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 Title:
 Characterization of the Bauschinger effect in sheet metal undergoing large strain reversals in bending
 Creator:
 Hanzon, Drew Wyatt ( author )
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 University of Colorado Denver
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 Department of Mechanical Engineering, CU Denver
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 bibliography ( marcgt )
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 Thesis (M.S.)University of Colorado Denver.
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 Includes bibliographic references.
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 Department of Mechanical Engineering
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 by Drew Wyatt Hanzon.
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Full Text 
CHARACTERIZATION OF THE BAUSCHINGER EFFECT IN SHEET METAL
UNDERGOING LARGE STRAIN REVERSALS IN BENDING
by
DREW WYATT HANZON
B.S., University of Colorado Denver, 2013
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado Denver in partial
fulfillment of the requirements for the degree of
Master of Science
Mechanical Engineering Program
2015
This thesis for the Master of Science degree by
Drew Wyatt Hanzon
has been approved for the
Mechanical Engineering Program
by
Luis Rafael Sanchez Vega. Chair
Peter Hoffman
Dana Carpenter
November 20, 2015
Hanzon, Drew Wyatt (MS, Mechanical Engineering Program)
Characterization of the Bauschinger effect in Sheet Metal undergoing Large Strain Reversals in
Bending
Thesis directed by Associate Professor Luis Rafael Sanchez Vega
ABSTRACT
This work consists on the quantification of sheet metal uniaxial stressstrain reversals from
pure bending tests. Bending strains to approximately 10% were measured by strain gages and
interferometry. Bendingunbending moments and strains were modeled and compared closely to the
experimental data. The reverse uniaxial stressstrains curves were determined from the optimal fit of
the model. Bauschinger effects were described by the reverse uniaxial response at the elastoplastic
range, between the elastic and the large strain, power fit ranges. Arc and straight line fittings on the
lnerlne scale proved accurate to describe the elastoplastic behavior. Reverse uniaxial data determined
for DP590 and DP780 steels and two Aluminum alloys showed significant Bauschinger effects with
distinct features. Lor the DP steels the magnitudes of the reverse compressive as curves compared
moderately higher, and merging to a power curve with parameters K, n previously defined by tension
testing. Bauschinger effects at small reversed strains were less pronounced for the aluminum alloys.
However, at higher strains the reverse elastoplastic response softened considerably, and during the
unbending span the magnitudes of the reverse compressive strains remained below the corresponding
K, n tensile values. The results showed pure bending as an efficient, simple to use technique to
generate as data for sheet metal at large reverse strains without the complicating restraining
hardware required by direct compression methods.
The form and content of this abstract are approved. I recommend its publication.
Approved: Luis Rafael Sanchez Vega
m
TABLE OF CONTENTS
CHAPTER
I. INTRODUCTION.....................................................1
II. LITERATURE REVIEW................................................3
III. EXPERIMENTAL METHODOLOGY ........................................6
Pure Bending Moment Device .....................................7
Strain Measurement using Eiterferometry ........................8
IV. CHACTERIZATION OF MATERIAL PROPERTIES ..........................10
LogLog Analysis of True Stress True Strain..................11
Parameterization of the ElastoPlastic Region..................12
V. PURE BENDING MOMENT MODEL ......................................14
VI. REVERSE BEND OPTIMIZATION.......................................18
VII. EXPERIMENTAL AND NUMERICAL MODEL RESULTS........................21
VIII. DISCUSSION.....................................................26
Dual Phase Steel Samples.......................................26
Aluminum Alloy Samples.........................................28
IX. CONCLUSIONS....................................................31
REFERENCES...........................................................32
APPENDIX
A. TENSILE TEST CURVE FITS FOR SAMPLE MATERIALS......................34
B. NUMERICAL MODEL (MATLAB PROGRAMING LANGUAGE)....................38
C. PLANE STRAIN PURE BENDING MODEL.................................54
IV
LIST OF TABLES
Table
1: Uniaxial tensile properties for test materials...............................................6
2: Descriptions and relationships for labeled regions in Figure 4...............................10
3: Tensile test fit parameters..................................................................21
4: Tensile test data standard deviation of the fit Sos in the Ins lno scale...................21
5: Reverse uniaxial o s fit parameters........................................................25
v
LIST OF FIGURES
Figure
1: Reverse strain behavior showing the Bauchinger effect...................................2
2: Description of Dr. Sanchez pure bending moment device....................................8
3: Interferometry measurements.............................................................9
4: Cyclic loading behavior................................................................10
5: Tensile Test lnerlne for DP590 steel...................................................11
6: lnerlne fits for DP590................................................................12
7: True StressTrue Strain fit for DP590 Steel.............................................13
8: Pure bending moment free body diagram used for reverse bend optimization................15
9 Flow diagram for random search optimization program......................................19
10: Comparison of final first bend stress profiles with models I and II....................20
11: DP590 steel pure moment bending/unbending curve........................................23
12: DP780 steel pure moment bending/unbending curve........................................23
13: Aluminum Alloy 1. Pure moment bending/unbending curve..................................24
14: Aluminum Alloy 2 pure moment bending/unbending curve..................................24
15: Uniaxial o  s tension and reverse best fit compression for DP590 Steel...............27
16: Uniaxial o  s tension and reverse best fit compression for DP780 Steel...............27
17: Uniaxial o  s tension and reverse best fit compression for Aluminum Alloy 1..........28
18: Uniaxial o s tension and reverse best fit compression for Aluminum Alloy 2...........29
19: Model with Isotropic Hardening and Bauschinger Factor..................................30
vi
NOMENCLATURE
Â£ Strain [] a Stress [Mpa]
V Poisson ratio [] Rc Arc fit radius []
n Power curve exponent [] K Power curve coefficient [Mpa]
r0 Lankford coefficient [] E Elastic modulus [Mpa]
h Thickness of sheet [m] M Bending moment [Nm]
w Width specimen [m] k Bend curvature [m1]
z Distance from neutral axis [m] S Stress State [Mpa]
s Standard deviation [] Â£p Plastic Strain []
Â£c Strain (end elastic fit) [] Â£k Strain (begin power curve) []
Â£a Strain (end arc fit) [] Â£f Strain (end power curve ) []
ERR Error (optimization algorithm) [] BF Bauschinger factor []
CON Convergence criterion [] UTS Ultimate tensile strength [Mpa]
UE Elongation at UTS [%] TE Total tensile elongation [%]
Subscripts
i Bend simulation step V Bend model layer
P Plastic deformation y Material yield point
1,2,3 Orthogonal principle directions i Intermediate power curve parameter
R Reverse loading f Plastic Flow
t Temporary model parameter b Bend test specimen
outer Surface layer of model/specimen d Tensile test specimen
Superscripts
e Elastic Range P Plastic range
Reverse bend property
Vll
CHAPTER I
INTRODUCTION
Since the original experiments conducted by Bauschinger [1], there has been ongoing
research to explain departures from isotropic hardening behavior due to plastic strain reversals.
Several mathematical models have been proposed for predicting material properties of sheet metals
undergoing small strain reversals, for which experimental data was obtained using uniaxial and
bending tests. However, the generation of large strain data for sheet metal under compression offers
experimental challenges. The tendency of sheet metal to buckle and wrinkle during compressive
loading, and friction at the interface between sheet and tools are major contributors to the
experimental uncertainty. These difficulties can be avoided by performing pure bending tests.
However, the experimental cyclic bending data must be properly correlated to uniaxial
tensile/compression behavior. This work consists of a novel analytical model based on tensile
uniaxial and pure bending experimental data at large reverse strain deformations. Strain gages and
interferometry techniques were used to track the tensile/compressive strains under bending. From the
experimental results, the uniaxial material properties were detennined for sheet metals undergoing
large tensile/compressive strains.
The Bauschinger effect can be defined as a yielding of the material, after an initial plastic
deformation, at a significantly different stress in the reverse direction than would have been achieved
had the material been reloaded in the monotonic direction. In Fig. 1, the stressstrain curve will follow
the strain hardening power curve during tensile monotonic loading to SBp, followed by unloading to ep
and monotonic reloading to .S'/ w ill return to the same curve. Reloading the material in the reverse,
compressive direction from ep results in yielding at SR, where S^R  = Sf andSR =Â£ SfR. The
difference between these values is referred to as die Bauschinger effect. As SfR SR > 0, the
Bauschinger effect vanishes.
1
True Stress [MPa]
Figure 1: Reverse strain behavior showing the Bauchinger effect
2
CHAPTER II
LITERATURE REVIEW
While the tensile curve to Sep is readily available from uniaxial tension tests, the
determination of the exact configuration upon reversal from ep to SR requires testing the sheet metal
under imiaxial compression. Since sheet metals would buckle under such compression, the sheet
samples need to be laterally restrained. This approach has been pursued by several researchers.
Tozawa tested sheet metal under compression using stacks of glued sheet to prevent or delay buckling
[2], Using this technique, he carried out experiments under combined stress in order to define the
Yield loci for various materials. Kuwabara et Al. designed an inplane compression test and
prevented buckling by sandwiching the sheet between combshaped dies and compressing it between
rigid and parallel platens to sandwich the sheet [3], Yoshida et Al bonded sets of five sheets and
laterally restrained the set to prevent buckling [4], They reported large strain reversals of up to 10%
when the antibuckling device was used. Boger at Al designed an inplane compression test using
solid flat platens for buckling constraint smaller than the sample length and specimens tailored to
avoid bucking outside the constrained region [5], They reported optimized experimental conditions in
order to maximize the uniformity and the magnitude of the compressive strains. Additional versions
of uniaxial compression, as well as a brief discussion on the advantages and disadvantages of other
approaches such as reverse shear, and cyclic torsion and can be found in references [4], and [5],
The generation of representative sheet metal experimental data under uniaxial compression
face challenges common to the methods above. Ideally, the sample would uniformly compress axially
and expand radially under compressive stresses. Allowing for unifonn radial expansion would require
frictionless, flat compressive platens, as opposed to gripped ends. Additionally, buckling modes
developing under axial compression are not uniform, with the segment around the midlength of the
sample more prone to buckling. Assessing the compressive stresses caused by the antibuckling plates
would require estimation of the effective contact area at the small section first prone to buckling. This
transverse stress may be expected to be significantly higher than average, and the uniaxial condition
3
would need reevaluation. Also alignment must be preserved during testing. Any departure from true
axial loading would result in additional moment offsets. Overall sliding friction from antibuckling
supports of any kind acting on contact surfaces adds to the underlying uncertainty of the experimental
results. It should be noted that roller backed platens do not prevent sliding friction between the sheet
samples and the plates, given the deformation gradient in the samples. From Boger [5], friction
coefficients were estimated to be negligible, in the range of 0.060.09. However, as the os slope
decreases with strain, the effects of friction become more relevant and cannot be neglected.
Bending tests benefit from providing momentcurvature, and momentstrain information
useful to industrial forming practices. Bending is a common sheet metal forming mode, and its
associated forming issues notably, springback, is related to bending residual stresses. Outer layers of
tension and compression stresses develop during bending, and reverse bending result in opposite
stress states. Various forming operations such as flow around comers and draw beads are common
examples of stressstrain reversals due to cyclic bending. Therefore, it has been of practical relevance
to study the Bauschinger effect by means of bending tests. Several bending test devices have been
reported, which can be classified in two major groups; (1) threepoint bending devices, and (2) pure
bending moment devices. Recently, Zang et al used a threepoint bending test to measure the
Bauschinger effect from the bending springback profile [6], They used prestrained sheets to measure
the reverse transient and permanent softening behavior for DP780 steel sheets. Eggertsen and
Mattiasson used a three point test to study the bendingunbending behavior for various steel sheet
grades [7], They applied inverse modeling approaches to determine the main parameters towards
improved springback predictions. Threepoint bending introduces transverse shear and contact effects
at the middle pin, where the maximum bending takes place. The span between bending points is
typically large; as are needed to accommodate the geometry of the bending apparatus Large strains,
on the order of 10%, require bending radii around five times the thickness. This results in very small
spans for thin sheets, around 1 mm thick. Furthermore, as the span decreases, the transverse shear and
4
contact effects increase at the location of interest. These characteristics make the threepoint bending
test most useful experimental testing at small strains, and/or thicker materials.
Pure bending moment techniques do not introduce transverse shear or contact effects. Four
point bending devices, for instance, apply pure bending moments on the central section of the rig.
These devices are not typically suited for pure bending of thin sheet metal due to flexural effects at
the pins. Instead, pure bending is applied using rigid jaws guided by specialized loading mechanisms.
Duncan et Al. designed a bending test rig that mounted on a tensile test machine and made use of the
uniaxial crosshead force to produce a pure bending couple [8]. The changing geometry during testing
resulted in nonlinear loaddisplacement curves, but accurate pure bending moment measurements
were reported at the elasticplastic region. A more recent variation of a cross head mounted device
was reported by Weiss et Al [9]. The test specimen was mounted at the ends of pinjointed arms. A
bending moment was imparted on the sample as the arms rotate during cross head motion. The
deflection was measured by an LVDT installed at the midsection. Due to the fact that an axial force
passes through the sample axis, the test is not rigorously pure bending. Also, as in three point
bending, span requirements limit the level of strain that can be imparted to thin sheet specimens.
5
CHAPTER III
EXPERIMENTAL METHODOLOGY
This study included four testing materials; two dual phase (DP590 and DP780) sheet steels
and two aluminum alloys (All, A12). Experimental data was collected from uniaxial tensile and pure
bending moment tests. As validation of the method is the primary intent, data collection was limited
to the rolling direction and strain rate effects were factored out by testing under quasistatic
conditions. Mechanical properties of each material derived from traditional tensile testing are given in
Table 1.
Table 1: Uniaxial tensile properties for test materials
0.2% YS UTS UE TE K n r0
Material [Mpa] [Mpa] [%] [%] [Mpa]
DP590 320.3 567.2 18.9 24.3 1038.7 0.228 2.11
DP780 415.3 720.7 13.0 19.6 1105.7 0.143 1.89
AA1 129.7 290.0 22.1 27.9 448.4 0.254 0.92
AA2 164.9 281.4 20.9 27.5 533.1 0.259 1.12
Pure bend tests differ from uniaxial testing in various aspects. Under pine bending,
tensile/compression strain gradients develop through the thickness, with maximum values at the outer
layers. Bending induced residual stresses developed upon unloading, and the neutral axis has been
reported to shift [10], For wide specimens, aside a small anticlastic curvature developing at the edges
during bending, the width remains approximately constant. The resultant plane strain condition differs
from uniaxial flow, complicating the correlation between the tests.
However, a properly designed pure bending test has important advantages. The layers under
compression are supported by the layers under tension, and thereof large compression strains can be
reached on the concave side without reinforcing antibuckling devices. Shear stresses are negligible,
and there are no contact or friction effects at the location of study. Additionally the bend test areas are
6
open and accessible to monitoring; bending moment, strain and 3D geometry can be collected
accurately using sensors and optical techniques.
Pure Bending Moment Device
Pure bending moment experiments conducted here followed a procedure previously
developed by Sanchez [11], Specimens were bent to small radii by incrementally increasing the
applied moment until the outer most fibers of the material experienced strains of the required
magnitude (up to a maximum of around 15 %).
Dr. Sanchez pure bending device developed from its original form, first published in 1988
[19] to the current form shown in Fig.2. It consists of two grips, one fixed rigidly to the base and the
other free to rotate parallel to the axis of the bend and to translate on an XY plane orthogonal to this
same axis. All moving parts were supported by bearings to reduce frictional effects. The bending
moment was transmitted under quasistatic conditions by a bending couple translating with the
carriage (Fig. 2b). The load was reversed to unbend and reversebend the sample. The moment was
measured directly by a sensor mounted underneath the moving grip. During the performance of the
test, the grip was rotated to a predetermined angle while the output of the strain and moment sensors
was simultaneously monitored. Given that grip effects may contribute to the angle of rotation, grip
displacements were not considered for strain calculations. Strain measurements were taken at a small
area at the center of the bending span. Low level strains were monitored using foil strain gages (Fig.
2d). Although gages up to 10% strain are commercially available, they require curing oven time.
Testing under quasistatic conditions was practically performed using 03% strain gages with lead
wires attached; which were easier to use, faster to install and more economical. The strain range may
be extended by gluing new gages at progressive steps with the device under load. However, extensive
testing showed that the process could be simplified to one gage for low strains, with the final strain
independently determined by interferometry.
7
(a) Pure Bending Moment Device
(b) Carriage riding on linear berings
(c) Pulley Mechanism
Figure 2: Description of Dr. Sanchez pure bending moment device
8
Strain Measurement using Interferometry
Interferometry was used to track 3D geometries on a 1,9mm by 2.0 mm test area on both
surfaces of each specimen. The strains at the outer layers were measured from inscribed marks (Fig.
3a). By fdtering cylindricity and tilt, the 2D flat distance between the marks could be measured (Fig.
3 b). As described in [11], strains can also be determined from the sample thickness and profde radii
obtained from 3D profile measurements (Figs. 3c, 3d). Final strain uncertainties were on the order of
+ 0.003 strain.
X Profile
X: 1.543 mm
f X: 0.438 [Y: 8.3 u 1 nm ") f X: 1.! *_J 80 mm Oum J li
yn iV^y/VV
H.............H...................................................
023 <323 0.40 080 0.80 100 100 1.40 180 180 2530 223 2.40
(b) 2D Measurement of Distance
(d) 2D Convex Profile
Figure 3: Interferometry measurements. See also reference [11]
9
CHAPTER IV
CHACTERIZATION OF MATERIAL PROPERTIES
Despite eventual uncertainties due to the experimental techniques, published estimates of the
cyclic true stressstrain curves for fonnable aluminum and steel sheets share several common trends
(Fig 4):
Table 2: Descriptions and relationships for labeled regions in Figure 4.
Labeled Region StressStrain Region Constitutive Equation
0 A Linear range a = Ee
AC Elastoplastic range Unknown
CD Large plastic strains 17 = (KCD)Â£nCD
DA Unloading/reverse loading linear a = Ee
AC Elastoplastic transient Unknown
CD Large plastic strain reversal a = (Kc.D)encD'
EF Repeated cycles at same level of strain shows saturation
Figure 4: Cyclic loading behavior
10
LogLog Analysis of True Stress True Strain
Similar large reverse elastoplastic transients are shown by early and current researchers;
Caulk and Naghdi [12], and Chaboche [13], in their study of the cyclic response for stainless steels.
Yoshida and Uemori in their studies of in plane cyclic tensioncompression [14] and springback
simulation [15], Eggertsen and Mattiasson [7] in their experimental forcedisplacement diagrams. The
constitutive models developed by these researchers varied in complexity, as needed to fit the
materials responses.
Figure 4 evidences the need for an accurate representation of the transient between the elastic
OA, and the power curve CD sections. A close fit at AC was accomplished on the lnerlne scale
following a procedure proposed by Dr. Sanchez. OA and CD are straight lines in the lnerlne scale, as
shown in Figure 5 for DP590 Steel. The experimental data at the transient region curved from the
elastic line at AB and intersected the power line at a sensibly constant slope at BC.
7
6.5
6
5.5
3 5
c
J 4.5
4
3.5
3
2.5
2
9 8 7 6 5 4 3 2 1
: D
j C
j B
j A r
j //
: ^ cperimen tal Tensi le Testin g Data
. j y
Elastic Fit (OA)
fower tit feu]
Ln (e)
Figure 5: Tensile Test lnerlne for DP590 steel
11
Parameterization of the ElastoPlastic Region
This behavior leads to a simple, but close fit to the experimental data through an arc of circle
AB and a straight line BC on the lnerlne scale (Fig. 6). The center point of the circle was located by
offsetting both the elastic and intermediate straight line fits by an arbitrary distance r and finding the
intersection of the two lines. A circular curve of radius Rc, centered at this intersection was then
generated between points perpendicular to the offset lines and intersecting the original fits. The
circular arc of radius Rc was then adapted to find a best fit to the experimental data.
\m(s)
Figure 6: lncrlne fits for DP590
The slope of the fit showed a small discontinuity at the intersection between the lines (point
C). This effect was small enough that further modeling complexities were deemed unnecessary,
however an additional arc fit in this small region would eliminate any edge in the a e curve. The fit
accurately characterized the material properties throughout the entire os curve as shown in Fig. 7.
The fit was adequate for both the steel and aluminum samples tested in this experiment (Table 2).
This approach lends itself to an automated adaptation routine for the optimization fit of the reverse
bending moment experimental data.
12
True Stress (o) [MPa]
800
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
True Strain (e) []
Figure 7: True StressTrue Strain fit for DP590 Steel
13
CHAPTER V
PURE BENDING MOMENT MODEL
Modeling of pure bending of sheet metal involves shifting of the neutral plane under hoop
and radial stresses. A pure bending model including these parameters as well as through thickness
anisotropy was developed by Dr. Sanchez. A summary of the model is shown in appendix C.
Modeling bending/unbending requires Bauschinger strain reversal curves. During the first
bend, strain reversals occur due to shifting of the neutral plane. Since the Bauschinger curve is
unknown at the first bend, the solution was based on an iterative scheme, where the proposed
Bauschinger curve was first determined by the experimental data using an approximate approach. The
quality of fit by the approximate Bauschinger curve was verified at posteriori (See Fig. )
The approximate approach takes into account that, at the plastic strain levels reached (10%),
plastic behavior predominates. The shifting of the neutral plane is expected small at the ratios
t! 2
~ 10 used on this study. For the materials tested, the experimental data showed negligible
R
thickness changes during bending/unbending. The concave and convex strains were of similar
magnitude, and the surface profiles shared the same center radii. Therefore, cross planes remained
plane. Additionally the radial stresses developed by the bending were considered small enough to be
ignored.
14
Figure 8: Pure bending moment free body diagram used for reverse bend optimization
The true strain as a function of curvature (k, = 1/r,) and the distance of the layer (r) from the
neutral axis is given in Equation (1):
Â£ijV = ln(l + ktz,,)  \ < z < \ (1)
For the following analysis die primary stress and strain directions will be oriented with the
direction experiencing normal forces due to the bend as 1 (oi), the direction parallel to the axis of the
bend as 2 (o2) and the dimension through the thickness of the sheet as 3 (o3). Within the elastic
range, at the initial stage of the bend, the stress developed in each layer obeys Hooks Law as shown
in Equation (2).
o1 = Ee1, a2 = 0, o3 = 0; Â£2 = ~VÂ£, Â£3 = ~VÂ£\ (2)
As the outer layers enter plastic deformation, the sheet bends and stretches under plane strain with
Â£2=0. Given the large width/thickness ratio for sheet metal, the elastic core is constrained by the
plastic layers under plane strain and
15
G2V(Ji n
 =0; o2= vax
(3)
Fe ~
Â£2
Oi
(4)
(5)
e VffiVO2 . s. ffi V g
3 E y J E 1v 1
Assuming rotational symmetry, r0 = r90 = r where the Lankford coefficient was determined at
15% tensile strain the plastic strains are obtained from the flow rules:
h=<6>
Under plane strain along the width the instantaneous deformation is given by:
ds? = dsn + ds? = 0
(7)
However, due to the large width to thickness ratio this condition can be reduced to Eq. (8)
and (9):
ds2 0;
r
z 1+r
1+r _
Oi = , 0
1 Vl+2r
Where a, s are determined from tensile test data fit to sections below:
Section OA defined by the linear fit
(8)
(9)
a = Es
Elastoplastic section AC defined by:
Arc fit AB:
(10)
(fig (lnÂ£lnÂ£0)2+ln
= ey
where (lne0, lner0) are the coordinates of the center of the arc in the lne lner scale.
Intermediate fit BC:
bc = KbcÂ£71bc
Large strains power fit CD
bc = KcdÂ£71cd
(11)
(12)
(13)
16
From the constitutive relationship obtained by the true stressstain curve, the moment needed
to attain a bend of arbitrary radius r, can be found from the standard bending moment equation for a
unit width and thickness h given in Equation (13).
h/
Mi = Sh/ iz dz
/ 2
(14)
The sheet is monotonically bent from flat to a bending moment AT,. Upon unbending, the
residual stress in each layer is calculated by (14) where Mf is the final moment, hb is the thickness of
the sample, zn is the distance of layer // from the neutral axis and the width is a unit length.
R,r) 1,77
12 Mf
zv
(15)
17
CHAPTER VI
REVERSE BEND OPTIMIZATION
For the reverse bend the parameters that define the elastoplastic region of the true stressstrain
curve are iterated to produce a bending moment that conforms to the measured reverse bend moment
using a random search optimization program. In this algorithm the circle radius as well as the
intermediate power curve coefficient and exponent are adjusted by a given step size to produce a new
true stressstrain curve at each iteration. The program then calculates the error of the resulting reverse
bending moment curve with the experimental data obtained from the pure bending moment device.
The error is then compared to the error from the previous loop iteration or the error given by the
original constitutive curve for the first iteration. If the resulting error exceeds the previously
determined error the direction of the step is reversed. This is done individually for each of the
variables defining the Elastoplastic region. The new bending moment curve is determined based on
the complete set of new parameters describing the reverse stress strain curve. The resulting error is
checked against the condition for convergence (CON). If the condition is not met, the step size is
reduced and the new parameters form the basis for the next iteration; provided the resulting total error
is less than the previous loop. A flow diagram of the program is given in Fig. 11. The program output
material properties are:
From the tensile test fits: E, (Inrr0, In sn), Rr,KBC, nBC,KCD,nCD
From optimal fits to the reverse pure bending test: (In
18
Figure 9: Flow diagram for random search optimization program
19
Figure compares the through thickness hoop stresses between models I and II. Model I
(Appendix C) includes neutral plane shifting and radial stresses. Model II corresponds to the
approximate solution, where the Bauschinger effect during bending was not included.
Neglecting elastic volumetric changes in Model I, the stress reversals were calculated using
the reverse Bauschinger curve obtained by the approximate solution. Section AB in the Figure
corresponds to the strain reversal zone during bending.
The radial stresses in Model I, determined using the approximate Bauschinger curve, do not
necessarily satisfy continuity. The discrepancy on radial stresses at A, of around 6 %, provides a
measure of the uncertainty using the approximate solution.
This discrepancy between models was small to translate in a meaningftd change on the
Bauschinger results using the approximate approach.
Bending Stress [MPa]
Figure 10: Comparison of final first bend stress profiles with models I and II
20
CHAPTER VII
EXPERIMENTAL AND NUMERICAL MODEL RESULTS
Table 3 shows the fits to the uniaxial tensile test data. These included the Elastoplastic (AC),
and the large plastic (CD) regions.
Table 3: Tensile test fit parameters
Material Elasto Plastic Fit Farge Plastic Strain Fit (CD)
Arc Fit (AB) Fit (BC)
Rc Fn(Mpa) K Fn(Mpa) K (Mpa) n K Fn(Mpa) K (Mpa) n
DP590 1.45 6.324 557.6 0.098 6.946 1038.7 0.228
DP780 1.3 7.369 1585.8 0.243 7.008 1105.7 0.143
AA1 1.25 5.562 260.3 0.124 6.106 448.4 0.254
AA2 0.95 5.788 326.4 0.127 6.279 533.1 0.259
The uniaxial tensile data standard deviations of the fit ( Scs) are shown in Table 4. For all
materials, the straight section of the Elastoplastic fit (section BC) was comparable to the standard
deviation of the power equation fit (section CD). The arc fit AB was comparable to the fit to the
elastic data OA. Overall, the deviations were small; although closer fits were noted for sections at
larger strains BC, and CD.
Table 4: Tensile test data standard deviation of die fit Sos in the Ins lno scale
Material Elastic Fit Elastoplastic fit Farge Plastic fit (CD)
Arc fit (AB) Fit (BC)
DP590 0.035 0.018 0.004 0.007
DP780 0.022 0.040 0.006 0.004
AA1 0.012 0.014 0.007 0.003
AA2 0.015 0.010 0.007 0.004
Figures 1215 compare the bendingunbending experimental and model data. For the bending
section OD, the comparison was based in two independent tests. The model was based on the tensile
21
test parameters in Table 3, as well as the experimentally determined elastic modulus, and the
experimental data on the pure bending test. Under unbending section DD the model was fitted to the
experimental data, from where the reverse tensile test parameters were calculated using the uniaxial
testing data as the starting point.
The figures show the pure bending model plotted against the compressive outer layer true
strain. The range for the strain gage was significantly larger when glued to the outer layer under
compression. Therefore, strain gage data was only acquired at the compressive side which
corresponds to the concave profile under bending and to the convex profile under unbending. Strain
gage tracking covered the Elastoplastic range OC and the initial power fit. Testing was performed
quasistatically, and stepped strain gages were sometimes glued under load for longer tracking.
However, monotonic bending at large strains was mainly dependent on the power equation and was
very predictable. Since the final strain at D was measured independently, strain gage data at low
strains prove adequate during bending. Prior to unloading, a second strain gage was glued with the
sample under maximum bending moment for the steel samples. The final strain at D was determined
from interferometer readings at O and gage data DO. Data under unbending OD was obtained
following a similar procedure.
22
6
Outer Fiber True Strain (e) []
Figure 11: DP590 steel pure moment bending/unbending curve. Interferometer readings taken at O,
O and O.
0.02 0 0.02 0.04 0.06 0.08 0.1
Outer Fiber True Strain (s) []
Figure 12: DP780 steel pure moment bending/unbending curve
23
Figure 13: Aluminum Alloy 1. Pure moment bending/unbending curve
Outer Fiber True Strain (e) []
Figure 14: Aluminum Alloy 2 pure moment bending/unbending curve
24
The calculated fits for the uniaxial o s reverse curve are shown in Table 4. The parameters
define the reverse uniaxial curves resulting at the minimum ERR values (Fig. 9) calculated from the
reverse M eouter curve.
Table 5: Reverse uniaxial o s fit parameters
Elasto Plastic Fit
Material Arc Fit (AB) Fit (BC) Farge Plastic Strain fit (CD)
Rc Fn(Mpa) K Fn(Mpa) K (Mpa) n K Fn(Mpa) K (Mpa) n
DP590 3.105 6.682 798.1 0.141 6.946 1038.7 0.228
DP780 0.657 7.594 1986.7 0.269 7.008 1105.7 0.143
AA1 1.003 5.744 312.3 0.099 6.106 448.4 0.254
AA2 0.482 5.887 360.5 0.095 6.279 533.1 0.259
25
CHAPTER VIII
DISCUSSION
At bending sections OD (Figs 1215), the M eouter fits and the experimental bending moment
data were obtained by independent approaches. The experimental moment was determined from the
pure bending device, and the calculated moment from the equation (9) using tensile test data
according to its parametrical fit (Table 2). For unbending section OD, the uniaxial parameters were
determined from the best fit of the pure bending model to the reverse experimental M eouter data.
Fitting of Aluminum AA1, AA2, were closest, while DP590 and DP780 showed a small overestimate
at point D. This was attributed to anticlastic effects due to smaller pure bending sample widths
(necessitated by equipment limitations) for theses high strength materials.
Figs 1619 show the uniaxial o 8 curves under monotonic (OD) and reverse loading (OD).
A direct correlation is appreciated between the o 8 fitting sections and theM ecurves. The Elasto
plastic radial arc fit (AB) was characterized by a small strain zone, serving as a transition between the
elastic range (OA) and the intermediate power fit (BC). With increasing bending, the Elastoplastic
contributions became less significant as large strains developed from C to D. Section CD was more
closely modeled by the tensile test power equation parameters K. n, given in Table 1. During
unbending, the intermediate power fit BC was significant, as shown by uniaxial fits with large BC
Elastoplastic reversals.
Dual Phase Steel Samples
The uniaxial reverse fit for DP590 and DP780 are shown in Figs 16 and 17. Both steels
evidence an initially significant Bauschinger effect at the small strain section defined by the arc fit.
The reverse response was close to the original tensile curve. Section BC expanded to BC upon
reversal, with end point C joining the original power fit CD. Differences in the Elastoplastic
response are appreciated in Tables 2 and 4. Intermediate KBC, and nBC (557.6Mpa, 0.098) for DP590
increased significantly upon reversal (798.1Mpa, 0.141) but significantly below the original K, n
26
values at CD (1038.7Mpa, 0.228). This trend was opposite for DP780, with KBC, and nBC larger than
K. n values at CD.
Figurel5: Uniaxial o s tension and reverse best fit compression (primed labels) for DP590 Steel.
Figure 16: Uniaxial o s tension and reverse best fit compression for DP780 Steel. Uabels refer to
the reverse compression curve
27
Aluminum Alloy Samples
The reverse curves for Aluminum alloys 1 and 2 showed similar behavior between them
(Figs. 18, 19); but showed significant differences to steel in the Elastoplastic response. For AA
sheets die reverse curve consisted of a large span defined by the intennediate power fit BC. The
reverse small strain point B was significantly higher than B. However, the intermediate power fit BC
softened considerably upon reversal, with reverse nBC values significantly smaller dian for CD (0.009
and .254 for AA1). Point C upon reversal extended close to full unbending to flat, with no
discernible CD section.
True Strain (e)
Figure 17: Uniaxial o 8 tension and reverse best fit compression for Aluminum Alloy 1. Primed
labels refer to the reverse compression curve
28
350
0 0.02 0.04 0.06 0.08 0.1
True Strain (sx)
Figure 18: Uniaxial o 8 tension and reverse best fit compression for Aluminum Alloy 2. Primed
labels refer to the reverse compression curve
Figs 1219 show the tendency of the materials towards reaching stress levels upon reversal
similar to end point D and further defonn following the original K. n values. That was the case for
DP590 and DP780. The AA1. AA2 sheets followed that tendency, but trends merging to the original
K, n values were inconclusive. Nevertheless, the intennediate KBC, and nBC power fits were shown
significant to the reverse response for all materials tested.
Fig. 20 shows the experimental data compared to reversals based on a fraction BF of the
isotropic hardening curve, with BF = Slf S, (Figl). This approach was deficient, as the predicted
curves did not follow the overall reverse response. Comparison to Fig 12 shows how the inclusion of
the Elastoplastic response resulted in a significant improvement of the fitting.
29
0 0.02 0.04 0.06 0.08 0.1 0.12
Outer Fiber True Strain (e) []
Figure 19: Model with Isotropic Hardening and Bauscliinger Factor
Given the tendency of sheet metals to saturate with repetitive cycles (Fig. 4), the reversal
effects are expected more pronounced at the first bendingunbending. For model validation purposes,
the experimental testing was confined to that range in this study.
Simplifications to the procedure may be carried out depending on the features of the
experimental curve. For the thin sheet materials tested in this study, profile radii and strain data from
interferometry located the neutral axis at the center of sheet within experimental error [11], For these
cases, simpler bending models based on a midneutral axis may suffice. The experimental collection
of strain bending/unbending data may be reduced to two gages. During unbending, point D can be
approximated as an extrapolation of the data OA obtained from a gage glued at O. This
simplification applies for unloading cases with DO approximately collinear to the reverse elastic
range OA. As noticed by other researchers
Additional research may refine the correlation between pure bending and uniaxial testing.
Since straining under pure bending locates at an unsupported and accessible section, the test is lends
to adaptation of other contact/noncontact measurement techniques. Residual stresses, for instance,
can be assessed on the unbent to flat samples using X rays diffraction, as reported by Kassner et Al.
[18].
30
CHAPTER IX
CONCLUSIONS
The reverse uniaxial stressstrain curves can be obtained from simple to perform pure
bending and tensile tests. The reverse curves, determined by the best fit to the experimental moment
outer strain data, were distinct for the sheet materials tested. For DP steels, reverse stresses merged to
the originally monotonic curve at midrange strains. Aluminum alloys showed an initially higher
reverse stress response, followed by significant softening. Contrary to DP steel sheets, reverse
stresses for aluminum alloys did not merge to the original curve during the unbending span.
Bauschinger effects were quantified as the material reverse response in the Elastoplastic
range. This range was defined between the elastic and the large strain sections, represented by straight
lines in the In <7 In S scale. Close standard deviations of the fits were obtained in this range by
fitting an arc to the elastic line, followed by an intermediate linear fit. The intermediate fit best
described large reverse strain transients occurring upon reversal.
The determination of the reverse uniaxial curve from unbending data was indirect and
required modeling. However, the relationships between pure bending and uniaxial loading become
better known with improved experimental techniques. Pure bending naturally applies large
compression strains to unsupported sheet metals and it is simple to perform. These represent major
advantages over direct uniaxial compression of sheet metal, which requires restraining.
31
REFERENCES
[1] J. Bauschinger, Uber die Veranderung der Elastizitatsgrenze und des Elastizitatsmoduls
Verschiedener Metalle, Civilingenieur 27 (1881), 289347
[2] Tozawa, Y. Plastic Deformation Behavior Under Conditions of Combined Stress, Mechanics of
Sheet Metal Forming, Koistinen, Wang eds., Plenum Press, (1978) 179211
[3] Kuwabara T., Morita, Y. Takahashi, S. Elastic Plastic Behavior of sheet metal subjected to in
plane reverse loading. Proceedings of Plassticityl995, Dynamic Plasticity and Structural
Behavior, Gordon and Breach.
[4] Yoshida F., Uemori, T., Fujiwara, K. Elasticplastic behavior of steel sheets under inplane cyclic
tensioncompression at large strain, International Journal of Plasticity 18 (2002) 633659
[5] Boger RK, Wagoner RH, Barlat F, Fee MG, Chung K. Continuous, large stain,
tension/compression testing of sheet materials. International Journal of Plasticity 2005: 21:
23192343.
[6] Zang, S.F., Fee, M.G., Sun, F., Kim, J.H., Measurement of the Bauschinger behavior of sheet
metals by threepoint bending springback test with prestrained strips, International Journal of
Plasticity 59 (2014) 84107.
[7] Eggertsen, P.A., Mattiasson, K., on the modeling of the bendingunbending behavior for accurate
springback predictions, International Journal of Mechanical Sciences 51 (2009) 547563
[8] Duncan JF, Ding SC, Jiang WF. Momentcurvature measurement in thin sheetpart 1:
equipment. International Journal of Mechanical Sciences 1999: 41: 249260.
[9] Weiss, M., Wolfkamp, H., Rolfe, B.F., Hodgson, P.D., Hemmerich, E., Measurement of bending
properties in strip for roll forming, IDDRG 2009 International Conference, 13 June 2009,
Golden CO, USA.
[10] Hill, R., The mathematical theory of Plasticity, Oxford University Press, Fondon, 1950.
[11] Sanchez FR, Peterson S, Simonsen CG, Satar A. An interferometer based experimental
technique to evaluate large strains and spring back on sheet metal. Proceedings of the ASME
2013 International Mechanical Engineering Congress and Exposition 2013: 15.
[12] Caulk, D.A., Naghdi, P.M., On the hardening response in small deformation of metals, Journal
of Applied Mechanics, 45 (1978) 755764
[13] Chaboche, J.F., Constitutive equations for cyclic plasticity and cyclic viscoplasticity,
International Journal of Plasticity 5 (1989) 247302
[14] Yoshida F., Uemori, T., A model of large strain cyclic plasticity describing the Bauschinger
effect and workhardening stagnation, International Journal of Plasticity 18 (2002) 661686
32
[15] Yoshida F., Uemori, T., A model of large strain cyclic plasticity and its application to
springback simulation, International Journal of Mechanical Sciences 45 (2003) 16871702
[16] Eggertsen, P.A., Mattiasson, K., On constitutive modeling for springback analysis, International
Journal of Mechanical Sciences 52 (2010) 804818
[17] L.R.Sanchez, A Theoretical Experimental Method to Determine Bauschinger Effects on
Sheet Metal under Bending Reversals, Proceedings of the International Deep Drawing
Research Group, (IDDRG 2009) International Conference, 13 June 2009, Golden, CO, USA,
pp 373384
[18] Kassner,M.E., Geantil, P., L.E. Levine, L.E., Larson, B.C., Backstress, the Bauschinger Effect
and Cyclic Deformation, Materials Science Forum Vols. 604605 (2009) pp 3951 online at
http: //www. scientific .net
[19] A. Rosenburger, K.J. Weinmann, L.R. Sanchez "The Bauschinger Effect of Sheet Metal Under
Cyclic Reverse Pure Bending". The Annals of the College International pour LEtude de
Production Mecanique (CIRP). Japan 1988.
[20] Dadras, P., Majlessi, S. A. Plastic Bending of Work Hardening Materials, Transactions
of the ASME, vol 104 (1982) 224230
33
APPENDIX A
TENSILE TEST CURVE FITS FOR SAMPLE MATERIALS
DP780 Steel
True Strain (rx) []
DP780 Steel
Log of True Strain [Loge(Â£x)]
34
DP590 Steel
DP590 Steel
35
Log of True Stress [Loge(cx)] True Stress (cx) [psi]
NMF Aluminum
True Strain (rx) []
NMF Aluminum
Log of True Strain [Loge(Â£x)]
36
Log of True Stress [Loge(cx)] True Stress (cx) [psi]
AEDT Aluminum
True Strain (rx) []
AEDT Aluminum
9 8 7 6 5 4 3 2 1
Log of True Strain [Loge(Â£x)]
APPENDIX B
NUMERICAL MODEL (MATLAB PROGRAMING LANGUAGE)
Main Program
% Pure Bending Moment Theoretical Model
% Main Program
% towards Master of Science Thesis
% University of Colorado Denver
% Drew Hanzon
clear all
l g, g_ a. o_ g. g. <
t a. o. g. o. o. g. <
t g. o. g. g. g. g. <
f 5 9 SS S 5 5
5 0 O'S'O O 0 OT> O Q'O O O O'O'O O OTS'O 0 OT> 00'S O O O'O'O 0 O'^i'OO 0 OT> 00'S O O O'O'O 0 OTS'O Q 0 OT> 00'S o o o'o'o o oJS'oo 0 oT> oo
Constants and Simulation Parameters
iQOOOOQOOOOl
iQOOOOQOOOOl
iQOOOOQOOOOl
iQOOOOQOOOOl
CLElim = 0.05;
Tensile Testing Sample Geometry
to
wO
0.0412;
0.5;
% Sample Thickness [in.]
% Sample Width [in.]
% Pure Bending Moment Sample Geometry
tb = 0.0412;
wb = 1.530;
% Sample Thickness [in.]
% Sample Width [in.]
% Material Identifier
W = 3;
% Material Properties
str = 'DP590LB_Fracture.txt';
str2 = 'DP590LB_Moment.txt';
str3 = 'DP590LB_MomentRev.txt'
epsi = 0.00008;
epsy = 0.00090552;
epsi = 0.0025112;
epsp = 0.0095712;
epscf = 0.12;
epsRf = 0.102;
rA =2.1;
nu = 0.3;
zE = CLElim*(tb);
Rroe = tb/(2*(exp(0.1)1));
(Instron)
Sanchez Device)
Sanchez Device)
% Tensile Testing Raw Data
% Bending Moment Data (Dr.
r% Reverse Moment Data (Dr.
% Beginning of elastic modulus fit
% Yield Strain of Material
% Beginning of Intermediate Power Curve
% Beginning of Power Curve (True Strain)
% End of Power Curve
% Final true strain at outer most layer
% Anisotropy Factor []
% Possion Ratio []
[
[
[
[
[
[
% Radius of circle for loglog data fit
r = 1.45; % Circle radius
% Circle Radius
N = 300;
% Number of points in stressstrain curve
% Simulation Parameters
NL = 200;
NS = 100;
NR = 100;
% Number of Layers
% Number of Steps in First Bend Simulation
% Number of Steps in Second Bend Simulation
9999~S9S93S
9999~S999SSQ
9999~S9S93SQ
i Q 0 O O O
Determination of Material Properties from TT Data
% Functions to find Material Properties from Raw Tensile Testing Data
[epsQ sigQ Te Ts]
0
U
strl
[M SE]
Circ_Fit(str,tO,wO,epsi,epsy,epsI,epsp,epscf,r,N,W);
MatPropCurve(Te,Ts,epsi,epsy,epsp,epscf,epsl,1);
E=0(1);Y0=O(2);K=0(3);n=0(4);K2=0(5);n2=0(6);
find(str=='_' ) ;
str (1: (Ul));
MomentExp(str2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%% Initialization of Simulation Objects and Variables %%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Instantiates object for each layer of simulation
SigY = E*epsy^Y0
tL = tb/NL; %
tl = 0;
for i = 1:NL
tl
NA
zT
z =
L (i)
L (i) t (1) =
end
Thickness of each layer
tl + tL;
tb/2 ;
tl NA;
zT (tL/2);
Layer(z,tL,NS,NR,SigY);
tL;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%% Simulation of First Bend %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Construction of Curvature Radius Vector
Rmax = le5;
Rmin = (tb/2)/epsRf;
R = logspace(loglO(Rmax),loglO(Rmin),NS);
Momentl = zeros(1,NS);
Max = length(epsQ);
for j = 1:NS
for i = 1:NL
L(i) = L(i).Strain(epsy,R(j),nu,j);
L(i) = L(i).Stress(epsQ,sigQ,epsy,rA,Max,K, n, j);
Momentl(j) = Momentl(j) + L(i).sigl(j)*L(i).z*L(i).tL;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%% Determination of Residual Stresses/Strain %%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Cl = 12*Momentl(end)/tbA3;
for i = 1:NL
L(i) = L(i).ResStress(Cl);
L(i) = L(i).ResStrain(E);
end
39
9999999999
9999 9999999
9999 9999999
Reverse StressStrain Optimization
^^^^^^^^^^^^^^^^^^999999999999999999999999999999999999^^^^^^^^^^^^^^^^^^
ooooo'oooooooooooooooo'oooooooooooooooo'oooooooooooooooo'oooooooooooooooo'ooo
lOOOOOOOOOO^
19999999999':
lOOOOOOOOOO^
19999999999'
[MoR StrR]
Rmax
RO
MomentO
Max
for j = 1:
for i
= MomentExp(str3);
= 50;
= logspace(loglO(Rmax),loglO(Rmin0.01) NR) ;
= zeros(1,NR);
= length(epsQ);
NR
= 1: NL
L (i)
L (i)
MomentO(j)
end
L(i).OptStrain(RO(j),j ) ;
L(i) .OptStress(epsQ,sigQ,rA,Max, K, n,j);
MomentO(j) + L(i) .sigO(j)*L(i) .z*L (i) .tL;
end
epsO = L(end).epsO L(end).epsO(l);
ERR = Test_Moment(StrR,MoR,epsO,MomentO,wb);
% Optimization
rR = r ; K2R = K2; n2R = n2 ;
Con = 10; Cmax = 30; cnt = 0;
negR = 1; negK = 1; negn = 1;
ERRR = ERR; ERRK = ERR; ERRn = ERR;
step = 0.84 ; stpl = step;
while ERR > Con && cnt < Cmax && step > le10
g, o Radius Test
rRl = rR+negR*step*rR;
[epsO sigO] = Test_Curve(E,Y0,K,n,K2R,n2R,rRl,epscf,StrR(1),N);
MomentO = zeros(1,NR);
for j = 1:NR
for i = 1:NL
L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j);
MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL;
end
end
epsLO = L(end).epsO L(end) .epsO (1);
ERRT = Test_Moment(StrR,MoR,epsLO,MomentO,wb);
if ERRT > ERRR
negR = negR;
rRl = rR+negR*step*rR;
end
ERRR = ERRT;
% Intermediate Coefficient Test
K2R1 = K2R+negK*step*K2R;
[epsO sigO] = Test_Curve(E,Y0,K,n,K2R1,n2R,rR,epscf,StrR(1), N);
MomentO = zeros(1,NR);
for j = 1:NR
for i = 1:NL
L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j);
MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL;
end
end
epsLO = L(end).epsO L(end) .epsO (1);
ERRT = Test_Moment(StrR,MoR,epsLO,MomentO,wb);
if ERRT > ERRK
negK = negK;
K2R1 = K2R+negK*step*K2R;
end
ERRK = ERRT;
% Intermediate Exponent Test
n2Rl = n2R+negn*step*n2R;
[epsO sigO] = Test_Curve(E,Y0,K,n,K2R,n2Rl,rR,epscf,StrR(1),N);
40
dp c\o op
MomentO = zeros(1,NR);
for j = 1:NR
for i = 1:NL
L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j);
MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL;
end
end
epsLO = L(end).epsO L(end) .epsO (1);
ERRT = Test_Moment(StrR,MoR,epsLO,MomentO,wb);
if ERRT > ERRn
negn = negn;
n2Rl = n2R+negn*step*n2R;
end
ERRn = ERRT;
% New Stress Strain Curve
[epsO sigO] = Test_Curve(E,YO,K,n,K2R1,n2Rl,rRl,epscf,StrR(1),N)
MomentO = zeros(1,NR);
for j = 1:NR
for i = 1:NL
L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j);
MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL;
end
end
epsLO = L(end).epsO L(end) .epsO (1);
ERRV = Test_Moment(StrR,MoR,epsLO,MomentO,wb);
if ERRV < ERR
rR = rRl; K2R = K2R1; n2R = n2Rl;
ERR = ERRV
end
ERR
ERRV
step = stpl (cnt/Cmax)*stpl
cnt = cnt+1
clear figure(1)
figure(1)
plot(epsLO,wb*MomentO,'r','LineWidth',1.3)
hold on
plot(StrR,MoR,'*k')
title ( Reverse Bend Optimazation ')
xlabel(' Reverse Strain [ ] ')
ylabel(' Moment [ lbfin ] ')
drawnow
hold off
end
Q = Circ_Opt (E, YO, K, n, K2R,n2R, rR) ;
xC=Q(l); yC=Q(2); epsyR=Q(3); eps2R=Q(4);
epspR=Q(5); thl=Q(6); th2=Q(7);
% Elastic Curve
epsE = linspace(Te(1),epsyR,N);
sigE = zeros(1,N);
for i = 1:N
sigE(i) = E*epsE (i) ''YO;
end
% Power Curve 1
epsP = linspace(epspR,epscf,N);
sigP = zeros(1,N);
for i = 1:N
sigP(i) = K*epsP(i)^n;
end
% Intermediate Power Curve
eps2 = linspace(eps2R,epspR,N);
sig2 = zeros(1,N);
for i = 1:N
sig2(i) = K2R*eps2(i)^n2R;
end
% Circle Fit
th = linspace(thl+pi,th2+pi,N);
Fx = zeros(1,N);
Fy = zeros(1,N);
for i = 1:N
Fx(i) = rR*cos(th(i))+xC;
Fy(i) = rR* sin(th(i))+yC;
end
if epspR > epscf
% Intermediate Power Curve
eps2 = linspace(eps2R,epspR,ceil(epspR)*N);
sig2 = zeros(1,ceil (epspR)*N);
for i = 1:N
sig2 (i) = K2R*eps2 (i) An2R;
end
epsP = eps2(end);
sigP = sig2(end);
end
epsO = [epsE exp(Fx) eps2 epsP]
sigO = [sigE exp(Fy) sig2 sigP]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%% Reverse Bend Simulation %%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Rmax = 50;
Moment2 = zeros (1,NR);
for j = 1:NR
for i = 1:NL
L(i) = L(i) .RevStrain(R(j) j) ;
L(i) = L(i).RevStress(epsO,sigO,rA,Max,K,n,j);
Moment2(j) = Moment2(j) + L(i).sigR(j)*L(i).z*L(i).tL;
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%% Code for Plotting and Visualization %%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
[M SE] = MomentExp(str2);
Z = zeros(1,NL);
epsr = zeros(1,NL);
MomentNA = 0;
MomentComp = 0;
for i = 1:NL
if L(i).z < zE && L(i).z > 0
L(i) = L(i).StressNA(epsO,sigO,epsy,r,Rroe,Max,K,n);
else
L(i) = L(i).StressNA(epsQ,sigQ,epsy,r,Rroe,Max,K,n);
end
MomentNA = MomentNA + L(i).sigNA*L(i).z*L(i).tL;
end
42
1: NL
for i =
L(i) = L(i).StressNA(epsQ,sigQ,epsy,r,Rroe,Max,K,n);
MomentComp = MomentComp + L(i).sigNA*L(i).z*L(i).tL;
end
for i = 1:NL
epsr(i) = L(i).epsFB;
Z(i) = L(i).z;
end
th = linspace(0,2*pi,N);
Cx = zeros(1,N);
Cy = zeros(1,N);
for i = 1:N
Cx(i) = rR*cos(th(i))+xC;
Cy(i) = rR*sin(th(i))+yC;
end
epsT = logspace(loglO(Te(1)),loglO(epscf),3*N);
xl = log(Te);
yl = log(Ts);
x2 = log(eps2);
y2 = log(sig2);
xE = log(epsE);
yE = log(sigE);
xP = log(epsP);
yP = log(sigP);
figure(1)
plot(Te,Ts, '* k', 'MarkerSize',4)
hold on
plot(epsE,sigE,'r','LineWidth',1.2)
plot(exp(Fx),exp(Fy), ' g', 'LineWidth',1.2)
plot(eps2,sig2, ' y', 'LineWidth',1.2)
plot (epsP,sigP, 'm', 'LineWidth',1.2)
hold off
xlabel(' Strain ( \epsilon ) ')
ylabel(' Stress ( \sigma ) ')
str2 = sprintf(' %s: Tensile Testing Plot ', strl);
legend ('Experimental Data','Elastic Fit','Circle Fit',...
'First Power Fit','Second Power Fit')
title (str2)
grid
figure(2)
loglog(Te,Ts, '* k', 'MarkerSize',4)
hold on
loglog(epsE,sigE,'r','LineWidth',1.2)
loglog(exp(Fx),exp(Fy),'m','LineWidth',1.2)
loglog(eps2,sig2, ' g', 'LineWidth',1.2)
loglog(epsP,sigP, ' y', 'LineWidth',1.2)
hold off
xlabel(' True Strain ( \epsilon ) [] ')
ylabel(' True Stress ( \sigma ) [psi] ')
str2 = sprintf(' %s: LogLog Plot ', strl);
legend ('Experimental Data','Elastic Fit','Circle Fit',...
'First Power Fit','Second Power Fit')
title (str2)
grid
43
figure(3)
plot (xl,yl, '*k', 'MarkerSize' 4)
hold on
plot(xE,yE,'r','LineWidth',1.2)
plot(x2,y2, ' g', 'LineWidth',1.2)
plot (xP,yP, 'y', 'LineWidth',1.2)
plot (Cx, Cy, in' LineWidth' ,1.2)
hold off
axis egual
xlabel(' Log of Strain In( \epsilon ) ')
ylabel(' Log of Stress In( \sigma ) ')
str2 = sprintf(' %s: Tensile Testing LogLog Plot ', strl);
legend ('Experimental Data','Elastic Fit', 'First Power Fit',...
'Second Power Fit','Circle Fit')
title (str2)
grid
44
Class Definition for Layer Objects
classdef Layer
properties
z
tL
sigm
sigr
epsFB
sigl
epsl
sigR
epsR
sigO
epsO
R
end
Location relative to natural axis
Original Thickness of Layer
Maximum Flow Stress
Residual Stress after First Bend
Resultant Plastic Strain after First Bend
First Bend Stress (Transverse with Bend)
First Bend Strain (Transverse with Bend)
First Bend Stress (Transverse with Bend)
Reverse Bend Strain (Transverse with Bend)
Optimization Bend Stress (Transverse with Bend)
Optimization Bend Strain (Transverse with Bend)
Residual strain of layer after first bend
methods
% Constructor Method
function obj = Layer(z,t,NS,NR,SigY)
obj sigm = SigY;
obj sigr = 0;
obj epsFB = 0;
obj z = z;
obj tL = t;
obj t = zeros(1 , NS) ;
obj sigl = zeros (1 , NS) ;
obj epsl = zeros(1 , NS) ;
obj sigR = zeros (1 , NS) ;
obj epsR = zeros(1 , NS)
obj sigO = zeros(1, NR) ;
obj epsO = zeros (1, NR) ;
obj .R = 0;
end
% Method for Calculating First Bend Layer Strain
function obj = Strain(obj,roe,i)
if obj.z < 0
neg = 1;
else
neg = 1;
end
obj.epsl(i) = neg*log (1+abs (obj.z)/roe);
end
function obj = Stress(obj eps,sig,epsy,r,Max,K,n,i)
RF = ((1+r)/sqrt(l+2*r));
C = (2/sqrt(3));
epsC = abs(obj.epsl (i));
if obj.epsl(i) <0
neg = 1;
else
[in.]
[in.]
[psi]
[psi]
[]
[psi]
[]
[psi]
[]
[psi]
[]
[]
neg = 1;
end
if epsC >= epsy
epsC = C*abs (obj.epsl(i));
end
LS = find(eps <= epsC,1,'last');
if epsC <= eps (1)
xO = 0;
xl = eps(1);
yO = 0;
yl = sig(1);
obj.sigl(i) = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
45
elseif LS > Max1
obj.sigl(i) = neg*RF*K*epsC/'n;
else
xO = eps(LS);
xl = eps (LS+1);
yO = sig(LS);
yl = sig(LS+1);
obj.sigl(i) = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
end
if abs(obj.sigl(i)) >= abs(obj.sigm)
obj.sigm = obj.sigl(i);
end
end
% Method for Calculating Residual Stress
function obj = ResStress(obj,Coef)
obj.sigr = obj.sigmCoef*obj.z;
end
% Calculation of Plastic Strain after Release of First Bend Moment
function obj = ResStrain(obj,E)
obj.epsFB = obj.sigr/E;
obj.R = obj.epsl(end)(obj.sigl(end)/E);
end
% Method for Calculating Strain on Reverse Bend
function obj = RevStrain(obj,roe, i)
if obj.z < 0
neg = 1;
else
neg = 1;
end
obj.epsR(i) = obj.epsFBneg*log(1+abs(obj.z)/roe);
end
function obj = RevStress(obj,eps,sig,r,Max,K,n,i)
RF = ( (1+r)/sqrt(l+2*r));
epsC = abs(obj.epsR(i));
C = (2/sqrt(3));
if obj.epsR(i) < 0
neg = 1;
else
neg = 1;
end
epsC = (C*epsC);
LS = find(eps <= epsC,1,'last');
if epsC <= eps (1)
xO = 0;
xl = eps(1);
yO = 0;
yl = sig(1);
obj.sigR(i) = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
elseif LS > Max1
obj.sigR(i) = neg*RF*K*epsC/sn;
else
xO = eps(LS);
xl = eps (LS + 1);
yO = sig(LS);
yl = sig(LS+1);
obj.sigR(i) = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
end
end
% Method for Calculating Strain for Optimization Routine
function obj = OptStrain(obj,roe,i)
if obj.z < 0
neg = 1;
else
46
neg = 1;
end
obj.epsO(i) = obj.epsFBneg*log(1+abs(obj.z)/roe);
end
function obj = OptStress(obj,eps,sig,r,Max,K,n,i)
RF = ((1+r)/sqrt(l+2*r));
epsC = abs(obj.epsO(i));
C = (2/sqrt(3));
if obj.epsO(i) < 0
neg = 1;
else
neg = 1;
end
epsC = (C*epsC);
LS = find(eps <= epsC,1,'last');
if epsC <= eps (1)
xO = 0;
xl = eps(1);
yO = 0;
yl = sig(1);
obj.sigO(i) = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
elseif LS > Max1
obj.sigO(i) = neg*RF*K*epsCAn;
else
xO = eps(LS);
xl = eps (LS+1);
yO = sig(LS);
yl = sig(LS+1);
obj.sigO(i) = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
end
end
function obj = StressNA(obj,eps,sig,epsy,r,roe,Max,K,n)
if obj.z < 0
neg = 1;
else
neg = 1;
end
obj.epsNA = neg*log(1+abs(obj.z)/roe);
RF = ((1+r)/sqrt(l+2*r));
C = (2/sqrt(3));
epsC = abs(obj.epsNA);
if epsC >= epsy
epsC = C*abs(obj.epsNA);
end
LS = find(eps <= epsC,1,'last');
if epsC <= eps (1)
xO = 0;
xl = eps(1);
yO = 0;
yl = sig(1);
obj.sigNA = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
elseif LS > Max1
obj.sigNA = neg*RF*K*epsCAn;
else
xO = eps(LS);
xl = eps (LS + 1);
yO = sig(LS);
yl = sig(LS+1);
obj.sigNA = neg*RF*(y0+(ylyO)*((epsCxO)/(xlxO)));
end
end
end
end
47
Function for Equation Fitting of Tensile Testing Data
function [epsQ sigQ Te Ts] = Circ_Fit(str,tO,wO,epsi,epsy,epsl,epsp,epscf,r,N,W)
U = find(str=='_');
strl = str(1:(Ul));
[eta sig]
J
[eps sig Te Ts]
0
RawData(str,tO,wO);
MatPropElastic(epsi,epsy,eta,sig);
True_S_S(J(l),eta,sig);
MatPropCurve(Te,Ts,epsi,epsy,epsp,epscf,epsi,1);
E=0(1);Y0=O(2);K=0(3);n=0(4);K2=0(5);n2=0(6);
thPl = atan(1);
thP2 = atan((l/n2));
xEO = log(epsy)+r*cos(thPl);
yEO = log (E*epsy/'Y0) +r*sin (thPl) ;
x20 = log(epsp)+r*cos(thP2);
y20 = log(K2*epsp/'n2)+r*sin (thP2)
bl = yE0l*xE0;
b2 = y2 0n2 *x20;
xC = (b2bl)/(ln2);
yC = n2 *xC+b2;
xEl = log(epsy);
yEl = log (E*epsy/'Y0) ;
x21 = log(epsp);
y21 = log (K2*epsp/'n2) ;
blT = yEll*xEl;
b2T = y21n2*x21;
bPE = yC(1)*xC;
bPI = yC((l/n2))*xC;
xPE = (blTbPE)/ ( (l)l);
xPI = (b2TbPI)/(((l/n2))n2);
xPP = (log(K2)log(K))/(nn2) ;
% Elastic Curve
epsE = linspace(Te(1),exp(xPE),N);
sigE = zeros(1,N);
for i = 1:N
sigE(i) = E*epsE(i)AY0;
end
% Power Curve 1
epsP = linspace(exp(xPP),epscf,N);
sigP = zeros(1,N);
for i = 1:N
sigP(i) = K*epsP(i)^n;
end
% Power Curve 2
eps2 = linspace(exp(xPI),exp(xPP),N);
sig2 = zeros(1,N);
for i = 1:N
sig2(i) = K2*eps2(i)An2;
end
xl = log(Te);
yi = log(Ts);
x2 = log(eps2)
Y2 = log(sig2)
xE = log(epsE)
48
yE
xP
yP
= log(sigE);
= log(epsP);
= log(sigP);
th = linspace(0,2*pi,N);
Cx = zeros(1,N);
Cy = zeros(1,N);
for i = 1:N
Cx(i) = r*cos(th(i))+xC;
Cy(i) = r*sin(th(i))+yC;
end
thl = atan ( (yE (end)yC) / (xE (end)xC) ) ;
th2 = atan((y2(1)yC)/(x2(1)xC));
th = linspace(thl+pi,th2+pi,N);
Fx = zeros(1,N);
Fy = zeros(1,N);
for i = 1:N
Fx(i) = r*cos(th(i))+xC;
Fy(i) = r*sin(th(i))+yC;
end
epsQ = [epsE exp(Fx) eps2 epsP]
sigQ = [sigE exp(Fy) sig2 sigP]
Exl = log(epsE(1));
Ex2 = log(epsE(end));
Eyl = log(0.006894759086775369*sigE(l));
Ey2 = log(0.006894759086775369*sigE(end));
Cxi = Fx(1) ;
Cx2 = Fx(end);
Cyl = log(0.006894759086775369*exp(Fy(1)));
Cy2 = log(0.006894759086775369*exp(Fy(end)));
Ixl = log(eps2(1));
1x2 = log(eps2(end));
Iyl = log(0.006894759086775369*sig2(1));
Iy2 = log(0.006894759086775369*sig2(end));
Pxl = log(epsP(1));
Px2 = log(epsP(end));
Pyl = log(0.006894759086775369*sigP(1));
Py2 = log(0.006894759086775369*sigP(end));
EXFile = 'PB_Graphs.xlsx';
B = [Exl Eyl Ex2 Ey2 Pxl Pyl Px2 Py2];
C = [Cxi Cyl Cx2 Cy2 Ixl Iyl 1x2 Iy2];
if I == 1
xlswrite(EXFile,B,2, 'C 4 4')
xlswrite(EXFile,C,2,'C54')
elseif W == 2
xlswrite(EXFile,B,2,'C45')
xlswrite(EXFile,C,2,'C55')
elseif W == 3
xlswrite(EXFile,B,2,'C42')
xlswrite(EXFile,C,2,'C52')
elseif W == 4
xlswrite(EXFile,B,2,'C43')
xlswrite(EXFile,C,2,'C53')
end
end
49
Function for Reading Raw Tensile Testing Data
% Pure Bending Moment Theoretical Model
% Function Call to Find Material Properties from Instron Raw Data
% Towards Master of Science Thesis
% University of Colorado Denver
% Drew Hanzon
function [eta sig] = RawData(str,tO,wO)
A = wO*tO; % Cross Sectional Area [in.^2]
d = importdata(str);
sig = zeros((length(d)/2),1);
eta = zeros((length(d)/2),1);
for i = 1:(length(d)/2)
j = i 2 1 ;
sig(i) = d(j)/A;
eta (i) = d (j +1) ;
end
end
Function to Determine Material Properties
function 0 = MatPropCurve(Teps,Tsig,epsi,epsy,epsp,epscf,epsl,sel)
z = epsi;
LS = find(Teps <= z,l,'last');
z = epsy;
LE = find(Teps <= z,l,'last');
z = epsp;
PS = find(Teps <= z,l,'last');
z = epscf;
PE = find(Teps <= z,l,'last');
z = epsi;
IS = find(Teps <= z,l,'last');
% Linear Region Fit
SumX=0; SumY=0; SumXY=0; SumX2=0; N = LELS+1;
for i = LS:LE
Tempeta = Teps(i);
SumX = SumX +log(Tempeta);
SumY = SumY +log(Tsig(i));
SumXY = SumXY+log(Tempeta)*log(Tsig(i));
SumX2 = SumX2+log(Tempeta)^2;
end
YO = (SumX*SumYN*SumXY) / (SumX/s2N*SumX2 ) ;
E = exp ( (SumX*SumXYSumX2*SumY) / (SumX/s2N*SumX2 ) ) ;
% Power Curve Fit
SumXLN=0; SumYLN=0; SumXYLN=0; SumX2LN=0; N = PEPS+1;
for i = PS:PE
Tempeta = Teps(i);
SumXLN = SumXLN +log(Tempeta);
SumYLN = SumYLN +log(Tsig(i));
SumXYLN = SumXYLN+log(Tempeta)*log(Tsig(i));
SumX2LN = SumX2LN+log(Tempeta)^2;
end
m = (SumXLN*SumYLNN*SumXYLN)/(SumXLFT 2N*SumX2LN) ;
Ktemp = (SumXLN*SumXYLNSumX2LN*SumYLN)/(SumXLN/'2N*SumX2LN);
K = exp(Ktemp);
50
% Intermediate Curve Fit
SumXLN=0; SumYLN=0; SumXYLN=0; SumX2LN=0; N = PSIS+1;
for i = IS:PS
Tempeta = Teps(i);
SumXLN = SumXLN +log(Tempeta);
SumYLN = SumYLN +log(Tsig(i));
SumXYLN = SumXYLN+log(Tempeta)*log(Tsig(i)) ;
SumX2LN = SumX2LN+log(Tempeta)A2;
end
ml
Ktempl
K1
0
(SumXLN*SumYLNN*SumXYLN)/(SumXLNA2N*SumX2LN);
(SumXLN*SumXYLNSumX2LN*SumYLN)/(SumXLNA2N*SumX2LN);
exp(Ktempl);
[ E YO K m K1 ml ] ;
end
Function for Importing Pure Bending Moment Device Data
function [MomentExp StrainExp] = MomentExp(str2)
d = importdata(str2);
MomentExp = zeros((length(d)/2),1);
StrainExp = zeros((length(d)/2),1);
for i = 1:(length(d)/2)
j = i 2 1 ;
MomentExp(i) = d(j);
StrainExp(i) = d(j+l);
end
end
Function for Converting Tensile Testing Data to True Stress True Strain
% Pure Bending Moment Theoretical Model
% Function Call to Find Material Properties from Instron Raw Data
% Towards Master of Science Thesis
% University of Colorado Denver
% Drew Hanzon
function [eps sig Teps Tsig] = True_S_S(ET,eta,sig)
N = length(eta);
% Account for initial Stress in Strain Vector
eps = zeros(1,N);
eps(l) = sig(l)/ET;
for i = 2:N
eps(i) = eps(i1)+(eta(i)eta(i1));
end
Teps = zeros(1,N);
Tsig = zeros (1,N);
for i = 1:N
Teps(i) = log(1+eps(i));
Tsig(i) = sig(i)*exp(Teps(i));
end
end
51
Error Calculation for Reverse Bend Optimization
function [Sum] = Test_Moment(StrR,MoR,epsO,MomentO,wb)
Sum = 0;
M2 = zeros(1,length(StrR));
for i = 1:length(StrR)
LS = find(epsO >= StrR(i),1last');
xO = epsO (LS);
xl = epsO(LS+1);
yO = MomentO(LS);
yl = MomentO(LS+1);
M2(i) = wb*(y0+(ylyO)*((StrR(i)xO)/(xlxO)));
Sum = Sum + abs(MoR(i)M2(i));
end
end
Function to Generate the Constitutive Relationship from Iterative Parameters
function [epsO sigO] = Test_Curve(E,YO,K, n, K2,n2, r, epscf, Tel, N)
Q = Circ_Opt(E,YO,K,n,K2,n2 r) ;
xC=Q(l); yC=Q(2); epsyR=Q(3); eps2R=Q(4);
epspR=Q(5); thl=Q(6); th2=Q(7);
% Elastic Curve
epsE = linspace(Tel,epsyR,N);
sigE = zeros(1,N);
for i = 1:N
sigE(i) = E*epsE(i)"Y0;
end
% Power Curve 1
epsP = linspace(epspR,epscf,N);
sigP = zeros(1,N);
for i = 1:N
sigP(i) = K*epsP(i)An;
end
% Intermediate Power Curve
eps2 = linspace(eps2R,epspR,N);
sig2 = zeros(1,N);
for i = 1:N
sig2(i) = K2*eps2(i)"n2;
end
% Circle Fit
th = linspace(thl+pi,th2+pi,N);
Fx = zeros(1,N);
Fy = zeros(1,N);
for i = 1:N
Fx(i) = r*cos(th(i))+xC;
Fy(i) = r*sin(th(i))+yC;
end
if epspR > epscf
eps2 = linspace(eps2R,epspR,(2*N1));
sig2 = zeros(1,(2*N1));
for i = 1:(2*N1)
sig2(i) = K2*eps2(i)^n2;
end
epsP = eps2(end);
sigP = sig2(end);
end
epsO = [epsE exp(Fx) eps2 epsP]
sigO = [sigE exp(Fy) sig2 sigP]
end
52
Function for Circle Fit in Optimization Method
function Q
epsy =
epsp =
thPl =
thP2 =
xEO =
yEO =
x20 =
y20 =
bl
b2
xC
yC =
= Circ_Opt(E,YO,K, n, K2 n2 r)
0.001;
0.003;
atan(1);
atan((l/n2));
log(epsy)+r*cos(thPl);
log(E*epsyAY0)+r*sin(thPl) ;
log(epsp)+r*cos(thP2);
log (K2*epspAn2 ) +r*sin (thP2 ) ;
yE0l*xE0;
y2 0n2 *x20;
(b2bl)/(ln2);
n2 *xC+b2;
xEl = log(epsy);
yEl = log (E*epsy/'Y0) ;
x21 = log(epsp);
y21 = log(K2*epspAn2);
blT = yEll*xEl;
b2T = y21n2*x21;
bPE = yC(1)*xC;
bPI = yC((l/n2))*xC;
xPE = (blTbPE)/ ( (1)1) ;
xPI = (b2TbPI)/(((l/n2))n2);
xPP = (log(K2)log(K))/(nn2);
thl = atan((log(E*exp(xPE)AY0)yC)/(xPExC));
th2 = atan((log(K2*exp(xPI)An2)yC)/(xPIxC));
epsy = exp(xPE);
epsp2 = exp(xPI);
epsp = exp(xPP);
Q = [xC yC epsy epsp2 epsp thl th2];
end
53
Appendix C
PLANE STRAIN PURE BENDING MODEL
Developed by Luis Rafael Sanchez Vega
The pure bending model follows a development similar to the model published by P. Dadras
and S.A. Majlessi [20] with the differences below:
1. In this model the neutral plane was located to satisfy j"
neutral plane was determined from continuity in the radial stresses.
2. This model accounts for strain reversals (due to neutral plane movement) through an
approximate procedure based on experimental data. Ref [20] defines arbitrary
Bauschinger curves for completeness of the analytical model.
3. In this model, continuity of the radial stresses are not automatically satisfied, and are
used as a criteria to assess the adequacy of the approximate Bauschinger curve.
4. This model includes through thickness anisotropy. Ref. [20] considers isotropic
conditions.
Nomenclature similar to Ref. [20] was used, with Zones I (R> Rcj, and III (R Rnj under
monotonic tension and compression, respectively. Layers in Zone II (Rn
strain reversals. The neutral plane shifts during loading from the original midplane to Rn.
54
Under plane strain,
lir f7= rr^r^0 \j\ + 2r (1)
With <7 defined by the tensile test fits:
Arc fit AB: Rr~ (lnÂ£lnÂ£0)2+ln
Intermediate fit BC: aBC = KBcÂ£nBC (2.2)
Large strains fit CD: aBC = KcDÂ£7lCD (2.3)
Equilibrium under plane strain gives: d
Solving: da = ^l + 2r dR r 1 + r R (4)
For Zones I and III, ar can be integrated from (4), with a given by the corresponding eq.
(2), and strains s expressed as function of R:
s =
1 + r
y/l + 2 r
(5)
Ze
(6)
Where Rs is the radius for the segment at the original length.
Layers in Zone II reverse continuously as the neutral radius shifts to its minimum at Rn. The
range is given by the layer originally at the midsection (sq =0) which displaces toRq and
by the layer compressed to Rn, with sq =ln() an instant before reversal. The strain
RS
reversals range is:
55
(7)
0 = ln(~") ^ s0rev < ln(^)
Kn KS
And the strains in the corresponding layers an instant before reversal are:
lnA<% Sln(^) = 0
Ks Ks
Eqs. (7), (8) define the location of the strain reversals for each layer in Zone II. The a values
corresponding to the reversal can be obtained by interpolations from the proposed reverse
compression curves.
56

Full Text 
PAGE 1
CHARACTERIZATION OF THE BAUSCHINGER EFFECT IN SHEET METAL UNDERGOING LARGE STRAIN REVERSALS IN BENDING by DREW WYATT HANZON B.S., University of Colorado Denver, 2013 A thesis submitted to the Faculty of the Graduate School of the University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science Mechanical Engineering Program 2015
PAGE 2
ii This thesis for the Master of Science degree by Drew Wyatt Hanzon has been approved for the Mechanical Enginee ring Program by Luis Rafael Sanchez Vega, Chair Peter Hoffman Dana Carpenter November 20 2015
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iii Hanzon, Drew Wyatt ( MS Mechanical Engineering Program ) Characterization of the Bauschinger effect in Sheet Metal undergoing Large Strain Reversals in B ending Thesis directed by Associate Professor Luis Rafael Sanchez Vega ABSTRACT This work consists on the quantification of sheet metal uniaxial stress strain reversals from pure bending tests. Bending strains to approximately 10% were measu red by strain gages and interferometry. Bending unbending moments and strains were modeled and compared closely to the experimental data. The reverse uniaxial stress strains curves were determined from the optimal fit of the model. Bauschinger effects were described by the reverse uniaxial response at the elasto plastic range, between the elastic and the large strain, power fit ranges. Arc and straight line fittings on the ln ln scale p roved accurate to describe the e lasto plastic behavior. Reverse uniaxial data determined for DP590 and DP780 steels and two Aluminum alloys showed significant Bauschinger effects with distinct features. For the DP steels the magnitudes of the reverse compressive curves compared moderately higher, and merging to a power curve with parameters K, n previously defined by tension testing. Bauschinger effects at small reversed strains were less pronounced for the aluminum alloys. However, at high e r strains the reverse e lasto plastic response softened considerably, and during the unbending span the magnitudes of the reverse compressive strains remained below the corresponding K, n tensile values. The results showed pure bending as an efficient, sim ple to use technique to generate data for sheet metal at large reverse strains without the complicating restraining hardware required by direct compression methods. The form and content of this abstract are approved. I recommend its publication. Approved: Luis Rafael Sanchez Vega
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iv TABLE OF CONTENTS CHAPTER I. I NTRODUCTION ................................ ................................ ................................ ......................... 1 II. LITERATURE REVIEW ................................ ................................ ................................ .............. 3 III. EXPERIMENTAL METHODOLOGY ................................ ................................ ....................... 6 Pure Bending Moment Device ................................ ................................ ................................ 7 Strain Measurement using Interferometry ................................ ................................ ................ 8 IV. CHACTERIZATION OF MATERIAL PROPERTIES ................................ ............................. 10 Log Log Analysis of True Stress True Strain ................................ ................................ ...... 11 Parameterization of the Elasto Plastic Region ................................ ................................ ........ 1 2 V. P URE BENDING MOMENT MODEL ................................ ................................ .................... 14 VI. REVERSE BEND OPTIMIZATION ................................ ................................ .......................... 18 VII. EXPERIMENTAL AND NUMERICAL MODEL RESULTS ................................ .................. 21 VII I DISCUSSION ................................ ................................ ................................ .............................. 2 6 Dual Phase Steel Samples ................................ ................................ ................................ ....... 2 6 Alum inum Alloy Samples ................................ ................................ ................................ ....... 2 8 IX CONCLUSIONS ................................ ................................ ................................ ......................... 3 1 REFERENCES ................................ ................................ ................................ ................................ ..... 3 2 APPENDIX A. TENSILE TEST CURVE FITS FOR SAMPLE MATERIALS ........... B. NUMERICAL MODEL (MATLAB PROGRAMING LANGUAGE 38 C. PLANE STRAIN PURE BENDING MODEL 4
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v LIST OF TABLES Table 1: Uniaxial tensile properties for test materials ................................ ................................ ..................... 6 2: Descriptions and relationships for labeled regions in Figure 4. ................................ ...................... 10 3: Tensile test fit parame ters ................................ ................................ ................................ ................ 21 4: Tensile te ................................ .......... 2 1 ................................ ................................ ............................... 25
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vi LIST OF FIGURES Figure 1: Reverse strain behavior showing the Bauchinger effect ................................ ................................ .... 2 2: Description of Dr. Sanchez pure bending moment device ................................ ................................ 8 3: Interf erometry measurements ................................ ................................ ................................ ........... 9 4: Cyclic loading behavior ................................ ................................ ................................ ................... 10 5: Tensile Test ln ln for DP590 steel ................................ ................................ ............................... 11 6: ln ln fits for DP590 ................................ ................................ ................................ ...................... 12 7: True Stress True Strain fit for DP590 Steel ................................ ................................ .................... 13 8: Pure bending moment free body d iagram used for reverse bend optimization ............................... 15 9 Flow diagram for random search optimization p rogram . 10: Comparison of final first bend stress profiles with m odels I and II .... 20 11 : DP590 steel pure moment bending/ unbending curve ................................ ................................ ... 23 1 2 : DP780 steel pure moment bending/unbending curve ................................ ................................ .... 23 1 3 : Aluminum Alloy 1. P ure moment bending/unbending curve ................................ ........................ 24 1 4 : Aluminum Alloy 2 pure moment bending/unbending curve ................................ ......................... 24 1 5 on for DP590 Steel. ................................ .... 27 1 6 n for DP780 Steel ................................ ..... 27 1 7 reverse best fit compression for Aluminum Alloy 1 .......................... 28 1 8 um Alloy 2 ........................... 29 19 : Model with Isotropic Hardening and Bauschinger F actor ................................ .............................. 30
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vii NOMENCLATURE Strain [ ] Stress [M p a] Poisson ratio [ ] R C Arc fit radius [ ] n Power curve exponent [ ] K Power curve coefficient [M p a] r 0 Lankford coefficient [ ] E Elastic modulus [M p a] h Thickness of sheet [m] M Bending moment [N m] w Width specimen [m] k Bend curvature [m 1 ] z Distance from neutral axis [m] S Stress State [M p a] s Standard deviation [ ] p Plastic Strain [ ] c Strain (end elastic fit) [ ] k Strain (begin power curve) [ ] a Strain (end arc fit) [ ] f Strain (end power curve ) [ ] ERR Error (optimization algorithm) [ ] BF Bauschinger factor [ ] CON Convergence criterion [ ] UTS Ultimate tensile strength [M p a] UE Elongation at UTS [%] TE Total tensile elongation [%] Subscripts i Bend simulation step Bend model layer p Plastic deformation y Material yield point 1,2,3 Orthogonal principle directions I Intermediate powe r curve parameter R Reverse loading f Plastic Flow t Temporary model parameter b Bend test specimen outer Surface layer of model/specimen d Tensile test specimen Superscripts e Elastic Range p Plastic range Reverse bend property
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1 CHAPTER I INTRODUCTION Since the original experiments conducted by Bauschinger [1 ], there has been ongoing research to explain departures from isotropic hardening behavior due to plastic strain reversals Several mathematical models have been proposed for predicti ng material properties of sheet metals undergoing small strain reversals fo r which experimental data was obtained using uniaxial and bending tests H owever the generation of large strain data for sheet metal under compression offers experimental challen ges. The tendency of sheet metal to buckle and wrink l e during compressive loading and friction at the interface between sheet and tools are major contributors to the experimental uncertainty. These difficulties can be a voided by performing pure bending t ests. However, the experimental cyclic bending data must be properl y correlated to uniaxial tensile/compression behavior This work consists of a novel analytical model based on tensile uniaxial and pure bending experimental data at large reverse strain de formations. Strain gages and interferomet ry techniques were used to track the tensile/compressive strains under bending From the experimental results, the uniaxial material properties were determined for sheet metals undergoing large tensile/compressive s trains The Bauschinger effect can be defined as a yielding of the material after an initial plastic deformation, at a significantly different stress in the reverse direction than would have been achieved had the material been reloaded in the monotonic di rection. In Fig.1, the stress strain curve will follow the strain hardening power curve during tensile monotonic loading to S followed by unloading to p and monotonic reloading to S f will return to the same curve Reloading the material in the reverse compressive direction from p results in yielding at S R where and T he difference b etween these values is referred to as the Bauschinger effect. As the Bauschinger effect vanishes.
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2 Figure 1 : Reverse s train b ehavior showing the Bauchinger e ffect
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3 CHAPTER II LITERATURE REVIEW While the tensile cur ve to S is readily available from uniaxial tension tests, the determination of the exact configuration upon reversal from p to S R requires testing the sheet metal under uniaxial compression. Since sheet metals would buckle under such compression, the sh eet samples need to be laterally restrained. This approach has been pursued by several researchers. Tozawa tested sheet metal under compression using stacks of glued sheet to prevent or delay buckling [2]. Using this technique, he carried out experiments u nder combined stress in order to define the Yield loci for various materials. Kuwabara et Al. designed an in plane compression test and prevented buckling by sandwiching the sheet between comb shaped dies and compressing it between rigid and parallel plate ns to sandwich the sheet [3] Yoshida et Al bonded sets of five sheets and laterally restrained the set to prevent buckling [4]. They reported large strain reversals of up to 10% when the anti buckling device was used. Boger at Al designed an in plane comp ression test using solid flat platens for buckling constraint smaller than the sample length and specimens tailored to avoid bucking outside the constrained region [5]. They reported optimized experimental conditions in order to maximize the uniformity and the magnitude of the compressive strains. Additional versions of uniaxial compression, as well as a brief discussion on the advantages and disadvantages of other approaches such as reverse shear, and cyclic torsion and can be found in references [4], and [5]. The generation of representative sheet metal experimental data under uniaxial compression face challenges common to the methods above. Ideally, the sample would uniformly compress axially and expand radially under compressive stress es Allowing for u n iform radial expansion would require frictionless, flat compressive platens, as opposed to gripped ends. Additionally, b uckling modes developing under ax ial compression are not uniform, with the segment around the mid length of the sample more prone to buc kling Assessing the compressive stresses caused by the anti buckling plates would require estimation of the effective contact area a t the small s ection fi r st prone to buckling. This transverse stress may be expected to be significantly higher than average and the uniaxial condition
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4 would need re evaluation. Also a lignment must be preserved during testing. Any departure from true axial loading would result in additional moment offsets. Overall sliding friction from anti buckling supports of any kind acting on contact surfaces adds to the underlying uncertainty of the experimental results. It should be noted that roller backed platens do not prevent sliding friction between the sheet samples and the plates, given the deformation gradient in the samples. From Boger [5], friction coefficients were estimated to be negli gi ble, in the range of 0.06 0.09. H owever, as the slope decreases with strain, the effects of friction become more relevant and cannot be neglected. Bending tests benefit from providing mome nt curvature, and moment strain information useful to industrial forming practices. Bending is a common s heet metal forming mode, an d its associated forming issues notably springbac k, is related to bending residual stresses Outer layers of tension and co mpression stresses develop during bending and reverse bending result in opposite stress states. V arious forming operations such as flow around corners and draw beads are common examples of stress strain reversals due to cyclic bending Therefore, it has b een of practical relevance to study the Bauschinger effect by means of bending tests. Several bending test devices have been reported, which can be classified in two major groups; (1) three point bending devices, and (2) pure bending moment devices. Recent ly, Zang et al used a three point bending test to measure the Bauschinger effect from the bending springback profile [6]. They used pre strained sheets to measure the reverse transient and permanent softening behavior for DP780 steel sheets. Eggertsen and Mattiasson used a three point test to study the bending unbending behavior for various steel sheet grades [7]. They applied inverse modeling approach es to determine the main parameters towards improved springback predictions. Three point bending introduce s transverse shear and contact effects at the middle pin, where the maximum bending takes place. The span between bending points i s typically large; as are needed to accommodate the geometry of the bending apparatus Large strain s, on the order of 10% requ ire bending radii around five times the thickness. This results in very small spans for thin sheets around 1 mm thick. Furthermore, as the span decreases, the transverse shear and
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5 contact effects increase at the location of interest These characteristics make the three point bending test most useful experimental testing at small strains, and/or thicker materials. Pure bending moment techniques do not introduce transverse shear or contact effects. Four point bending devices for instance, apply pure bend ing moments on the central section of the rig. These devices are not typically suited for pure bending of thin sheet metal due to flexural effects at th e pins Instead, pure bending is appli ed using rigid jaws guided by specialized loading mechanisms Dunc an et Al designed a bending test rig that mounted on a tensile test machine and made use of the uniaxial crosshead force to produce a pure bending couple [8 ]. The changing geometry during testing resulted in non linear load displacement curves, but accura te pure bending moment measurements were reported at the elastic plastic region. A more recent variation of a cross head mounted device was reported by W eiss et Al [9 ]. The test specimen wa s mounted at the ends of pin join t ed arms. A bending moment wa s imp arted on the sample as the arms rotate duri ng cross head motion. T he deflection wa s measured by an LVDT installed at the mid section. Due to the fact that an axial force passes through the sample axis, the test is not rigorously pure bending. Also, as in t hree point bending, span requirements limit the level of strain that can be imparted to thin sheet specimens
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6 CHAPTER III EXPERIMENTAL METHODOLOGY This study included four testing materials; two dual phase (DP590 and DP780) sheet ste els and two aluminum alloys (Al1, Al2) Experimental data was collected from uniaxial tensile and pure bending moment tests As validation of the method is the primary intent, data collection was limited to the rolling direction and strain rate effects wer e factored out by testing under quasi static conditions. Mechanical properties of each material derived from traditional tensile testing are given in Table 1. Table 1 : U niaxial tensile properties for test m aterials Material 0.2% YS UTS UE TE K n r 0 [M p a] [M p a] [%] [%] [M p a] DP590 320.3 567.2 18.9 24.3 1038.7 0.228 2.11 DP780 415.3 720.7 13.0 19.6 1105.7 0.143 1.89 AA1 129.7 290.0 22.1 27.9 448.4 0.254 0.92 AA2 164.9 281.4 20.9 27.5 533.1 0.259 1.12 Pure bend tests differ from uniaxial testing in various aspects. Under pure bending, tensile/compression strain gradients develop through the thickness, with maximum values at the outer layer s Bending induced r esidual s tresses develop ed upon unloading and the neutral axis has been reported to shift [10 ]. For wide specimens, aside a small anticlastic curvature developing at the edges during bending, the width remains approximately constant. The resultant plane strain condition differs from uniaxial flow complicating the correl ation between the tests. However, a properly designed pure bending test has important advantages. The layer s under compression are supported by the layers under tension and thereof large compression strains can be reached on the concave side without rein forcing anti buckling devices. Shear stresses are negligible, and there are no contact or friction effects at the location of study Additionally t he bend test area s are
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7 open and accessible to monitoring; bending moment, strain and 3D geome try can be colle cted accurately using sensors and optical techniques. Pure Bending Moment Device P ure bending moment experiments conducted here follow ed a procedure previously developed by Sanchez [11 ]. Specimens were bent to small radii by incrementally increasing the a pplied moment until the outer most fibers of the material experienced st rains of the required magnitude ( up to a maximum of around 15 %). Dr. Sanchez pure bending device developed from its original form, first published in 1988 [19 ] to the current form s hown in Fig.2 It consists of two grips, one fixed rigidly to the base and the other free to rotate parallel to the axis of the bend and to translate on a n X Y plane orthogonal to this same axis Al l moving parts were supported by bearings to reduce frict ional effects The bending moment was transmitted under quasi static conditions by a bending couple translating with the carriage (Fig. 2b ). The load was reversed to unbend and reverse bend the sample. The moment was measured directly by a sensor mounted underneath the moving grip. During the performance of the test, the grip was rotated to a predetermined angle while the output of the strain and moment sensors was simultaneously monitored. Given that grip effects may contribute to the angle of rotation, g rip displacements were not considered for strain calculations Strain measurements were taken at a small area at the center of the bending span. Low level strains were monitored using foil strain gages (Fig. 2d). Although gages up to 10% strain are commerc ially available, they require curing oven time. Testing under quasi static conditions was practically performed using 0 3% strain gages with lead wires attached; which were easier to use, faster to install and more economical. The strain range may be exte nded by gluing new gages at progressive steps with the device under load. However, extensive testing showed that the process could be simplified to one gage for low strains, with the final strain independently determined by interferometry.
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8 Figure 2 : Description of Dr. Sanchez p ure bending moment d evice F F x y x z Weighs Weighs
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9 Strain Measurement using Interferometry Interferometry was used to track 3D geometries on a 1.9mm by 2.0 mm test area on both surfaces of each specimen. The strains at the out er layers were measured from inscribed marks (Fig. 3a). By filtering cylindricity and tilt, the 2D flat distance betwee n the marks could be measured (Fig. 3 b) As described in [11], strains can also be determined from the sample thickness and profile radi i obtained from 3D profile measurements (F ig s 3c 3d ). Final strain uncertainties were on the order of + 0.003 strain. (a) 3D Concave Profile (b) 2D Measurement of Distance (c) 2D Concave Profile (d) 2D Convex Profile Figure 3 : Interfero metry measurements. See also reference [11]
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10 CHAPTER IV CHACTERIZATION OF MATERIAL PROPERTIES Despite eventual uncertainties due to the experimental techniques, published estimates of the cyclic true stress strain curves for formable aluminum and steel she ets share several common trends (Fig 4): Table 2 : Descriptions and r elationships fo r labeled regions in Figure 4 Labeled Region Stress Strain Region Constitutive Equation O A Linear range A C Elasto plastic range Un known C D Large plastic strains (K CD ) D Unloading/reverse loading l inear Elasto plastic transient Unknown Large plastic strain reversal C D ) Repeated cycles at same le vel o f strain shows saturation Figure 4 : Cyclic loading behavior
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11 Log L og Analysis of True Stress True Strain Similar large reverse elasto plastic transients are shown by early and current researchers; Caulk and Naghdi [12], and Chaboche [13], in their study of the cyclic response for stainless steels. Yoshida and Uemori in their studies of in plane cyclic tension compression [14 ] and springback simulation [15 ]. Eggertsen and Mattiasson [7 ] in their experimental force displacement diagrams. The constitut ive models developed by these researchers varied in complexity as needed to fit the materials responses Fig ure 4 evidences the need for an accurate representation of the transient between the elastic OA, a nd the power curve CD sections A close fit at AC was accomplished on the ln ln scale following a procedure proposed by Dr. Sanchez. OA and CD are straight lines in the ln ln scale, as shown in Figure 5 for DP590 Steel. The experimental data at the transient region curved from the elastic line at AB and intersected the power line at a sensibly constant slope at BC. Figure 5 : Tensile Test ln ln for DP590 steel
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12 Parameterization of the Elasto Plastic Region This behavior leads to a simple, but close fit to the experimental data through an arc of circle AB and a straight li ne BC on the ln ln scale (Fig. 6) The center point of the circle was located by offsetting both the elastic and intermediate straight line fits by an arbitrary distance r and finding th e intersection of the two lin es. A circular curve of radius R C cen tered at this intersection was then generated between points perpendicular to the offset lines and intersecting the original fits. The circular arc of radius R C was then adapted to find a best fit to the experimental data. Figure 6 : ln ln fits for DP590 The slope of the fit showed a small discontinuity at the intersection between the lines (point C). This effect was small enough that further modeling complexities were deemed unnecessary, however an additional arc fit in this small region would elim inate any edge in the curve The fit accurately characterized the material properties throughout the entire curve as shown in Fig. 7. The fit was adequate for both the steel and aluminum samp les tested in this experiment (Table 2) This approach lends itself to an automated adaptation routine for the optimization fit of the reverse bending moment experimental data.
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13 Figure 7 : True Stress True Strain fit for DP590 Steel
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14 CHAPTER V PURE BENDING MOMENT MODEL Modeling of pure bending of sheet metal involv es shifting of the neutral plane under hoop and radial stresses. A pure bending model including these parameters as well as through thickness anisotropy was developed by Dr. Sanchez. A s ummary of the model is shown in appendix C. Modeling bending /unbending require s Bauschinger strain reversal curves During the first bend, strain reversals occur due to s hifting o f the neutral plane. Since the Bauschinger curve is unknown a t the first bend, the solution wa s based on an iterative scheme, where the proposed Ba uschinger curve wa s first determined by the experimental data using a n approximate approach. The quality of fit by the approximate Bauschinger curve was (See Fig. ) The approximate approach takes into account that, at the plastic strain levels reached (10%), plastic behavior predominates. T he shifting of the neutral plane is expected small at the ratios used on this study. For the materials tested, the experimental data showed negligible thickness changes during bending/unbending The concave and convex strains were of similar magnitude, and the surface profiles shared the same center radii. Therefore, cross planes remained plane Additionally the radial stresses developed by the bending were considered s mall enough to be ignored.
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15 Figure 8 : Pure bending moment free body diagram used for reverse bend optimization The true strain as a function of curvature ( k i = 1/r i neutral axis is given in Equation (1): (1) For the following analysis the primary stress and strain directions will be oriented with the direction 1 ), the direction parallel to the axis of the 2 3 ). Within the elastic range at the initial stage of the bend, the stress develope in E quation ( 2 ). (2) As the outer layers enter plastic deformation the s heet bends and stretches under plane strain with Given the large width/thick ness ratio for sheet metal, the elastic core is constrained by the plastic layers under plane strain and
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16 (3) (4) (5) Assuming rotational symmetry r 0 = r 90 = r where the Lankford coefficient was determined at 15 % tensile strain the plastic strains are obtained from the flow rules: (6) Under plane str ain along the widt h the instantaneous deformation is given by: (7) However, due to the large width to thickness ratio this condition can be reduced to Eq. (8) and (9): (8 ) (9 ) Where are determined from tensile test data fit to sections below: Section OA defined by the linear fit (10 ) Elasto plastic section AC defined by: Arc fit AB: (11 ) w here (ln 0 ln 0 ) are the coordinates of the center of the arc in the ln ln scale. Intermediate fit B C : (12 ) Large strains power fit CD (13 )
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17 From the consti tutive relationship obtained by the true stress stain curve, the moment needed to attain a bend of arbitrary radius r i can be found from the standard bending moment equation for a unit width and thickness h given in Equation (13 ). (14 ) The sheet is monotonically bent from flat to a bending moment M f Upon unbending, t he resi dual stress in each laye r is calculated by (14 ) where M f is the final moment, h b is the thickness of the sample, z is the distance of layer from the neutral axis and the width is a unit length (1 5 )
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18 CHAPTER VI REVERSE BEND OPTIMIZATION For the reverse bend the parameters that define the e lasto plastic region of the true st ress strain curve are iterated to produce a bending moment that conforms to the measured reverse bend moment using a random search optimization program In this algorithm the circle radius as well as the intermediate power curve coefficient and exponent a re adjusted by a given step size to produce a new true stress strain curve at each iteration The program then calculates the error of the resulting reverse bending moment curve with the experimental data obtained from the pure bending moment device The error is then compared to the error from the previous loop iteration or the error given by the original constitutiv e curve for the first iteration If the resulting error exceeds the previously determined error the direction of the step is reversed. This i s done ind ividually for each of the variables defining the Elasto plastic region. T he new bending moment curve is determined based on the complete set of new parameters describing the reverse stress strain curve. T he resulting error is checked against the condition for convergence (CON). If the condition is not me t the step size is reduced and the new parameters form t he basis for the next iteration; provided the resulting total error is less than the previous loop. A flow diagram of the p rogram is given in Fig. 11 The program output material properties are: From the tensile test fits: From optimal fits to the reverse pure bending test:
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19 Figure 9: Flow diagram for random search optimization p rogra m
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20 Figure compares the through thickness hoop stresses between models I and II. Model I (Appendix C) includes neutral plane shifting and radial stresses. Model II corresponds to the approximate solution, where the Bauschinger effect during bending was not included. Neglecting elastic volumetric changes in Model I, the stress reversals were calculated using the reverse Bauschinger curve obtained by the approximate solution. Section AB in the Figure corresponds to the strain reversal zone during bending The radial stresses in Model I, determined using the approximate Bauschinger curve, do not necessarily satisfy continuity. The discrepancy on radial stresses at A, of around 6 %, provides a measure of the uncertainty using the approximate solution. This discrepancy between models was small to translate in a meaningful c hange on the Bauschinger results using the approximate approach. Figure 10 : Comparison of final first bend stress p rofile s with models I and II 0 0.2 0.4 0.6 0.8 1 1.2 1000 800 600 400 200 0 200 400 600 800 1000 Sample Thickness [mm] Bending Stress [MPa] Simplified Model Complete Model Complete Model Radial Stresses
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21 CHAPTER VII EXPERIMENTAL AND NUMERICAL M ODEL RESULTS Table 3 shows t he fits to the uniaxial tensile test data. These included the Elasto plastic (AC), and the large plastic (CD) regions. Table 3 : Tensile test fit parameters Material Elasto Plastic Fit Large Plastic St rain Fit (CD) Arc Fit (AB) Fit (BC) R C Ln(M p a) K Ln(M p a) K (M p a) n K Ln(M p a) K (M p a) n DP590 1.45 6.324 557.6 0.098 6.946 1038.7 0.228 DP780 1.3 7.369 1585.8 0.243 7.008 1105.7 0.143 AA1 1.25 5.562 260.3 0.124 6.106 448.4 0.254 AA2 0.95 5.788 326. 4 0.127 6.279 533.1 0.259 The uniaxial tensile data standard deviations of the fit ( ) are shown in Table 4 For all materials, the straight section of the Elasto plastic fit (section BC) was comparable to the standard deviation of the power equation fit (section CD). The arc fit AB was comparable to the fit to the elastic data OA. Overall, the deviations were small; although closer fits were noted for section s at larger strains BC, and CD. Table 4 : Tensi le test data standard d Material Elastic Fit Elasto plastic fit Large Plastic fit (CD) Arc fit (AB) Fit (BC) DP590 0.035 0.018 0.004 0.007 DP780 0.022 0.040 0.006 0.004 AA1 0.012 0.014 0.007 0.003 AA2 0.015 0.010 0.007 0.004 Figures 12 15 compare the bending unbending e xpe rimental and model data. For the bending section O D, the comparison wa s based in two independent tests. The model was based on the te nsile
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22 test parameters in Table 3 as well as the experimentally determi ned elastic modulus, and the experimental data on the pure bending test. U nder unbending section D the model was fitted to the experimental data from where the reverse tensile test parameters were calculated using the uniaxial testing data as the start ing point The figures show the pure bending model plotted against the compressive outer layer true strain. The range for the strain gag e was significantly larger when glued to the outer layer under compression. Therefore, strain gage data was only acquir ed at the compressive side which corresponds to the c oncave profile under bending and to the convex profile under unbending. Strain gage tracking covere d the Elasto plastic range OC and the initial power fit. Testing was performed quasi statically, and ste pped s train gages were sometimes glued under load for longer tracking. However, monotonic bending at large strains was mainly dependent on the power equation and was very predictable Since the final strain at D was measured independently, strain gage d ata at low strains prove adequate during bending. Prior to unloading, a second strain gage was glu ed with the sample under maximum bending moment for the steel samples The final strain at D was determined f gage data was obtained following a similar procedure.
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23 Figure 11 : DP590 steel pure moment bending/unbending c urve. Interferometer readings taken at O, Figure 12 : DP780 steel p ure moment bending/unbending c urve
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24 Figure 13 : Aluminum Alloy 1. P ure moment bending/unbending c urve Figure 14 : Al uminum Alloy 2 pure moment bending/unbending c urve
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25 The calculated fits for the uniaxial reverse curve are shown in Table 4 The parameters define the reverse uniaxial curves resulting at the minimum ERR values (Fig. 9) calculated from the reverse M outer curve. Table 5 Material Elasto Plastic Fit Large Plastic Strain Fit (CD) Arc Fit (AB) Fit (BC) R C Ln(M p a) K Ln(M p a) K (M p a) n K Ln(M p a) K (M p a) n DP590 3.105 6.682 798.1 0.141 6.946 1038.7 0.228 DP780 0.657 7.594 1986.7 0.269 7.008 1105.7 0.143 AA1 1.003 5.744 312.3 0.099 6.106 448.4 0.254 AA2 0.482 5.887 360.5 0.095 6.279 533.1 0.259
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26 CHAPTER VIII DISCUSSION At bending sections OD (Figs 12 15 ) the M outer fits and the experimental bending moment data were obtained by independent approaches. The experimental moment was determined from the pure bending device, and the calculated moment from the equation (9) using tensile test data according to its parametrical fit (Table 2). the uniaxial parameters were determined from the best fit of the pure bending model to the reverse experimental M outer data Fitting of Aluminum AA1, AA2, were closest, while DP590 and DP780 showe d a small overestimate at point D This was attributed to anticlastic effects due to smaller pure bending sample widths ( necessitated by equipment limitations) for theses high strength materials. Figs 16 19 show the uniaxial curves under monotonic (O A direct correlation is appreciated between the fitting sections and the M curves The Elasto plastic radial arc fit (AB) was characterized by a small strain zone, serving as a transition between the elastic range ( OA) and the intermediate power fit (BC). With increasing bending, t he Elasto plastic contributions became less significant as large strains developed from C to D. S ection CD was more closely modeled by the tensile test power equation parameters K. n, give n in Table 1. During u nbending was significant, as shown by uniaxial fits with large Elasto plastic reversals. Dual Phase Steel Samples The uniaxial reverse fit for DP590 and DP780 are shown in Figs 16 a nd 17 Both st eels evidence an initially significant Bauschinger effect at the small strain section defined by the arc fit. T he reverse response was close to the original tensile curve. Section BC expanded upon reversal ower fit CD. Differences in the Elasto plastic response are appreciated in Table s 2 and 4 Intermediate K BC and n BC (557.6M p a, 0.098) for DP590 increased significantly upon reversal (798.1M p a, 0.141) but significantly below the original K n
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27 values at CD (1038.7M p a, 0.228). This trend was opposite for DP780, with K BC and n BC larger than K n values at CD. Figure 15 : Figure 16 e best fit compression for DP780 Steel. Labels refer to the reverse compression curve
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28 Aluminum Alloy Samples The reverse curves for Aluminum alloys 1 and 2 showed similar behavior between them (Figs. 18, 19 ); but showed significant differences to steel i n the Elasto plastic response For AA sheets the reverse curve consisted of a large span defined by the intermediate power fit BC. The softened conside rably upon reversal with reverse n BC values significantly sma ller than for CD ( 0.0 0 9 and .254 for AA1) full unbending to flat, with no Figure 17 Primed l abels refer to the reverse compression curve
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29 Figure 18 labels refer to the reverse co mpression curve Figs 12 19 show the tendency of the materials towards reaching stress levels upon reversal similar to end point D and further deform following the original K n values. That was the case for DP590 and DP780. The AA1, AA2 sheets followed th at tendency, but trends merging to the original K n values were inconclusive. Nevertheless, the intermediate K BC and n BC power fits were shown significant to the reverse response for all materials tested. Fig. 20 shows the experimental data compared to reversals based on a fraction BF of the isotropic hardening curve, with BF = S R /S f (Fig 1 ). This approach was deficient, as the predicted curves did not follow the overall reverse response. Comparison to Fig 12 shows how the inclusion of the Elasto pla stic response resulted i n a sign ificant improvement of the fitting.
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30 Figure 19 : Model with Isotropic Hardening and Bauschinger Factor Given the tendency of sheet metals to saturate wi th repetitive cycles (Fig. 4), the reversal effects are expected more pronounced at t he first bending unbending For model validation purposes, the experimental testing was confined to that range in this study Simplifications to the procedure may be carried out depending on the features of the experimental curve. For the t hin sheet materials tested in this study, profile radii and strain data from interferometry located the neutral axis at the center of sheet within experimental error [ 11 ]. For these cases, simpler bending models based on a mid neutral axis may suffice. The experimental collection of strain bending/unbending data may be reduced to two gages During unbending, point D can be approximated as an e simplification applies for unloading cases with As noticed by other researchers Additional research may refine the correlation between pure bending and uniaxial testing. Since straining under pure bending locates at an unsupported and acce ssible section, the test is lends to adaptation of other contact/non contact measurement techniques Residual stresses, for instance, can be assessed on the unbent to flat samples using X rays diffraction, as reported by Kassner et Al [ 18 ].
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31 CHAPTER IX CO NCLUSIONS The reverse uniaxial stress strain curve s can be obtained from simple to perform pure bending and tensile tests. The reverse curve s determined by the best fit to the experimental m oment outer strain data, were distinct for the sheet materials te sted. F or DP steels, reverse stresses merged to the originally monotonic curve at mid range strains. Aluminum alloys showed an initially higher reve rse stress response, followed by significant softening. Contrary to DP steel sheets, reverse stresses for al uminum alloys did not merge to the original curve during the unbending span. Bauschinger effects were quantified as the material reverse response in the Elasto plastic range. This range was defined between the elastic and the large strain sections, repr esented by straight lines in the scale. Close standard deviations of the fits were obtained in this range by fitting an arc to the elastic line, followed by an intermediate linear fit. The intermediate fit best described large re verse strain transients occurring upon reversal. The determination of the reverse uniaxial curve from unbending data was indirect and require d modeling. H owever, the relationships between pure bending and uniaxial loading become better known with improved ex perimental techniques Pure bending naturally applies large compression strains to unsupported sheet metals and it is simple to perform. These represent major advantages over direct uniaxial compression of sheet metal which requires restraining
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32 REFERENCES [1] J. Bauschinger, Uber die Veranderung der Elastizitatsgrenze und des Elastizitatsmoduls Verschiedener Metalle, Civilingenieur 27 (1881), 289 347 [2] Tozawa, Y. Plastic Deformation Behavior Under Conditions of Combined Stress, Mechanics o f Sheet Metal Forming, Koistinen, Wang eds., Plenum Press, (1978) 179 211 [3] Kuwabara T., Morita, Y. Takahashi, S. Elastic Plastic Behavior of sheet metal subjected to in plane reverse loading. Proceedings of Plassticity1995, Dynamic Plasticity and Struc tural Behavior, Gordon and Breach. [4] Yoshida F., Uemori, T., Fujiwara, K. Elasticplastic behavior of steel sheets under in plane cyclic tension compression at large strain, International Journal of Plasticity 18 (2002) 633 659 [5] Boger RK, Wagoner RH, Barlat F, Lee MG, Chung K. Continuous, large stain, tension/compression testing of sheet materials. International Journal of Plasticity 2005: 21: 2319 2343. [6] Zang, S.L., Lee, M.G., Sun, L., Kim, J.H., Measurement of the Bauschinger behavior of sheet metals by three point bending springback test with pre strained strips, International Journal of Plasticity 59 (2014) 84 107. [7] Eggertsen, P.A., Mattiasson, K., on the modeling of the bending unbending behavior for accurate springback predictions, Inte rnational Journal of Mechanical Sciences 51 (2009) 547 563 [8] Duncan JL, Ding SC, Jiang WL. Moment curvature measurement in thin sheet part 1: equipment. International Journal of Mechanical Sciences 1999: 41: 249 260. [9] Weiss, M., Wolfkamp, H., Rolfe, B.F., Hodgson, P.D., Hemmerich, E., Measurement of bending properties in strip for roll forming, IDDRG 2009 International Conference, 1 3 June 2009, Golden CO, USA. [10] Hill, R., The mathematical theory of Plasticity, Oxford University Press, London, 1 9 50. [11] Sanchez LR, Peterson S, Simonsen CG, Satar A. An interferometer based experimental technique to evaluate large strains and spring back on sheet metal. Proceedings of the ASME 2013 International Mechanical Engineering Congress and Exposition 2013 : 1 5. [12] Caulk, D.A., Naghdi, P.M., On the hardening response in small deformation of metals, Journal of Applied Mechanics, 45 (1978) 755 764 [13] Chaboche, J.L., Constitutive equations for cyclic plasticity and cyclic viscoplasticity, International J ournal of Plasticity 5 (1989) 247 302 [14] Yoshida F., Uemori, T., A model of large strain cyclic plasticity describing the Bauschinger effect and workhardening stagnation, International Journal of Plasticity 18 (2002) 661 686
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33 [15] Yoshida F., Uemori, T., A model of large strain cyclic plasticity and its application to springback simulation, International Journal of Mechanical Sciences 45 (2003) 1687 1702 [16] Eggertsen, P.A., Mattiasson, K., On constitutive modeling for springback analysis, Internati onal Journal of Mechanical Sciences 52 (2010) 804 818 Experimental Method to Determine Bauschinger Effects on Research Group, (IDDRG 200 9) International Conference,1 3 June 2009, Golden, CO, USA, pp 373 384 [18] Kassner,M.E., Geantil, P., L.E. Levine, L.E., Larson, B.C., Backstress, the Bauschinger Effect and Cyclic Deformation, Materials Science Forum Vols. 604 605 (2009) pp 39 51 online at http://www .scientific.net [19] A. Rosenburge r, K.J. Weinmann, L.R. Sanchez "The Bauschinger Effect of Sheet Metal Under Cyclic Reverse Pure Bending". Production Mecani que (CIRP) Japan 1988. [20] Dadras, P., Majlessi, S. A. Plastic Bending of Work Hardening Materials, Transactions of the ASME, vol 104 (1982) 224 230
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34 APPENDIX A TENSILE TEST CURVE FITS FOR SAMPLE MATERIALS 0 100 200 300 400 500 600 700 800 900 1000 0.00E+00 5.00E02 1.00E01 1.50E01 2.00E01 2.50E01 3.00E01 True Stress ( T ) [MPa] True Strain ( T ) [ ] DP780 Steel Experimental Tensile Testing Data Elastic Fit Circle Fit Intermediate Power Fit Power Fit 0 1 2 3 4 5 6 7 8 9 8 7 6 5 4 3 2 1 0 Log of True Stress [Log e ( T )] Log of True Strain [Log e ( T )] DP780 Steel Experimental Tensile Testing Data Elastic Fit Circle Fit Intermediate Power Fit Power Fit
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35 0 100 200 300 400 500 600 700 800 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 True Stress ( T ) [MPa] True Strain ( T ) [ ] DP590 Steel Experimental Tensile Testing Data Elastic Fit (OA) Arc Fit (AB) Intermediate Power Fit (BC) Power Fit (CD) 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 9 8 7 6 5 4 3 2 1 Log of True Stress [Log e ( T )] Log of True Strain [Log e ( T )] DP590 Steel Experimental Tensile Testing Data Elastic Fit (OA) Arc Fit (AB) Intermediate Power Fit (BC) Power Fit (CD) Circle Fit Arc Fit Center
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36 0 5000 10000 15000 20000 25000 30000 35000 0 0.02 0.04 0.06 0.08 0.1 0.12 True Stress ( T ) [psi] True Strain ( T ) [ ] NMF Aluminum Experimental Tensile Testing Data Elastic Fit Circle Fit Intermediate Power Fit Power Fit 5 6 7 8 9 10 11 9 8 7 6 5 4 3 2 1 0 Log of True Stress [Log e ( T )] Log of True Strain [Log e ( T )] NMF Aluminum Experimental Tensile Testing Data Elastic Fit Circle Fit Intermediate Power Fit Power Fit
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37 0 5000 10000 15000 20000 25000 30000 35000 40000 0 0.02 0.04 0.06 0.08 0.1 0.12 True Stress ( T ) [psi] True Strain ( T ) [ ] AEDT Aluminum Experimental Tensile Testing Data Elastic Fit Circle Fit Intermediate Power Fit Power Fit 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 9 8 7 6 5 4 3 2 1 Log of True Stress [Log e ( T )] Log of True Strain [Log e ( T )] AEDT Aluminum
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38 APPE NDIX B NUMERICAL MODEL (MATLAB PROGRAMING LANGUAGE) Main Program % Pure Bending Moment Theoreti cal Model % Main Program % towards Master of Science Thesis % University of Colorado Denver % Drew Hanzon clear all %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% Constants and Simulation Parameters %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CLElim = 0.05; % Tensile Testing Sample Geometry t0 = 0.0412; % Sample Thickness [in.] w0 = 0.5; % Sample Width [in.] % Pure Bending Moment Sample Geometry tb = 0.0412; % Sample Thickness [in.] wb = 1.530; % Sample Width [in.] % Material Identifier W = 3; % Material Properties str = 'DP590LB_Fracture.txt' ; % Tensile Testing Raw Data (Instron) str2 = 'DP590LB_Moment.txt' ; % Bending Moment Data (Dr. Sanchez Device) str3 = 'DP590LB_MomentRev.txt' ; % Reverse Moment Data (Dr. Sanchez Device) epsi = 0.00008; % Beginning of elastic modulus fit [ ] epsy = 0.00090552; % Yield Strain of Material [ ] epsI = 0.0025112; % Beginning of Intermediate Power Curve [ ] epsp = 0.0095712; % Beginning of Power Curve (True Strain) [ ] epscf = 0.12; % End of Power Curve [ ] epsRf = 0.102; % Final true strain at outer most layer [ ] rA = 2.1; % Anisotropy Factor [ ] nu = 0.3; % Possion Ratio [ ] zE = CLElim*(tb); Rroe = tb/(2*(exp(0.1) 1)); % Radius of circle for log log data fit r = 1.45; % Circle radius % Circle Radius N = 300; % Number of points in stress strain curve % Simulation Parameters NL = 200; % Number of Layers NS = 100; % Number of Steps in First Bend Simulation NR = 100; % Number of Step s in Second Bend Simulation %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% Determination of Material Properties from TT Data %%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%
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39 % Functions to find Material Properties from Raw Tensile Testing Data [epsQ sigQ Te Ts] = Circ_Fit(str,t0,w0,epsi,epsy,epsI,epsp,epscf,r,N,W); O = MatPropCurve(Te,Ts,epsi,epsy,epsp,epscf,epsI,1); E=O(1);Y0=O(2); K=O(3);n=O(4);K2=O(5);n2=O(6); U = find(str== '_' ); str1 = str(1:(U 1)); [M SE] = MomentExp(str2); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%% Initialization of Simulation Objects and Variables %%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Instantiates object for each layer of simulation SigY = E*epsy^Y0; tL = tb/NL; % Thickness of each layer t1 = 0; for i = 1:NL t1 = t1 + tL; NA = tb/2; zT = t1 NA; z = zT (tL/2); L(i) = Layer(z,tL,NS,NR,SigY); L(i).t(1) = tL; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% %%%%%%%%%% Simulation of First Bend %%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Construction of Curvature Radius Vector Rmax = 1e5; Rmin = (tb/2)/epsRf; R = logspace(log10(Rmax), log10(Rmin),NS); Moment1 = zeros(1,NS); Max = length(epsQ); for j = 1:NS for i = 1:NL L(i) = L(i).Strain(epsy,R(j),nu,j); L(i) = L(i).Stress(epsQ,sigQ,epsy,rA,Max,K,n,j); Moment1(j) = Moment1(j) + L(i).sig1(j)*L( i).z*L(i).tL; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% Determination of Residual Stresses/Strain %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C1 = 12*Moment1(end)/tb^3; for i = 1:NL L(i) = L(i).ResStress(C1); L(i) = L(i).ResStrain(E); end
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40 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Reverse Stress Strain Optimization %%%%%%%%%%%%%% %%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [MoR StrR] = MomentExp(str3); Rmax = 50; RO = logspace(log10(Rmax),log10(Rmin 0.01),NR); MomentO = zeros(1,NR); Max = length(epsQ); for j = 1:NR for i = 1:NL L(i) = L(i).OptStrain(RO(j),j); L(i) = L(i).OptStress(epsQ,sigQ,rA,Max,K,n,j); MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL; end end epsO = L(end).epsO L(end).epsO(1); ERR = Test_Moment(St rR,MoR,epsO,MomentO,wb); % Optimization rR = r; K2R = K2; n2R = n2; Con = 10; Cmax = 30; cnt = 0; negR = 1; negK = 1; negn = 1; ERRR = ERR; ERRK = ERR; ERRn = ERR; step = 0.84; stp1 = step; while ERR > Con && cnt < Cmax && step > 1e 10 % Radius Test rR1 = rR+negR*step*rR; [epsO sigO] = Test_Curve(E,Y0,K,n,K2R,n2R,rR1,epscf,StrR(1),N); MomentO = zeros(1,NR); for j = 1:NR for i = 1:NL L(i) = L(i).OptStress(e psO,sigO,rA,Max,K,n,j); MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL; end end epsLO = L(end).epsO L(end).epsO(1); ERRT = Test_Moment(StrR,MoR,epsLO,MomentO,wb); if ERRT > ERRR negR = negR; rR 1 = rR+negR*step*rR; end ERRR = ERRT; % Intermediate Coefficient Test K2R1 = K2R+negK*step*K2R; [epsO sigO] = Test_Curve(E,Y0,K,n,K2R1,n2R,rR,epscf,StrR(1),N); MomentO = zeros(1,NR); for j = 1:NR for i = 1:NL L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j); MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL; end end epsLO = L(end).epsO L(end).epsO(1); ERRT = Test_Moment(StrR,MoR,epsLO,MomentO,wb); if ERRT > ER RK negK = negK; K2R1 = K2R+negK*step*K2R; end ERRK = ERRT; % Intermediate Exponent Test n2R1 = n2R+negn*step*n2R; [epsO sigO] = Test_Curve(E,Y0,K,n,K2R,n2R1,rR,epscf,StrR(1),N);
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41 MomentO = zeros(1,NR); for j = 1:NR for i = 1:NL L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j); MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL; end end epsLO = L(end).epsO L(end).epsO(1); ERRT = Test_Moment(StrR,MoR,e psLO,MomentO,wb); if ERRT > ERRn negn = negn; n2R1 = n2R+negn*step*n2R; end ERRn = ERRT; % New Stress Strain Curve [epsO sigO] = Test_Curve(E,Y0,K,n,K2R1,n2R1,rR1,epscf,StrR(1),N); MomentO = zeros(1,NR); fo r j = 1:NR for i = 1:NL L(i) = L(i).OptStress(epsO,sigO,rA,Max,K,n,j); MomentO(j) = MomentO(j) + L(i).sigO(j)*L(i).z*L(i).tL; end end epsLO = L(end).epsO L(end).epsO(1); ERRV = Test_Moment(StrR,M oR,epsLO,MomentO,wb); if ERRV < ERR rR = rR1; K2R = K2R1; n2R = n2R1; ERR = ERRV end ERR ERRV step = stp1 (cnt/Cmax)*stp1 cnt = cnt+1 clear figure(1) figure(1) plot(epsLO,wb*MomentO, r' 'Lin eWidth' ,1.3) hold on plot(StrR,MoR, '*k' ) title( Reverse Bend Optimazation ) xlabel( Reverse Strain [ ] ) ylabel( Moment [ lbf in ] ) drawnow hold off end Q = Circ_Opt(E,Y0,K,n,K2R,n2R,rR); xC=Q(1); yC=Q(2); epsyR=Q(3) ; eps2R=Q(4); epspR=Q(5); th1=Q(6); th2=Q(7); % Elastic Curve epsE = linspace(Te(1),epsyR,N); sigE = zeros(1,N); for i = 1:N sigE(i) = E*epsE(i)^Y0; end % Power Curve 1 epsP = linspace(epspR,epscf,N); sigP = zeros(1,N);
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42 for i = 1:N sigP(i ) = K*epsP(i)^n; end % Intermediate Power Curve eps2 = linspace(eps2R,epspR,N); sig2 = zeros(1,N); for i = 1:N sig2(i) = K2R*eps2(i)^n2R; end % Circle Fit th = linspace(th1+pi,th2+pi,N); Fx = zeros(1,N); Fy = zeros(1,N); for i = 1:N Fx(i) = rR*cos(th(i))+xC; Fy(i) = rR*sin(th(i))+yC; end if epspR > epscf % Intermediate Power Curve eps2 = linspace(eps2R,epspR,ceil(epspR)*N); sig2 = zeros(1,ceil(epspR)*N); for i = 1:N sig2(i) = K2R*eps2(i)^n2R; end epsP = eps2(end); sigP = sig2(end); end epsO = [epsE exp(Fx) eps2 epsP]'; sigO = [sigE exp(Fy) sig2 sigP]'; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%% Reverse Bend Simulation %%%%%%%%%%%%%%%%%%%%%% %%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Rmax = 50; Moment2 = zeros(1,NR); for j = 1:NR for i = 1:NL L(i) = L(i).RevStrain(R(j),j); L(i) = L(i).RevStress(epsO,sigO,rA,Max,K,n,j); Moment2(j) = Moment2(j) + L(i).sigR(j)*L(i).z*L(i).tL; end end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% Code for Plotting and Visualization %%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [M SE] = MomentExp(str2); Z = zeros(1,NL); epsr = zeros(1,NL); MomentNA = 0; MomentComp = 0; for i = 1:NL if L(i).z < zE && L(i).z > 0 L(i) = L(i).StressNA(epsO,sigO,epsy,r,Rro e,Max,K,n); else L(i) = L(i).StressNA(epsQ,sigQ,epsy,r,Rroe,Max,K,n); end MomentNA = MomentNA + L(i).sigNA*L(i).z*L(i).tL; end
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43 for i = 1:NL L(i) = L(i).StressNA(epsQ,sigQ,epsy,r,Rroe,Max,K,n); MomentComp = MomentCo mp + L(i).sigNA*L(i).z*L(i).tL; end for i = 1:NL epsr(i) = L(i).epsFB; Z(i) = L(i).z; end th = linspace(0,2*pi,N); Cx = zeros(1,N); Cy = zeros(1,N); for i = 1:N Cx(i) = rR*cos(th(i))+xC; Cy(i) = rR*sin(th(i))+yC; end epsT = logspac e(log10(Te(1)),log10(epscf),3*N); x1 = log(Te); y1 = log(Ts); x2 = log(eps2); y2 = log(sig2); xE = log(epsE); yE = log(sigE); xP = log(epsP); yP = log(sigP); figure(1) plot(Te,Ts, '*k' 'MarkerSize' ,4) hold on plot(epsE,sigE, r' 'LineWidth ,1.2) plot(exp(Fx),exp(Fy), g' 'LineWidth' ,1.2) plot(eps2,sig2, y' 'LineWidth' ,1.2) plot(epsP,sigP, m' 'LineWidth' ,1.2) hold off xlabel( Strain ( \ epsilon ) ) ylabel( Stress ( \ sigma ) ) str2 = sprintf( %s: Tensile Testing Plot str1); legend ( Experimental Data' 'Elastic Fit' 'Circle Fit' ... 'First Power Fit' 'Second Power Fit' ) title(str2) grid figure(2) loglog(Te,Ts, '*k' 'MarkerSize' ,4) hold on loglog(epsE,sigE, r' 'LineWidth' ,1.2) loglog(exp(Fx),exp(Fy), m' 'LineWidth' ,1.2) logl og(eps2,sig2, g' 'LineWidth' ,1.2) loglog(epsP,sigP, y' 'LineWidth' ,1.2) hold off xlabel( True Strain ( \ epsilon ) [ ] ) ylabel( True Stress ( \ sigma ) [psi] ) str2 = sprintf( %s: Log Log Plot str1); legend ( 'Experimental Data' 'Elastic Fit' 'Cir cle Fit' ... 'First Power Fit' 'Second Power Fit' ) title(str2) grid
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44 figure(3) plot(x1,y1, '*k' 'MarkerSize' ,4) hold on plot(xE,yE, r' 'LineWidth' ,1.2) plot(x2,y2, g' 'LineWidth' ,1.2) plot(xP,yP, y' 'LineWidth' ,1.2) plot(Cx,Cy, m' 'LineWidth' 1.2) hold off axis equal xlabel( Log of Strain ln( \ epsilon ) ) ylabel( Log of Stress ln( \ sigma ) ) str2 = sprintf( %s: Tensile Testing Log Log Plot str1); legend ( 'Experimental Data' 'Elastic Fit' 'First Power Fit' ... 'Second Power Fit' 'Circle Fit' ) title(str2) grid
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45 Class Definition for Layer Objects classdef Layer properties z % Location relative to natural axis [in.] tL % Original Thickness of Layer [in.] sigm % Maximum Flow Stress [psi] sigr % Residual Stress after First Bend [psi] epsFB % Resultant Plastic Strain after First Bend [ ] sig1 % First Bend Stre ss (Transverse with Bend) [psi] eps1 % First Bend Strain (Transverse with Bend) [ ] sigR % First Bend Stress (Transverse with Bend) [psi] epsR % Reverse Bend Strain (Transverse with Bend) [ ] sigO % Optimization Bend Stress (Transverse with Bend) [psi] epsO % Optimization Bend Strain (Transverse with Bend) [ ] R % Residual strain of layer after first bend [ ] end methods % Constructo r Method function obj = Layer(z,t,NS,NR,SigY) obj.sigm = SigY; obj.sigr = 0; obj.epsFB = 0; obj.z = z; obj.tL = t; obj.t = zeros(1,NS); obj.sig1 = zer os(1,NS); obj.eps1 = zeros(1,NS); obj.sigR = zeros(1,NS); obj.epsR = zeros(1,NS); obj.sigO = zeros(1,NR); obj.epsO = zeros(1,NR); obj.R = 0; end % Method for Cal culating First Bend Layer Strain function obj = Strain(obj,roe,i) if obj.z < 0 neg = 1; else neg = 1; end obj.eps1(i) = neg*log(1+abs(obj.z)/roe); end function obj = Stress(obj,eps,sig,epsy,r,Max,K,n,i) RF = ((1+r)/sqrt(1+2*r)); C = (2/sqrt(3)); epsC = abs(obj.eps1(i)); if obj.eps1(i) < 0 neg = 1; else n eg = 1; end if epsC >= epsy epsC = C*abs(obj.eps1(i)); end LS = find(eps <= epsC,1, 'last' ); if epsC <= eps(1) x0 = 0; x1 = eps(1); y0 = 0; y1 = sig(1); obj.sig1(i) = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0)));
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46 elseif LS > Max 1 obj.sig1(i) = neg*RF*K*epsC^n; else x0 = eps(LS); x1 = ep s(LS+1); y0 = sig(LS); y1 = sig(LS+1); obj.sig1(i) = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0))); end if abs(obj.sig1(i)) >= abs(obj.sigm) obj.sigm = obj.sig1(i); end end % Method for Calculating Residual Stress function obj = ResStress(obj,Coef) obj.sigr = obj.sigm Coef*obj.z; end % Calculation of Plastic Strain after Release of First Bend Moment func tion obj = ResStrain(obj,E) obj.epsFB = obj.sigr/E; obj.R = obj.eps1(end) (obj.sig1(end)/E); end % Method for Calculating Strain on Reverse Bend function obj = RevStrain(obj,roe,i) if obj.z < 0 neg = 1; else neg = 1; end obj.epsR(i) = obj.epsFB neg*log(1+abs(obj.z)/roe); end function obj = RevStress(obj,eps,sig,r,Max,K,n,i) RF = ((1+r)/sqrt(1+2* r)); epsC = abs(obj.epsR(i)); C = (2/sqrt(3)); if obj.epsR(i) < 0 neg = 1; else neg = 1; end epsC = (C*epsC); LS = find(eps <= epsC,1 'last' ); if epsC <= eps(1) x0 = 0; x1 = eps(1); y0 = 0; y1 = sig(1); obj.sigR(i) = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0))); elseif LS > Max 1 obj.sigR(i) = neg*RF*K*epsC^n; else x0 = eps(LS); x1 = eps(LS+1); y0 = sig(LS); y1 = sig(LS+1); obj.sigR(i) = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0)) ); end end % Method for Calculating Strain for Optimization Routine function obj = OptStrain(obj,roe,i) if obj.z < 0 neg = 1; else
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47 neg = 1; end obj.epsO(i) = obj.epsFB neg*log(1+abs(obj.z)/roe); end function obj = OptStress(obj,eps,sig,r,Max,K,n,i) RF = ((1+r)/sqrt(1+2*r)); epsC = abs(obj.epsO(i)); C = (2/sqrt(3)); if o bj.epsO(i) < 0 neg = 1; else neg = 1; end epsC = (C*epsC); LS = find(eps <= epsC,1, 'last' ); if epsC <= eps(1) x0 = 0; x1 = eps(1); y0 = 0; y1 = sig(1); obj.sigO(i) = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0))); elseif LS > Max 1 obj.sigO(i) = neg*RF*K*epsC^n; else x0 = eps(LS); x1 = eps(LS+1); y0 = sig(LS); y1 = sig(LS+1); obj.sigO(i) = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0))); end end function obj = StressNA(obj,eps,sig,epsy,r,roe,Max,K,n) if obj.z < 0 neg = 1; else neg = 1; end obj.epsNA = neg*log(1+abs(obj.z)/roe); RF = ((1+r)/sqrt(1+2*r)); C = (2/sqrt(3)); epsC = abs(obj. epsNA); if epsC >= epsy epsC = C*abs(obj.epsNA); end LS = find(eps <= epsC,1, 'last' ); if epsC <= eps(1) x0 = 0; x1 = eps(1); y0 = 0; y1 = sig(1); obj.sigNA = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0))); elseif LS > Max 1 obj.sigNA = neg*RF*K*epsC^n; else x0 = eps(LS); x1 = eps(LS+1); y0 = sig(LS); y1 = sig(LS+1); obj.sigNA = neg*RF*(y0+(y1 y0)*((epsC x0)/(x1 x0))); end end end end
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48 Function for Equation Fitting of Tensile Testing Data function [epsQ sigQ Te Ts] = Circ_Fit(st r,t0,w0,epsi,epsy,epsI,epsp,epscf,r,N,W) U = find(str== '_' ); str1 = str(1:(U 1)); [eta sig] = RawData(str,t0,w0); J = MatPropElastic(epsi,epsy,eta,sig); [eps sig Te Ts] = True_S_S(J(1),eta,sig) ; O = MatPropCurve(Te,Ts,epsi,epsy,epsp,epscf,epsI,1); E=O(1);Y0=O(2);K=O(3);n=O(4);K2=O(5);n2=O(6); thP1 = atan( 1); thP2 = atan( (1/n2)); xE0 = log(epsy)+r*cos(thP1); yE0 = log(E*epsy^Y0 )+r*sin(thP1); x20 = log(epsp)+r*cos(thP2); y20 = log(K2*epsp^n2)+r*sin(thP2); b1 = yE0 1*xE0; b2 = y20 n2*x20; xC = (b2 b1)/(1 n2); yC = n2*xC+b2; xE1 = log(epsy); yE1 = log(E*epsy^Y0); x21 = log(epsp); y21 = log(K2*epsp^n2); b1T = yE1 1*xE1; b2T = y21 n2*x21; bPE = yC ( 1)*xC; bPI = yC ( (1/n2))*xC; xPE = (b1T bPE)/(( 1) 1); xPI = (b2T bPI)/(( (1/n2)) n2); xPP = (log(K2) log(K))/(n n2); % Elastic Curve e psE = linspace(Te(1),exp(xPE),N); sigE = zeros(1,N); for i = 1:N sigE(i) = E*epsE(i)^Y0; end % Power Curve 1 epsP = linspace(exp(xPP),epscf,N); sigP = zeros(1,N); for i = 1:N sigP(i) = K*epsP(i)^n; en d % Power Curve 2 eps2 = linspace(exp(xPI),exp(xPP),N); sig2 = zeros(1,N); for i = 1:N sig2(i) = K2*eps2(i)^n2; end x1 = log(Te); y1 = log(Ts); x2 = log(eps2); y2 = log(sig2); xE = log(epsE);
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49 yE = log(sigE); xP = log(epsP); yP = log(sigP); th = linspace(0,2*pi,N); Cx = zeros(1,N); Cy = zeros(1,N); for i = 1:N Cx(i) = r*cos(th(i))+xC; Cy(i) = r*sin(th(i))+yC; end th1 = atan((yE(end ) yC)/(xE(end) xC)); th2 = atan((y2(1) yC)/(x2(1) xC)); th = linspace(th1+pi,th2+pi,N); Fx = zeros(1,N); Fy = zeros(1,N); for i = 1:N Fx(i) = r*cos(th(i))+xC; Fy(i) = r*sin(th(i))+yC; end epsQ = [epsE exp(Fx) eps2 epsP]'; sigQ = [sigE exp(Fy) sig2 sigP]'; Ex1 = log(epsE(1)); Ex2 = log(epsE(end)); Ey1 = log(0.006894759086775369*sigE(1)); Ey2 = log(0.006894759086775369*sigE(end)); Cx1 = Fx(1); Cx2 = Fx(end); Cy1 = log(0.00689 4759086775369*exp(Fy(1))); Cy2 = log(0.006894759086775369*exp(Fy(end))); Ix1 = log(eps2(1)); Ix2 = log(eps2(end)); Iy1 = log(0.006894759086775369*sig2(1)); Iy2 = log(0.006894759086775369*sig2(end)); Px1 = log(epsP(1)); Px2 = l og(epsP(end)); Py1 = log(0.006894759086775369*sigP(1)); Py2 = log(0.006894759086775369*sigP(end)); EXFile = 'PB_Graphs.xlsx' ; B = [Ex1 Ey1 Ex2 Ey2 Px1 Py1 Px2 Py2]; C = [Cx1 Cy1 Cx2 Cy2 Ix1 Iy1 Ix2 Iy2]; if W == 1 xls write(EXFile,B,2, 'C44' ) xlswrite(EXFile,C,2, 'C54' ) elseif W == 2 xlswrite(EXFile,B,2, 'C45' ) xlswrite(EXFile,C,2, 'C55' ) elseif W == 3 xlswrite(EXFile,B,2, 'C42' ) xlswrite(EXFile,C,2, 'C52' ) elseif W == 4 xlswrite(EXFile,B,2, 'C43' ) xlswrite(EXFile,C,2, 'C53' ) end end
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50 Function for Reading Raw Tensile Testing Data % Pure Bending Moment Theor et ical Model % Function Call to Find Material Properties from Instron Raw Data % Towards Master of Scien ce Thesis % University of Colorado Denver % Drew Hanzon function [eta sig] = RawData(str,t0,w0) A = w0*t0; % Cross Sectional Area [in.^2] d = importdata(str); sig = zeros((length(d)/2),1); eta = zeros(( length(d)/2),1); for i = 1:(length(d)/2) j = i*2 1; sig(i) = d(j)/A; eta(i) = d(j+1); end end Function to Determine Material Properties function O = MatPropCurve(Teps,Tsig,epsi,epsy,epsp,epscf,epsI,sel) z = epsi; LS = fi nd(Teps <= z,1, 'last' ); z = epsy; LE = find(Teps <= z,1, 'last' ); z = epsp; PS = find(Teps <= z,1, 'last' ); z = epscf; PE = find(Teps <= z,1, 'last' ); z = epsI; IS = find(Teps <= z,1, 'last' ); % Linear Region Fit SumX=0; SumY=0; SumXY=0; SumX2=0; N = LE LS+1; for i = LS:LE Tempeta = Teps(i); SumX = SumX +log(Tempeta); SumY = SumY +log(Tsig(i)); SumXY = SumXY+log(Tempeta)*log(Tsig(i)); SumX2 = SumX2+log(Tempeta)^2; end Y0 = (SumX*SumY N*SumXY)/(SumX^2 N*SumX2); E = exp((SumX*Sum XY SumX2*SumY)/(SumX^2 N*SumX2)); % Power Curve Fit SumXLN=0; SumYLN=0; SumXYLN=0; SumX2LN=0; N = PE PS+1; for i = PS:PE Tempeta = Teps(i); SumXLN = SumXLN +log(Tempeta); SumYLN = SumYLN +log(Tsig(i)); SumXYLN = SumXYLN+log(Tempeta)*log (Tsig(i)); SumX2LN = SumX2LN+log(Tempeta)^2; end m = (SumXLN*SumYLN N*SumXYLN)/(SumXLN^2 N*SumX2LN); Ktemp = (SumXLN*SumXYLN SumX2LN*SumYLN)/(SumXLN^2 N*SumX2LN); K = exp(Ktemp);
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51 % Intermediate Curve Fit SumXLN=0; SumYLN=0; SumXYLN=0; Sum X2LN=0; N = PS IS+1; for i = IS:PS Tempeta = Teps(i); SumXLN = SumXLN +log(Tempeta); SumYLN = SumYLN +log(Tsig(i)); SumXYLN = SumXYLN+log(Tempeta)*log(Tsig(i)); SumX2LN = SumX2LN+log(Tempeta)^2; end m1 = (SumXLN*SumYLN N*SumXYLN) /(SumXLN^2 N*SumX2LN); Ktemp1 = (SumXLN*SumXYLN SumX2LN*SumYLN)/(SumXLN^2 N*SumX2LN); K1 = exp(Ktemp1); O = [E Y0 K m K1 m1]; end Function for Importing Pure Bending Moment Device Data function [MomentExp StrainExp] = MomentExp(str2) d = importdata(str2); MomentExp = zeros((length(d)/2),1); StrainExp = zeros((length(d)/2),1); for i = 1:(length(d)/2) j = i*2 1; MomentExp(i) = d(j); StrainExp(i) = d(j+1); end end Function for Converting Tensile Testing Data to True St ress True Strain % Pure Bending Moment Theoretical Model % Function Call to Find Material Properties from Instron Raw Data % Towards Master of Science Thesis % University of Colorado Denver % Drew Hanzon function [eps sig Teps Tsig] = True_S_S(ET,eta, sig) N = length(eta); % Account for initial Stress in Strain Vector eps = zeros(1,N); eps(1) = sig(1)/ET; for i = 2:N eps(i) = eps(i 1)+(eta(i) eta(i 1)); end Teps = zeros(1,N); Tsig = zeros(1,N); for i = 1:N Teps(i) = log(1+eps(i)); T sig(i) = sig(i)*exp(Teps(i)); end end
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52 Error Calculation for Reverse Bend Optimization function [Sum] = Test_Moment(StrR,MoR,epsO,MomentO,wb) Sum = 0; M2 = zeros(1,length(StrR)); for i = 1:length(StrR) LS = find(epsO >= StrR(i),1, 'last' ); x0 = epsO(LS); x1 = epsO(LS+1); y0 = MomentO(LS); y1 = MomentO(LS+1); M2(i) = wb*(y0+(y1 y0)*((StrR(i) x0)/(x1 x0))); Sum = Sum + abs(MoR(i) M2(i)); end end Function to Generate the Constitutive Relationship from Iterative Parameters func tion [epsO sigO] = Test_Curve(E,Y0,K,n,K2,n2,r,epscf,Te1,N) Q = Circ_Opt(E,Y0,K,n,K2,n2,r); xC=Q(1); yC=Q(2); epsyR=Q(3); eps2R=Q(4); epspR=Q(5); th1=Q(6); th2=Q(7); % Elastic Curve epsE = linspace(Te1,epsyR,N); sigE = zeros(1,N); for i = 1:N sigE(i) = E*epsE(i)^Y0; end % Power Curve 1 epsP = linspace(epspR,epscf,N); sigP = zeros(1,N); for i = 1:N sigP(i) = K*epsP(i)^n; end % Intermediate Power Curve eps2 = linspace(eps2R,epspR,N); sig2 = zeros(1,N); for i = 1:N sig2(i) = K2*eps2(i)^n2; end % Circle Fit th = linspace(th1+pi,th2+pi,N); Fx = zeros(1,N); Fy = zeros(1,N); for i = 1:N Fx(i) = r*cos(th(i))+xC; Fy(i) = r*sin(th(i))+yC; end if epspR > epscf eps2 = linspace(eps2R,epspR,(2*N 1)); sig2 = zeros(1,(2*N 1)); for i = 1:(2*N 1) sig2(i) = K2*eps2(i)^n2; end epsP = eps2(end); sigP = sig2(end); end epsO = [epsE exp(Fx) eps2 epsP]'; sigO = [sigE exp(Fy) sig2 sigP]'; end
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53 Function for Circle Fit in Optimization Method function Q = Circ_Opt(E,Y0,K,n ,K2,n2,r) epsy = 0.001; epsp = 0.003; thP1 = atan( 1); thP2 = atan( (1/n2)); xE0 = log(epsy)+r*cos(thP1); yE0 = log(E*epsy^Y0)+r*sin(thP1); x20 = log(epsp)+r*cos(thP2); y20 = log(K2*epsp^n2)+r*sin(thP2); b1 = yE0 1 *xE0; b2 = y20 n2*x20; xC = (b2 b1)/(1 n2); yC = n2*xC+b2; xE1 = log(epsy); yE1 = log(E*epsy^Y0); x21 = log(epsp); y21 = log(K2*epsp^n2); b1T = yE1 1*xE1; b2T = y21 n2*x21; bPE = yC ( 1)*xC; bPI = yC ( (1/n2))*xC; xPE = (b1T bPE)/(( 1) 1); xPI = (b2T bPI)/(( (1/n2)) n2); xPP = (log(K2) log(K))/(n n2); th1 = atan((log(E*exp(xPE)^Y0) yC)/(xPE xC)); th2 = atan((log(K2*exp(xPI)^n2) yC)/(xPI xC)); epsy = exp(xPE); epsp2 = exp(xPI); epsp = exp(xPP); Q = [xC yC epsy epsp2 epsp th1 th2]; e nd
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54 Appendix C P LANE S TRAIN P URE B ENDING M ODEL Developed by Luis Rafael Sanchez Vega The pure bending model follows a development similar to the model published by P. Dadras and S.A. Majlessi [20] with the differences below: 1. In this model the neutral plane was located to satisfy In Ref. [20] the neutral plane was determined from continuity in the radial stresses. 2. Thi s model accounts for strain reversals (due to neutral plane movement) through an approximate procedure based on experimental data. Ref [20] defines arbitrary Bauschinger curves for completeness of the analytical model. 3. In this model, continuity of the rad ial stresses are not automatically satisfied, and are used as a criteria to assess the adequacy of the approximate Bauschinger curve. 4. This model includes through thickness anisotropy. Ref. [20] considers isotropic conditions. Nomenclature similar to Re f. [20] was used, with Zones I ( R> R c) and III ( R < R n) under monotonic tension and compression, respectively. Layers in Zone II ( R n < R < R c ) experience strain reversals. The neutral plane shifts during loading from the original mid plane to R n
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55 Under plane strain, (1) With defined by the tensile test fits: Arc fit AB: (2.1) Intermediate fit BC : (2.2) Large strains fit CD : (2.3) Equilibrium under plane strain gives: (3) Solving: (4) For Zones I and III, can be integrated fr om (4), with given by the corresponding eq. (2), and strains expressed as function of R : (5) (6) Where is the radius for the segment at the original length Layers in Zone II reverse continuously as the neutral radius shifts to its minimum at R n The range is given by the layer originally at the mid section ( which displaces to and by the layer compressed to R n with an instant before reversal. The strain reversals range is:
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56 (7) And the strains in the corresponding layers an instant before reversal are: (8) Eqs. (7), (8) define the location of the strain reversals for each layer in Zone II. The values corresponding to the rever sal can be obtained by interpolations from the proposed reverse compression curves.

