USING ELASTIC BUCKLING THEORY
TO BETTER UNDERSTAND THE FABRICATION OF TWISTED COIL POLYMER ACTUATORS FOR USE IN UPPER LIMB PROSTHETICS
B.S., Worcester Polytechnic Institute, 2017
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 Department of Bioengineering
Â©2019 BEN PULVER
ALL RIGHTS RESERVED
This thesis for the Master of Science degree by Ben Pulver
has been approved for the Department of Bioengineering by
Richard ff. Weir, Chair Christopher M. Yakacki Kendall S. Hunter
Date: June 28, 2019
Pulver, Ben (M.S., Department of Bioengineering)
Using Elastic Buckling Theory to Better Understand the Fabrication of Twisted Coil Polymer Actuators for Use in Upper Limb Prosthetics Thesis directed by Dr. Richard ff. Weir
Upper limb amputation is a problem that affects a significant number of people in the United States and the current prosthetic options are not adequately addressing this issue. A major barrier to improved prosthetic devices, particularly prosthetic hands, is the lack of good actuator alternatives to conventional electric motors. Twisted coil polymer actuators (TCPAs) made using nylon 6,6 have recently been suggested as a viable alternative but they suffer from slow cycle times. TCPAs made using liquid crystal elastomers (LCEs) are suggested here to address this issue. To create these though, the fabrication process for making TCPAs must be better understood. This work models the fabrication process using elastic buckling theory and presents experimental data for both nylon TCPAs and LCE TCPAs. The model is able to predict the necessary tension and number of twists required to reliably make nylon TCPAs of different lengths and radii. Extending the model to predict these parameters for LCEs showed much less success due to the violation of a number of assumptions in the model. Despite this,
LCEs are shown to form TCPAs under the right conditions. Overall, this work provides a strong foundation for further work in modeling the TCPA fabrication process and shows the first evidence of the potential of LCEs as TCPAS.
The form and content of this abstract are approved. I recommend its publication.
Approved: Richard ff. Weir
Thank you to my committee members for helping me navigate a diverse array of topics for this research. Thank you to the members of the BioMechatronics Development Lab for providing assistance and feedback along the way. Thank you to the members of the Smart Materials and Biomechanics lab, particularly Sabina Ula, for their help in working with liquid crystal elastomers. Thank you to my friends and family for supporting me and providing me with much needed breaks.
TABLE OF CONTENTS
DECLARATION OF ORIGINAL WORK...........................Error! Bookmark not defined.
TABLE OF CONTENTS.................................................................vi
CHAPTER I: INTRODUCTION............................................................1
1.2 Summary of Goals............................................................2
1.2.1 Specific Aim 1:.........................................................2
1.2.2 Specific Aim 2:.........................................................2
CHAPTER II: BACKGROUND.............................................................3
2.1 Upper Limb Loss Statistics..................................................3
2.2 Upper Limb Prosthetic Options...............................................3
2.2.4 Limitations of Current Devices..........................................4
2.3 Overview of Actuators in Commercial Upper Limb Prosthetics..................5
2.3.1 DC Electric Motors......................................................6
2.4 Smart Actuators as an Alternative...........................................7
2.4.1 Shape Memory Materials and Fluidic Muscles..............................7
2.5 Mechanics and Dynamics of a Twisted Fiber in Tension.......................15
2.5.1 Elastica Theory........................................................16
2.5.2 Modern Elastic Buckling Theory.........................................19
2.5.3 Mechanics of Yarns.....................................................25
CHAPTER III: MODEL DEVELOPMENT....................................................27
3.1 Experimental Equipment and Sample Creation.................................28
3.1.1 Fabrication Rig........................................................28
3.1.2 Sample Creation/Preparation............................................31
3.2.1 Confirmation of Key Assumptions........................................32
3.3 Minimum Load Determination.................................................36
3.4 Torque Generation During Twisting...............................................39
3.4.1 Length Contraction..........................................................39
3.4.2 Torque Independence from Tensile Load.......................................42
3.4.3 Predicting Torque as a Function of Twists...................................44
3.5 Predicting Fiber Failure........................................................48
CHAPTER IV: MODEL VALIDATION WITH NYLON................................................51
4.1 Experimental Methods............................................................51
4.1.1 Test Procedures.............................................................51
4.1.2 Data Collection.............................................................52
4.1.3 Data Analysis...............................................................53
4.2.1 Different Tensile Loads.....................................................55
4.2.2 Different Lengths...........................................................57
4.2.3 Different Radii.............................................................59
4.2.4 Maximum Load Before Failure.................................................61
CHAPTER V: MODEL EXTENSION TO LCES.....................................................67
5.1 Sample Creation.................................................................67
5.1.2 Synthesis Procedure.........................................................68
5.1.3 Preparing Samples for Twisting..............................................69
5.2 Confirming Assumptions and Empirical Models.....................................70
5.2.1 Confirmation of Key Assumptions.............................................70
5.2.2 Confirming Empirical Models.................................................75
5.3.1 Different Tensile Loads.....................................................81
5.3.2 Different Lengths...........................................................83
5.3.3 Different Radii.............................................................86
5.3.4 Maximum Load Before Failure.................................................87
CHAPTER VI: CONCLUSION AND FUTURE WORK.................................................93
6.2 Future Work.....................................................................94
CHAPTER I: INTRODUCTION
Upper limb loss is expected to be affecting approximately 2.2 million Americans in 2020 , Between 27% and 56% of people with upper limb loss use a prosthesis of some kind , Part of the reason for this low device adoption has to do with the quality and functionality of current devices. Many of the issues facing current prosthetic devices, such as rigidity, poor durability, and weight, can be traced back to the reliance on conventional electric motors and transmissions as actuators , ,
Many smart materials have been proposed as potential soft actuators to address this issue due to their better biomimicry and potential for high power density; these include: shape memory polymers, shape memory alloys, and fluidic muscles , Of the shape memory polymers, twisted coil polymer actuators (TCPAs) are some of the most exciting due to their low cost and high work capacity , Unfortunately they suffer from slow cycle times due to heat dissipation issues  and thus there is a need to develop a TCPA that uses a non-thermal stimulus. One such promising candidate is the liquid crystal elastomer (LCE) due to its easy synthesis, tunable mechanical properties, and ability to respond to UV light , In order to use an LCE as a TCPA, the fabrication process for TCPAs has to be much better characterized.
Elastic buckling theory provides a strong potential method of modeling the TCPA fabrication process. Significant work has been done to generate analytical models of elastic rods undergoing similar general loading to that found in TCPA fabrication -, Some experimental work has also been done to verify these models , , but this work is limited to slightly different specific loading conditions than what is found in TCPA
fabrication. Therefore, there is a clear need to test these existing models in order to begin to develop a method of prescribing the necessary parameters to create TCPAs using different materials.
1.2 Summary of Goals
The goal of this research was to perform experimental work to validate existing elastic buckling models under the specific loading conditions found in TCPA fabrication in order to better prescribe the necessary parameters to consistently create TCPAs. This would allow for TCPAs to more reliably be created out of traditional materials, such as nylon-6,6, but more importantly for other materials, such as LCEs, to be used. The ability to create TCPAs using LCEs has the potential to create light based soft actuators for use in prosthetic hands.
1.2.1 Specific Aim 1:
The first aim of this research was to develop the necessary equations and models to predict the required tensile load and number of twists for making TCPAs using nylon 6,6 monofilament of different lengths and diameters.
1.2.2 Specific Aim 2:
The second aim of this research was to apply the model developed in Specific Aim 1 to the fabrication of TCPAs using LCE fibers.
CHAPTER II: BACKGROUND
2.1 Upper Limb Loss Statistics
In 1996 it was estimated that 185,000 people in the United States have a limb amputated, either upper or lower [11 ]. At the time, approximately 1.2 million Americans were living with a limb amputation [11 ]. By 2005 this number had grown to 1.6 million, with approximately 34% of those being upper limb amputations , Ziegler-Graham and co-authors used this data and information on the distribution of amputations, causes of amputations, and population trends to build a model to estimate the number of people living with an amputation in the U.S. by 2050 [1 ]. By 2050 the number of Americans living with an amputation is expected to grow to 3.6 million , If the 34% distribution of upper limb amputations remains constant, it may decrease due to the increasing prevalence of lower limb amputations due to cardiovascular disease, then 1.2 million Americans will be living with an upper limb amputation by 2050. Even if this number is underestimated, it demonstrates that limb amputation is a growing issue that must be addressed. It should be noted that although the proportion of upper limb amputations is currently at roughly 34%, Edwards and Osterman have reported that the above elbow amputation proportion is much lower, at roughly 10-15% , This shows that more proximal amputations are rarer and these are the kind of amputations that require full hand prostheses. Even considering this, upper limb loss is currently a significant issue in the United States and will continue to be for the foreseeable future.
2.2 Upper Limb Prosthetic Options
Approximately 27%-56% of people with upper limb amputations use a prosthesis , There are three major prosthetics options for those that choose to use a prosthesis: cosmetic, body-powered, and myoelectric. Cosmetic devices typically have limited or no
functionality and are primarily designed simply to restore the appearance of an amputated limb. Body-powered devices are typically simple grippers that are actuated using a cable system by the user. Myoelectric devices can be simple grippers or full multi-articulating hands that use powered actuators and electromyogram (EMG) based control systems. The focus of this paper is on prosthetic hands so although there any many prosthetic elbows and shoulders, they will not be discussed. An excellent review of current devices and future technologies is provided by , They note that despite significant advancements in the field, simple body-powered and myoelectric grippers, such as the Ottobock AxonHook, are still the most commonly used , An in depth review of the performance characteristics of commercial and research myoelectric hands is provided by , At least in terms of research and funding, the trend is definitely towards more sophisticated myoelectric devices.
2.2.4 Limitations of Current Devices
While there have been many advancements in the field, there are still many
barriers to device adoption and retention. Cordelia and co-authors  performed an extensive literature review to summarize the findings of seven large studies - on upper limb prosthetic device use and limitations. The key findings, presented as user needs, are found in Table 1. Additional columns have been added to categorize each user need as to whether it is primarily associated with the control of the device or the mechanical design of the device.
Since all but one of the user needs in Table 1 are in some way related to the mechanical design of the device, it is interesting that there is often significant emphasis placed on control being the limiting factor in device adoption. This is in part due to the limitations placed on device design by the available control schemes. Many of the
needs related to mechanical design are more specifically related to the actuators used in myoelectric devices, primarily DC electric motors and transmissions. As evidenced by Table 1 there is a clear need to improve the mechanical design of prosthetic devices and one major way to do this is to develop better actuators.
Table 1: Summary of user needs as determined by , Each need is categorized broadly as to whether it is primarily related to the control of the device or the mechanical design of the device.
User Need Control Related Design Related
Improved ability to perform ADLs âœ“ âœ“
Sensory feedback âœ“ âœ“
Battery and electrode reliability X âœ“
Improved grip strength X âœ“
Less reliance on visual cues for performing actions âœ“ X
High degree of anthropomorphism X âœ“
More stable grasps âœ“ âœ“
Better ability to change the orientation of grasped objects âœ“ âœ“
Independent finger movement âœ“ âœ“
Improved performance of the thumb, index, and middle finger âœ“ âœ“
Improved efficiency at grasping small objects âœ“ âœ“
Improved heat dissipation X âœ“
Reduced motor noise X âœ“
Improved prosthetic wrists âœ“ âœ“
Improved variety and compatibility of cosmetic gloves X âœ“
More natural open hand posture X âœ“
Improved device duration X âœ“
2.3 Overview of Actuators in Commercial Upper Limb Prosthetics
Due to the importance of actuators in the functioning of a prosthetic device and
the need to improve their performance, it is useful to review what kinds of actuators are currently used in commercial prosthetic devices. Del Cura and co-authors provide a
good comprehensive review of actuators and actuation mechanisms used in upper limb prosthetics , They broadly characterize actuators into the following groups: conventional, hydraulic, and non-conventional ,
2.3.1 DC Electric Motors
Falling into the conventional category are various types of DC electric motors including brushed and brushless motors, micromotors, and servomotors ,
Commercial hands and fingers that use electric motors include the bebionic hand , the Michelangelo hand , the i-Digit/i-Limb system , the Taska hand , the Vincent hand , Myoelectric Speed Hands , System Electric Greifer , AxonHook , ElectroHand 2000 , and Transcarpal Hand , The main reason electric motors are so ubiquitous is that they have good power density, low power consumption, use simple control circuitry, have repeatable actuation, are fast, and are well understood , However, the major issues are that they are heavy, rigid, not very anthropomorphic, and can be costly , While the issue of anthropomorphism can often be superficially addressed using a plastic or silicone shell over the prosthetic hand, because of the rigidity of electric motors the hand can still not mimic the flexibility and conformability of the human hand.
While hydraulic and pneumatic prosthetic devices were popular devices in the mid-20th century, they have largely fallen out of fashion , The only commercially available prosthetic hand that uses hydraulics is the Hy5 MyHand , Hydraulics are rarely used now due to their lower efficiency, higher energy consumption, lower power density, difficulty of precise control as compared to electric motors, and are prone to
leaks , They also suffer from a similar issue in terms of anthropomorphism due to their rigid structure.
2.4 Smart Actuators as an Alternative
Due to the issues outlined above with electric motors and hydraulic actuators,
there has been a lot of interest in developing new types of actuators. The primary issue most of these novel actuators attempt to address is the rigidity of traditional actuators, seeking to better mimic the flexibility and conformability of the human hand. While stiff actuators are good for control purposes, soft actuators can allow for better impulse responses and shock resistance. This section seeks to provide a brief discussion of a few different actuator technologies and a more in depth discussion of two actuator technologies that will be the focus of this paper.
2.4.1 Shape Memory Materials and Fluidic Muscles
Materials that exhibit shape memory can be deformed and then the original
shape can be recovered through the application of a particular stimulus. These materials are typically some type of polymer or a metal alloy. Electroactive polymers are a class of polymers that undergo a shape change in response to a change in electrical current , They have been used as artificial muscles in grippers and robotic arms, but suffer from limited strength and high actuation voltages , Dielectric elastomers are a particular type of electroactive polymer that have been explored for use in upper limb prosthetics , The primary issues impeding their adoption are durability, poor control, high voltage requirements, susceptibility to contamination, and limited strength , Contractile gels are another type of shape memory polymer that undergo volume changes in response to a stimulus, often pH changes or electric current , Unfortunately, they suffer from many of the same issues as electroactive polymers ,
Shape memory alloys (SMAs) are a type of metallic alloy that exhibit two solid state phases, the austenitic and martensitic phases, which can be transitioned between under the application of a stimulus , They are typically made of a nickel-titanium alloy or a copper based alloy -, The primary issues with SMAs in terms of prosthetic applications are efficiency, speed, difficulty of control, and power consumption , Despite these issues, SMAs have been explored in a variety of prosthetic applications -, While SMAs do hold some promise for prosthetic applications, the slow cycle time due to heat dissipation is a significant issue that cannot be solved by simply changing the material composition.
Although hydraulic and pneumatic actuators are rarely used in commercial devices, there has been a resurgence of research interest. Most of this focus is on non-conventional pneumatic actuators but some work has been done on minimal gas consumption traditional pneumatics , The non-conventional actuators generally fall under the umbrella of fluidic muscles, an extension of the traditional McKibben muscle. Soft pneumatic fluidic muscles have been used to create multi-articulating prosthetic hands with some success -, Liquid based fluidic muscles have also been explored for prosthetics, orthotics, and robotics , While these advances show some promise, the issues of power consumption and size still need to be addressed.
This section only briefly touched on many of the exciting advances in smart materials relevant to prosthetic applications. The following sections though, provide a more thorough review of the two smart actuator technologies of primary interest for this paper.
220.127.116.11 Liquid Crystal Elastomers
Liquid crystal elastomers (LCEs) are a type of polymer network with both crystalline and elastomeric properties that incorporates liquid crystals (mesogens) and can respond to different stimuli including heat and light , , LCEs have a long range liquid crystal order relating to the orientation of the mesogens, a short range liquid crystal order relating to the position of the mesogens, and a network architecture relating to how the mesogens are connected, see  for a more in depth review of LCE liquid crystal phases and network architectures. Both the structure of the LCE and the types of mesogens, spacers, and cross-linkers used to connect the mesogens determine the material and mechanical properties of the LCE. Depending on the synthesis procedure, the LCE can be formed in an unordered or polydomain state, meaning that the mesogens are not aligned together in any particular direction, or in a monodomain state that preserves mesogen alignment , , Regardless of the initial order of the mesogens, the LCE can transition between an ordered and unordered state upon the application of a stimulus , , LCEs have been shown to exhibit up to 400% recoverable strain , This transition between phases is what drives shape change, or actuation, of the material. In order to achieve quick and repeatable actuation, the LCE must be entirely monodomain, meaning that the mesogens must be directionally aligned or ordered ,The axial to cross-section anisotropy introduced by this ordering of the mesogens is what drives the shape change but it also means that the monodomain LCE will exhibit anisotropic material properties ,
Much of the recent work with LCEs has been on developing novel synthesis procedures and chemistries to allow for the easy creation of large batches of material, complicated shapes, and to allow for locking in of a monodomain state , In particular,
the two-stage thiol-acrylate click reactions developed by the Yakacki lab group have greatly expanded the viability of LCEs as actuators. These reactions allow for the easy swapping out of spacer molecules, mesogens, and cross-linkers as well as the ability to synthesize the LCE in one single reaction and then use a secondary cross-linking step to lock in a permanent mondomain if desired , -, These changes allow for significant control over the material and mechanical properties. Most of these reactions use a thermally responsive mesogen, typically RM257 or RM82, and thus the shape change is driven by a temperature change , The downside of thermal actuation is a slow cycle time as the material needs to cool back down below the transition temperature before a new contraction is induced. This issue presents a big challenge in terms of implementing LCEs as soft actuators for prosthetic applications. Despite this issue, LCEs have been explored for a number of other soft actuator applications -,
Luckily, by using a different mesogen, such as azobenzene, an LCE can be photo-responsive -, Using UV light to cause cis-to-trans isomerization of azobenzene has been shown to generate up to 400% recoverable strain in LCEs , Using light as the stimulus may possibly be able to avoid the slow cycle times of thermally responsive LCEs since UV light induces a conformational change in the mesogen instead of a change in the liquid crystal order. Unfortunately, the actuation speed is also influenced by the elastic modulus; increasing elastic stiffness increases coupling but slows the orientation dynamics and thus in order to achieve high speed actuation there will be a tradeoff with material stiffness , Another problem is the penetration of light into the LCE. Corbett and Warner have shown that there is a photo-
bleaching effect where, as light starts to penetrate it increases the transmittance of the
LCE and thus allows further penetration , However, this effect has only really been
demonstrated on thin sheets of azobenzene-doped LCE and thus may present a
problem when attempting to uniformly actuate other geometries. Overall, LCEs present
a promising candidate for use in prosthetic applications due to their ease of synthesis,
tunability of material and mechanical properties, and ability to be light responsive.
18.104.22.168 Twisted Coiled Polymer Actuators
Twisted coil polymer actuators (TCPAs) are a type of contractile actuator
consisting of one or more fibers twisted and coiled, see Figure 1, that contracts when
heat is applied , , TCPAs are made by hanging a weight on the end of the fiber
and then twisting the fiber while preventing the fiber ends from rotating , As the fiber
twists, the strain in the fiber grows until it reaches a critical amount at which the fiber
changes from a straight fiber to a coiled fiber.
Figure 1: (A) 0.71mm diameter nylon 6,6 monofilament. (B) TCPA made of the filament in (A)
The two most common materials for the fibers used for TCPAs are nylon 6,6
monofilament and carbon nanotube yarns, with nylon far and away being cheaper and easier to manufacture , , TCPAs work because the fiber materials used have large thermal anisotropy and thus contract in length and expand in radius when heated; when the fiber is coiled these two effects compound to produce more contraction in length by bringing the coils closer together , , This effect is particularly useful
since a straight nylon monofilament only contracts about 4% when heated compared to the roughly 30% contraction of a nylon TCPA , To produce the largest tensile actuation, fibers that have large thermal anisotropy, large thermal expansion coefficients, and lower tensile moduli are most useful; nylon 6,6 has been shown to strike the best balance of these properties , Fibers with higher moduli, such as polyethylene and Kevlar, will contract less but will sustain higher loads, thus there is a trade-off between load capacity and actuator stroke , One of the major advantages of TCPAs is that their actuation properties have been shown to be scale invariant , , Another advantage is that when multiple TCPAs are used in conjugation with each other, the force and strain are additive just like in natural muscle fiber recruitment , These advantages have made TCPAs a popular choice for artificial muscles in robotics and prosthetics applications -,
While TCPAs are inexpensive to make and have the mechanical properties appropriate for use as an artificial muscle, their primary limitation is cycle time. Since TCPAs are actuated using heat, there is a heat dissipation problem that prevents them from being able to produce full contractions in quick succession , , Various groups have tried to address this by using active cooling mechanisms such as hydrothermal actuation , , , TEC/Peltier devices , hydrogel coatings , and forced air ; while these generally improve the cycle time they are still nowhere near the cycle time necessary to be relevant for prosthetic applications. Even if these cooling mechanisms can significantly improve the cycle time, including these mechanisms makes the actuator heavy and bulky and thus not much better than DC electric motors and transmissions.
Another major issue with TCPAs is that their fabrication is not well characterized or understood, the focus has typically been on modeling and understanding actuation, as evidenced by , -, , What is known is that if a particular tensile load is applied to the fiber, then at a certain amount of twists, the fiber will coil. Anecdotal evidence is often presented describing that with too little load the fiber tangles and with too much it snaps , , Aziz and co-authors as well as Shafer and co-authors have both presented extensive analyses of the mechanism behind the torsional actuation that drives the linear actuation in TCPAs, but there is almost no analysis of the mechanisms behind the formation of the twisted coils , , Since the focus of most papers on TCPAs is actuation, the authors often do an inadequate job of explaining the experimental procedure used to fabricate TCPAs, usually not describing how the correct tensile load was determined , , -, The closest to an actual analysis of the fabrication parameters is [59, eqn. 5] and [5, eqn. 1],
Where: rc is the critical torsion o is the tensile stress E is the elastic modulus r is the fiber radius G is the shear modulus /? is the pitch angle
P = tan 1(2nrr) Equation 2
Where: r is the torsion
r is the fiber radius /? is the pitch angle
The issue with Equation 1 is that it involves the tensile load and thus in order to know how many twists per initial length are needed to induce coiling you need to know what tensile load to apply. Finding this tensile load experimentally through trial and error is what most researchers do currently. However, if researchers would like to explore materials other than nylon monofilament then there really is no starting point and significant effort is wasted in guessing the correct tensile load. Additionally, understanding why the fiber tangles, coils, or snaps at various loads could provide valuable insight into developing TCPAs with better actuation properties.
There is a clear need to develop a TCPA based on a fiber that does not require heat as the stimulus. Such a TCPA would potentially not suffer the cycle time issues that current TCPAs do and be more useful in prosthetic applications. As discussed in Section 22.214.171.124, LCEs present a promising candidate for this new fiber as they can be synthesized using azobenzene as the mesogen and thus contract in response to UV light. However, in order to begin to look at LCEs in the context of TCPAs, the fabrication process for TCPAs needs to be better characterized, see Figure 2. A mathematical analysis of the mechanics during fabrication is needed such that equations and procedures can be developed to actually calculate the appropriate tensile loads and twists to induce coiling in fibers of different materials.
Well Understood Poorly Understood Well Understood
(w/ Heat as Stimulus)
Figure 2: Outline of procedure to use LCEs as pre-cursor fibers for TCPAs. Green boxes denote parts of the process that are well understood and the red box indicates the part that is poorly understood, the twist insertion.
2.5 Mechanics and Dynamics of a Twisted Fiber in Tension
The fabrication procedure for making TCPAs involves applying an axial tensile load and torque at one end of the fiber while blocking the rotation of the other end of the fiber. Thus a fiber during this process is subjected to an axial tension and moment couple, Figure 3. This loading configuration has been extensively studied, though not in the context of TCPAs. Due to the thin nature of the fiber and the large deformations it undergoes during twisting, the best way to understand the fabrication process is using elastic buckling theory and treating the fiber as an elastic line (also known as an elastica).
Figure 3: Free-body diagram of a fiber during TCP A fabrication
Table 2: Variables used in all equations and models
T Tension (N)
M Torque (Nm)
P Fiber helix angle during twisting (rads)
T Fiber torsion in twists/initial length (rads/m)
c Torsional stiffness (Pa*m4)
B Bending stiffness (Pa*m4)
r Fiber radius (m)
i Second moment of area (m4)
Xc Unitless â€œcritical valueâ€ for determining loop â€œpop-outâ€
E, Et, E0 Elastic modulus, elastic modulus during twisting (Pa), Initial elastic modulus (Pa)
a Tensile normal stress (Pa)
D Axial end shortening normalized to initial fiber length (unitless)
Co Unitless parameter
J Polar moment of inertia (m4)
P Helix radius (m)
G Shear modulus (Pa). Defined by G = E/2(1 + v)
R End rotations normalized to initial fiber length (unitless)
6 Fiber twists normalized to length (rads/m)
V Poissonâ€™s ratio
Lf, Lo Instantaneous length (m), initial length (m)
Tshear Shear stress (Pa)
Â«t> Fiber twists (rads)
2.5.1 Elastica Theory
Elastica theory focuses on understanding the behavior of solid materials under loads by analyzing the strain , This study has a long history including contributions from noted scientists such as Galileo, Hooke, Euler, Kirchhoff, Navier, Kelvin, and many others , The problem of an elastic rod under tension and torsion was first tackled by Euler but it was not until Kirchhoff developed his theory of thin rods that elastic theory
could be applied for large deformations , Kirchhoff s key contribution is what is known as the â€œkinetic analogâ€ that relates the equilibrium equations for the centerline deformation of a thin rod subjected to only terminal loads to the equilibrium equations of a heavy rigid body rotating about a fixed point (essentially a spinning top) , Love provides a proof of Kirchhoff s â€œkinetic analogâ€ for the case of a thin rod under tension and torsion [68, Article 260],
Figure 4: Coordinate system used throughout this section. Note that 0 in the source figure has been replaced with /3. The e coordinate system is the global coordinate system and the d system is for the fiber cross-section, with d3 normal to the cross-section. Adapted from [7, fig. 1]Â© Kluwer Academic Publishers 2000.
The â€œkinetic analogâ€ can be used to understand the buckling of thin rods under tension and torsion. Buckling occurs when the rod loses its elastic stability and transitions from the twisted but straight geometry into some deformed geometry. Buckling is thus the proper term to describe the transition of straight fibers into coils during the fabrication of TCPAs. Love [68, Article 270] presents one of the first solutions to this problem [68, eqns. 36-37], for both the tension and torque required to buckle an elastic rod into a helix (otherwise referred to as a coil):
Where: T is the tension
C is the torsional stiffness r is the torsion /? is the pitch angle r is the fiber radius B is the bending stiffness
M = Czsin(P) + Bcos^ Equation 4
Where: M is the torque
C is the torsional stiffness r is the torsion /? is the pitch angle r is the fiber radius B is the bending stiffness
While these equations are useful, equations to determine the pitch angle and torsion are not given in terms of rotations and without this it is not possible to solve these equations. Love does however give a simpler condition for when the rod will reach the instability point based on a linear eigenvalue analysis of the equilibrium equations [68, Article 272, eqn. 44], Note that this equation has been slightly rearranged by letting the length go to infinity and changing the compression to tension; here M is the torque,
B is the bending stiffness, and T is the tension:
M2 = 487 Equation 5
Where: M is the torque
B is the bending stiffness T is the tension
Equation 5 was later derived by Greenhill using a different approach  and then again by Timoshenko and Gere , According to Equation 5, when M/^BT
reaches 2 the rod will lose elastic stability and buckle; this value is referred to as the critical load throughout the rest of this paper. While Equation 5 is useful for
understanding when an elastic rod will buckle, it does not provide any insight into what the post-buckling geometry will be. For fabricating TCPAs the desired buckling shape is a helical coil, but as discussed in Section 126.96.36.199 this does not always occur.
2.5.2 Modern Elastic Buckling Theory
Further analysis of the problem of a rod under tension and torsion was not done
until the late 1900s when the advent of computers allowed for the complicated systems of differential equations to be solved numerically. Coyne provides the first new comprehensive analysis of this problem, in the context of determining why undersea cabling forms loops and tangles and how to remove them , Using elastic theory and numerical integration he determines the different cable geometries given different
values of the dimensionless scaling factor Cc = -L^= [9, fig. 2],
The larger the value of Cc, the straighter the fiber; as Cc decreases a loop begins to form and this loop tightens as Cc decreases further , The buckling instability point was found to be when Cc = 0.707 ,While Coyne does not focus on helical geometries post-buckling, he provides a comprehensive analysis of determining the tension required to remove a loop that has formed in a cable. Fie particularly focuses on the importance of preventing slack from developing, as this is a primary cause of loop formation , The condition for popping out of the loop is given by [9, eqn. 72]:
Where: xc is a non-dimensional parameter r is the radius E is the elastic modulus / is the second moment of area T is the tension
This equation can be rearranged and solved for xc by multiplying the tension by the cross-sectional area to get the tensile stress (the rand area term from this cancel out /) to produce Equation 7:
Xr = ---
Where: xc is a non-dimensional parameter E is the elastic modulus
This equation can then be used to calculate the tensile stress required to â€œpop-outâ€ a loop in a cable. Through numerical integration, Coyne provides a relationship between xc and the dimensionless parameter Cc [9, eqn. 82]:
Xc = 1. 3 Equation 8
Where: Ccis a non-dimensional parameter
By using the Cc values from Figure 5 it is possible to generate a range of xc values to plug into Equation 7 to then calculate the minimum tensile load required to pull out a loop in a cable. Since Cc is normalized to the material and geometric properties this provides a useful approach determining the â€œpop-outâ€ stress for different fibers.
Figure 5: Equilibrium configurations of cable, obtained by numerical integration for parameter Cc from 0.1 to 0.9. Coordinates are normalized by El/T). Adapted from [9, fig. 2], Â© IEEE 1990
Coyneâ€™s analysis is expanded upon and generalized by Thompson and Champneys , They look at the more general case of trying to determine how the loading of an elastic rod determines the exact type of buckling it will undergo, broadly characterized as localized buckling (tangling) or helical buckling (coiling) [8, fig. 1],
M T -0â€”
Figure 6: Spatial localization of a stretched and twisted rod: (a) the trivial undeflected straight configuration; (b) the helical form which bifurcates from the trivial solution; (c) to (d) continuous quasistatic localization, observable under rigid loading; (d ) to (e) a dynamic jump, observed under rigid loading, (f) the final writhing that follows the self-contact at (e). Adapted from [8, fig 1] Â© JSTOR 1996.
The elastica theory and â€œkinetic analogâ€ are expanded upon by applying the dynamic phase-space analogy  to examine not only the elastic stability but also the dynamic stability of the system , Under this analysis, buckling is treated as a Hamiltonian-Hopf bifurcation where one solution is the helical solution suggested by Love  and the other solution is the localized solution suggested by Coyne ; [8, figs 7,9,10] show various plots of this bifurcation behavior. They identify three distinct possible post-buckling geometries: the Hi helix, the H3 helix, and the L3 localized solution , The H3 helix is the helix described by Love and represents what would be considered a full helical coil whereas the Hi helix is more like a sine wave superimposed over a straight fiber, similar to Figure 6c. Although they also use
Equation 5 to define the buckling point, they provide some new equations for the
normalized axial shortening, D, [8, eqn. 4.21] and normalized end rotation, R, [8, eqn. 4.22) that allow for a better understanding of how different loading conditions affect buckling:
n - o M
Lf â€” L â€” -^== Equation 9
Where: D is the normalized axial shortening M is the torque T is the tension B is the bending stiffness
R _ Vbt i 4bt
\fBT ~ B C
Where: R is the normalized end rotation M is the torque T is the tension B is the bending stiffness C is the torsional stiffness
These equations are used to evaluate different loading conditions, one of which is described as â€œpre-fixed dead tension with controlled rotationâ€; this is the exact loading condition used during fabrication of TCPAs. As the length of the rod increases, the localized mode becomes more energetically favorable [8, fig. 12], Through experiments on steel wire and rubber rods, they conclude that Hi is the first post-buckling geometry to appear, but is quickly replaced by the l_3 geometry as this is more energetically favorable; the H3 helix does not make an appearance , While these results are interesting, none of the described experiments were conducted under pre-fixed tension and controlled rotation so it is possible that the H3 helix only appears under those conditions. Another possible reason for the non-appearance of the H3 helix is that the
boundary conditions and finite length of the samples violates some of the assumptions of the model, as suggested by ,
The analysis of Thompson and Champneys was reformulated and expanded on by Van der Heijden and Thompson , This new analysis makes the following assumptions about the rod and loading conditions :
1. Linear elastic behavior
2. Rod is inextensible (no axial length change)
3. Rod is under only tension and torsion, no shear
4. Rod initially starts straight
5. Quasi-static loading
6. Isotropic (equal bending stiffnesses in the cross-section)
Starting from these assumptions, a coordinate system was created to describe the geometry (Figure 4), this was used to generate the relevant strain equations, the strain equations were combined to produce the equilibrium equations [7, eqns 18-20], and the equilibrium equations were solved using the â€œkinetic analogyâ€; see  for the full derivation. Ultimately, Equation 5 is again arrived at as the equation governing when buckling occurs , With this more rigorous analysis, they are able to better classify the buckling behavior as a supercritical Hamiltonian pitchfork bifurcation [7, fig. 4],
Figure 7: Overview of the solutions from the Hamiltonian pitchfork bifurcation arising from [7, eqn. 31]. Adapted from [7, fig. 4]. Â© Kluwer Academic Publishers 2000.
This analysis also introduces the concept of linking number, twist, and writhe to describe how the different post-buckling geometries are created , Twist refers to inplane deformations and writhe refers to out-of-plane deformations. The derivations in [7, Section 3] show that since the linking number must be preserved, once buckling occurs any additional twist input into the system must be converted to writhe and this writhe is what gives rise to the helical and localized buckling geometries , The other major contribution from this analysis is an equation that gives the radius of the post-buckling helix [7, eqn. 59]:
p = â€” sin(/?)( 1 + cos(fi)) Equation 11
Where: p is the coil radius
B is the bending stiffness M is the torque /? is the pitch angle
This is important because it has been shown that the ratio of fiber radius to coil radius in TCPAs is important for determining the load and stroke capacity ,
The analysis of  was tested experimentally using nitinol wires in  under loading conditions where the rotation and axial position were controlled while measuring the force and torque. Since these loading conditions do not match the conditions for fabricating TCPAs, only limited information can be learned. The relevant results are that with large rotations, there is the most deviation from the model and the authors note that this could be related to the boundary conditions or the breaking down of the assumptions of linear elasticity and non-shearability , Since fabrication of TCPAs involves very large numbers of rotations, these issues may also be encountered.
2.5.3 Mechanics of Yarns
Snarling is the term used in textile manufacturing to describe the localized buckling (tangling) discussed in the above sections. Many studies have looked at this behavior in the context of textiles as well at looking at the effects of tension and torsion on yarns -, Typically the focus though is on the removal of snarls and not on stabilizing a particular type of buckling. Hearle and Yegin compared experimental data on snarl removal for rubber and nylon fibers to a theoretical model that assumed linear behavior , They found that at low torsional stresses both materials showed good agreement with the model but large deviations occurred at large torsional stresses , For nylon, they note that twist induced snarls are a form of plastic deformation that cannot be totally recovered and that the viscoelastic properties of nylon under torsion cause large deviations from the linear model , Bennett and Postle studied the effects of tensile stress on torque in multifilament yarns and showed that there is a linear relationship between the tensile load applied to a yarn and the residual torque created
by it , This is caused by the twisting and movement of filaments within the yarn , Rao and Farris provide analytical and experimental evidence for the weakening of yarn modulus and tensile strength due to twisting for high performance yarns such as Kevlar and Spectra , They do note that twist also negatively affects the properties of nylon fibers, but that the low anisotropy of nylon means this affect can easily be accounted for by multiplying the elastic modulus by cos(/?)2 , While all of these works were done on multifilament yarns, the issues they bring up are important to be aware of as some of these effects may also be present for monofilaments.
CHAPTER III: MODEL DEVELOPMENT
The primary goal of this research was to be able to prescribe the loading parameters to induce helical buckling in nylon fibers and then see if this model can be extended to LCEs. As described in Section 188.8.131.52, TCPAs are fabricated by twisting a fiber under tension while preventing end rotation of the fiber. The 2 important fabrication parameters in this setup are the tensile load and the number of rotations. The tensile load dictates whether the fiber undergoes localized buckling (tangling) or helical buckling (coiling). The number of rotations dictates when the buckling will occur as buckling occurs when the critical load, Equation 5, reaches 2 and the rotations drive the torque part of this equation. The analytical work described in Sections 2.5.1 and 2.5.2 provides the foundation for modeling the fabrication process for TCPAs. The missing pieces that need to be developed are identifying the appropriate minimum tensile load to induce helical buckling and modeling the torque generated during twisted. Thus, the model focuses on three things: predicting the minimum tensile load needed to induce coiling, predicting the torque at buckling, and predicting the twists at buckling. Of additional interest is the modeling of fiber failure as snapping during fabrication has been noted.
Since the model was developed for nylon TCPAs, this chapter will only present the data and equations related to that. When the model is extended to LCEs in Chapter V, the data and equations relevant to the adapted model will be included. The model used to prescribe the loading parameters is a mixture of different equations adapted from previous works on elastic buckling as well some novel empirical models. Due to the inclusion of empirical models it is necessary to describe how the data were collected
and how samples were created in order to generate the data for those parts of the model. Thus, this chapter starts with a brief materials and methods section before proceeding to describing more of the model.
3.1 Experimental Equipment and Sample Creation
3.1.1 Fabrication Rig
In order to be able to consistently create TCPAs and to measure various properties during the fabrication process, a fabrication rig was created. The rig was designed to facilitate the measuring of the torque generated during twisting as well as the number twists imparted to the fiber. It was also designed to be as simple and as low cost as possible. Figure 8 shows the rig in its final form.
A bipolar stepper motor (SM-42BYG011-25 from Digi-key) was attached to the upper cross-member in the center. The stepper motor was controlled using an Arduino Uno with a half h-bridge driver chip (SN754410NE from Digi-key). A custom 3D printed grip was used to attach to the output shaft of the stepper motor and then clamp the end of the fiber. A similar custom grip was used to clamp the other end of the fiber and to attach to the crosshead. The crosshead is simply a wooden dowel with at attachment point for a hanging weight. Ball bearings in a bag were used to apply tension to the fiber as this allowed fine control of the tensile load. A 1500g load cell (TE Connectivity FS2050-0000-1500-G from Digi-key) was attached to one end of the crosshead in order to measure the force generated at the contact point between the crosshead and vertical support. The distance between the load cell center and center of the crosshead was 66.7mm and thus a torque could be calculated. A passive low pass filter circuit was added to filter the signal from the load cell.
Figure 8: Fabrication rig in final form with nylon monofilament prepared for twisting.
Multiple different custom grips for the fiber ends were experimented with to find
one that produced consistent results and minimized stress concentrations. Figure 9 shows the different grips used. The grips in Figure 9a simply clamp the fiber end between two smooth flat plates. They were easy to use and could fit many different fiber diameters but introduced stress concentrations at the clamping location. The grips in Figure 9b were based on filament style grips used in tensile testing and attempt to isolate the stress concentration effects of clamping by first having the fiber wrap many times around a mandrel. While this works very well in strict tensile testing, the introduction of twisting causes slack to develop in the fiber sections wrapped around the mandrel and this ultimately leads to the fiber tangling at all tensile loads. The grips in Figure 9c eliminated clamping altogether and used friction to hold the fiber in place. The
fiber end is woven back and forth through the plastic a couple times in one direction and then is woven back and forth a couple of times in a perpendicular direction before exiting through the center of the grip. This proved to work very well in terms of reducing stress concentrations but could not easily be adapted to different fiber diameters. Additionally, due to the nature of the weaving it was much harder to control things like final fiber length once both ends were secured in the grips. The grips in Figure 9d were purchased from McMaster-Carr with the intention being that clamping on more surfaces would reduce the stress concentration effects produced by the grips in Figure 9a. Unfortunately, the collet in these grips had angled teeth as opposed to concave teeth and thus actually cut into the fibers when clamped and tended to cut through the fibers during twisting.
Figure 9: Different grips experimented with to secure the fiber ends. Top row are grips that interface with the stepper motor output shaft and the bottom row are grips that interface with the crosshead. (A) Final grip design, fibers are clamped between the two flat plates of the grip. (B) Similar to filament grip used for tensile testing with nylon (see Section 184.108.40.206), where the fiber wraps around a mandrel multiple times before the end is secured between two flat plates. (C) Fiber end weaves in and out of plastic grip multiple times in two perpendicular layers to provide enough friction to avoid clamping entirely. (D) Miniature collet style grip used to adapt drill presses for very small drill bits (McMaster #: 30505A5). These grips are clamped into the grips in (A) to then interface with the fabrication rig.
Ultimately the grips seen in Figure 9a were chosen as these were the easiest to use, produced the most consistent results in terms of a fibers buckling behavior at the same load, and were the most adaptable to wide variations in fiber diameter. Flowever,
further study of the grip-fiber interface would be useful as the boundary conditions applied at the fiber ends play an important role in determining the buckling behavior.
3.1.2 Sample Creation/Preparation
Two different sizes of nylon monofilament were used for experiments, 0.71mm diameter (Berkley Trilene Model# BGQS50C-15 BG1) and 0.55mm diameter (Berkley Trilene Model # XTFS30-15). The diameter of each type of monofilament was measured along approximately 0.5m of length to get an actual measurement of the diameter to be used in calculations as opposed to the nominal diameter given in the product specifications. The 0.71mm diameter monofilament was the correct size while the 0.55mm diameter monofilament actually measured slightly smaller at 0.54mm.
Twisting fibers was performed by first calculating the desired tensile load in grams based on the initial unloaded fiber diameter and desired tensile stress. Anytime tensile stress is mentioned in proceeding sections it is also calculated in reference to the initial unloaded fiber diameter. Ball bearings were added to the bag to reach the desired load and the bag was then attached to the crosshead. A section of monofilament was then cut and secured in both grips. The grips were then attached to the stepper motor output shaft and the crosshead. At this point the fiber was positioned in the fabrication rig and under tension. The length of this tensioned fiber was measured and recorded. Twisting was then initiated until the fiber tangled, snapped, or completed coiling. At this point the number of revolutions and force measured during twisting were recorded in a spreadsheet.
As discussed in Section 2.5.2, the model developed by Van der Heijden and Thompson  uses the following assumptions:
1. Linear elastic behavior
2. Rod is inextensible (no axial length change)
3. Rod is under only tension and torsion, no shear
4. Rod initially starts straight
5. Quasi-static loading
6. Isotropic (equal bending stiffnesses in the cross-section)
Assumptions 3, 4, and 6 are easily met by using circular cross-sections fixed in the fabrication rig as shown in Figure 8. Assumptions 1,2, and 5 required some testing to see how well they would apply. In addition to these assumptions, it was assumed that the tensile load and torque were independent.
3.2.1 Confirmation of Key Assumptions
220.127.116.11 Linear Elasticity
Tensile testing was conducted to determine the stress-strain response of nylon monofilament using a Mark-10 ESM303 Motorized Test Stand with a Series7 Force Gauge. A 1500N load cell (Mark-10 MR01 -300) was used with a miniature filament grip (Mark-10 G1078).
Five samples each were tested at three different strain rates: 5mm/min, 50mm/min, and 100mm/min. The different strain rates were tested to see how strain rate affected the elastic modulus and ultimate tensile strength, as in general viscoelastic materials such as polymers tend to be stiffer and stronger at higher strain rates , Both 0.71mm and 0.54mm nylon monofilament were tested. Approximately 1N of preload was applied for each test. Data were collected at 20Flz.
Figure 10 shows the stress-strain response for 0.71mm and 0.54mm nylon monofilament at the three different strain rates. The behavior is broadly linear with the
exception of the â€œtoe regionâ€ at low strains; this is consistent with the behavior shown by  and , However, within this â€œtoe regionâ€ the behavior is quite linear and there is very little difference between the strain rates at these low strains. Although the behavior is not perfectly linear, it is well within reason to consider this behavior linearly elastic for the purposes of modelling. The mean and standard deviations for the elastic modulus for both the â€œtoe regionâ€ and the rest of the curve are found in Table 3. Since twisting is done with a dead tensile load, the 5mm/min â€œtoe regionâ€ elastic modulus was used for modeling purposes.
0 20 40 60 80 100
0 2 4 6 8 10 - +5 mm/min (0.71mm) +5 mm/min (0.54mm) +50 mm/min (0.71mm) +50 mm/min (0.54mm) + 100 mm/min (0.71mm) +100 mm/min (0.54mm)
Figure 10: Stress vs. strain for 0.71mm and 0.54mm diameter nylon monofilament at different strain rates. Error bars represent +/- one standard deviation for 5 samples. There is very little variation in modulus in the low stress range for different strain rates, while this variation grows as the stress increases. There are two regions that are highly linear; there is a â€œtoe regionâ€ at about <10% strain and then a region after this toe region. Inset shows zoomed in view of the â€œtoe regionâ€
Table 3: Mean and standard deviation for elastic modulus for 0.71mm and 0.54mm diameter nylon monofilament at 5mm/min
Diameter (mm) Mean Toe Region Modulus (GPa) Std. Dev. Toe Region Modulus (GPa) Mean Non-Toe Region Modulus (GPa) Std. Dev. Non-Toe Region Modulus (GPa)
0.71 1.08 0.0412 0.8523 0.0828
0.54 1.12 0.038 0.9625 0.0070
18.104.22.168 Strain Rate Independence
Assumption 5 states that buckling occurs under quasi-static conditions and thus it is important to know whether the viscoelastic effects exhibited by these polymers at
different strain rates influence the buckling behavior. Therefore, fibers were twisted at different strain rates on the custom fabrication rig. Five samples each were tested at 10 different strain rates, ranging nominally from 10 to 100 rpm. All tests were conducted with 0.71 mm diameter monofilament at 20MPa of tension. For each sample the initial length was measured prior to the initiation of twisting.
Figure 11 shows the critical load, Equation 5, for the 10 different strain rates. The critical load is consistent between different strain rates but buckling appears to occur slightly later in the lower strain rate samples. This can be explained by accounting for the fact that the lower strain rate samples were also much shorter. Looking at Figure 12, it is clear that even at the same strain rate shorter samples tend to buckle slightly later than longer samples.
2 0.5 0
0 1000 2000 3000 4000
Figure 11: Mean critical load for 0.71mm diameter nylon monofilament at 20MPa of tension and different strain rates. Each line represents the mean of 5 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error.
This qualitative analysis is supported by the quantitative analysis of Figure 13 that shows the mean and median critical load. In both of those cases, the distribution of the mean and median of the highest strain rate is inclusive of the mean and medians of
all the other strain rates. This indicates that there is no strain rate dependence at these strain rates. Further, the distribution of the means and medians is random.
Figure 12: Torque during twisting for 0.71mm nylon monofilament at different lengths. The tensile load for all samples was 20MPa. Each plot represents 5 samples. There is a slight shift from right to left of the buckling point as the length increases, this shift explains the shift in the strain rate plots from Figure 11 as the slower strain rate plots were also samples with shorter lengths than the higher strain rate samples.
1.05rad/s 2.09 rad/s 3.33 rad/s 4.17 rad/s 5.19 rad/s 6.18rad/s 7.14 rad/s 8.06 rad/s 9.01 rad/s 9.1 rad/s
Figure 13: (Top) Mean critical load at buckling for 0.71mm diameter nylon monofilament at different strain rates. Error bars represent the between sample variance and all experimental sources of error. All samples show okay agreement with the expected critical value and the distribution around the critical value is random. The error bar for 9.1 rad/s is inclusive of the means for all other samples indicating that there is no clear difference in critical load at these different strain rates. (Bottom) Median critical load. Each box represents 5 samples. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
3.3 Minimum Load Determination
Experiments were conducted to attempt to identify the minimum tensile load
needed to induce coiling and relate it to material and geometric properties. Using Equation 7 and Equation 8, and the values for Cc in Figure 5, tensile stress values were generated for 0.71 mm and 0.54mm diameter nylon monofilament. Originally the Cc value range was 0.1 to 0.707 but this was changed to 0.1 to 0.195 as Cc values higher than 0.195 produce tensile stresses close to the threshold for failure as determined by the Tresca failure criteria, see Section 3.5. The basic theory was that since higher tensile loads have been shown to pull a loop out and localized buckling starts with loop formation, Equation 7 and Equation 8 should be able to be used to find the right tensile load that allows for self-contact but prevents further loop formation. All tests were conducted on the custom fabrication rig at 100rpm. For the 0.71mm diameter nylon the following tensile stresses were tested: 2.35MPa, 5MPa, 10MPa, 12.5MPa, 13.7MPa, 14.5MPa, 15MPa, 15.5MPa, 16MPa, 16.5MPa, 17MPa, 20MPa, and 25MPa. For the 0.54mm diameter nylon the following tensile stress were tested: 12MPa, 12.6MPa, and 13MPa. The smaller range of values for the 0.54mm monofilament was because those tests were to confirm the findings from the 0.71 mm tests as opposed to searching for the correct Cc value. For all of these tensile loads except 25MPa, 5 samples were collected each for three different length ranges. The length ranges were categorized as short (~1-2in), medium (~5-1 Oin), and long (>1 Oin). The different length ranges were included to assess to examine whether the length effects described in Section 2.5.2 were present.
For each test, just the initial length and the result of twisting were recorded. The twisting results were categorized into broad categories: coil, tangle, tangle then coil,
snap, and other. Samples that coiled created a uniform helical coil along the entire length of the fiber. Samples that tangled underwent some variation of localized buckling and never succeeded in creating helical coils, see Figure 14. Samples that tangled but then coiled underwent localized buckling near the grips but then quickly transitioned to helical buckling. Samples that snapped failed prior to buckling. Samples in the other category typically either tangled and then snapped or had some combination of tangling and coiling along the fiber length.
Figure 14: Example of 0.71mm nylon monofilament undergoing localized buckling (tangling) at a tensile load below the critical threshold
The data generated from the above described experiments were used to determine the lowest possible tensile load at which pure coiling can be observed. This load is then used to calculate a critical value (xc), using Equation 7. This critical value can be used to calculate minimum loads for other types of similar fibers. From Figure 15, it is clear that 12.5MPa is when pure coiling is first observed and this corresponds to a critical value of 0.15 and a Cc value of 0.14. Coiling can be initiated below this threshold but only if the fiber has first undergone localized buckling at the grips. The variation in buckling behavior at different lengths is likely a result of the boundary conditions applied by the grips and some asymmetries and misalignments in the
fabrication rig that become more influential with longer fibers. The higher prevalence of tangling at longer lengths does support the claims of  that localized buckling is energetically preferred at longer lengths. Further investigation into this phenomenon would help to better characterize the fabrication process.
Figure 15: Buckling behavior of 0.71mm diameter nylon monofilament at different loads. Pure coiling does not occur until 12.5MPa which corresponds to a critical value of 0.15. Successful coiling is not achieved at longer lengths until 17MPa, and 20MPa seems to have the most overall success.
The critical value of 0.15 determined above was then used to calculate the critical tensile load for the 0.54mm nylon which had a slightly different modulus. The critical tensile stress found there was 12.6MPa. Figure 16 shows this to be accurate as pure coiling is not initiated until 12.6MPa. While the critical value of 0.15 seemed to hold for calculating the minimum tensile load, it did not work well for longer fibers. Thus 20MPa and its corresponding critical value of 0.19 was selected as a suitable tensile load for all further experiments.
14 'w' 12
= ID â€œD
Â£ 6 (/) c
â™¦ Tangle then Coil
0 100 200 300 400
Figure 16: Buckling behavior of 0.54mm diameter nylon monofilament at different tensile loads. No coiling occurs below the critical load of 12.6MPa.
3.4 Torque Generation Purina Twisting
The critical load to induce buckling is a function of the fiber modulus and radius
as well as the tension and torque applied. Since all previous work on TCPAs has used a dead tension, the only real variable to control is the torque via twisting the fiber. Thus, in order to be able to prescribe how much tension you need to apply and how many twists need to be added, it is critical to be able to accurately model the torque generated by twisting a fiber. The most basic way to do that is to use the torsion formula :
M = GIO Equation 12
Where: M is the torque
G is the shear modulus
/ is the second moment of area
6 is the rotations normalized to length
Small deformation models would consider 0 to be normalized by the initial length but since the number of rotations applied in this application is very large, using the initial length is not applicable and thus the contraction in length must be accounted for.
3.4.1 Length Contraction
The contraction/extension behavior of nylon in response to twisting was investigated to determine how to properly model torque generation as a function of
twists. Whether the fiber contracts or extends is also important for knowing whether the radius expands or contracts as this also impacts the torsion equation.
Tests were conducted using the custom fabrication rig. Five samples of 0.71mm diameter nylon monofilament were tested at 20MPa of tension and a nominal strain rate of 100rpm. A tape measure was attached to the vertical support of the fabrication rig and a video camera (Plugable 250x USB 2.0 Digital Microscope) was positioned such that the crosshead and tape measure were clearly visible. Video footage was recorded until just after buckling occurred. The initial length of each sample was measured prior to initiating twisting. The video was processed frame by frame to extract the change in position of the crosshead.
Figure 17 (left) shows the results of the above described experiment. These data were fit with a single term sinusoidal curve as seen in Figure 17 (right). A trigonometric function was selected as the best fit due to the pitch angle of the fiber chains being related to the length through the tangent function, see Equation 2.
LF = Lo(sin(Ar(0/Lo) + B) Equation 13
Where: Lf is the instantaneous length Lo is the initial length A is a fit parameter r is the fiber radius B is a fit parameter
Figure 17: (Left) Ratio of change in length for 0.71mm diameter nylon monofilament up until buckling. There are 5 total samples represented. (Right) Same data fit with a single term sinusoid, automatically generated using MATLAB. The mean correlation coefficient is -0.9103.
Although coefficients for the frequency (Ar), where r is the radius, and phase shift (6) could be automatically calculated and optimized for each set of data this would not be useful for a generalized model. To get a better idea of how A and B affect the fit of the curve, a grid of fits was generated varying both A and B. This grid is in Figure 18. As the frequency increases the curvature gets greater as expected. It would be expected that fibers of a larger diameter would contract in length faster since the pitch angle is a function of the radius. Thus, it makes sense to make the frequency coefficient a function of the radius as shown in Equation 13, 0.9 was found to be the best value for A through least squares regression. As for the phase shift, a value of approximately 1.4 seems to provide the best fit and this is equal to the relationship between the elastic modulus and shear modulus for a material with a Poissonâ€™s ratio of 0.4 such as nylon. The best and most generalized fit was therefore found to be the following equation.
L = L0(sin(0.9r((f)/L0) + 2(1+ u)) Equation 14
Where: Lf is the instantaneous length Lo is the initial length r is the fiber radius
A = A = A = A = A =
Figure 18: Variation in frequency (A) and phase shift parameters (B) for sinusoidal fit to length change. Each plot represents 5 samples. The x-axis for each plot is twists (rads/m) from 0 to 2500 and the y-axis is the percentage change in length as current length/initial length ranging from 0 to 1.1. Visually the best fit is found when A is close to 1 and B is close to 1.5; the quantitative best fit was found at A = 0.9, and B = 1.4.
The residuals for this fit are shown in Figure 19. From this data it is clear that nylon contracts in length and expands in radius during twisting and thus both of these effects must be accounted for in modeling the torque. If unaccounted for then assumption 2 will be violated.
0 500 1000 1500 2000 2500
Figure 19: Residuals from fit in Figure 17 (right). There is no clear pattern in the residuals indicating a good fit. The lone value on the far left is not an outlier; there just arenâ€™t any data points in that region to fill out the plot.
3.4.2 Torque Independence from Tensile Load
It is important to know whether the tensile load affects the torque. One reason is
that if they are not independent then prescribing fabrication parameters becomes much harder as the relationship between torque and tension would need to be modeled as well. The second is that the composite loading model for a cylinder under tension and
torsion, which is used to model failure in Section 3.5, assumes that they are independent. While it is reasonable to assume they are independent some researchers have shown that at least in multifilament and multi-ply yarns, applying tension can introduce a torque into the yarn , Since a common way to model a monofilament is as a yarn comprised of many polymer chains, this same behavior may be present as well.
Tests were conducted on the custom fabrication rig. Five 0.71mm diameter nylon monofilament samples were tested at 2.35MPa, 5MPa, 10MPa, 15MPa, 20MPa, and 25MPa all at a nominal strain rate of 100rpm. The initial length of each sample was recorded.
Figure 20 and Figure 21 show the torque and critical load for the test described above. The behavior to be expected is that the shape of the torque and critical load curves would be the same but that the buckling point would shift up and to the right on the torque curve as the tensile load increases. The reason for this is that since the critical load needs to reach 2 in order to induce buckling, as the tension increases the torque must also increase since tension is in the denominator of Equation 5. The right shift is because torque is generated as a function of rotational strain and thus to generate a higher torque, more rotational strain is required. This expected behavior is exactly what is seen in Figure 20 and Figure 21 and thus it is clear that the torque is independent of the tensile load. Only the samples at 15, 20, and 25MPa coiled, the other samples all tangled.
Figure 20: Torque during twisting for 0.71mm diameter nylon monofilament at different tensile loads. The y-axis for each plot goes from 0 to 0.05 Nm. As expected as the number of twists increases, the torque increases. As the tensile load increases the buckling point, where the torque reaches its maximum, shifts towards the right and shifts up.
Figure 21: Mean critical load for 0.71mm diameter nylon monofilament at different tensile loads. Each line represents the mean of 5 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error. The same buckling point shift as in Figure 18 is observed and for samples that did coil the critical load is consistent. The high variation in the 5MPa and 2.35MPa loads is primarily due to the load cell not staying in contact with the vertical upright during the entire process as the tensile load is not high enough to stabilize the fiber during twisting. Also note that the large peak at the right of the 5MPa curve is not actually where buckling occurred, buckling occurred at the smaller peak around 2,000 rads/m
3.4.3 Predicting Torque as a Function of Twists
Starting with the basic torsion equation, Equation 12, additions were made to
account for the length contraction, radius expansion, and possible weakening of the fiber modulus due to twisting. Figure 22 shows the progression from the simplest model to more complex models that incorporate more of these factors. The instantaneous length is incorporated by using Equation 14. The radius expansion is incorporated by
using the length change and Poissonâ€™s ratio to calculate the change in radius. As seen in Figure 22 neither the basic torsion equation nor the torsion equation with the instantaneous length and radius do a good job of matching the shape of the torque curve or the buckling point. The next addition is to include a factor to account for a possible weakening of the elastic modulus due to twisting. This effect has been noted in many yarns and fibers including nylon , , One such factor is from :
Et = EqCOS(
Where: Et is the elastic modulus during twisting Eo is the initial elastic modulus /? is the pitch angle
This factor is included in Figure 22c and clearly does a poor job of modeling the torque. This is interesting because  present evidence that this equation actually captures the modulus change for nylon quite well, so likely something else is missing from the model. Flearle and co-authors have suggested a more complete model of this modulus weakening that includes the effects of transverse forces in the yarn :
Et = E0(- +-cos(/?)2 + 3 cot(P)2ln(cos(P))) Equation 16
Where: Et is the elastic modulus during twisting Eo is the initial elastic modulus /? is the pitch angle
This adjusted factor was included in Figure 22d and does a very poor job of modeling the torque. Figure 22e uses a novel factor:
Et = 1.15E0sin(P) Equation 17
Where: Et is the elastic modulus during twisting Eo is the initial elastic modulus /? is the pitch angle
This proved to do a pretty good job at modeling the shape and buckling point in the torque curve. The 1.15 coefficient was found by calculating the mean correlation coefficient for all 5 samples of data in Figure 22 using different coefficient values and selecting the coefficient that generated the highest mean correlation coefficient. This maximum correlation coefficient was 0.9431.
Figure 22: Fitting of various functions for predicting the torque generated by 0.71mm diameter nylon monofilament at 20MPa. Each prediction stops when the torque reaches a value to satisfy the critical load of 2. In each plot the black dots are the experimental data for 5 samples and the red lines are the modeled torque. (A) uses the standard torsion equation with the initial length. (B) uses the standard torsion equation but uses the instantaneous length during twisting. (C) uses the same as above but replaces the elastic modulus with Equation 15. (D) replaces the elastic modulus with Equation 16 instead. (E) replaces the elastic modulus with Equation 17. The y-axis for each plot goes from 0 to 0.04 Nm.
Although the fit in Figure 22e is pretty good, a look at the residuals in Figure 23
shows that a better fit could probably be created but it would likely require an in-depth
study to characterize all the small effects of material properties and microscopic
geometry. Due to the small size of the residuals and the fact that they are mostly
distributed around zero, this fit is considered acceptable.
Figure 23: Residuals from the fit in Figure 20e. There is somewhat of a pattern in the residuals as up until about 1000 rads/m they are mostly random but then they follow somewhat of a parabola until spiking in the end. The spike at the end indicates the critical load for the data is lower than that of the model.
Overall the residuals are still very low.
While this model for the torque worked well for this load and this diameter of nylon monofilament it was important to verify that it also worked for different tensile loads and diameters. Figure 24 shows the same data as Figure 20 but this time with the theoretical torque plotted as well. The model does a pretty good job capturing the shape of the curve for all of the tensile loads and also does a good job of capturing the buckling point, though it tends to be a little high. Since it seemed to work best at 20MPa and that was the load used for other tests, the model was deemed acceptable.
Figure 24: Torque during twisting for 0.71mm diameter nylon monofilament at different tensile loads. The y-axis goes from 0 to 0.05 Nm. The model accurately predicts the torque generation behavior for different tensile loads even though the model does have the tensile load as a parameter. This again indicates that the torque is independent of the tensile load. Deviation from the model for the lower tensile loads is primarily due to the load cell not staying in contact with the vertical upright during the entire twisting process.
Figure 25 shows data for 5 samples of 0.71 mm and 0.54mm diameter nylon monofilament tested at 20MPa. The inclusion of the radius to scale the frequency
coefficient in Equation 14 and the inclusion of the radius in Equation 12 seem to do a good job of accounting for the effect of radius on the torque. It does overestimate the torque slightly for the 0.54mm diameter data and underestimate the critical load location but this could be due to the uncharacteristic dip in the data where the data drops below the initial offset. Looking at how well the model predicts torque generation at other diameters would be of interest in future work.
0.04 â– â–
z o_________;... .........,_____,______
Figure 25: Torque during twisting for 0.71mm and 0.54mm diameter nylon monofilament at 20MPa of tension. The model does a decent job of predicting the torque generation behavior for different radii. Although the model seems to overshoot the data for the 0.54mm diameter nylon, there is an uncharacteristic dip in the data.
3.5 Predicting Fiber Failure
Fiber failure was predicted using the Tresca failure criteria which states that when the principal stress exceeds the ultimate shear strength of the material, the material should fail , While technically Tresca and other similar theories are for the yielding of a material, they are also commonly applied for failure by substituting the ultimate strength for the yield strength. The loading condition for the fiber is seen in Figure 3 and features a torque couple and an axial tension. These two forces create a shear stress and a normal stress respectively, Figure 26.
Figure 26: Simple planar stress diagram
The principal stresses generated from these are:
4 ^ shear*
Where: o12 are the principal stresses o is the normal stress Tshearis the shear stress
The maximum allowable shear stress according to the Tresca theory is thus:
_ r7 i 2
Tmax ~ -J 4 ' ^shear Equation 19
Where: xmax is the maximum shear stress a is the normal stress Tshearis the shear stress
Since the shear stress is just a function of the torque and the torque required to reach the buckling point is determined by Equation 5, it is ultimately the tensile stress that determines when failure will occur. This is not saying that the failure is due to the tensile stress, just that the tensile stress is what drives the Equation 19 due to the buckling criteria. Substituting in Equation 5 into Equation 19 and simplifying terms give the
following expression for the maximum allowable tensile stress before failure should occur:
a = 2Vw2 + 64 E2 - 8E Equation 20
Where: rmax is the maximum shear stress o is the normal stress E is the elastic modulus
When the tensile stress is equal to this equation failure should occur just prior to buckling. The ultimate shear stress (USS) according to Tresca theory is equal to 80% of the ultimate tensile stress (UTS) ,
CHAPTER IV: MODEL VALIDATION WITH NYLON
The equations and models developed in Chapter III were tested to see how well they could predict the critical load and buckling point for nylon monofilament. The test conditions included looking at the effects of different tensile loads, different lengths, and different radii.
4.1 Experimental Methods
4.1.1 Test Procedures
22.214.171.124 Different Tensile Loads
The same procedure was followed as described in Section 3.4.2.
126.96.36.199 Different Lengths
The effect of sample length on the buckling behavior was investigated to determine whether when normalized to initial length fibers of the same nominal length buckle at the same number of twists and the same torque. All tests were conducted on the custom fabrication rig. Five samples each were tested at 4 different nominal lengths, 25mm, 50mm, 75mm, and 100mm. All tests were conducted with 0.71mm diameter monofilament at 20MPa of tension and a nominal strain rate of 100rpm. For each sample the initial length was measured prior to the initiation of twisting to account for small variations resulting from slightly different clamping.
188.8.131.52 Different Radii
One of the factors the model includes is the fiber radius, so experiments were conducted to investigate how well the model could predict the torque generation and buckling behavior of fibers made of the same material but with different diameters. All tests were conducted on the custom fabrication rig. Five samples of 0.71mm and 0.54mm diameter nylon monofilament were tested at a tensile load of 20MPa and a
nominal strain rate of 100rpm. The initial length of each sample was recorded prior to twisting.
Although the minimum tensile load required to induce helical buckling is of most importance, it is also useful to know the maximum possible tensile stress before the fiber fails. Equation 20 was used to determine the maximum allowable tensile stress. For 0.54mm diameter nylon monofilament, this stress came out to be approximately 40MPa so 5 samples each were tested at 40MPa, 45MPa, and 50MPa at a nominal strain rate of 100rpm. All tests were conducted on the custom fabrication rig. The initial length of each sample was recorded prior to twisting.
4.1.2 Data Collection
For every experiment the following data were collected: fiber diameter unloaded, fiber diameter under tension, initial fiber length under tension, tensile load, force during twisting, and number of revolutions. Of these, the force during twisting is the most difficult to accurately measure and thus the load cell was characterized in order to be able to include appropriate error bars in figures. The noise band, noise frequency, and frequency spectrum during twisting were all investigated. Figure 27 shows the noise band and frequency spectrum of the noise during a static zero load test.
101 â– â– ' -â– â€”â€”
5 I I â€”Noise Limitsl
0 10 20 30 40 50
50 100 150
Figure 27: (Left) Raw data from load cell with zero load. The noise band is +/- 9 grams. Although this maximum noise band is pretty large most of the data is +/-1.5 grams. (Right) Frequency spectrum of the zero load noise on the load cell. The peak is at 60Hz representing AC electrical interference, but otherwise the noise is random.
Besides zero load noise, it was also important to characterize the frequency spectrum of the force data during twisting. Noise in this data is a product of both the load cell zero load noise as well as noise created by the load cell wobbling while contacting the vertical support of the fabrication rig. Figure 28 shows the frequency spectrum for an example twisting test with 0.71mm diameter nylon monofilament at 20MPa of tension and a nominal strain rate of 100rpm.
Figure 28: Frequency spectrum for filtered force measurement during twisting with 0.71mm diameter nylon monofilament at 20MPa tension and a nominal strain rate of 100rpm. The 60Hz peak from AC noise has been filtered out and the high frequency noise has also been filtered out. The peak at just below 2Hz matches up well with the oscillatory frequency (as captured on video) of the fabrication rig crosshead due to a slight asymmetry and misalignment in the rig.
4.1.3 Data Analysis
Data collected from a twisting experiment was processed in MATLAB using
custom scripts. The first step was to filter the data to remove noise; this was done using a low-pass zero-phase shift filter and then a 10th order median filter. The cutoff frequency for the low pass filter was set to be equivalent to roughly one measurement
per every 10 degrees of rotation on the stepper motor. Figure 29 shows the filtering sequence for an example set of data.
0 1 .............
0 1000 2000 3000 4001 0
1000 2000 3000
1000 2000 3000
Figure 29: (Left) Raw data sample for 0.71mm diameter nylon at 20MPa of tension. (Center) Sample data passed through low-pass filter with zero offset corrected. (Right) Sample data passed through low-pass filter and 10th order median filter with zero offset corrected
Once the data had been filtered, the buckling point was then identified using a custom peak finding algorithm that searches for the first peak in the filtered data that is larger than all other previous peaks, is larger than the next peak, and occurs at a location equal to at least 30% of the maximum number of twists. This algorithm ensured that peaks generated by the load cell first coming into contact with the vertical support are ignored.
If data were being compared to the model then the relevant equations from the model were evaluated and appropriate comparisons were made. The peak values found using the custom peak finding algorithm were compared to the peaks values generated by the model. The peaks in the model were identified when the critical load, Equation 5, reached the critical value of 2. Box plots and bar charts were generated to compare the number of twists, torque, and critical load at buckling between the experimental data and model. For the twists and torque, a paired t-test was carried out at a significance level of 0.05 with the null hypothesis being that there is no difference between the data and the model. In this case, showing no statistical significance is good. While the t-test provides some quantitative measure of how accurate the model is, the very small sample size limits the power of this metric.
For the critical load, further plots were generated by re-sampling data sets in order to average together data points at equivalent twist values. These plots allowed for easy comparison between test conditions including all experimental sources of error as well as between sample variance. However, this re-sampling did not always capture the buckling point well.
4.2.1 Different Tensile Loads
Data from the torque independence tests described in Section 3.4.2 were used to also evaluate how well the model performed in terms of predicting the torque and number of twists at which buckling would occur. Figure 30 shows examples of the final buckling geometries for these tensile loads. Figure 31 shows a bar chart and box plot of the mean and median respectively of the critical load for different tensile loads. For the 3 loads, 15MPa, 20MPa, and 25MPa, that were high enough to induce coiling, the data show good agreement with the model and all three loads produced similar results. The data sets that did not have coiling occur, show worse agreement with the model and this is mostly because the tensile loads were not high enough to stabilize the crosshead during data collection and thus the recorded force from the load cell is less accurate. While the critical load does seem to increase with increasing tensile load, it levels off once the load is high enough to induce coiling.
Figure 30: Buckling behavior for 0.71mm diameter nylon monofilament at: (A) 2.35MPa (B) 5MPa (C) 10MPa (D) 15MPa (E) 20MPa (F) 25MPa
3---------â– --------â– --------.--------.--------â– --------â– --------1 3
Figure 31: (Left) Mean critical load for 0.71mm diameter nylon monofilament at different tensile loads. Each tensile load has 5 samples and the error bars represent the between sample variance and all experimental sources of error. Samples at 2.35MPa, 5MPa, and 110MPa all tangled while samples at 15MPa, 20MPa, and 25MPa all coiled. (Right) Median critical load. Each box represents 5 samples. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
Figure 32 shows a comparison with the model for the torque and number of twists at buckling. Overall, the model does quite well at predicting both the torque and the twists, and it is generally better for the samples that coiled as opposed to samples that tangled. The model does better at predicting the twists at buckling.
Figure 32: (Left) Mean torque at buckling for 0.71mm diameter nylon monofilament at different tensile loads. The mean torque for the experimental data is within one standard deviation of the mean torque for the model only for 10MPa. (Right) Mean twists (rads/initial length) at buckling for 0.71mm diameter nylon monofilament at different tensile loads. The mean twists are much closer to the model for tensile loads in which coiling was induced. The error bars represent +/- one standard deviation. Asterisks indicate statistically significant differences at the 0.05 level.
4.2.2 Different Lengths
Due to the observed effect described in Section 184.108.40.206 with shorter lengths tending to buckle later and the evidence that localized buckling is preferred at longer
lengths , further analysis of this phenomenon was required. Figure 33 shows the average critical load for the 4 different lengths tested. The right shift of the buckling
point as the length decreases is clearly observed. Figure 34 shows that this shift does not affect the actual value of the critical load very much. The longer lengths do tend to have a slightly lower mean and median critical load but the distributions for both overlap and thus there is no real difference.
Figure 33: Mean critical load as a function of twists per initial length for 0.71mm diameter nylon monofilament at 20MPa of tension at different lengths. Each line represents the mean of 5 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error.
25mm 50mm 75mm 100mm 25mm 50mm 75mm 100mm
Figure 34: (Left) Mean critical load for 0.71mm diameter nylon monofilament at 20MPa of tension and different lengths. Each length has 5 samples and the error bars represent the between sample variance and all experimental sources of error. The 75mm and 100mm samples show better agreement with the model, but all the errors are roughly the same. (Right) Median critical load. Each box represents 5 samples. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
Figure 35 shows comparison with the model for the torque and number of twists at buckling. These data show that the torque at buckling at longer lengths tends to be
slightly lower and that the twists at buckling tend to be slightly lower for longer lengths. The effect on the twists is more pronounced, with the longer samples shower better agreement with the model. This suggests that the boundary conditions and alignment and symmetry of the fabrication rig can significantly affect the buckling behavior. Thus, in future work it will be very important to have a more sophisticated fabrication rig with tighter tolerances and to characterize the effects of different grips.
^ 0.01 0
25mm 50mm 75mm 100mm 25mm 50mm 75mm 100mm
Figure 35: (Left) Mean torque at buckling for 0.71mm diameter nylon monofilament at 20MPa of tension at different lengths. (Right) Mean twists. The standard deviations are very tight for the 75mm and 100mm samples and show strong agreement with the model. The error bars represent +/- one standard deviation. Asterisks indicate statistically significant differences at the 0.05 level.
4.2.3 Different Radii
Figure 36 shows the average critical load for 0.71mm and 0.54mm diameter nylon monofilament samples at 20MPa of tension. As expected, the torque increases more rapidly in the larger diameter samples and thus the buckling point occurs sooner. Due to the re-sampling required to generate this plot, the 0.54mm diameter samples appear to have a much lower critical load but looking at Figure 37, the mean and median appear to actually be very close to 2. In fact, the 0.54mm samples show better agreement with the model than the 0.71 mm samples, though the distribution of the mean and median for the 0.54mm diameter samples is larger.
Figure 36: Mean critical load as a function of twists per initial length for 0.54mm and 0.71mm diameter nylon monofilament at 20MPa of tension. Each line represents the mean of 5 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error.
5 2 c
Figure 37: (Left) Mean critical load for 0.71mm and 0.54 diameter nylon monofilament at 20MPa of tension. Each diameter has 5 samples and the error bars represent the between sample variance and all experimental sources of error. (Right) Median critical load. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
Figure 38 shows comparison with the model for the torque and number of twists
at buckling. The 0.54mm diameter samples show better agreement with the model for
the torque at buckling while the 0.71 mm diameter samples show better agreement with the model for the twists at buckling. Overall, the agreement for both batches of samples
is pretty good.
Figure 38: (Left) Mean torque at buckling for 0.71mm and 0.54mm diameter nylon monofilament at 20MPa of tension.. The mean torque for the experimental data is within one standard deviation of the mean torque for the model for the 0.54mm diameter but not for the 0.71mm diameter. (Right) Mean twists at buckling. The mean twists for the experimental data is within one standard deviation of the mean torque for the model for the 0.71mm nylon but not for the 0.54mm. The error bars represent +/- one standard deviation. Asterisks indicate statistically significant differences at the 0.05 level.
4.2.4 Maximum Load Before Failure
Figure 39 shows the Tresca stress plotted as a function of twists for the three different loads tested. This shows how poor this failure model is for this particular situation as there is no clear separation in terms of Tresca stress for the samples that snapped versus those that did not. While Table 4 shows that a higher percentage of samples failed at the higher tensile loads above the critical threshold, the actual stress data does not explain why the samples failed. The samples that failed tended to do so exactly at the buckling point as expected and this buckling point was consistent between samples that failed and those that did not. Since the buckling point is consistent, the deformed geometry of each sample is consistent. Further investigation is needed to better understand the failure mechanism.
Figure 39: Tresca stress for 0.54mm diameter nylon monofilament at 40MPa (red), 45MPa (blue), and 50MPa (black). The magenta horizontal line is the UTS at 750MPa and the green horizontal line is the USS as 80% of the UTS, 600MPa. Solid lines in each plot plot represent fibers that coiled while dashed lines represent fibers that snapped. This model performs very poorly at predicting when failure occurs as no solid lines should be above USS and all dashed lines should be above it.
Table 4: Percent of samples failed for 0.54mm diameter nylon monofilament. For each tensile load, 5 samples were tested.
Tensile Load (MPa) Percentage of Samples Failed (%)
The use of elastic buckling theory combined with the empirical models for length contraction and torque generation provide a good foundation for beginning to understand the fabrication process of TCPAs using nylon monofilament. The core assumptions of the model appear to be true enough to describe the behavior of nylon. This work also provides insight into the many complicated issues that arise when trying to characterize the behavior of nylon during the TCPA fabrication process.
The results in Section 3.3 show that the work of  provides a good framework for approaching the problem of determining the minimum tensile load required to induce coiling. While the approach was partly empirical the calculated critical value did carry over to samples of a slightly different nylon monofilament (different diameter and modulus). This does indicate that at least for similar materials this critical value provides a good starting point for determining the minimum tensile stress. The tensile stresses found through this approach also agree with the trial and error results from , , The other nice result from this approach is that it provides a range of critical values that can be tested. This is particularly important in expanding this work to other materials as for new materials there are not published works upon which to base tensile loads.
However, the inconsistency in buckling behavior shown in Figure 15 demonstrates that the clamping conditions and fabrication rig symmetry and alignment have a big impact. The fact that longer samples tended to tangle or tangle first before coiling
indicates that the analytical results of  are supported. The torque plots in Section 3.4 also support the claims of  in regards to coiling being an energy minimizing process and the conversion of twists to writhe. The torque grows until buckling occurs and then dramatically drops and stays relatively constant. The initial drop indicates a reduction in strain energy and the constant torque post buckling indicates that additional twists are not increasing the torque, but being converted to helical coils.
The results presented in Section 4.2 show that the model is fairly good at predicting the behavior exhibited by nylon under different tensile loads, different lengths, and different diameters. Under all of these conditions, nylon does seem to buckle when the critical load reaches 2. For loads that are less than the minimum required to induce coiling, the critical load tends to be a little less than 2. This could be due to there not being enough tension to stabilize the crosshead and thus the load cell not being in good contact with the vertical support of the fabrication rig. For the samples where coiling did occur, the critical load is very consistent and slightly above 2. Part of the reason for this is that the identification of the buckling point is done based on the force output of the load cell and the peak that occurs in this data is from when the fiber snaps into the coil formation, the H3 helix; however, buckling according to the model occurs slightly before this when the fiber loses elastic stability and changes shape into the Hi helix. Thus, the fiber is passing through the critical value, and the Hi helix, where buckling first occurs and then snapping into the H3 helix. The difference in timing between these two points is very small though. Whereas Thompson and Champneys  were unable to observe or stabilize the H3 helix, the results of these experiments clearly show that under dead tension and controlled rotation, the H3 helix can be stabilized. Figure 40 shows this
characteristic coiling formation sequence, including the Hi and H3 helices. Another reason for deviations from the model is that TCPA fabrication takes place at very high torsional strains and as discussed in Section 2.5.2 and 2.5.3, both elastic buckling models and yarn models tend to perform worse at higher rotational strains due to the introduction of non-linear effects , .
Figure 40: Example 0.71mm nylon monofilament at 20MPa undergoing the characteristic coiling sequence. (A) Straight and twisted fiber. (B) H1 helix formation (C) H3 helix formation
In terms of predicting the twists and torque at buckling, the model does well. Although a statistically significant difference was found in some cases, the high level of sample variability and small sample size mean this result is not super useful. Looking at the figures presented in Chapter IV though, it is clear that the model scales across different tensile loads, lengths, and diameters to provide a good estimation of when buckling will occur. The better performance of the model at longer fiber lengths supports the claims of Thompson and Champneys , This also makes sense because in order to create Equation 5 from [68, eqn 42], the length was let go to infinity. Additionally, the length to diameter ratio for the nylon samples is very high and thus the infinite length assumption is more valid. The two primary limiting factors to the accuracy of the model are the inaccuracy of the torque generation equation and the clamping effects and
misalignment effects introduced by the fabrication rig. A more sophisticated model of the torque generated during twisting that better captures the curvature of the torque data and the location of the buckling point would improve the model. A study of exactly how the elastic modulus changes as a function of twists would be of interest in improving Equation 17. Redesigning the fabrication rig with tighter tolerances and to use a torsion load cell would likely also improve the repeatability of these experiments. An in-depth study of the grip-fiber interface would also help reduce variability. Multiple researchers have noted the sensitivity of elastic buckling to boundary conditions , but the clamping of fibers during TCPA fabrication has largely been ignored in published works. These results clearly demonstrate that how the fiber ends are fixed plays a critical role in whether the fiber snaps, tangles, or coils.
The results in Section 4.2.4 show that the simple model used in Section 3.5 to determine fiber failure is inadequate. While Table 4 does show that the number of samples that failed increased with increasing the tensile load, the reason for this was unclear based on looking at the Tresca stress values. The Tresca criteria seems to give a good starting point estimate as to when failure might occur, and this value is far enough above the minimum tensile load to allow for the creation of TCPAs at a large number of tensile loads. As with the formation of coils, the failure of the fiber is also greatly impacted by the clamping conditions. Before abandoning the Tresca criteria it would be useful to test multiple different nylon samples with different moduli and radii on a test setup that better controls for the clamping conditions. A good starting point for improving the failure model would be look at the work of Hudspeth and co-authors which looked at bi-axial shear and tension induced failure in Dyneema SK76 fibers ,
One of the key takeaways from this work with nylon was the importance of the fabrication rig being symmetric and aligned properly. Due to the presence of other sources of error during data collection, this issue was not discovered until after the data had been collected and analyzed. Figure 41 shows data for 0.71mm and 0.54mm diameter nylon monofilament at 20MPa, with the load cell contacting the left and then right uprights of the fabrication rig; the crosshead was simply rotated to accomplish this. The difference in the force values demonstrates that the rig was in fact asymmetric; it was tilted by 2 degrees. This tilt caused a counter torque to be applied by the hanging load that reduced the measured torque. Once the rig was straightened, the torque was re-calibrated and found to be correct. Isolating this effect allowed for the missing torque to be calculated and added back into the collected data and thus the data presented is more accurate. However, just based on the impact of this very small misalignment it is possible that other smaller misalignments and asymmetries could have gone unnoticed and introduced more variability to the data.
Figure 41: Force measured during twisting for 0.71mm diameter nylon monofilament (top) and 0.54mm nylon monofilament (bottom) at 20MPa with the load cell contacting the right (red) and left (blue) upright of the fabrication rig. There are three samples for each diameter and load cell location. The left side clearly measures a lower force indicating that the fabrication rig is asymmetric.
CHAPTER V: MODEL EXTENSION TO LCES
Chapter IV showed that the model developed was able to predict the necessary fabrication parameters to create TCPAs for nylon monofilament to an acceptable level of accuracy. As noted in Section 220.127.116.11, LCEs have a number of advantages over nylon as a potential precursor fiber for TCPAs. However, since no work has been done with LCEs in regards to TCPAs the first step was to characterize the fabrication of TCPAs using LCEs. This Chapter extends the model developed in Chapter IV to see if it applies to LCEs and can be used to prescribe the fabrication parameters. This extension involves re-verifying assumptions and empirical models and adjusting as necessary and then completing the same validation tests as described in Chapter IV.
5.1 Sample Creation
As the focus of this part of the study was on the fabrication of TCPAs from LCE fibers, a simplified thiol-acrylate synthesis routine was used for sample creation. No excess acrylate and no photoinitiators were included as the second stage photo-cross-linking was not needed since locking in a monodomain state was not important.
Isotropic polydomain nematic elastomer (i-PNE) main chain samples were selected as these could easily be synthesized at room temperature and the maintenance of mesogen order was not considered important for this application.
Chemical Name Amount (g) Description Supplier
RM257 (4-bis-[4-(3-acryloyloxypropypropyloxy) benzoyloxy]-2-methylbenzene) 6 Acrylate monomer (mesogen) Wilshire Technologies (Princeton NJ, USA)
BHT (butylated hydroxytoluene) 0.16 Free radical absorber to increase working time Sigma-Aldrich (St. Louis, MO, USA)
Toluene 1.8 Solvent to dissolve mesogen Sigma-Aldrich (St. Louis, MO, USA)
EDDET (2,2-(ethylenedioxy) diethanethiol) 1.6161 Di-functional spacer thiol monomer Sigma-Aldrich (St. Louis, MO, USA)
PETMP (pentaerythritol tetrakis(3-mercaptopropionate)) 0.3822 Tetra-functional cross-linking thiol monomer Sigma-Aldrich (St. Louis, MO, USA)
TEA (triethylamine) 0.08 Catalyst Sigma-Aldrich (St. Louis, MO, USA)
5.1.2 Synthesis Procedure
LCE samples were synthesized using the procedure described in  for making i-PNE samples with a 15%mol ratio of thiol groups between PETMP and EDDET. Refer to  for a full description of the procedure but in short the procedure was as follows:
1. Prepare mold, see Figure 42, by applying vacuum grease around all edges and then clamping mold halves together using binder clips
2. Combine RM257, BHT, and Toluene in glass vial
3. Place vial in oven at 80Â°C for approximately 15 mins or until RM257 is completely dissolved
4. Add EDDET and PETMP
5. Vortex solution
6. Add TEA
7. Vortex solution
8. Place vial in oven at 80Â°C and apply vacuum (-500 mmHg) until all air bubbles have been removed
9. Use glass pipette to transfer solution from vial into mold
10. Let solution polymerize in mold for approximately 24hrs at room temperature
11. Remove samples from mold and place samples in oven at 80Â°C for approximately 24hrs or until toluene is evaporated. This step can also be done under vacuum to speed up the process.
Figure 42: Two-part LCE fiber mold 3D printed from PLA plastic. Mold contains cylindrical cavities of 6mm, 5mm, 4mm, 3mm, 2mm, and 1mm diameter that are 6in long. Only the 6mm and 5mm diameter cavities were used for sample creation.
5.1.3 Preparing Samples for Twisting
Twisting of fibers was performed by first calculating the desired tensile load in
grams based on the nominal fiber diameter and desired tensile stress. Ball bearings were added to the bag to reach the desired load and the bag was then attached to the crosshead. The LCE fiber was then cut if needed and secured in both clamps, the clamps were tightened such that the clamp faces were brought within roughly %â€ of each other. The clamps were then attached to the stepper motor output shaft and the crosshead. At this point the fiber was positioned in the fabrication rig and under tension. The length of this tensioned fiber was measured and recorded. Twisting was then initiated until the fiber tangled, snapped, or completed coiling. At this point the number of revolutions and force measured during twisting were recorded in a spreadsheet. Due
to the limited supply of LCE fibers, samples were then typically removed from the fabrication rig, released from the grips, and then their initial untwisted shape was reset by applying heat with a heat gun so that they could be tested again. A similar resetting technique has been used before with success ,
5.2 Confirming Assumptions and Empirical Models
Due to the different material and mechanical properties of LCEs, it was
necessary to confirm both the assumptions of the model and the empirical models generated in Chapter IV. Assumptions 3 and 4 were again easy to confirm. Assumption 6 regarding isotropy was not tested, the theory being that applying sufficient tension to align the mesogens would create sufficient isotropy. The anisotropy introduced by mesogen alignment  is more in regards to axial vs. cross-sectional symmetry as opposed to asymmetry within the cross-section. Assumptions 1,2, and 5 were again tested
5.2.1 Confirmation of Key Assumptions
18.104.22.168 Linear Elasticity
Tensile testing was conducted to determine the stress-strain response of LCE fibers using a Mark-10 ESM303 Motorized Test Stand with a Series7 Force Gauge. A 250N load cell (Mark-10 MR01 -50) was used with a wedge grip (Mark-10 G1061 -1). Two samples each were tested at the same three strain rates, 5mm/min, 50mm/min, and 100mm/min. Approximately 1N of pre-load was applied for each test and the radius at this pre-load was used to calculate the area. Data were collected at 20Hz. Testing was conducted at room temperature, approximately 20Â°C.
Figure 43 shows the stress-strain response of the LCE fibers. The behavior is very linear outside of the significant â€œtoe regionâ€ which is from about 0% to 80% strain.
This â€œtoe regionâ€ is typical of LCEs and represents the transition of the material from the polydomain state into the ordered monodomain state , The plateau in the â€œtoe regionâ€ represents this transitionary period and the highly linear region beyond this represents the fiber in the monodomain state; this is strain hardening behavior , For the purposes of modeling the modulus was taken outside the â€œtoe regionâ€ and at 5mm/min, the modulus was 0.3866MPa Â± 0.037MPa. This modulus is about half that reported by Traugutt and co-authors though they tested at 12mm/min and 30Â°C so a higher modulus in their results is expected , In the monodomain region it is absolutely applicable to say that the LCE fibers are behaving linearly elastically.
Figure 43: Stress-strain data for LCE fibers at 5mm/min, 50mm/min, and 100mm/min. A large â€œtoe regionâ€ is observed below 80% strain but is followed by a very linear region. The sharp changes in the 50mm/min and 100mm/min are due to one of the two samples failing well before the other. Inset is a zoomed in view of the â€œtoe regionâ€.
22.214.171.124 Strain Rate Independence
Three samples each were tested at the same 10 strain rates as described in Section 126.96.36.199. Each sample was subjected to 0.15MPa of tension based on an initial diameter of 5.75mm. Note that any time a diameter is given in this chapter it is referring to the diameter of the unloaded LCE fiber. Once the sample was clamped into the fabrication rig, the initial length and new loaded diameter were recorded.
Figure 44 shows the average critical load for the above described tests. The critical load is far above the expected value of 2 for all strain rates. There is no significant shift in the buckling point as these samples were all roughly the same length. With the exception of the 5.19 rad/s samples, the other samples for the other strain rates all exhibit the same behavior and thus there is little indication of a dependency on strain rate.
Â® 4 'c
1.05 rad/s â€”2.09 rad/s â€”1â€”3.14 rad/s â€”[â€”4.18 rad/s â€”Iâ€”5.19 rad/s â€”Iâ€”6.18 rad/s â€”Jâ€”7.15 rad/s â€”Jâ€” 8.09 rad/s â€”9.04 rad/s â€”[â€”9.4 rad/s
200 400 600 800
Figure 44: Mean critical load for 5.75mm diameter LCE at 0.15MPa of tension at different strain rates. Each line represents the mean of 5 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error. There is very little difference in the critical load at buckling for different strain rates. The critical load does appear much higher than the theoretical value of 2. There is no right shift in the buckling point as all samples were roughly the same length.
Figure 45 shows the mean and median critical load for these samples. All samples have a critical load well above the expected value of 2 but there is no clear pattern in the critical loads indicating no strain rate dependence. Flowever, unlike with the nylon samples in Figure 13, the highest strain rate is not inclusive of the medians of all the other strain rates, though its distribution does overlap with all the other distributions. All of this together indicates that there is no significant strain rate dependence.
1.05 rad/s 2.09 rad/s 3.14 rad/s4.18 rad/s 5.19 rad/s 6.18 rad/s 7.15 rad/s 8.09 rad/s 9.04 rad/s 9.4 rad/s
Figure 45: (Top) Mean critical load at buckling for 5.75mm diameter LCE at 0.15MPa of tension at different strain rates. Error bars represent the between sample variance and all experimental sources of error. All samples show poor agreement with the model but reasonably good agreement with each other. (Bottom) Median critical load for 5.75mm diameter LCE at different strain rates. Each box represents 3 samples. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
188.8.131.52 Torque Independence
Five 5.81 mm diameter samples were tested at 0.025MPa, 0.045MPa, 0.065MPa, 0.085MPa, and 0.105MPa. The initial length and loaded diameter of each sample were recorded. Figure 46 shows the torque as a function of twists for these different tensile loads. The torque clearly increases with increasing tensile load and the buckling point does shift to the right, but the shape of the curve is the same for each load. Even though increasing the tensile load does reduce the radius, since the LCE stretches, this
effect seems to not have a big impact on the torque. This reduction in radius should be accounted for though.
1000 Twists (rads/m)
Figure 46: Torque during twisting for different tensile loads for 5.81mm diameter LCE fibers. The y-axis for each plot goes from 0 to 0.025 Nm. As expected, as the number of twists increases, the torque increases. As the tensile load increases, the buckling point, where the torque reaches its maximum, shifts towards the right and shifts up. Although the buckling point moves, the torque behavior is consistent for each tensile load, simply scaling to the tensile load.
Figure 47 shows the critical load for these samples as a function of twists. The right shift of the buckling point is more clearly seen here. There is also a vertical shift of the buckling point and this is unexpected and does indicate that the torque may not be completely independent of the tensile load as the increased torque is not negated by the increased tensile load in Equation 5.
Figure 47: Mean critical load as a function of twists per initial length for 5.81mm diameter LCE fibers at different tensile loads. Each line represents the mean of 5 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error. The buckling point shifts up and to the right with increasing tensile load. The right shift is expected, but the vertical shift is not as the increased torque should be cancelled out by the increased tensile load.
5.2.2 Confirming Empirical Models
184.108.40.206 Minimum Load Determination
For the LCEs the same procedure was used as described in Section 3.3 to generate the possible tensile stresses. However, in this case the Cc values ranged from 0.1 all the way to 0.707 because the tensile stress at 0.707 was right around the tensile stress where failure was predicted using the Tresca failure criteria. The following tensile stresses were tested on 5.57mm diameter LCE fibers: 0.0043MPa, 0.03MPa, 0.06MPa, 0.087MPa, 0.085MPa, 0.09MPa, 0.1MPa, 0.15MPa, and 0.19MPa. For all of these tensile stresses only a single sample was collected, except for 0.085MPa and 0.087MPa which had 2 and 0.15MPa and 0.19MPa which had 5 samples. Due to the limited number of available samples, no length ranges were tested for these different loads like with the nylon monofilament.
Table 5 shows the buckling behavior for the samples tested. The data suggest that a tensile load of 0.087MPa is the minimum load necessary to induce helical buckling. This corresponds to a critical value, xc, of 0.67 and a Cc value of 0.55. This critical value is much higher than that of nylon and thus the model is not consistent between these two materials. Even though 0.087MPa seemed to be the minimum load, 0.15MPa produced the most repeatable results and thus was selected as the tensile load for further tests. Note that this load also puts the samples into the linear region of the stress-strain curve above the â€œtoe-regionâ€.
Table 5: Buckling behavior for 5.57mm diameter LCE samples at different tensile loads. The first tensile load to produce pure coiling is 0.087MPa and 0.15MPa is the most consistent.
Tensile Load (MPa) Buckling Behavior
0.085 Tangle, Tangle then coil
0.087 Coil, Coil
0.15 Coil, Coil, Coil, Coil, Coil
0.19 Snap, Coil, Coil, Coil, Snap
220.127.116.11 Length Contraction
Three 5.75mm and 4.8mm diameter fibers were tested at a 0.15MPa tensile load
and a nominal strain rate of 100rpm. Video footage was recorded until just after buckling occurred. The initial length of each sample and loaded diameter were measured prior to initiating twisting.
Unlike nylon which only contracted during twisting, the LCE fibers exhibited both contraction and extension with extension dominating the behavior, as seen in Figure 48. This means that the LCE fiberâ€™s radius contracts during twisting. Interestingly, the 4.8mm diameter samples which were much shorter initially exhibit both more contraction and more extension than the larger diameter samples. However, as expected, the larger diameter samples have a more rapid increase in length. Due to this new mixed behavior, it is no surprise that the original sinusoidal model for length contraction, Equation 14, does a terrible job of capturing this behavior, as evidenced in Figure 48. These results also indicated that the modelâ€™s assumption of inextensibility is
not valid; this is doubly true because the LCE fibers stretch out under the applied tensile loads before twisting even begins.
Figure 48: (Left) Ratio of change in length for 5.75mm diameter LCE fibers (black) and 4.8mm diameter LCE fibers (blue) up until buckling. There are 3 total samples represented for each diameter. (Right) Data fit with original sinusoid fit (red) for length change, Equation 14, with solid being for the 5.75mm diameter and dashed for the 4.8mm diameter. This model does not describe the extension and contraction of the LCE accurately.
A single sinusoid term model for this new behavior was found to be
unacceptable. The simplest and best model was found to be a quadratic model where
the first coefficient was related to the fiber radius. This new model is:
lF = L0((-l. 227 * 104r3 + 70. 55r2 - 0.133r + 8.25 * 10_5)(^-)2 - 0.00018((-^-) +
1) Equation 21
Where: Lf is the instantaneous length Lo is the initial length r is the fiber radius
The coefficients for the polynomial to make the first coefficient of the quadratic were found by performing a polynomial fit of the sample radii and coefficients generated from the original quadratic fit. The second coefficient in the quadratic was taken as the mean of the corresponding coefficients from the original quadratic fitting. The final coefficient was set to 1 as the fiber should start at 100% of its original length. This new quadratic fit is seen in Figure 49. While it is not nearly as nice as the way Equation 14 fit
the nylon data it is able to capture the wildly different behaviors of the two different diameter samples fairly well. While the residuals in Figure 49 are pretty poor overall, as seen in Figure 22 the instantaneous length only has a small effect on the torque generation equation so even an okay model of the length change should be acceptable. Further investigation would be needed to better model and understand the length change behavior of the LCE fibers during twisting.
CD 0 D
;g cn 0)
Figure 49: (Left) Ratio of change in length for 5.75mm diameter LCE fibers (black) and 4.8mm diameter LCE fibers (blue) up until buckling. There are 3 total samples represented for each diameter. Red is the new quadratic model for the length change, Equation 21. The average correlation coefficient is 0.8139. (Right) Plot of the residuals. The residuals for the 5.75mm diameter samples are all very small and randomly distributed whereas the residuals for the 4.8mm samples are large and exhibit clear patterns. Although this fit is not very good, as evidenced by Figure 20, the instantaneous length makes only a small difference in the torque generation equation and thus an okay fit is acceptable.
18.104.22.168 Torque Generation
The empirical model used to predict torque as a function of twists for nylon was tested on LCE fibers. The model was compared to data for 5.81mm diameter samples tested at 0.105MPa of tension. A similar progression of models was tested as in Figure 22. Figure 50 shows these different models. Figure 50a uses the torsion equation with initial length and radius, Equation 12. Figure 50b uses the torsion equation but with instantaneous radius and length, as calculated using Equation 21. Figure 50c uses the instantaneous radius but the initial length. Figure 50d uses the same torque model as used for nylon. Figure 50d uses the instantaneous radius, initial length, and changes the modulus to:
Et = 2.875E0sin(l. 750) Equation 22
Where: Et is the elastic modulus during twisting Eo is the initial elastic modulus /? is the pitch angle
0 500 1000 1500
Figure 50: Fitting of various functions for predicting the torque generated by 5.81mm diameter LCE fibers at 0.105MPa of tension. Each prediction stops when the torque reaches a value to satisfy the critical load of 2. In each plot the black dots are the experimental data for 5 samples and the red lines are the modeled torque. (A) uses the standard torsion equation with the initial length. (B) uses the standard torsion equation but uses the instantaneous length and radius during twisting as calculated using Equation 21. (C) uses the instantaneous radius but the initial length. (D) uses the same torque model as used for the nylon. (E) replaces the elastic modulus with Equation 22 and uses the instantaneous radius but initial length. The y-axis for each plot goes from 0 to 0.025 Nm. The shaded region indicates the region over which buckling occurred.
Even accounting for the different length change relationship for the LCE, the original torque model does a very poor job modeling the torque, both not capturing the shape of the curve and not capturing the buckling point. The new model does a better job of capturing the shape and buckling point, but is still not very good as evidenced by the residuals in Figure 51.
0 200 400 600 800
Figure 51: Residuals for torque equation in Figure 50e. The equation shows good agreement initially but then proceeds to get worse. Although the residuals get larger, the distribution is still mostly random, with most of them being negative.
While this new model for the torque worked okay for this load and this diameter of fiber it was important to verify that it also worked okay for different tensile loads and diameters. Figure 52 shows the same data as Figure 46 but this time with the theoretical torque plotted as well. The model does an okay job capturing the shape of the curve for all of the tensile loads and also does a good job of capturing the critical load, though it tends to be a little high particularly at lower tensile loads.
â€”0.025MPa '' r1 i
â€”0.045MPa I I i wi i
â€”0.065MPa 1 1 1
â€”0.085MPa 1 1 1â€”
â€”0.105MPa 1 1 1
0 500 1000 1500 Twists (rads/m)
Figure 52: Torque as a function of twists for different tensile loads for 5.81mm diameter LCE fibers. The y-axis for each plot goes from 0 to 0.025Nm. The model does an okay job of predicting the buckling point, particularly for the higher tensile loads, but does not do as well with predicting the torque at buckling. The shaded regions indicate roughly when buckling occurred.
Figure 53 shows data for 3 samples of 4.8mm and 5.75mm diameter LCE fibers
tested at 0.15MPa of tension. Since this model uses the initial length, the only inclusion of radius is in the initial torsion equation and this clearly is not enough to accurately scale the torque for the different diameters. The model greatly over-shoots the buckling
point for the 4.8mm samples and is also underestimating the torque. While it captures the buckling point quite well for the 5.75mm samples, the torque is greatly underestimated. All of this indicates that much more work needs to be done to characterize the torque generation during twisting for these LCE fibers.
~ 0 0
? 0.02 H 0.01 0
0 200 400 600 800 1000 1200 1400
Figure 53: Torque as a function of twists for 4.8mm and 5.75mm diameter LCE fibers at a tensile load of 0.15MPa. While the model does adapt to the change in diameter, the change is quite minimal. The buckling point for 4.8mm diameter is over-shot as the model predicts it to occur at about 800 twists whereas it actually occurs at closer to 600 twists. The model is much closer at predicting the buckling point for the 5.75mm diameter fibers but underestimates the torque.
The same validation tests that were performed with nylon were also performed with the LCE fibers.
5.3.1 Different Tensile Loads
Data from the torque independence tests described in Section 22.214.171.124 were used to also evaluate how well the model performed in terms of predicting the torque and number of twists at which buckling would occur. Figure 54 shows some examples of the final buckling geometry from these tests. While the 0.45MPa and 0.065MPa examples shown are coils, only 1 and 2 samples respectively from those test groups actually coiled while the rest tangled. Coiling was not observed at these lower loads in further tests on other synthesis batches so 0.087MPa was still determined to be the minimum tensile load. Figure 55 shows a bar chart and box plot of the mean and median respectively of the critical load for different tensile loads. All of the samples show about
the same level of agreement with the model, with samples that induced coiling having higher critical loads and samples that did not coil having lower critical loads. The agreement with the model is not as good as with nylon but is still okay. However, there is a clear trend of increasing the critical load with increasing tensile load and thus at higher tensile loads such as the 0.15MPa used in the tests later in this section, there is expected to be less agreement with the model. Unlike with nylon, there does not appear to be a leveling off of critical load once the tensile load is high enough to induce coiling.
Figure 54: Buckling results for 5.81mm diameter LCE fibers at: (A) 0.025MPa. (B) 0.045MPa. (C) 0.065MPa. (D) 0.085MPa (E) 0.105MPa
0.025MPa 0.045MPa 0.065MPa 0.085MPa 0.105MPa
Figure 55: (Left) Mean critical load for 5.81mm diameter LCE fibers at different tensile loads. Each tensile load has 5 samples and the error bars represent the between sample variance and all experimental sources of error. Samples at 0.025MPa, 0.045MPa, and 0.065MPa all tangled, with the exception of 1 sample at 0.045MPa and 2 samples at 0.065MPa that coiled, while samples at 0.085MPa and 0.105MPa coiled. (Right) Median critical load. Each box represents 5 samples. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
Figure 56 shows comparison with the model for the torque and number of twists at buckling. The model does a poor job of predicting the torque and twists at buckling, underestimating both. Interestingly, even though the torque at buckling increases with increasing tensile load as expected, the twists at buckling do not show as clear of an increasing trend. The 0.045MPa samples are the clear exception to this trend.
Figure 56: (Left) Mean torque at buckling for 5.81mm diameter LCE fibers at different tensile loads. (Right) Mean twists. The error bars represent +/- one standard deviation. Asterisks indicate statistically significant differences at the 0.05 level.
5.3.2 Different Lengths
Samples were tested at 4 different nominal lengths, 25mm, 50mm, 75mm, and
120mm. Each length had the following number of samples respectively: 3, 3, 5, 2. Each sample was subjected to 0.15MPa of tension based on an initial diameter of 5.75mm.
The nominal strain rate was 100rpm. Once the sample was clamped into the fabrication rig the initial length and new diameter were recorded.
Figure 57 shows the average critical load at the 4 different lengths. The 50mm, 75mm, and 120mm samples seem to show pretty consistent behavior whereas the 25mm samples have a higher critical load and the buckling point occurs further right. Just as with the nylon, better agreement is found at longer lengths.
0 200 400 600 800 1000 1200
Figure 57: Mean critical load as a function of twists per initial length for 5.75mm diameter LCE fibers at 0.15MPa of tension and at different nominal lengths. Each line represents the mean of 3, 3, 5, and 2 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error. A slight right shift in the buckling point is seen, but this is less pronounced than with the nylon.
Figure 58 shows that there is a trend of increasing critical load as the sample length decreases. This indicates that the boundary conditions applied by the grips as well as asymmetries and misalignments in the fabrication rig play an important role in determining the buckling behavior. Additionally the longer lengths make the infinite length assumption more valid due to an increasing length to diameter ratio.
25mm 50mm 75mm 120mm
Figure 58: (Left) Mean critical load for 5.75mm diameter LCE fibers at 0.15MPa of tension and at different nominal lengths. There are 3, 3, 5, and 2 samples for each length respectively and the error bars represent the between sample variance and all experimental sources of error. The 120mm samples show the best agreement with the model. (Right) Median critical load at different lengths. Each box represents 3, 3. 5, and 2 samples respectively. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
Figure 59 shows comparison with the model for the torque and number of twists at buckling. The agreement with the model is better and variance is lower for samples of a longer length. This data shows that the torque at buckling at longer lengths tends to be slightly lower and that the twists at buckling tend to be slightly lower for longer lengths, although the 75mm samples are clearly an exception. For both the torque and the twists, the experimental data has large variances for each length and the only length for which this variance does not include the model value is the 75mm length. As with the nylon, this indicates the boundary conditions applied by the grips and the alignment and symmetry of the fabrication rig is very important.
-p- 600 ~c/5
| 200 S
25mm 50mm 75mm 120mm
Figure 59: (Left) Mean torque at buckling for 5.75mm diameter LCE fibers at 0.15MPa of tension and at different nominal lengths. The mean torque for the experimental data is within one standard deviation of the mean torque for the model for all but the 75mm length. (Right) Mean twists (rads/initial length) at different lengths. The mean twists for the experimental data is within one standard deviation of the mean torque for the model for all but the 75mm length. The error bars represent +/- one standard deviation. Asterisks indicate statistically significant differences at the 0.05 level; no differences were found.
5.3.3 Different Radii
Three samples of 4.8mm and 5.75mm diameter samples were tested at 0.15MPa and a nominal strain rate of 100rpm. The initial loaded length and loaded diameter were recorded.
Figure 60 shows the average critical load for 4.8mm and 5.75mm diameter samples at 0.15MPa of tension. Interestingly, the smaller diameter samples buckle sooner and at a slightly lower critical load; this is the opposite of the behavior exhibited by nylon in Figure 34. Figure 61 shows that the difference in critical load is actually pretty negligible, as both diameter samples have a mean of approximately 4. The smaller diameter samples do show a larger variance though.
I ] 1 II â– Model â–¡Experimental \ li
25mm 50mm 75mm 120mm
â– Model FI Experimental |-
Figure 60: Mean critical load as a function of twists per initial length for 4.8mm and 5.75mm diameter LCE fibers. Each line represents the mean of 3 samples every 50 rads/m with error bars representing the between sample variance and all experimental sources of error.
Figure 61: (Left) Mean critical load for 4.8mm and 5.75mm diameter LCE fibers. Each diameter has 3 samples and the error bars represent the between sample variance and all experimental sources of error. (Right) Median critical load. The red line indicates the median critical load, the upper and lower black lines indicate the minimum and maximum critical load. The upper and lower blue lines represent the 75th and 25th quartiles. The notches represent a 95% confidence interval for the median, overlapping notches between groups represents that the median critical load for those groups is not significantly different.
Figure 62 shows comparison with the model for the torque and number of twists at buckling. Neither diameter shows very good agreement with the model for the torque at buckling but the 5.75mm samples do show very good agreement for the twists at buckling. Figure 62 also confirms that the smaller diameter samples did indeed buckle slightly sooner than the larger diameter samples.
4.8mm 5.75mm 4.8mm 5.75mm
Figure 62: (Left) Mean torque at buckling for 4.8mm and 5.75mm diameter LCE fibers. (Right) Mean twists at buckling. The mean twists for the experimental data is within one standard deviation of the mean twists for the model for the 5.75mm diameter but not for the 4.8mm diameter. The error bars represent +/-one standard deviation. Asterisks indicate statistically significant differences at the 0.05 level.
5.3.4 Maximum Load Before Failure
This test was not repeated for the LCE samples as the failure model was shown
to not provide very good prediction for nylon, see Section 4.2.4, and the predicted failure stress, 0.13MPa, for the LCE was below the stress found to work very well for inducing coiling.
Extending the model developed in Chapter III to apply to LCEs ultimately showed that this material is sufficiently different from nylon that many of the assumptions the model makes simply do not apply. While the linear elasticity and strain rate independence assumptions hold up okay, there does seem to be some dependence of torque on tensile load. The linear elasticity assumption may be more of an issue though since the tensile loads used are very close to the highly non-linear â€œtoe regionâ€. Performing more tensile tests to confirm the modulus across synthesis batches would help determine exactly where this â€œtoe-regionâ€ is located. The assumption of isotropy may also not be totally accurate and would require further investigation to verify. The alignment introduced by the tension may well be disrupted during twisting resulting in anisotropy. If such a disruption occurs it is not enough to change the opacity of the samples during twisting. Simply accounting for the different modulus and different radii was insufficient to scale the model to this new material. This is interesting because while the model equations do not predict the behavior very well the actual behavior of the fibers is very similar in terms of tangling at lower tensile loads, buckling into the Hi and then H3 helices at sufficiently high tensile loads, and then snapping at very high tensile loads. This characteristic coil formation sequence is shown in Figure 63.
Figure 63: Example of 5.75mm LCE fiber undergoing the characteristic coiling process at 0.15MPa. (A) Straight fiber. (B) Straight but twisted fiber. (C) H1 helix formation. (D) H3 helix formation
The results in Section 126.96.36.199 show that the length change of the LCE fibers during twisting is much different than that of nylon. Nylon only contracted in length and tended to buckle at around 80% of the initial length. The LCE fibers exhibited contraction initially and then extension, extending up to 3 times the initial length in some cases. This behavior was also highly dependent on fiber diameter, with larger diameter fibers showing much less extension. Due to this new behavior, Equation 14, used to model the length change did not apply and the new model, Equation 21, was only barely acceptable. This behavior also likely means the assumption of inextensibility the model makes is not valid. While the nylon fibers did change length, the change was quite minimal, whereas the change is significant with the LCEs.
With the length change relationship not carrying over to the new material the torque generation model also did not apply. As evidenced in Section 188.8.131.52, even accounting for the new length change relationship did not provide a good model of the torque. The adapted model does a much better job but still performs pretty poorly in
comparison to how well the original model did with nylon. Both the new length change behavior and change in the torque generation model demonstrate that the LCE fibers cannot be modeled as simply as the nylon fibers. Work must be done to characterize the architecture of the LCE network to see how it changes under torsion.
The critical value of 0.15 determined for nylon did not carry over for the LCE fibers. Despite this, the new critical value of 0.67 was still within the range of possible values originally generated and thus the methodology to find the minimum tensile load is found to be quite useful. This approach did allow for a rapid narrowing down of the appropriate tensile load.
With the torque generation model not being very good, it was expected that the results in Chapter V would not be very good and this proved to be the case. The results in Section 5.3.1 show that the critical load tends to increase with increasing tensile load. This is due to the torque term growing more than the tensile term in Equation 5. That equation also uses the loaded fiber radius prior to twisting and thus if it were to be replaced with the instantaneous radius the critical load would be even larger since the radius decreases during twisting. Although the critical values are not that much larger than the expected value of 2, the fact that there is a large difference between the critical values for the 0.85MPa and 0.105MPa loads indicates that the increasing trend is unlikely to stabilize as it did with nylon. Since the critical load is higher than the expected value the predictions for torque and twists at buckling are greatly underestimated.
The results in Section 5.3.2 show that the model works best for longer fibers, supporting the claims of  and agreeing with the results from the nylon testing. While
there is some agreement for the shorter fibers, the sample variance is very large. The critical load, torque at buckling, and twists at buckling are all closest to the model for the 120mm samples. This is actually consistent with the results in Section 4.2.2 that show strongest agreement with the model for the 100mm nylon fibers. Deviation from the model at the shorter lengths is again in large part likely due to the influence of the clamping conditions. Due to the softness of the LCE fibers, the clamps basically flatten the fiber ends and thus the fiber geometry between the clamps is not perfectly circular. This flattening effect may also contribute to model error since it assumes a perfectly circular cross-section in order to meet the isotropy condition. The deviation from the model at shorter lengths could also be more pronounced with the LCEs than with the nylon because the ratio of length to fiber diameter is smaller for the LCE fibers.
The results in Section 5.3.3 show consistent critical loads between the two different diameter fibers, but both critical loads are quite a bit higher than the expected value. The prediction of torque is very inaccurate and this is no surprise as Figure 53 showed that the torque model scaled to different diameters pretty poorly. Interestingly, despite the critical loads being higher and the torque being underestimated, the twists at buckling are actually decently accurate for the 5.75mm diameter. This is due to the fact the torque model does a good job of capturing the buckling point for the 5.75mm diameter samples as evidenced by Figure 53, even though the torque is underestimated.
Although many parts of the model did not carry over well to modeling the LCE fibers, the actual behavior of the LCE fibers indicates that the LCE fibers can be made into TCPAs, the fabrication process simply requires a more sophisticated model.
Understanding how twisting the LCE fiber changes the structure of the polymer network and how this change affects the torque generated is the most important part of the model that needs to be changed in order to get better predictions.