCu model is shown in figure 5.11 ; material properties are highlighted AlSSTCu Development Model Figure 5.11 The AlSSTCu model was intended to provide the basis for development and testing of a workable 3body contact problem. As testing proceeded it became apparent this approach to the 3body contact problem had serious difficulties. The mesh, material properties, and 3body contact definition was able reach a solutionbut only with difficulty. 111is model was characterized by high singularity and convergence ratios indicating the matrix became illconditioned and the solution inaccurate. Figure 94
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5.12 shows the x displacement of the AlSSTCu model at t = 300 s. Thermoelastic deformation is magnified 500 times Figure 5.12 shows that the mesh has lost integrity along the SSTCu interface. The AlSSTCu model was then modified by changing the material properties of the SST region shown in figure 5.11. Thermal conductivity was lowered (kMgoequals approximately .001 to .1 W/m C), an arbitrary elastic modulus was assigned lower than kssr. but higher than kAI The designation of the three contact bodies were reordered accordingly. This model, which is not discussed in detail here, was unsuccessful in characterizing the thermal field known to exist in the composite from experimental measurements. The 3body contact model was discarded when, after numerous attempts and modifications, it failed to produce successful simultaneous displacement and thermal fields The next attempt took the mesh shown in figure 5.11 and eliminated the elements in the SST region. This approach essentially located a thermally nonconductive region without elements in place of the void in the AlMgOCu composite. This reduced the original 3body contact problem to a 2body contact problem. The mesh for this model is shown in figure 5.13. Mesh, xDisplacement Field, and Deformation of AlSSTCu Model, t = 300 s Figure 5.12 95
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Successful Axisymmetric Mode/for Composite with Void Figure 5 .13 In terms of mesh integrity and analysis of the known thermal field, the model in figure 5.13 successfully calculated the thermal response of the AlMgOCu bonded composite. Success of the model was predicated on the assumption the MgOfilled void was essentially nonconductive thermally. This model was subsequently chosen as the basis for analysis of the AlMgOCu bonded composite The temperature field resulting from a 4W load applied to the aluminum surface (qcoupled = 1200 W/m2 ) is shown in figure 5.14 fort= 300 s. The temperature field in figure 5.14 is described graphically by contours. Though the film conductance between the Aland Cu is set at le6 (essentially infinite), a very small discontinuity can be detected at the interface of the composite. This was regarded as insignificant. 96
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Temperature Contours for AlMgOCu Model, t = 300 s Figure 5.14 The temperature gradient can be seen to move around the void in figure 5.14. The magnitude of temperature difference through the composite is small due to the high thermal conductivities of aluminum and copper. In conclusion, the models presented in sections 5.4 and 5.5 were considered accurate in that (as will be seen in the last chapter) they successfully predicted the simultaneous displacement and thermal fields for both the A and B bonded composites. 97
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6.0 Assessment by TVH Interferometery of Diffusion Bonded Composites 6 .1 Introduction The goal of this work is to determine if TVH can characterize the difference between good and poor diffusion bonds assessing the transient thermoelastic response of those bonds. lbis chapter summarizes the results of three two comparisons of two composite cylindersone designated A and the other designated B. Figure 6.1 shows these two bonded composites. The A specimen is an Al Cu composite bonded as optimally as possible. The B specimen is an AlCu composite possessing an unbonded region at the interface of the two parent metals. As explained in chapter 1, the unbonded region in B is a cylindrical void 38.1 mm in diameter and 1.5 mm deep positioned at the interface of the composite extending into the copper lbis void was fJJ.led with 1.5 g of magnesia (MgO) to create an additional thermal barrier at the unbonded region. The cylinders shown in figure 6.1 were photographed as machined for the second experiment. These experiments were designed to determine if the void could be detected influencing the thermoelastic response of the B cylinder compared to the same response in the A cylinder. A and B were tested in two configurations. The only difference between the two configurations was the thickness of the A and B cylinders. In the first experiment, each cylinder was 23.87 mm thick equally divided between the AI and the Cu. In the second experiment, this thickness was reduced to 12.7 mm again equally divided between the AI and the Cu. The diameters of A and B were unchanged at 61.72 mm in both experiments. A and B Bonded Composites as used for Experiment No. 2 Figure 6.1 The amount of metal surrounding the void in B was reduced for the second experiment. If no difference in the thermoelastic response between A and B could be detected in the first experiment, 98
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when the thickness of A and B was at a maximum of24 mm, the second experiments might reveal a difference as the A and B were simultaneously reduced in size and mass. 6.2 Preliminary Tests Preliminary experiments were performed on .00635 mthick homogenous cylinders of AI, Cu, and Sb potted into Macor collars. The objective was to determine if the TVH could observe deformations on a heated metal cylinder and a bonded composite plate. Concern existed during these preliminary tests that convectiondriven air currents created by the heated samples could interfere with the interferometry. These frrst tests also helped determine what kind of heat load could be used without deforming the test cylinders to excess thereby defeating the TVH' s ability to quantify the deformation. It was also necessary to examine what degree of symmetry of defonnation resulted from the carefully controlled conditions of the experiment. Symmetrical deformation was expected and was regarded as a good indication the experiments were well controlled and likely to be repeatable. Figure 6.2 shows the interferometric fringe pattern observed for an homogenous .00635 m thick copper plate. The plate bas reached thermal equilibrium. The relatively straight fringe patterns in figure 6.2 indicate the copper cylinder has tilted in its RTV potting. The Macor collar surrounding the Cu cylinder also has deformed as evidenced by between one and two fringes forming on its surface. This was typical of the Macor collar at late times t for most subsequent experiments. The flat fringe pattern denotes the test cylinder is deforming little or not at all relative to itself as it is heated. SteadyState Deformation of Homogenous Cu Test Cylinder Figure 6.2 Metal reflectivity proved to be a difficulty. The relatively polished surface of the AI, Cu, and Sb test pieces tended to scatter the laser light used by the TVH. This scattering resulted in obscuring the 99
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fringe pattern or, at times, disrupting it altogether. 1bis problem was addressed by painting the 1VHobserved surface of each test cylinder with flat white paint. Since the 1VH creates a reference state by eliminating all observed deformations at a given time (t = 0), the paint did not interfere with the 1VH's ability to assess deformation of the observed surface relative to itself for all times t > 0. The results of the test shown in figure 6.2 provided assurance that 1) the 1VH could observe defonnations on the surface of test cylinders, 2) the heat load supplied to a test cylinder had to be between 2 and 8 watts for cylinders of .00635 m thickness or more to create a visible fringe pattern without creating excessive deformation 3) convectiondriven air currents were not a major problem, and 4) the painted surface of the test cylinders successfully eliminated excessive reflection of the 1VH laser without interfering with fringe pattern development. Figure 6.3 shows the first test of a bonded AlCu composite. The circular fringe pattern is typical of every bonded composite observed during testing. The patch seen a the center of the observed surface is the thermocouple used to measure temperature. As can be seen in figure 6.3, the circular fringe pattern is not centered on the observed surface. 1bis offset is due to the orientation of the test sample relative to the 1VH. The offcenter pattern is an indication the test cylinder is not perfectly normal to the 1VH laser. OffCenter Fringe Pattern, AlCu Bonded Composite Figure 6.3 Adjustment of the TVH laser compensated for this slightly offnormal orientation. 1bis compensation centered the fringe pattern as seen in figure 6.4. This compensation resulted in moving the Macor collar to the offnormal position previously occupied by the test cylinder. This small compensatory translation resulted in the fringe pattern appearing on the Macor collar as also seen in figure 6.4. 100
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Fringe Pattern Centered by 1VH Compensation Figure 6.4 These preliminary tests showed that the TVH could be used to make nearrealtime thermoelastic measurements of homogenous and bonded metal plates Experimentation now moved to comparison of A and Bin their most massive, thickwalled geometry. 6.3 Results of Experiment No. 1 The f1rst comparison of A and B was done in a configuration 23.86 mm thick with a diameter of61.7 mm. Each metal in the composite was 11.93 mm thick Both AlCu bonded cylinders were machined to identical dimensions. A and B were left in a maximum material condition. A and B were not potted in Macor collars for experiment no. 1; Macor stock was not available in sufficient thickness to make the collars. Cylinder A had two thermocouples attached; cylinder B had one thermocouple attached. A and B were placed on their sides resting on a vise An 8 W load was applied giving a qcoupled of approximately 2100 W/m2 Experiment no. 1 was conducted for 300 seconds with interferometry taken twiceonce at t = 150 sand once at t = 300 s. These times and temperatures are indicated in figure 6.5. Temperatures for A and B nearly coincide within the limits of measurement error The LANL TVH has the capability to take and record interferometry at any given time 10 seconds. Adjustments required to center the fringes bad to be made immediately prior to recording the interferometry. 101
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36 80 100 Temperature Data for Experiment No. 1 Figure 6.5 Time (s) Figures 6.6 and 6.7 show the defonnations of cylinder A at times 1 = 155 sand 1 = 300 s. These defonnations characterize cylinder A's thermoelastic response to qcoupted = 2100 W/m2 The observed surface is copper; in figures 6.5 and 6.6, the resistance heater is attached to the opposing aluminum face. The bowing of cylinder A at t = 155 s is seen in figure 6.5. Magnitude of the displacement is indicated by 4 fringes .1.u, i.e., the displacement of the center of the cylinder relative to the edge of the cylinder, equals approximately 1 micron at 1 = 155 s The bowing of cylinder A at t = 300 sis seen i n figure 6 .6. Magnitude of the displacement is indicated by between 6 and 7 fringes. at 300 s equals approximately 1.75 microns. Cylinder B was also loaded with a qcoupted = 2100 W/m2 Interferometry of the resulting thermoelastic response also was taken at 1 = 150 sand 1 = 300 s. Interferometry forB at 1 = 150 sand 1 = 300 s appears in figures 6 6 and 6.7, respectively. at 150 s is indicated by four fringes; .1.u therefore equals approximately 1 micron. at 300 s indicated by almost 7 fringes, equals approximately 1.75 microns. 102
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Thermoelastic Response of Cylinder A; t = 155 s, = 2100 Wlm2 Figure 6.6 Thermoelastic Response of Cylinder A; t = 300 s, = 2100 Wlm2 Figure 6.7 103
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Thermoelastic Response of Cylinder B; t = 150 s qcoupled = 2100 Wlm2 Figure 6.8 Thermoelastic Response of Cylinder B; I= 300 s, qcoupled = 2100 Wlm2 Figure 6.9 104
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Given the same materials, dimensions, heat fluxes, boundary conditions, and initial temperatures, the transient thermoelastic response of A and B were essentially identical in experiment no. 1. It was theorized the high thermal conductivities of alwninwn and copper permitted even the transient temperature response of A and B to be equal at the observed face of each cylinder. Additionally, it was believed the comparatively thick walls surrounding the void in B masked any possible affect on deformation, even when observed at the submicron level. It was concluded without further analysis that TVH could not detect an unbonded region in cylinder B under the conditions of experiment no.1. 6.4 Results of Experiment No. 2 6.41 Comparison of Modeling and Experimental Data for Cylinder A Bonded composites A and B were then machined to a thickness of 12.7 mm. 'Ibis brought the void in B to within approximately 4.8 mm of the observed surface. A and B as pictured in figure 6.1 were potted into Macor collars for experiment no. 2. The heat load used for experiment no. 2 was changed to 1 W due to the reduced mass of A and B. This proved unsuccessful, however, when thermal and thermoelastic equilibrium was reached at about t = 200 s. The heat load was then increased to 4W with satisfactory results. Had the heat load not been reduced from the 8 W used in experiment no. 1, the thermoelastic displacements observed by the TVH would have been too great to quantify on the submicron level. A and B were again heated from the aluminum face by an adhesive resistance heater for 300 seconds. As in experiment no. 1, one interferometric measurement was taken at approximately t = 155 s and another at t = 300 s. A MARC FEA analysis of the static coupled response was done to discover if the transient thermoelastic response of A and B could be predicted accurately under the conditions of experiment no. 2. Two models were used. The ftrSt model was based on the discussion in section 5.4. It modeled cylinder A given the dimensions of A in experiment 2. The second model was that discussed in section 5.5 and modeled cylinder B. Guideline boundary conditions and material properties were used for both analyses. qcaicuJated for experiment 2 was 1336 W/m2 based on the 4 W heat load described above. qcoupled was placed at 1200 W/m2 for reasons discussed in chapter 3. If the thermoelastic response is to be modeled correctly, then the temperature and xdisplacement fields must be calculated simultaneously and accurately. The temperature field for A was calculated for a location at the center of the observed surface, i.e., the copper face. The initial temperature for A measured during the experiment was 27 C. Figure 6.10 shows the temperature curve for T(r,z,t) at r = 0, z = .00635 m, and 0 S t 300 s. 'Ibis may be compared to figure 6.11 which shows the data for temperature at the same location accrued during experiment no. 2. The ftrSt interferometry for A during experiment no. 2 was taken at t = 155 s. 105
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T(r=O, Z = .00635,1) Calculated for Cylinder A, 0 t 300 s, qcoupled = 1200 s Figure 6.10 Temperature at J' t = 155 s I __, v l"' J 'f Temperature at t = 300 s .. I T(r=O, z = .00635,1) Measured for Cylinder A: q c oupled = 1200 Wlm2 Figure 6.11 Time (min) The MARC calculation for temperature and the measured data show good agreement. The xdisplacement field was calculated from the same model. Displacement in the x direction was calculated for two locations: the center of the observed surface and the edge of the observed surface Figure 6.12 shows the changes in xdisplacement for the center (node 378) and the edge (node 738) of 106 I
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the bonded composite. Experimental observations are available at three times: t = 0 s (the reference state), t = 155 s, and xDisplacement as a Function of Time for Cylinder A: 0 5 t 300 s, Qcoupled = 1200 Wlm2 Figure 6.12 at 1 = 300 s. As can be seen from figure 6.12, the displacement across the observed surface, i.e., L1u, at t = 300 sis calculated to be 3.6 JJID at the edge (node 738) minus 1.42 J.Un at the center (node 378). flu is therefore calculated to be approximately 2.2 J.LII1 at t = 300 s. This should appear as between 8 and 9 fringes on the interfereometty taken at t = 300 s. Similarly, flu at 1 = 155 sis calculated to be approximately 1.38 Jliil, the difference between 2.2 J.LII1 at node 738 and 0.82 J1IIl at node 378. A flu of approximately 2.2 JJID should appear as between 5 and 6 fringes on interferometry taken at t = 155 s. These predicted values for flu can be compared with the inteferometry taken as cylinder A was tested for experiment 2. Figure 6.13 shows L1u as measured by the TVH for A at t = 155 s; figure 6.14 shows flu measured by the TVH at approximately t = 300s. 107
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Measured !J.u for Cylinder A, t = 155 s, Qcoupled = 1200 Wlm2 Figure 6.13 Measured !J.ufor Cylinder A; t = 300 s, Qcoupkd = 1200 Wlm2 Figure 6.14 Six interferometric fringes are visible in figure 6.13 for cylinder A at t = 155 s 'Ibis represents a measured .1u of 1.5 J.llD which compares with the calculated .1u of 1.38 J.llD. Figure 6.14 shows 10 interferometric fringes. 'Ibis represents a measured l!u of 2.5 J.U11 which compares with the calculated l!u of 2.2 J.llD. The MARC model is able to predict both the temperature field and the 108
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displacement field simultaneously with considerable accuracy. This shows the MARC static coupled model effectively predicts the transient thermoelastic response of cylinder A. The interferometric fringes seen in figures 6.13 and 6.14 tend to be more widely spaced towards the center of test cylinder A and more closely packed towards the edge of the cylinder. This indicates du changes more quickly at the edge of the test cylinder and less quickly at the center. Figure 6.15 gives a predicted contour band plot of x displacement over all of cylinder A at t = 300 s. The rate of change of xdisplacement is seen to be greater towards the edge of the bonded cylinder. The rate of change of xdisplacement decreases towards the center of the bonded cylinder. Tbis is further evidence the MARC coupled static model for cylinder A is accurate. xDisplacementfor c ylinder A; l = 300 s, q co upld = 1200 Wlm2 Figure 6.15 642 Comparison of Modeling and Experimental Data for Cylinder B Cylinder B contains the thermal void. It was in the dimensional configuration pictured in figur e 6.1 and potted into a Macor collar for all testing during experiment no. 2. All conditions and thermal loads used for cylinder A were recreated as closely as possible for cylinder B The MARC model used for predicting temperature and xdisplacement is that described in section 5.5 figure 5.13. As before with cylinder A, the lVH observed the copper side of cylinder B. 109
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Macor collar mounting the cylinder was placed in its holding fixture, and a reference state was taken be the interferometer effectively zeroing out all displacements at t = 0. The 4 Wheat load was started at t = 0. lbermocouple 2 (Unused) I Time (Min.) I Temperature Data for Cylinder B, Experiment 2;. qcoupled = 1200 Wlm2 Figure 6.16 MARC Temperature Curve for Cylinder B, Experiment No. 2; qcoupted = 1200 W/m2 Figure 6.17 110
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MARC Temperature Curve for Cylinder B, Experiment No.2; Qcoupled = 1200 Wlm2 Figure 6.17 This represented a Qcoupied of approximately 1200 w/m2 Temperature data was taken for the period 0 t 300 s. Interferometery was taken at t = 0, t = 167 s, and t = 300s. The temperature data appear in figure 6.16; temperatures at t = 167 s and t = 300 s are marked. The MARC model for temperature appears in figure 6.17 and runs 0.5 to 1.0 degree hotter than the measured data. Displacement was then predicted for cylinder B. The prediction made by the MARC model of xdisplacement as a function of time appears in figure 6.18. Displacement at t = 300 s at the edge of the cylinder (node 738) is calculated to be 3.7 J.I.Dl; displacement at the center of cylinder B (node 378) is calculated to be1.25 J.I.Dl. tlu across the observed face of cylinder B is therefore calculated to have a magnitude of 2.45 J.I.Dl at t = 300 s. The difference in displacement across the observed face of cylinder B as calculated should appear as 10 fringes at t =300 s when cylinder B should have a temperature of 32.4 C. This can be compared to the actual interferometry in figure 6.19. xDisplacement as a Function of Time for Cylinder B; 0 t 300 s, = 1200 Wlm2 Figure 6.18 111
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Measured ilufor Cylinder B, Experiment No.2; t = 300 s, Qcoupkd = 1200 Wlm2 Figure 6.19 Ten fringes appear on Figure 6.19 with the tenth fringe only partially formed. This is again good agreement between the prediction of the static coupled MARC model for cylinder B and experimental data. In addition, comparison of figures 6.14 and 6.19 show almost no measurable difference in flu between cylinders A and Batt= 300 sunder a 4 Wheat load, i.e., qcoupted = 1200 W/m2 In addition to a nearly identical number of fringes, the similarity in transient thermoelastic response between these cases also extends to the relative spacing of the fringes. The spacing of fringes in cylinder B is predicted by the MARC model for cylinder B. Figure 6.20 shows this predicted spacing which is also nearly identical to the fringe spacing predicted for and observed in cylinder A. 112
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xDisplacementfor Cylinder A; t = 300 s, Qcoupld = 1200 Wlm2 Figure 6.20 Thus the predictions of the numerical model and the measured data are in agreement for both cylinders A and B at the end of the transient response period, 0 t 300 s. Interferometry was taken in experiments 1 and 2 at t = 0 s, t = 300 s, and at t = 150 s. The intermediate measurement was taken to help confirm the calculated displacements for times 0 t 300 s were essentially linear as predicted by the MARC models in figures 6 12 and 6 18. It is useful to investigate the response at t = 150 sin detail. Interferometry was taken in experiment no. 2 at t = 167 s The MARC model for cylinder B was used to calculate Ll.u in the cylinder for times 0 t 167 s. The FEA calculation was done in 300 iterations to give good accuracy for the exact timer= 167 s. The result of this calculation is given in figure 6.21. Displacement across the observed face at t = 167 sis calculated as1.6 This is compared to the measured xdisplacement between center and edge shown in figure 6.22 i.e., Ll.u = 6 fringes or approximately 1.4 This result indicates good agreement exists between prediction and measurement at the intermediate time oft= 167 s 113
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Calculation of ..1u, Cylinder B. Experiment No. 2: 0 1 167 s Figure 6.21 l!llerjerometric Measurement of ..1u, Experiment No. 2. Cylinder B: 1 = 167 s Figure 6.22 114
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6.5 Conclusion and Areas for Future Research Experiments 1 and 2 were designed to assess if the LANL TVH could detect a difference in the transient thermoelastic response of a good and a poor diffusion bond in an AB comparison of those bonds Boundary conditions and material properties were correctly understood. The static coupled FEA model which incorporated them accurately predicted the thermoelastic response of test composites A and B. The following conclusions are supported by the experimental data and subsequent modeling of that data: 1. TV Laser Holographic Interferometry can effectively characterize the transient thermoelastic response of a bonded metallic composite but failed to detect a known unbonded condition. 2. The high thermal conductivities of the metals used in the test composites may have prevented observation of subtle thermoelastic responses indicative of an unhanded condition. Detection of voids in a diffusion bond may be possible in metals having low thermal conductivities. 3. It appears unlikely TV Laser Holographic Interferometry can be used to characterize the thermal resistance of a diffusion bond. This conclusion is certain with respect to diffusion bonds between metals possessing high thermal conductivity. 4. Static coupled thermoelastic theory provides an effective basis for modeling the transient thermoelastic response as observed and measured at the submicron level of deformation. 5. The structure of test coupon B, which incorporated a known unhanded region, may have prevented observation of a thermoelastic response characteristic of the unhanded condition. The stiffness of the solid region surrounding the void in the test coupon may have predominated the thermoelastic response of the observed surface. The following are areas for future research: 1. TVH and static coupled FEA analysis should be investigated as a means to characterize the transient thermoelastic response in a bonded composite made of low thermal conductivity metals. It may be possible to detect an unbonded condition between lowk metals or between metals representing a good impedance match of lowand highk metals. 2. The static coupled FEA analysis used in this work should be extended to calculate the transient thermoelastic strain rate of a bonded metal composite. Expected advances in software and hardware will give the TVH the capability to process continuous realtime interferometry in the future. Data could then be taken allowing measurement of a strain rates resulting from a transient thermoelastic response. Comparison of a predicted strain rate with a measured strain rate may provide the basis for characterization of an unbonded condition in a metallic bonded composite. 115
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DERIVATION OF FORMULA FOR T(r,z,t) Appendix 1 116
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Al.l Governing Equations Figure Al.l shows a side view of a finite cylinder with initial and boundary conditions. Coordinates for the cylinder are space variables r, q>, z, and t for time. The cylinder has a thickness of c and a radius of b. Symmetry in cp is assumed and so this variable is excluded from further consideration. The initial temperature of the cylinder, T0 (r ,z, t), is taken to be uniform throughout the cylinder. Three boundary conditions are present denoted by fJ. f2 and f3 ; they are located at z = 0, z = c, and r = b, respectively. Each of these three boundary conditions are of the third type At having the form dT k On + hT = hT_ (r, t), (ALl) where aT/an is the derivative along the outward normal, k is thermal conductivity, his the convective heat transfer coefficient, r is the representative spatial vector, and T_ is the ambient temperature surrounding the cylinder. Figure 1 shows the cylinder and associated initial and boundary conditions. T(r,z,t) = F(f,z) = T0 Uniform Initial Temperature Cylinder with Initial and Boundary Conditions Figure Al.l Since each boundary condition is positive (i.e., ft. f2 f3 > 0), formulation of T(r,z,t) requires solution of the nonhomogenous problem. This derivation uses the integral transform technique to change the governing equation, boundary conditions, and initial condition to a form without space variables. In this form, all equations can be solved as ordinary differential equations (0. D. E.) with appropriate AI Ozisik, M. Necati, Heat Conduction, 2nd Edition, John Wiley & Sons, New York, 1993. 117
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boundary conditions Once the 0. D. E. is solved it is transformed in reverse to obtain the function T(r, z, t). The governing equation is the Fourier equation in cylindrical coordinates 1 err(r, z, t) a dt 0 r < b, 0 z < c, t = 0 (Al.2) where T = T(r, z, t). The boundary conditions are given as equations (Al.3) through (Al.5): z=O, t>O (Al.3) err k + hzT = h(r, t) = hzT_ Z = C, t > 0 (A1.4) err k dr + = IJ(z, t) = r= b, t> 0 (Al.5) The initial condition is given by equation (Al.6): T(r, z, t) = F(r, z) 0 r < b, 0 z < c, t = 0 (A1.6) A1.2 Transform of the "r'' Variable The first spatial transform is for "r". The inversion and integraltransform formulae are equations (A1.7) and (Al.8) respectively: Ro(/Jm, r)_ T(r, z, t) = f:,. N(/3m) T(/Jm, z, t) (Al.7) T(fJm, Z, t) = r' R0
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where are eigenvalues, is a normalization integral, and Ro is an eigenfunction where v = 0 in the Bessel differential equation d2 + _!_ dRv + ({32 + v: )Rv = 0 Equation (A 1.2) is dr r dr r operated on by the operator r=O rR0 ({3m, r)dr which gives J b [a2T 1 ar) cP Jb T1 R0(/3m, r1 ) L2 + ;+ T1 R0(/3m, T1 )T(r1 z, t)dr1 r'=O or r dT u<. r =0 which, using the integraltransform equation (Al.8), gives 2Jb 2 a T(/3m1 Z1 t) r'=O r' Ro (/3m I r' )V T(r' I zl t)dr1+ az2 1 Cif({3m 1 Z1 t) a at (Al.lO) The term S:=o R0 (/3m I )V2T(r1 z. t)drl is evaluated using Green's theorem: + Jb R0 (/3m, r1 ) T :1 (r1 Ro (/3m 1 r1 ))]dr1 r'=O dr or (A1.11) Equation (Al.ll) becomes I +If i=l r'=b T1 R (fl T1 ) 0 Jinl f ( )d 1 k 3 z, t r (A1.12) 119
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where the integral in the summation is a surface integral. Again using equation (Al.8), equation (Al.12) simplifies to become fb f.l V2 d /32fJ r' Ro (/3mr' )I r'=O r' R0(Hn, r' ) T(r' z, t) r' = mT( m, Z, t) + /3 (z, t) k r'=b (Al.13) Introducing equation (Al.13) into equation (Al.9) gives 1 () a (}( T(fJm, z, t) (Al.l4) Similarly, equations (Al.3), (A1.4), and (Al.6) are operated on by the operator S:=o rR0 (/3m, r)dr, k Cfi({Jm, z, t) + h Z t) (}z I m = ft 0 (Al.IS) k Cfi({J';;; z, t) + h2T(/3m z, t) = fz 0 (Al.16) T(fJm, Z, t) = F(fJm, z) 0< z>c, t=O. (Al.l7) The system of equations (A1.2) through (A1.6) has had the space variable "r" removed resulting in the transformed system of equations (Al.14) through (Al.17). A1.3 Transform of the "z" Variable Equations (Al.14) through (Al.17) are now transformed with respect to the "z" variable using equations (Al.l8) and (Al.19) as the integral transform pair : Z(7Jp, z) =T(f3m, z, t) = N(1Jp) T(/3,.11p t) (Al.18) c r(/3,.. 11p r) = I z(11p. z)r(/3,.. z. t)dz' (Al.19) z'=O where 'Tlp are eigenvalues, N('Tlp) is a normalization integral, and Z(Tlp z) is an eigenfunction. Equation (Al.14) is operated on by the operator J:=o Z(71p z' )dz' giving 120
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2 Ic r' Ro (/3m' r' ) /3m z'=O Z(1Jp, z' )T(f3m, z' t)dz' + k r'=b (Al.20) 1 () Ic = Z(1Jp, z' )T(f3m, z' t)dz' a a' z'=O As before, the integral J:=o Z(1Jp, z' )V2T(f3m, .z' t)dz' is evaluated using Green's theorem: I Z(1Jp z') I Z(1Jp z') + /1(/3m t) k dz1+ /2(/3m t) k dz1 (A1.21) z'=O z'=c Substituting equation (Al.21) into equation (Al.20) and utilizing the transform equation (Al.19) gives equation (Al.22) 2 Jc 1 1 r1 Ro (/3m r1 ) /3m z '=O Z(1Jp, Z )T(f3m, .Z t)dz + k r'=b fJ(71p t) [ 2 .::. Z(11p z)l Z(1Jp z)l l + 1JpT(f3m,1Jp, t) + f!(/Jm, t) k z'=O + fz(/3m t) k z'=c (Al.22) 1 OT(/3m, 1Jp, t) = a at This can be reduced to an 0. D. E.: (Al.23) where 121
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(Al.24) and (Al.25) h(A,, I) = I:=o r' R0(/3m, r' )f1(r', t)dr' (Al.26) and (Al.27) The initial condition becomes at t= 0 (Al.28) The space variables are thus removed from the system making equation (Al.23) an ordinary differential equation. Solution of the O.D.E. then follows by employing an integrating factor: where (A1.29) 122
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At t == 0, the initial condition is Therefore, equation (29) becomes (Al.30) Therefore, solving for r(Pm, 11 p, t ). (Al.31) Equation (Al.31) is transformed using equations (Al.18), (Al.19), (Al.7), and (Al.8) : i' Z(7Jp z) .::. T(/Jm, z, t) == .LJ N( ) T(/Jm, 1Jp, t) p=! 7Jp i' i' Ro (/3m, r)Z(7Jp, z) .::. T(r, z, t) = .LJ .LJ N(a )N( ) T(/Jm, 7Jp, t) m=l p=l I'm) 7Jp The function T(r,z,t) is expressed as equation (Al.32) iiRo(/Jm, r)Z(7Jp, z) T(r, z. t) .LJ .LJ N(a )N( ) e m=l p=l I'm) 1Jp (Al.32) where b r' r' R0(/3m, r' )Z(7Jp z' )F(r', z' )dr' dz' 123
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and Combining equations (Al.32), (Al.33), and (Al.34) gives R0(/3m r)Z(17p Z) T(r, z, t) ...i ...i N( a )N( ) e m=l p=l fJm 17p {S:=o S:=o '1 R0(/3m, )Z(77p Z1 )F(r1 Z1 )dr1 dz1 + [bR (/3, b) f' ear [I' Z(17 Z1 )/ (Z1 k o m Jo z'=O P 3 (A1.35) + [ Z(1Jp,o)J; ear[f:=O R0(/3m, )/1(r1 + [z(17p c)J;ear[S:=o R0(/3m, )f2(r1 Since /1 = /2 = /3 = constant, equation (Al.35) simplifies to R0(/3m. r)Z(17p z) T(r, z, t) ...i ...i N(a )N( ) e m=l p=I #Jm 1Jp {I' Jb r1 R (/3, I r' )Z(77 Z1 )F(r1 1 Z1 )drl dzl z'=O r'=O 0 m p + [ bRo(f3mb) f (1ea(fJ;,+T/;)r)I' Z(17 'Zl )dzl k a(/3;. + 11;) 3 z'=O P (A1.36) + P f (1ea(l'm+T/p)r) r1 R (/3, T1 )dr1 [ 2 fb a(/3;. + 17;) I r'=O o m 2 fb + P f (1ea(l'm+T/p)t) T1 R (/3, T1 )dr1]} a(/3;. + 17;) 2 r '=O o m 124
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Combining terms, and since F(r', z') is also a constant, equation (Al.36) can be rewritten as .. .. Ro(f3m, r)Z(1Jp, z) ea(J3;,+11;,)t T(r, z, t) = L L N(/3m)N(1Jp) m=l p=l [ ( 1 e c + ( ) [bR.{JJ.b)J,f._ ,j.tz k p;, + 11; z 0 (Al.37) + z(1Jp,o)ttJ:=D r' Ro(/Jm, r')dr' + z(11p c)tzS:=o r' R0(/Jm, r')drJ]} A1.4 Values for Ro, Z, etc. Values for Z(llp,z), N(llp), and eigenvalues and llp for the given boundary conditions areA2 : R0(/3m r) = 10(/3mr) (Al.38) 1 2/3;, N(f3m) = [lo(f3mb)]\2(Hi + /3;,) (Al.39), (Al.40) where /3m are the positive roots of /3m 10 (/3mb) + H 3 J 0 (/3mb) = 0 The graph of this equation appears in figure Al.2: A2 Ozisik, M. Necati, Heat Conduction, 2nd Edition, John Wiley & Sons, New York, 1993, page 108. 125
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150 100 50 50 100 150 0 0 1 5 Positive Roots /3m J 0 (/3mb) + H 310 (/3mb) = 0 Figure Al.2 The graph in figure A 1 2 is an example of the eigenvalues for the antimony test cylinder. The associated eigenfunction and normalization integral for the vertical surfaces are equations (A1.41) and (Al.42) respectively: (A1.41) (A1.42) where H2 = I"J. k' (Al.42l), (A1.422) 126
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20 10 ..) ../ ..) ../ ..) ..1 f200 f400 { 600 I BOO { 1000 10 20 Using the Relationship J zv J vI (Pz)dz = zv J v (Pz), recognizing 11 (o) = 0, and substituting equations (Al.38), (Al.39), (Al.41) and (Al.42) into equation (Al.37) gives the analytical formula for T(r,z,t), equation (Al.43): (Al.43) 127
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BIBLIOGRAPHY 1 D. C. Williams, Optical Methods in Engineering Metrology Chapman & Hall, London, pg. 277. 2 Feduska, W., and Horigan, W. L., Weld. Met. Fabrication, 35(12),, December 1967, pgs. 483489. 3 Vaidyanath, L. R., Nicholas M.G., and Milner, D. R., British Welding Journal, 6(1), January 13, 1959. 4 Wiliford, C. F ;and Tylecote, R. F., British Welding Journal, 7(12), December 1960, pgs. 708712. s Schwartz, M. M., Metals Joining Manual, McGrawHill, 1979, pg 105. 6 Kazakov, N. F., "Diffusion Welding in a Vacuum," Moska, Izdvo, Mashionostroyniye, pp. 1332, 1968. 7 ASM Handbook, Vol. 2 ASM International, pgs. 11101113. 8 Holman, J.P., Heat Transfer, McGrawHill, New York, 1986, pgs. 635636. 9 ASM Handbook, Vol. 2, ASM International, pgs. 10991101. 10 D. E. Carlson, Handbuch der Physik, Volume Via/2, springerVerlag, 1972, pgs. 297312 11 William Alan Day, Heat Conduction Within Linear Thermoelasticity, Springer Tracts in Natural Philosophy, vol. 30, SpringerVerlag, 1985, pg 15. 12 Heat Conduction, M Necati Ozisik, 2nd Edition, Wiley & Sons, New York, 1993, pp. 4849. 13 B. A. Boley and J. H. Weiner, Theory ofThermal Stress e s, John Wiley & Sons, New York, 1970, pp. 4244. 14 Survey of Recent Developments in the Fields of Heat Conduction in Solids and ThermoElastcity, B. A Boley, Nuclear Engineering and Design,v 18 (1972), pp. 379380. 128
