Thermomechanics of Shape Memory Polymers and Composites

Transcription

1 University of Colorado, Boulder CU Scholar Mechanical Engineering Graduate Theses & Dissertations Mechanical Engineering Spring Theroechanics of Shape Meory Polyers and Coposites Qi Ge University of Colorado at Boulder, Follow this and additional works at: Part of the Engineering Mechanics Coons Recoended Citation Ge, Qi, "Theroechanics of Shape Meory Polyers and Coposites" (212). Mechanical Engineering Graduate Theses & Dissertations This Dissertation is brought to you for free and open access by Mechanical Engineering at CU Scholar. It has been accepted for inclusion in Mechanical Engineering Graduate Theses & Dissertations by an authorized adinistrator of CU Scholar. For ore inforation, please contact

2 THERMOMECHANICS OF SHAPE MEMORY POLYMERS AND COMPOSITES by Qi Ge B. Sc., Tongji Univeristy, China, 26 M. Sc. Zhejiang Univerity, China, 28 A thesis subitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillent of the requireent for the degree of Doctor of Philosophy Departent of Mechanical Engineering 212

3 This thesis entitled: Theroechanics of Shape Meory Polyers and Coposites written by Qi Ge has been approved for the Departent of Mechanical Engineering H. Jerry Qi Martin L. Dunn Date The final copy of this thesis has been exained by the signatories, and we Find that both the content and the for eet acceptable presentation standards Of scholarly work in the above entioned discipline.

4 iii Ge, Qi (Ph.D., Mechanical Engineering) Theroechanics of Shape Meory Polyers and Coposites Thesis directed by Associate Professor H. Jerry Qi and Professor Martin L. Dunn This dissertation presents studies with a cobination of experient, theory, and siulation on a broad range of polyeric aterial systes (both established and eerging), that exhibit the shape eory (SM) effect. The goals of the studies are to understand coplex theroechanical behaviors of shape eory polyers (SMPs), develop constitutive odels to describe the behaviors and use the developed odels to advance the developent and deployent of SMPs. The general approach used in the studies is independent of the aterial syste and consists of: i) saple fabrication and theroechanical experients including dynaic echanical analysis (DMA), stress-strain behaviors and theral strain tests; ii) developent of icroechanicsinspired constitutive odels incorporate large deforation and anisotropy; iii) coparison of experient and theory and paraetric studies. The studies have resulted in any ajor/key findings: a) for shape eory elastoeric coposites (SMECs) consisting of elastoeric atrix and crystallizable fibers, a sophisticated theroechanical constitutive was developed to describe the shape eory behaviors based on the elt-crystal transition of the fibers. b) For triple shape polyeric coposites (TSPCs) consisting aorphous SMP based atrix and crystallizable fibers, a constitutive odel separately considering the glassy transition of the atrix and the elt-crystal transition of fibers, was developed to describe triple shape eory behaviors of the coposites. c) For printed shape eory coposites (PSMCs), the fabrication approach for anisotropic shape eory elastoeric coposites (ASMECs) was developed. A theroechanical constitutive odel was also built to describe the fiber orientation dependent shape eory behaviors.

5 iv CONTENTS CHAPTER I. INTRODUCTION Background One-way and two-way shape eory effects Tripe shape eory behavior Fabrications for shape eory coposites Models for shape eory polyers The objective of this dissertation...7 II. SHAPE MEMORY ELASTOMERIC COMPOSITES Introduction Materials and experients Materials DMA experients Shape eory behavior Theral contraction/expansion experient Effects of theral rates Model description Overall odel Crystallization and elting Mechanical behavior...25

6 v Effective phase odel (EPM) Theral contraction/expansion Results Mechanical behaviors under uniaxial loading Coparison between experients and siulations Effect of theral rates Effect of volue fractions Conclusion...46 III. MECHANISMS OF TRIPLE-SHAPE POLYMERIC COMPOSITES DUE TO DUAL THERMAL TRANSITIONS Introduction Materials and theroechanical behaviors Materials DMA experients Stress relaxation tests Uniaxial tension tests Shape eory behavior Model description Overall odel Viscoelastic behavior for the atrix Mechanical behavior of the fiber network Strain energies Results Paraeter identification Coparison between odel siulations and experient...62

7 vi Stress and strain energy analyses for the triple-shape eory behavior Stress and strain energy analyses for the one-step-fixing shape eory behavior Conclusion...68 IV. PRINTED SHAPE MEMORY COMPOSITES Introduction Materials and experients PolyJet polyer jetting Fabrication of printed shape eory coposites DMA tests Uniaxial tension tests Shape eory behaviors Theral strain tests Model description Constitutive odel for anisotropic coposites Paraeter identification Models for theral strain Theroechanics during phase evolution Phase evolution rule Results Shape eory behaviors Paraetric study Effect of theral strain Conclusion...94 BIBLIOGRAPHY...95

8 vii APPENDIX A. PREDICTION OF TEMPERATURE-DEPENDENT FREE RECOVERY BEHAVIORS OF AMORPHOUS SHAPE MEMORY POLYMERS...1 B. THERMOMECHANICAL BEHAVIORS OF A TWO-WAY SHAPE MEMORY COMPOSITE ACTUATOR...117

9 1 CHAPTER I INTRODUCTION 1.1 Background Environental activated shape eory polyers (SMPs) have been investigated intensively because of their capability to recover a large predefored shape in response to an environental stiulus, such as heat (Lendlein and Kelch, 22, 25, Liu, et al., 27, Xie, 21), light (Jiang, et al., 26, Koerner, et al., 24, Lendlein, et al., 25, Li, et al., 23, Scott, et al., 26, Scott, et al., 25), oisture (Huang, et al., 25), agnetic field (Mohr, et al., 26) etc. Copared with shape eory alloys, SMPs is of low density, inexpensive to fabricate and able to exhibit larger deforation in excess of 11% (Lendlein and Kelch, 22, 25). These advantages allow SMPs to be used in any applications such as icrosyste actuation coponents, recoverable surface patterns, bioedical devices, aerospace deployable structure, and orphing structures (Lendlein and Kelch, 22, 25, Liu, et al., 26, Liu, et al., 24, Thoas and Lendlein, 22, Tobushi,Hara, et al., 1996, Willias, et al., 1955, Yakacki, et al., 27). For environental activated SMPs, the echanis can be either cheical or physical. For exaple, for the photo-induced SMPs, the shape eory (SM) effects are ostly achieved by changing cheical structure to create or eliinate additional crosslinking under certain wavelength UV irradiations (Jiang, et al., 26, Koerner, et al., 24, Lendlein, et al., 25, Li, et al., 23, Scott, et al., 26, Scott, et al., 25); for the water driven SMPs, water absorbed in SMPs binds the polyer and weakens the hydrogen bonding, leading the significant decrease of glass transition teperature (Huang, et al., 25); for the SMPs activated by agnetic field, the SM effect is realized by incorporating agnetic nanoparticle in SMPs and inductive heating of these copounds in alternating agnetic fields (Mohr, et al., 26). The latter two cases are considered as indirect therally activated behaviors. Copared to other environental activated

10 2 SMPs, the therally triggered SMPs are the priary focus in the SMP research field and the ost widely used and investigated (Yakacki, et al., 27). 1.2 One-way and two-way shape eory effects For therally activated SMPs, a typical four-step SM cycle is used to achieve the SM behavior as shown in Fig. 1: At Step 1 (S1), the SM cycle is initiated by deforing the aterial at a teperaturet H, where TH Ttrans and trans T can be glass transition teperature, crystallization teperature or other phase transition teperatures. Then at Step 2 (S2), the aterial is cooled down to a teperature TL Ttrans while aintaining the load causing the initial deforation. At Step 3 (S3), the prograed shape is fixed after unloading at T L. At Step 4 (S4), the SM effect is activated by increasing the teperature above T trans and the initial shape is recovered by heating back to T H. It is pointed out that once the aterial recovers, it requires an additional prograing step for any further recovery. Therefore, this is referred to as a one-way shape eory (1W-SM) effect. Fig. 1 Scheatic showing difference between the one-way and two-way shape eory effects.

11 3 The significant drawback of 1W-SM effects is the SM cycle is not repeatable: one ore prograing step is required to exhibit another SM cycle, which would be restricted in soe applications. Recently, the SMP with the capability of exhibiting the two-way shape eory (2W-SM) effect was developed (Chung, et al., 28). For the 2W-SM behavior, as shown in Fig. 1, the aterial is defored (defored shape I) under a constant weight at T H ; then cooling to T L with the sae load, the stretch increase due to phase transition such as stretch induced crystallization (SIC), the aterial reaches a new defored state (defored shape II). Once it is heated back to T H under the constant weight, the aterial recovers to the defored shape I. The process is repeatable to switch defored shape I and II by cooling to T L or heating to T H, as long as the constant load is on. 1.3 Tripe shape eory effect SMPs capable of fixing one teporary shape are referred to as dual-shape SMPs. Recently, the triple-shape SMPs were developed, which are able to fix two teporary shapes and recover sequentially fro one teporary shape to the other, and ultiately to the peranent shape under heating (Bellin, et al., 26, Bellin, et al., 27, Luo and Mather, 21). Fig. 2 illustrates steps to achieve the tripe-shape SM effects: At Step 1 (S1), the saple is initially defored at a teperature T H, where T H is higher than T trans I and T trans II. Then at Step 2 (S2), the teperature is decreased to T L1, while aintaining the load leading the initial deforation. At Step 3 (S3), the first teporary shape is fixed at T L1. At Step 4 (S4), the saple is defored differently fro S1 at T L1. At Step 5 (S5), the aterial is cooled to T L2, while aintaining the load causing the deforation at S4. At Step 6 (S6), after unloading, the second teporary shape is

12 4 fixed at T L2. At Step 7 (S7), as heating back to T L1, the aterial recovers to its first teporary shape. At Step 8 (S8), the aterial returns into its peranent shape by heating to T H. Fig. 2 Scheatics of steps to achieve the triple-shape SM effect. 1.4 Fabrications for shape eory coposites In theral activated SMPs, the ost coon echanis is the second order transition fro the rubbery state to the glassy state (Nguyen, et al., 28, Qi, et al., 28a), which is accopanied by a large change in aterial odulus fro a few MPa to a few GPa. This large change in odulus offers the advantage of easy prograing as the teporary shape is fored by deforing the aterial at a high teperature where the aterial odulus is low. However, in

13 5 soe applications, especially in bioedical applications such as bioietic sensing and actuation, soft aterials are desirable. Recently, Luo and Mather (29) reported a novel approach to develop a shape eory elastoeric coposite (SMEC) by ebedding a crystallizable theroplastic fiber network into an elastoeric atrix (Luo and Mather, 29). In the SMEC syste the elastoeric atrix provides rubber elasticity, while the fiber network serves as a reversible switch phase for shape fixing and recovery through the crystal-elt transition. Recently, based on the fabrication of shape eory elastoeric coposites (SMECs) (Luo and Mather, 29), Luo and Mather (Luo and Mather, 21) introduced a new and broadly applicable ethod for designing and fabricating triple shape polyeric coposites (TSPCs) with well controlled properties. In the TSPC, an aorphous SMP (epoxy with Tg 2 C 4 C, depending on the precise coposition) works as the atrix providing overall elasticity and provides one phase transition teperature and non-woven PCL (poly(ε-caprolactone)) icrofibers produced by electrospinning and incorporated into the atrix provides the other phase transition teperature due to the crystal-elt transition of PCL ( T 5 C). Copared with ethods reported previously, the approach for fabricating polyeric coposites is quite flexible, since one can tune the functional coponents separately to optiize aterial properties, opening up the potential to design for a variety of applications (Luo and Mather, 21). A natural paradig for SMECs is then to control the shape eory behavior via the architecture of the fibers; indeed Mather et al. has deonstrated anisotropic shape eory behavior by creating coposites with aligned electrospun fibers ebedded in an elastoeric atrix. Instead of fabricating coposites through electrospinning, we pursue the general idea of control of acroscopic shape eory behavior via control of the coposite architecture by introducing a new paradig enabled by recent advances in digital anufacturing of ultiaterial architectures printed shape eory coposites (psmcs). We present a ethod based on three-

14 6 diension (3D) printing technique to fabricate coposite aterials with precisely-controlled fiber architectures that lead to coplex, anisotropic shape eory behavior. Using the 3D printing technique, a kind of psmc lainas referred to as anisotropic shape eory elastoeric coposites were fabricated, which consists of an elastoeric atrix and oriented PCL fibers that undergo a elt-crystallization phase transforation during heating or cooling, and exhibits anisotropic echanical properties and shape eory behaviors. The 3D printing technique can be used to fabricate psmc lainates with uch ore coplex, but highly-tailorable, behavior. 1.5 Models for shape eory polyers In the design of SMP applications, theoretical odels are desired to predict the SM behaviors. In the past, several odels were developed. For exaple, 1D odel by Liu et al (Liu, et al., 26), 3D finite deforation odel by Qi et al (Qi, et al., 28a)and a constitutive theory by Chen et al (Chen and Lagoudas, 28a, b) for SMPs. These odels attribute the SM effect to the foration of phases during cooling which serve to lock the teporary shape and the loss of the phases during heating recover the peranent shape, and only consider phenoenological for aorphous SMPs but no physical eanings. In aorphous SMPs, the SM effect is due to the glass transition where the shape defored at high teperatures can be fixed at low teperatures as chain obility decreasing draatically and the aterial enters the nonequilibriu state (glassy state). Several odels were developed based on this concept and include the early odel by Tobushi (Tobushi,Hara, et al., 1996) and recent odels by Nguyen et al. (Nguyen, et al., 28) and Castro et al. (Castro, et al., 21). In sei-crystalline SMPs, the echanis of SM effect is due to the crystallization of aorphous phases, and the shape defored at high teperatures is fixed at low teperatures where crystalline phases fored and serve as crosslinks to lock the teporary shape. Several odels (Barot, et al., 28, Ma and Negahban, 1995a, Ma and Negahban, 1995b, Negahban, 1993, Negahban, et al., 1993, Westbrook,Parakh, et al., 21) were developed. For detailed descriptions of kinetics for non-isotheral crystallization and elting, for

15 7 the triple-shape eory effect with dual transition teperatures and for SMPs exhibiting anisotropic shape eory behaviors, to the author s best knowledge, none odels have been reported to data. 1.6 The objective of this dissertation The objective of y research is to develop rigorous experient-verified odels to describe thero-echanical phenoena in new eerging shape eory polyers and coposites. The general approach used in the studies is independent of the aterial syste and consists of: i) saple fabrication and theroechanical experients including dynaic echanical analysis (DMA), stress-strain behaviors and theral strain tests; ii) developent of icroechanics-inspired constitutive odels incorporate large deforation and anisotropy; iii) coparison of experient and theory and paraetric studies. The studies have resulted in any ajor/key findings: a) for shape eory elastoeric coposites (SMECs) consisting of elastoeric atrix and crystallizable fibers, a sophisticated theroechanical constitutive was developed to describe the shape eory behaviors based on the elt-crystal transition of the fibers. b) For triple shape polyeric coposites (TSPCs) consisting aorphous SMP based atrix and crystallizable fibers, a constitutive odel separately considering the glassy transition of the atrix and the elt-crystal transition of fibers, was developed to describe triple shape eory behaviors of the coposites. c) For printed shape eory coposites (PSMCs), the fabrication approach for anisotropic shape eory elastoeric coposites (ASMECs) was developed. A theroechanical constitutive odel was also built to describe the fiber orientation dependent shape eory behaviors. This dissertation priarily introduces the studies of the different shape eory coposites. In Chapter 2, we discuss the theroechanical behaviors of SMEC. In the chapter, experients of aterial properties and characterization of SM behaviors are introduced, a three diensional (3D) theroechanical constitutive odel with kinetics of non-isotheral

16 8 crystallization and elting is developed and results are presented by treating both atrix and fiber crystals as incopressible neo-hookean aterials. In Chapter 3, the theroechanical behaviors of TSPC is presented, including experients for echanical and theral responses, characterization of triple SM behaviors, a one diensional (1D) theroechanical constitutive odel with glass transition of the atrix and crystal-elt transition of fiber network, and analysis of stress distribution on coponents of the TSPC. Chapter 4 introduces the fabrication process, anisotropic theroechanical properties and anisotropic shape eory behaviors for ASMECs. A 3D anisotropic theroechanical constitutive odel was developed to capture the anisotropic shape eory behaviors. In appendix, we also introduce a siplified odel for prediction of teperature-dependent free recovery behaviors of aorphous SMPs and a theoretical odel for the two-way shape eory coposite actuator.

17 9 CHAPTER II SHAPE MEMORY ELASTOMERIC COMPOSITE (SMEC) 2.1 Introduction In SMPs triggered by teperatures, the undefored saple is heated to TH T, trans defored to the desired shape, and then cooled to TL T trans. After unloading at T L, the saple is fixed into the teporary shape. Heating back to T H deploys the saple and recovers to the peranent shape. The ost coon echanis in therally triggered SMP is the second order transition fro the rubbery state to the glassy state (Nguyen, et al., 28, Qi, et al., 28b), which is accopanied by a large change in aterial odulus fro a few MPa to a few GPa. This large change in odulus offers the advantage of easy prograing as the teporary shape is fored by deforing the aterial at a high teperature where the aterial odulus is low. However, in soe applications, especially in bioedical applications such as bioietic sensing and actuation, soft aterials are desirable. Recently, several attepts have been ade to create new soft and elastoeric SMPs. Rousseau and Mather (23) developed a sectic-c liquid crystalline elastoer (LCE) exhibiting the shape eory effects based on the sectic-c-to-isotropic transition of the LCE (Ingrid A. Rousseau and Patrick T. Mather, 23); Weiss et al (28) introduced a new type of SMP consisting of an elastoeric ionoer and low ass fatty acids (FA) or their salts, where ionoer provided a strong interolecular bond between the FA crystal and the polyer, and FA was used to produce secondary network, where T trans was related to the elting point of the FA (Weiss, et al., 28). However, these two aterials involves soe custo synthesis, which presents an inherent challenge for large scale production (Luo and Mather, 29). Recently, Luo and Mather (29) reported a novel approach to develop a shape eory elastoeric coposite (SMEC) by ebedding a crystallizable theroplastic fiber network into an elastoeric atrix (Luo and Mather, 29). In the approach by Luo and Mather (29), a poly (ε-

18 1 caprolactone) (PCL; M w = 65g/ol, Siga-Aldrich, St Louis, MO), is first electrospun into a fiber network (or at). This fiber network is then iersed into a silicone rubber (Sylgard 184, Dow Corning Corp., Midland, MI) pre-polyer solution, which is then cheically crosslinked to produce a SMEC syste: Sylgard serves as an elastoeric atrix and provides rubber elasticity, while the PCL fiber serves as a reversible switching phase for shape fixing and recovery through the crystal-elt transition. In a thero-echanical cycle, the SMEC is heated to T H T trans ( T trans is T, the elting teperature of PCL), defored at T H, then cooled to T L T while aintaining the external load. During cooling, PCL undergoes a phase change trans fro a polyer elt to a seicrystalline polyer. After unloading at T L, the teporary shape is fixed due to the crystallized PCL. Heating back to T H, where PCL elts and the stress stored in seicrystalline PCL is released, the SMEC recovers its peranent shape. Meanwhile, during the sae SM cycle, Sylgard behaves as an elastoer with its entropic elastic odulus proportional to the teperature. Because the operational teperatures for a SM cycle are uch higher than the glass transition teperatures for Sylgard and PCL, the odulus of SMEC is a few MPa at T H Ttrans and ~1 MPa at L trans T T. The approach developed by Luo and Mather for Sylgard/PCL SMEC provides a paradig for developing a wide array of sart polyer coposites with different cheistries that utilize elt-crystal transition in polyers to achieve the shape eory effect. It is therefore iportant to understand the thero-echanical behaviors and to develop corresponding aterial odels. In this paper, a three-diension (3D) constitutive odel is developed to describe the shape eory behavior of the novel SMECs. This odel includes a kinetic description of nonisotheral crystallization and elting. The assuption that the newly fored crystalline phase is in an undefored state is used to track the kineatics of evolving phases. In order to iprove the coputational efficiency, the effective phase odel (EPM) developed by the authors is adopted.

19 Materials and experients Materials The SMEC was prepared using a silicone rubber (Sylgard 184, Dow Corning Corp., Midland, MI) as the elastoeric atrix and a poly (ε-caprolactone) (PCL; M w = 65g/ol, Siga-Aldrich, St Louis, MO) as fiber reinforceents. A two-step process in Luo and Mather (Luo and Mather, 29) was followed: PCL was first electrospun fro a 15 wt% chlorofor/dmf (volue ratio=8:2) solution. The resulting nonwoven fiber at was then iersed in a two-part ixture of Sylgard 184 (base: agent ixing ratio = 1:1) and in order to ensure coplete infiltration of Sylgard 184 into the PCL fiber at, vacuu-assisted infiltration process was conducted. After reoving the excess Sylgard 184 resin on the surface, the infiltrated Sylgard/PCL was cured at roo teperature for ore than 48 hours. The orphology of the Sylgard/PCL SMEC was studied by scanning electron icroscopy (SEM). The surfaces of the asspun PCL fiber at (Fig. 3a) and Sylgard/PCL coposite (Fig. 3b) clearly show that the infiltration was coplete with all the original voids occupied by Sylgard 184, while the fiber structures were preserved. More details about the aterial saple preparation were described in Luo and Mather (29) (a) (b) Fig. 3 Scanning electron icroscope (SEM) iages showing: (a) the surface of as-spun PCL fiber at (fiber diaeter =.69±.36µ); (b) The surface of Sylgard/PCL coposite. (Luo and Mather, 29)

20 DMA experients DMA test was conducted using a dynaic echanical analyzer (Q8 DMA, TA Instruents). The Sylgard/PCL SMEC (a rectangular fil) was loaded by a dynaic tensile load at 1Hz and the teperature was lowered at a rate of 2 C/in fro 12 C to -145 C. Fig. 4a and b show the tensile storage odulus and tensile loss tangent tanδ of SMEC and neat Sylgard, respectively. The transition teperatures were deterined by the peaks of tanδ. For the neat Sylgard, there are two distinct transitions observed in the given teperature range, Syl g which are attributed to the glass transition ( T C ) and the elt-crystal transition Syl c ( T 65.9 C ) (Clarson, et al., 1985, Mark, 1999). In contrast, the SMEC has four transitions: the glass transition of Sylgard ( Syl g T C ), the elt-crystal transition of Sylgard Syl PCL ( T 65.9 C), the glass transition of PCL ( T 49.5 C ) and the elt-crystal transition c PCL c of PCL ( T 27.2 C ). Since the elt-crystal transition of Sylgard (-65.9 C ) and the glass transition of PCL (-49.5 C ) are very close, there is just one obvious peak of tanδ of SMEC observed around -5 C. At the teperature of 5 C, the SMEC has a storage odulus of 12.7MPa and a low tanδ value of.6, indicating that the aterial is soft and has low viscosity. The storage odulus of the coposite drops to 1.4MPa at 8 C (above the elting teperature of PCL), which is lower than that of neat Sylgard (3.16MPa) at this sae teperature. g

21 13 (a) Fig. 4 Dynaic Mechanical Analysis (DMA) of the Sylgard/PCL SMEC and neat Sylgard: (a) Storage odulus and (b) tanδ as functions of teperature. (b) Shape eory behavior To exhibit the SM behavior, the Sylgard/PCL SMEC undergoes a four-step theroechanical loading-unloading cycle, as illustrated in Fig. 5. In Step 1 (S1), the SMEC is isotherally predefored (stretched in this paper) at TH T (the elting teperature of PCL). Step 2 (S2) cools the aterial to TL Tc (the crystallization teperature of PCL) whilst the stretching force is aintained to be a constant. After unloading (Step 3, S3), the SMEC is fixed at (or prograed to) the teporary shape at T L. The SMEC can aintain this shape as long as the teperature does not change. In the final step (Step 4, S4), the SM effect is activated by raising teperature to T H and the initial shape is recovered.

22 14 Fig. 5 The scheatics of the free/constrained recovery experient. The left-side part shows the thero-echanical loading-unloading cycle for free recovery. The right-side part shows the constrained recovery cycle. The SM behaviors of the Sylgard/PCL SMEC were investigated on the DMA achine. They are shown in stress-teperature-strain (Fig. 6a) and strain-teperature plots (Fig. 6b), under three different iposed noinal stresses (.1MPa,.15MPa,.2MPa). In S1, the saple was stretched at 8 C ( T H ) by gradually raping the tensile stress at a loading rate.1mpa/in to the predefined value (.1MPa,.15MPa, and.2mpa in three different cases, respectively). In S2, the teperature was decreased to 5 C ( T L ) at a rate of 2 C/in while the noinal stress was fixed. Fig 6a and b show, when lowering the teperature, the strain initially increased for.15mpa, and.2mpa cases, while decreased for the.1mpa case. At the teperature around 3 C, the strain dropped by ~1% due to crystallization of PCL. In S3, the stress was quickly released at 5 C. Here, whilst ost of the deforation was fixed, a slight contraction of the saple can be observed. In S4, the unconstrained strain recovery was triggered by heating back to 8 C at a rate of 2 C/in.

23 15 In order to further qualify the SM behavior, the fixing ratio as following (Luo and Mather, 29, Rousseau, 28): R f and the recovery ratio R r are used R f u u p 1%, Rr 1%, (1) u where and u are the strains before and after unloading at T L, p is the peranent strain after heat induced recovery and is initial strain at T H. The calculated R f and R r are listed in Table 1 for the three different iposed noinal stress cases. As shown in Table 1, the recovery ratios R r for the three cases are very high (>98%), indicating that the aterial has a high capability of recovery to the initial shape. On the other hand, the fixing ratios R f are ~8%, which is lower than these (>97%) reported by Luo and Mather (Luo and Mather, 29). In fact, the fixity of SMEC can be tuned by changing the Sylgard/PCL coposition. This will be discussed later. (a) Fig. 6 Shape eory behaviors of Sylgard/PCL SMEC with different iposed stresses: (a) Stress-Teperature-Strain plot showing the 3D shape eory cycles. (b) Strain-Teperature plot. (b) Table1. List of R f and R r.1mpa.15mpa.2mpa R f 84.% 88.4% 88.8% R r 98.2% 99.2% 99.7%

24 16 The constrained recovery experient (scheatically shown in Fig. 5) was also conducted. The saple was isotherally stretched (S1) at T H, cooled (S2) to T L, finally heated back (S4) to T H. During cooling and heating, the external load was aintained. Note that the only difference between free recovery and constrained recovery is that the constrained recovery does not have the unloading step (S3). Fig. 7a and b show the results of the constrained recovery experient with the iposed noinal stress.15mpa. In S1, the saple was first stretched at 8 C ( T H ) by gradually raping the tensile stress to.15mpa. At S2, the teperature was decreased to 5 C ( T L ) at a rate of 2 C/in while the stress was fixed as a constant. In S4, the stretched saple was heated back to 8 C at a rate of 2 C/in. The strain increased initially when increasing teperature. At about 6 C, the strain decreased with increasing teperature. At 8 C, the saple was recovered to the stretched shape in S1. In Fig. 7b, coparing S2 and S4, the crystallization of PCL started about 2 C lower than its elting, which was due to the hysteresis in crystallization and elting of polyers. (a) Fig. 7 The constrained recovery of Sylgard/PCL SMEC: (a) Stress-Teperature-Strain plot. (b) Strain-Teperature plot. (b) Theral contraction/expansion experient

25 17 The theral contraction/expansion test (Fig. 4a) was conducted using the DMA achine. A sall stretching force (1 1-4 N) was applied to ensure the saple was straight. Teperature was decreased fro 8 C to -1 C with the rate of 2 C/in and heating back to 8 C with the sae rate. In Fig. 4, the aterial first contracted linearly with coefficient of theral expansion (CTE) / C, and then dropped by about 1% at a teperature range fro about 25 C to 15 C. After the nonlinear dropping, the aterial contract with a low CTE / C. The theral strain at 5 C is -2.11%, which will be used for paraeter fitting. When heating back fro -1 C to 8 C, the aterial expanded roughly linearly and then ascended abruptly about 1% due to elting at a narrow teperature range fro about 5 C to 6 C. After elting finished, the aterial expanded alost linearly and closed the loop. Fig. 8 The experiental result of the theral contraction/expansion test Effects of theral rates In order to investigate the effect of theral rates on SM behaviors, four different theral rates (1 C/in, 2 C/in, 5 C/in and 1 C/in) with a.15mpa noinal stress were applied to free recovery experients (Fig. 9a and b). In a previous study (Westbrook,Castro, et al., 21), it was found that the ain factors affecting the teperature distribution within the saple are the size of the saple and the contact between the saple and fixture. For a cylindrical saple (1 in diaeter and height) with iproved fixture design, the axiu

26 18 teperature difference within the saple is 5.3 C for a theral rate of 1 C/in). Since the saple used in this paper is a thin fil under a tensile testing setup, we expect that the teperature difference within the saple is very sall. For the 1 C/in case and 2 C/in case, the curves show siilar trend and about 9% of deforation can be fixed after unloading. For the 5 C/in case and 1 C/in case, because the crystallization did not finish copletely, the 5 C/in case just held about 6% teporary shape and the 1 C/in case just fixed about 4% (Fig. 9b). (a) Fig. 9 Shape eory behavior with different theral rates (1 C/in (red), 2 C/in (blue), 5 C/in (black) and 1 C/in (green)): (a) Stress-Teperature-Strain plot. (b) Strain- Teperature plot. (b) 2.3 Model description Overall odel In this section, a continuu thero-echanical constitutive odel frae for the Sylgard/PCL SMEC is developed by treating atrix and fiber network as a hoogenized syste of ultiple phases. The atrix is taken as a conventional elastoer and the fiber network is taken to be an aggregate of elt and crystalline regions; it is a elt at high teperature and is an elastic solid at low teperatures. The evolution of the fiber aggregate is described by existing theories of crystallization. The goal is to develop a odeling approach for this class of aterials. At this

27 19 point, soe levels of detailed understanding are sacrificed in favor of a siple way to describe the coplex thero-echanical phenoena and therefore several assuptions are ade: (i) Since PCL elts behave like liquids, it can only carry hydraulic pressure under copressive conditions. In addition, since the PCL sits an interconnected network, soe PCL ay leak out of the SMEC under hydraulic pressure. To siplify the odel, we assue that PCL do not carry loads in its elting state. Future studies should consider the load carrying capability of PCL elts. (ii) Since the elting teperature of PCL is uch higher than its glass transition teperature, the relaxation tie of PCL elts is fast at the teperature near its elting teperature. Therefore, it is assued that the external loading does not affect the orphology of PCL crystals. (iii) During crystallization, PCL crystals are fored in a stress-free (natural) configuration (Rajagopal and Srinivasa, 1998a, b). PCL crystals fored at different ties have different deforation history. (iv) During crystallization, the deforation transfer is siplified by an averaging schee that assues both Sylgard and PCL crystals will undergo a sall deforation increent. (v) Since deforation on PCL crystals are liited (~1% tensile at S2 and ~1% copressive at S3) in a shape eory cycle, inelastic deforation of PCL crystals is not considered. (vi) Considering icrostructure of the Sylgard/PCL syste, the concept of stress concentration factors is introduced. For the sake of siplicity, stress concentration factors are assued to be constants.

28 2 Fig. 1 1D analogy of the constitutive odel for the Sylgard/PCL SMEC: The block on the top represents the theral stretch due to theral expansion and contraction. The left spring represents Sylgard atrix as an elastoer with odulus NkT. The right ultiple springs represents PCL crystals fored at different tie. When PCL crystallized, the switch turns on and PCL with odulus μ; when PCL elted, the switch turns off and odulus of PCL is negligible. The odel decoposes the deforation of the aterial into theral deforation and echanical deforation (as shown in Fig. 1): M T T M G G G FG, F G, (2) where G is the total deforation gradient, G M is the echanical deforation gradient and gives rise to stress, G T is the deforation gradient due to theral expansion. For the convenience of odel description, we set M F G and use F to represent echanical deforation gradient. The echanical contribution fro Sylgard atrix is represented by a hyperelastic spring in Fig. 1, because it is always in the rubbery state with a low tanδ (=.6) during the teperature cycle. The PCL network changes its crystallinity with teperature: When T>T, PCL elts do not carry load; when T<T c, PCL crystallizes and can carry load. Since crystallization of PCL is a continuous and tie dependent process, we further divide PCL crystals into sall phases fored at different ties with volue fraction c v i with a deforation gradient of F i. More details will be described in the next section of the 3D generalized theory.

29 21 For the Sylgard/PCL syste, the siplest odel arranges stresses on Sylgard and PCL in parallel (as shown in Fig. 1) and the total Cauchy stress total σ is given by: total Syl PCL vsyl vsyl σ σ σ, (3) where Syl σ and PCL σ are the stresses in the Sylgard atrix and the PCL network, v Syl and v PCL are volue fractions of the Sylgard atrix and the PCL fibers, which are deterined by fabrication and v v 1. Syl PCL In reality, the stress distribution depends on the icrostructure and the geoetry of how two phases spatially arranged. Directly adding up the stresses in the two phases with respective volue fraction is not the case. Dunn (1998) proposed the concept of stress concentration factors, which can be used to redistribute stresses/strains in the coposite by considering the real icrostructure (Dunn, 1998). Here by introducing stress concentration factors, the total Cauchy stress total σ ay be rewritten as: σ v σ v σ, (4) total Syl PCL Syl Syl PCL PCL where Syl and PCL are stress concentration factors of the Sylgard atrix and the PCL fibers. Based on Dunn (1998), SylvSyl PCLvPCL 1. For the sake of convenience, SylvSyl and PCLvPCL are denoted by v Syl and v PCL Crystallization and elting Since the crystal-elt transition iparts the SM effect into the SMEC, it is essential for the odel to capture the kinetics of crystallization and elting of PCL. The classic theory to describe crystallization kinetics is the Avrai s phase transition theory (Avrai, 1941, 1939, 194). The original derivations by Avrai was siplified by Evans with the raindrop theory (Evans, 1945), in which one considers the wave (fronts of crystals) created by a drop falling down

30 22 at tie τ propagates fro τ to t. The probability that i-th waves pass over point P is given by Poisson s equation (Ozawa, 1971): E i e E pi, (5) i! where E is the expectancy and is equal to the average nuber of such wave fronts. In the crystallization process, a point in elt state is the point that is not passed over by any growing front. Therefore, the valid probability is that of i (Ozawa, 1971): p e E, (6) In the case of crystallization, the expectancy E is equal to the volue fraction of crystals without ipingeent V(t) (Sperling, 26). Therefore, the relative volue fraction of aorphous phase is (Sperling, 26): 1 ( ) V() t t e, (7) where () t is the relative degree of crystallinity. For Vt, () one first considers the case where the nuclei all develop iediately upon cooling the polyer to the teperature of crystallization then expands it to the case of sporadic nucleation (Sperling, 26). In the first case, the nucleation density is a constant. The volue growth rate is: n dv () t Gt AN, (8) dt t where N is the nucleation density, G is the growth rate of crystals (for the isotheral crystallization, G is a constant at a certain teperature ), A and n are constants related to the diension of the crystal. For one-diension rods, A d 2 /4, n 1, d is the thickness of crystal doain; for two-diension discs, A 2 d, n 2 ; for three-diension spheres, A 4, n 3. Integrating Eq. (8) over t fro to t c (the tie point during crystallization): n V ( t ) BN Gt, (9) c c

31 23 and B A n. For the sporadic nucleation case, the nucleation density is a function of tie Nt (). Fro to d, the nuber of newly fored nucleus per volue is dn( ) d and the volue d fraction of these crystalline phases at t (t>τ) is dn( ) n B G t d d to t c (the tie in crystallization), the volue fraction without ipingeent is: tc dn() n V ( tc) B G t c d d. (1) Eq. (1) describes the isotheral crystallization process. In reality, non-isotheral crystallization ight be ore coon. For exaple, in the case of this paper, the teperature is lowered continuously to a teperature lower than the crystallization teperature. As the nucleation density and the growth rate of crystals are function of teperature during the nonisotheral crystallization, the nucleation density and the growth rate are the function of teperature and tie, expressed as N[T(t)] and G[T(t)] respectively. Eq. (9) can be odified as: t n c dn T () tc. (11) V ( tc ) B G T s ds d d For G[T(t)], Hoffan-Lauritzen expression is widely adopted to describe the growth rate (Lauritzen and Hoffan, 1973): U K g G[ T( t)] G exp exp[ ], (12) RT ( t) T T( t) Tf where the pre-exponential factor of crystalline growth rate G is alost independent of teperature (Angelloz, et al., 2, Isayev and Catignani, 1997). The first exponential ter of Eq. (12) reflects the chain obility in the elt, where U is the diffusional activation energy for the transport of crystallizable segents at the liquid-solid interface (Liu, et al., 21), R is the universal gas constant, T is the teperature about T 3K (Angelloz, et al., 2, Isayev and g

32 24 Catignani, 1997). The second exponential ter contains the secondary nucleation effect on the crystal growth front (Angelloz, et al., 2). In the ter, 21), T T T t is the undercooling, where K g is the nucleation constant (Liu, et al., T is the equilibriu elting teperature; f 2T t T T t is a correction factor used for the variation of the heat of fusion with teperature (Angelloz, et al., 2). The evolution of the nuber of nuclei can be epirically described by the following equation (Angelloz, et al., 2, Isayev and Catignani, 1997): N[ T( t)] N exp T T( t), (13) where N is pre-exponential factor of nucleation density and an epirical paraeter. Based on the understanding of kinetics of crystallization, the elting kinetics can be developed. During the elting processing, the volue of the existing crystals decreases until it becoes zero. Assue elting starts at tie t t, the volue fraction at tie t ts t during s elting can be described by: tce dn T tce t n V ( t) B G T u du C T u du d d t s, (14) where t ce is the oent when crystallization ends; in general, tce ts. C(t) is the rate of decrease of crystal size; in this paper, it is treated as a constant in ters of a low heating rate (2 C/in) and narrow teperature range of elting (~35 C to ~65 C). By using Eq. (7) with Eqs. (11)-(14), the relative degree of crystallinity t can be calculated. However, in reality, polyer crystallinity cannot reach 1%, so the crystallinity of polyers should be expressed as: v( t) H( t), (15)

33 25 where H is the saturated crystallinity of a polyer when under a certain condition. According to Kweon et al. (Kweon, et al., 23), for the PCL fiber network, under siilar condition of this paper, H 25%. Fig. 11 shows the siulations of kinetics of crystallization and elting of PCL. It is seen the observable crystallization starts at ~3 C and finishes at ~2 C. During elting, observable volue change starts at ~35 C and finishes at ~65 C. Fig. 11 The siulation of kinetics of crystallization and elting Mechanical behavior As described and shown in Fig. 5, a SM cycle of SMEC involves four distinct theroechanical steps. During cooling and heating steps, echanical deforation can be coupled with crystallization/elting processes. It therefore becoes a natural choice to describe aterial behaviors under two thero-echanical scenarios: echanical deforation during crystallization and echanical deforation during elting. In these two cases, we consider the one-way coupling case where crystallization/elting can affect the concurrent echanical deforation. The coupling in the reverse direction where echanical deforation affects crystallization/elting kinetics is not considered. This assuption is valid for two reasons: First, during crystallization, the aterial deforation rate is uch larger than the relaxation tie of

34 26 polyer elts so that any nonequilibriu deforation can be relaxed and thus echanical deforation will not influence crystallization. Second, during elting, echanical deforation in the seicrystalline PCL is relatively sall so that the elting kinetics will not be influenced. For convenience we take both the atrix and fiber network to be hyperelastic solids. In our case the atrix, Sylgard, is in its rubbery state during a shape eory cycle, while the PCL crystals, although linear elastic, can be idealized as hyperelastic. As such there exist free energy functions so that the Cauchy stresses can be obtained as: Syl Syl Syl Syl 1 Syl W ( F ) Syl T σ F J F Syl F F CPCL CPCL W ( F ) σ F F F CPCL F 1 J CPCL CPCL CPCL CPCL, (16) where the superscripts Syl and C-PCL indicate the quantities associated with Sylgard and PCL crystals, respectively; W is the free energy density function; σ is the Cauchy stress tensor; F is the echanical deforation gradient. It is noted that atrix, CPCL σ is the Cauchy stress in the PCL seicrystalline phases Syl σ is stress contribution fro Sylgard As shown in Fig. 11, crystallization and elting are continuous processes and functions of teperature and tie. Our basic assuption here is that when a sall fraction of polyer crystals is fored, it is in a stress-free state. In order to satisfy the boundary conditions, either overall or locally, however, this sall fraction will defor iediately. This stress-free state for new born crystals phases was referred as natural configuration by Rajagapol (Rajagopal and Srinivasa, 1998a, b), and was used by any researchers to study crystallizing polyers (Ma and Negahban, 1995a, Ma and Negahban, 1995b, Negahban, 1993, Negahban and Winean, 1992), rubbers with therally induced scissoring and reforation (Tobolsky, 196, Winean, 1999, 25, Winean and Min, 2, 23, Winean and Shaw, 28), shape eory polyers (Barot and Rao, 26, Barot, et al., 28, Qi, et al., 28b), and light induced activated polyers (Long, et al., 29). In this session, we deal with echanics during crystallization. In addition, we

35 27 will provide an efficient calculation schee later. During a SM cycle, crystallization can occur during cooling (S2) and unloading at low teperature (S3). As indicated below, the ethod in this section is also applied to the case where there is no crystallization, such as the high teperature loading step (S1). To better illustrate the physics and the echanics, we use an increental approach. Considering at tie t t, all PCL crystals are elted, and echanical deforation gradient is F. Since only Sylgard carries load at this oent, F F, and Syl total Syl σ v Syl σ F. (17) After tie t t, crystallization starts. At t t t, a sall aount of crystal phase fors fro c PCL elts with a volue fraction of v1. As stated above, we assue that this newly fored crystal phase is initially undefored and stress-free. Siultaneously, a echanical deforation ay occur. One reason for this echanical deforation could be due to echanical loading change during crystallization. For exaple, in a SM cycle, crystallization process can easily extend into S3, which is the unloading step at the teperature below the crystallization teperature. Another reason could be attributed to odulus change of Sylgard due to its dependence on teperature (entropic elasticity). The deforation and load transfer fro atrix to polyer crystals can be very coplicated (Ma and Negahban, 1995a, Ma and Negahban, 1995b, Negahban, 1993, Negahban and Winean, 1992); for exaple, they ay depend on the shape, spatial distribution of the crystals. Here, we siplify this deforation transfer by assuing both Sylgard and PCL crystals will undergo a sall increental deforation gradient F 1. Then, the deforations in the Sylgard and the PCL crystal becoe and the total stress is F Syl F1F, 11 1 F F, (18) σ v σ FF v v σ F. (19) total Syl c C PCL 1 Syl 1 PCL 1 11

36 28 Here, the subscript in F 11 represents the deforation gradient at the tie of the 1 st tie increent for the crystal phases fored during the 1 st tie increent. In order to better deonstrate increental process before and during crystallization, scheatics are shown in Fig. 12. Fig. 12 Scheatics of echanics during crystallization: At t t, all PCL is elted and only c Sylgard carries load. During crystallization starts at t t t, a PCL crystalline phase with v1 is fored in a undefored and stress-free state (Step 1); then, a echanical increental deforation gradient F 1 is applied to both Sylgard and PCL crystal, as the changing external load or the entropic elasticity of Sylgard (Step 2). As the crystallization proceeds, ore sall volue fractions of PCL crystals for. Each of the has no deforation in its new-born state but defors after its birth due to the reasons discussed above. Following the above discussion, at tie t t t, deforation gradient in the crystalline phase fored at tie t t i t can be expressed as: L i i1 j ji F F F F, (2)

37 29 where n L eans the ultiplicative operation goes toward left, or i i1 n L ; i n 2 1 j i1 F is the increental deforation gradient at t t j t when the j-th crystal fors; F i represents the deforation gradient at tie t t t for the PCL crystals fored at tie t t i t. Note that F i has its reference configuration occupying the spatial configuration at tie t t ( i 1) t, so does F i Sylgard atrix is. The deforation gradient in the The total stress is Syl L F FjF F1 F. (21) j1 total Syl c C PCL vsyl 1 vpclvi i i1 σ σ F F σ F. (22) Note that Eq. (22) can also be applied to the case where there is no new crystalline phase fored c by siply setting. v i During elting, the crystalline phases vanish gradually. According to the kinetics description of elting process, the portion of crystal doains that grows at a later tie elts first (as they are closer to the crystal-elt boundary). Assuing at the tie when elting starts ( t ts ), the total nuber of crystal phases (which is equal to the total nuber of tie increents during crystallization) is e. At tie assue that within t, c v e c t ts t, v elts. Here, for the sake of brevity, we e volue fraction of crystals elts. Concurrently to elting, a sall elt increental deforation gradient F 1 is induced. This deforation, for exaple, could be due to shape recovery in the heating step of a SM cycle. Following the sae assuption for echanical deforation during crystallization, we assue that the increental deforation

38 3 elt gradient F 1 is applied to all the phases, except of e -th crystalline phase, which just elted and cannot carry load. Therefore, the total stress is e 1 total Syl elt c C PCL elt vsyl vpcl vi i i σ σ F F F σ F F, (23) Following the sae arguent, at tie elts and the deforation gradient is increased by c t ts n t, the crystal with volue fraction v n 1 elt F n. The total stress is: e n total Syl elt c C PCL elt vsyl n vpcl vi n i i1 n elt L elt 1n Fi i1 e σ σ F F F σ F F F., (24) Depending on boundary conditions, Eq. (24) can be solved for applied, (load control) or for total σ if is elt F n given (displaceent control). elt F n if a constant stress is Effective phase odel (EPM) The echanics discussed above is coputationally expensive as it requires the deforation gradient in each fraction of crystals to be updated in each tie increent and is stored as internal variables. This can quickly exhaust coputer CPU tie and eory resources. In order to enhance coputational efficiency, Long et al (21) developed an effective phase odel (EPM) to capture the echanics of aterial with evolving phases (Long, et al., 21). In the EPM, behaviors of the new phases fored at different tie are cobined into an effective phase with an effective deforation. The effective phase and its effective deforation are continuously updated to account for the response of each new phase that would norally develop under the general loading conditions (Long, et al., 21). In this paper, the EPM theory is used to iprove the efficiency.

39 31 During crystallization at tie t t t, there are PCL crystalline phases during crystallization beginning at t t. In the EPM, it is assued that the echanical deforation in these crystal phases can be represented by an effective phase with a cobined volue fraction of v c c vi and an effective deforation gradient i1 F. The total stress in SMEC is therefore: t t t At tie total Syl ( ) c C σ v σ F v v σ PCL ( F ), (25) Syl PCL 1, a new crystalline phase with volue fraction v c 1 is fored. Accopanying this new crystalline phase, the SMEC undergoes a further deforation with F 1 and the total stress is now calculated as: σ total Syl 1 ( 1 ) c C PCL ( 1 ) c C PCL v σ Syl F F v PCL v σ F F v σ 1 ( F 1). (26) t t t, we have Applying Eq. (25) to the tie 1 σ v σ ( F ) v v σ ( F ), (27) total Syl c C PCL 1 Syl 1 PCL 1 1 where F 1 F 1F and F 1 F 1F. Coparing Eqs. (26) and (27), we have v σ ( F F ) v σ ( F F ) v σ ( F ), (28) c CPCL c CPCL c CPCL For a given F 1 and with a calculation of Jacobian, Eq. (28) can be solved using Newton- Raphson schee for the deforation increent in the effective phase F 1. The sae process can be applied to elting and is not discussed here. In order to help readers better understand the EPM theory, scheatics are presented in Fig. 13.

40 32 Fig. 13 Scheatics of EPM: at t t t, there is an effective phase with volue fraction and effective deforation F. At t t 1 t, a PCL crystalline phase with v c 1 c v is fored in a undefored and stress-free state (Step 1); then, a echanical increental deforation gradient F 1 is applied to Sylgard, PCL effective phase and the newly fored PCL crystalline phase (Step 2); at Step 3, the newly fored PCL crystalline phase is erged into t t t, there the existing PCL effective phase. If directly applying the EPM theory to c is an effective phase with volue fraction v 1 and effective deforation F Theral contraction/expansion Theral contraction/expansion in SMEC consists of theral contractions/expansions of Sylgard and PCL network, where the latter can be further attributed to elts, crystals, and volue changes due to crystal-elt phase transition. Here, we assue theral contraction/expansion is always isotropic, therefore: where 3 G T J T I T I, (29) T is the linear stretch ratio due to theral contraction/expansion.

41 33 Base on the theral contraction/expansion experient in 2.4, there are three region of the theral strain: (i) the high teperature linear CTE ( 1 ) region where PCL is copletely elted; (ii) the nonlinear strain drop or rise ( PT ) region corresponding to the crystal-elt transition of PCL; (iii) the low teperature linear CTE ( 2 ) region where PCL is crystallized. At high teperatures, where PCL elts copletely and SMEC contracts or expands linearly with 1, theral stretch is: T 1, (3) 1 T T H where T H is the teperature where the experient starts. During the crystal-elt phase transition, where c v increasing fro towards the saturated crystallinity H and the theral deforation can be expressed by cobining the linear and nonlinear theral behavior: where c H PT T 1 1 T T v T H, (31) T is the equilibriu elting teperature in Eq. (12) where crystallization or elting starts and H is the saturated crystallinity in Eq. (15). c At low teperature, where PCL crystallizes copletely and v, SMEC contracts or expands linearly with 2, theral stretch is: T T T T T 1 (32) 1 H 2 r PT where T r is a reference teperature corresponding to the tie increent where the crystallinity reaches H. Fig. 14 shows the coparison between experiental data and siulation. Overall, the odel captures the theral contraction/expansion behavior. The largest error occurs during heating fro 1 C to 5 C: The odel predicts a linear expansion whilst the experient shows a slightly nonlinear behavior. H

42 34 Fig. 14 Coparison of experient and siulation of theral contraction and expansion. 2.4 Results Mechanical behaviors under uniaxial loading Before reporting results, we briefly describe the above odel with siple aterial behaviors and under uniaxial loading conditions. For Sylgard tested, the axiu stretch of the saple was with 1.15 and the stress-strain behavior did not exhibit the non-gaussian type of behaviors. For PCL seicrystalline polyer, as will be shown later, the strain is low. For the sake of brevity, we assue that both Sylgard and PCL seicrystalline polyers follow the incopressible neo-hookean behavior with the strain energy function as (Treloar, 1958): Syl CPCL W NkT ( ), W ( ), (33) 2 2 where i are the applied stretches in the three principal directions, N is the chain density, k is Boltzann s constant, T is teperature, is the shear odulus of seicrystalline PCL. Under the uniaxial loading condition and considering the aterial incopressibility, the Cauchy stress of each phase is:

43 35 W ( ) W ( ) Syl Syl 2 NkT CPCL PCL 2 1, (34) 1 where λ is the echanical stretch in the uniaxial stretching direction and it is defined as λ=1+ε, ε is the strain. Note that for Sylgard, the stress is also a function of teperature T. In addition, the total stretch of the aterial should include the theral contraction/expansion, i.e, total T. during the SM cycle can be solved by general odel and the EPM theory is used to iprove the coputational efficiency. Before crystallization, only Sylgard carries load. Therefore, v (, T ). (35) total Syl Syl During crystallization, at tie t t t, a sall aount of crystal phase fors fro elting phase with volue fraction c v and induces a sall stretch increent. Deforation in the crystalline phase fored at any t t i t is: i i1 j ji. (36) The total stress is given by: total Syl c C PCL vsyl 1 T vpcl vi i1 i1,, (37) where and T is the teperature at t t t. The above equation can be solved according to the boundary conditions applied. For the case of stretch controlled deforation, the stress can be calculated; for the case where the saple is subject to a constant external force, or a constant noinal stress s, and the Cauchy stress can be solved by following ethods. total becoes s 1,

44 36 As long as the tie increent is restricted to be sufficiently sall, can be rewritten as: 1, (38) where 1. The Cauchy stress for Sylgard and PCL crystalline phases ay be Taylor expanded to first-order in : Syl Syl Syl Syl ,, CPCL CPCL CPCL i1 i1 i1 i1 CPCL C PCL 1, (39) where. Inserting Eq. (39) into Eq. (37), the total Cauchy stress would be rewritten as: Therefore, 1 Syl c CPCL 1 s v, T v v 1 Syl 1 PCL i i1 i1. (4) 1 Syl c CPCL c CPCL vsyl 1, T 1 vpcl vi i 1 v 1 i1 is: 1 Syl c CPCL 1s vsyl 1, T vpclvi i 1 i1 1 Syl c C PCL c C PCL Syl 1, 1 PCL i i i1 1 v T v v v s (41) In order to better understand the process of nuerical ipleentation, a flowchart about calculation of stretch under stress control at tie t t t during crystallization for the general echanics theory is presented in Fig. 15.

45 37 Fig. 15 The flowchart shows the procedure to calculate stretch under stress control at tie t t t during crystallization by using the general theory. During elting, at tie t t n t, after several crystalline phases elt, the existing s crystal phases with volue fraction e n i1 v c i and the Sylgard atrix should defor elt n to satisfy the boundary conditions. Deforation of the crystalline phases fored at tie t t i t is:

46 38 n1 elt elt elt elt in n in1, in1 is 1n1, 1n1 j j1, (42) and the total stress is Siilarly with Eq. (37), e n total Syl elt elt c CPCL elt n vsyl n 1n1 s Tn vpcl vi n in i1, ( ). (43) elt n can be solved: e n elt Syl elt c CPCL 1 n1s s vsyl 1 n1s s, Tn vpcl vi i n1 elt i1 n 1 e n Syl elt c C PCL elt vsyl 1 n1s, Tn 1 vpcl vi i n1 1 n1s s i1. (44) Eq. (41) and Eq. (44) provide the solution of stretch increents during crystallization and elting. However the drawback of the general echanics odel is all deforations for the existing PCL crystalline phase should be stored as internal variables, which would quickly exhaust coputer CPU tie and eory resources. Here, EPM can be applied to iprove the efficiency. The total stress during crystallization at tie t t t is: total Syl 1, c C PCL 1 ( 1 ) c C PCL vsyl T vpcl v v ( ), (45) where v c 1 1 c vi is the volue fraction for the effective phase at 1 i1 t t t, 1 is t t t the effective deforation on the effective phase at crystalline phase with volue fraction stretch increent of Sylgard. 1. At t t t, a new c v is fored and the entire aterial defors by a sall due to the changing of conditions such as unloading or entropic elasticity can be solved siilarly with Eq. (37): Syl c CPCL 1s vsyl 1, T vpclv 1 1, 1 1 v T v v s Syl C PCL c C PCL Syl 1 1 PCL (46) According to EPM, at t t t, the newly fored phase can be erged into the existing effective phase. Therefore, total stress at t t t ay be rewritten as:

47 where one obtains: 39 total Syl 1, c C PCL vsyl T v PCL v ( 1 ), (47) is the increental deforation for the effective phase. Coparing Eqs. (45) and (47), v ( ) v ( ) v ( ), (48) c CPCL c CPCL c CPCL and can be calculated: v ( ) v ( ) v ( ). (49) c CPCL c CPCL c CPCL c CPCL v ( 1) 1 Siilar process can be applied to the elting process. The flowchart in Fig. 16 illustrates the calculation procedure of stretch under stress control at tie t t t during crystallization using the EPM theory.

48 4 Fig. 16. The flowchart shows the procedure to calculate stretch under stress control at tie t t t during crystallization by using the EPM theory Coparison between Experients and Siulations The experiental result fro the four-step thero-echanical cycle with.15mpa iposed stress was used to fit the odel paraeters. Paraeters are listed in Table 2. There are

49 41 totally 18 paraeters, 12 paraeters are obtained fro fabrication, experients, handbooks and literatures, and 6 paraeters are deterined by paraeter fitting. These paraeters were used for the rest of siulations in this section. Fig. 17 shows the odel fitting. The excellent agreeent between odel siulation and experient validates the odel capability of capturing the physics underlying the shape eory behavior. (a) Fig. 17 Model fitting to the experiental result for the case of iposed stress.15mpa. (a) Stress-Teperature-Strain plot. (b) Strain-Teperature plot. (b) The aterial paraeters identified by fitting the.15mpa curve were then used to predict the shape eory behaviors with.1mpa and.2mpa stresses. Fig. 18 shows the coparison between the odel prediction and the experient. The good agreeent between the odel prediction and the experient verifies the odel.

50 42 (a) Fig. 18 Coparisons between odel predictions (solid-dot line) and experiental results (solid line) for the cases of.1mpa (red) and.2mpa (black): (a) Stress-Teperature-Strain plot. (b) Strain-Teperature plot. (b) Table 2. List of aterial paraeters Paraeter Value Coposition Description (Fitting paraeter are described in italics) v Syl.87 Volue fraction of Sylgard (Deterined by fabrication) v PCL.13 Volue fraction of PCL (Deterined by fabrication) Kinetics of crystallization and elting G s -1 Pre-exponential factor of growth rate (Fro Liu et al (Liu, et al., 21)) U 17.3kJ ol -1 Diffusional activation energy (Fro Liu et al (Liu, et al., 21)) T 31K The equilibriu elting teperature (Fro Kweon et al. (Kweon, et al., 23)) T 193K The teperature about 3K lower than T g (T g obtained fro DMA) K g 358K 2 Nucleation constant (Fro Liu et al (Liu, et al., 21)) N Pre-exponential factor of nucleation density (Fitting paraeter) β.1349k -1 Epirical paraeter (Fro Angelloz et al (Angelloz, et al., 2)) C s -1 The elting contraction rate (Fitting paraeter) H.25 The saturated crystallinity (Fro Kweon et al. (Kweon, et al., 23)) Mechanical behaviors γ Syl.51 Stress concentration factor of Sylgard (Fitting paraeter) γ PCL 4.31 Stress concentration factor of PCL (Fitting paraeter) N Polyer crosslinking density (Fro DMA for neat Sylgard) µ 2.69MPa PCL crystal phase odulus (Fitting paraeter)

51 43 Theral contraction/expansion α C -1 CTE at high teperature (Measured fro experient) α C -1 CTE at low teperature (Measured fro experient) Δλ PT Theral strain due to phase transition (Fitting paraeter) The nuerical siulation of a constrained recovery experient as described above was also conducted to characterize the constrained recovery feature of the constitutive odel. Fig. 19 shows the coparison between the nuerical siulation and the experient. The odel predicts the constrained recovery of the aterial very well. Two peaks of strain at different teperature are also described due to hysteresis of crystallization and elting. (a) Fig. 19 Coparison between odel predictions and experients for the case of iposed stress.15mpa. (a) Stress-Teperature-Strain plot. (b) Strain-Teperature plot. (b) Effect of theral rates The constitutive odel with paraeters fitted by the.15mpa with 2 C/in case was used to investigate the effect of theral rates. The odel predicts the behaviors of other three theral rates (Fig. 2) with a very good agreeent with experients. Fig. 21 shows crystallization under different theral rates fro odel predictions. Clearly, crystallinity plays a key role in deterining the fixity upon unloading. For the slow cooling cases, such as 1 C/in and 2 C/in cases, crystallization occurs at a faster rate than the cooling rate and can reach the

52 44 saturated crystallinity, and therefore the best fixity can be achieved. For the fast cooling cases, crystallization occurs at a slower rate than the teperature rates, therefore, upon unloading, crystallinity only reach 5% and 2%, respectively, leading to a poor fixity: the 5 C/in case held about 6% teporary shape and the 1 C/in case just fixed about 4%. In addition, for the 5 C/in and 1 C/in cases, it is observed that crystallinity keeps growing at low teperature during heating, since the teperature is lower than the crystallization teperature. (a) Fig. 2 Coparison between siulations and experients of shape eory behavior with different theral rates (1 C/in (red), 2 C/in (blue), 5 C/in (black) and 1 C/in (green)): (a) Stress-Teperature-Strain plot. (b) Strain-Teperature plot. Solid lines represent siulations and solid-dot line represents experients. (b) Fig. 21 Siulations of kinetics under different theral rates ( C represents Cooling, H represents Heating ).

53 Effect of volue fractions As entioned in Section 2.3, the fixing capability of SMEC can be affected by changing the Sylgard/PCL coposition. The effect of volue fraction of PCL on the ability of fixing teporary shape was investigated here by the constitutive odel. Fig. 22 Strain-Teperature plot with different volue fraction of PCL with.15mpa. Fig. 22 shows odel predictions of different shape eory behaviors on.15mpa iposed stress by varying volue fraction of PCL fro 5% to 3%. At S1, the saple will be stretched ore with higher percentage of PCL, since higher percentage of PCL leads to softer SMEC stiffness at T H due to lower percentage of Sylgard. At S3, upon unloading, higher volue fraction of PCL yields less strain drop and better fixity. Therefore, the advantage of higher PCL fraction is the better fixity. However, higher PCL can also lead to softer aterial behavior at high teperatures. Table 3 lists the fixing ratios R f and the strains H at T H. Clearly, the fixing ratio increases with the volue fraction of PCL, nevertheless the strain H at T H increases, because the coposition of Sylgard decreases as increasing the one of PCL and due to Eq. (17), only Sylgard carries load at T H, the less Sylgard, the softer the aterial. Table 3. List of R f and ε H

54 46 v PCL 5% 13% 2% 3% R f 7.4% 88.4% 93.9% 95.4% ε H 9.73% 1.72% 11.71% 13.23% 2.5 Conclusion Shape eory behaviors of a novel SMEC are investigated in this paper. In the SMEC syste, Sylgard as an elastoeric atrix provides rubber elasticity and PCL serves as a reversible switch phase due to the crystal-elt transition, iparting shape eory ability into the syste. This paper investigates the theroechanical behaviors of SMEC and presents a first 3D constitutive odeling frae to describe the shape eory effects of SMECs. This odeling frae includes kinetics of non-isotheral crystal-elt transition of PCL, echanical and theral behaviors The kinetics for the crystal-elt transition utilizes the Evans theory and Hoffan- Lauritze expression for crystallization then expands Hoffan-Lauritze expression to consider elting. The kineatics of evolving phases is tracked by using the assuption that newly fored the crystalline phase is undefored and stress-free. In order to iprove the coputational efficiency, the effective phase odel (EPM) is adopted to capture the echanics of aterial with evolving phases. The odel accurately captures shape eory behaviors and is used to discuss effects of theral rates and volue fractions. The odel is also used to explore the effects of volue fraction and theral rates.

55 47 CHAPTER III MECHANISMS OF TRIPLE-SHAPE POLYMERIC COMPOSITES DUE TO DUAL THERMAL TRANSITIONS 3.1 Introduction Along with strong interests in dual-shape SMPs, such as SMECs which have two shapes, naely the teporary shape and the peranent shape, ulti-shape SMPs have started to attract increasing attention. To date, there are two approaches to achieve a ulti-shape eory effect (-SME). The first approach requires the SMP to have a wide teperature range of theroechanical transition. For exaple, Xie (Xie, 21) recently deonstrated that perfluorosulphonic acid ionoer (PFSA), a thero-plastic SMP with a broad glass transition teperature range fro 55 C to 13 C, could show -SME, if the teperature is increased during recovery in a staggered anner. Using a siple theoretical odel, we found that the physical echanis of this observed -SME is due to the ultiple relaxation odes of acroolecular chains, these relaxation odes being sensitive to teperature (Westbrook, et al., 211). At a particular teperature during recovery, soe relaxation odes are active (eaning their relaxation tie is coparable with lab tie scale) while soe other relaxation odes are not (eaning their relaxation tie is extreely long as copared to lab tie scale). If the teperature is increased in a staggered anner, the nuber of active relaxation odes also increases in a staggered anner, leading to the observed -SME. The second approach is to use ultiple transition teperatures to achieve -SME, ost notably, to use two distinct transition teperatures to obtain triple-shape eory effect (t-sme). Here, t-sme refers to fixing two teporary shapes and recovering sequentially fro one teporary shape to the other upon continuous heating, and ultiately to the peranent shape (Bellin, et al., 26, Bellin, et al., 27, Luo and Mather, 21, Xie, et al., 29). In a typical theroechanical triple-shape eory cycle (Fig. 2), at Step 1 (S1), the aterial is initially

56 defored at a teperature T, where T is higher than two phase transition teperatures ( H 48 Ttrans II TL 1 L1 H and ). At Step 2 (S2), the teperature is decreased to ( T T ), while aintaining the external load and fixing associated with the higher teperature transition begins. After trans I T trans I reoval of the external load, at Step 3 (S3), fixing of the first teporary shape at T L1 is revealed. At Step 4 (S4), the saple is defored further at T L1 (note: this need not be in the sae direction or in the sae plane as the first deforation). At Step 5 (S5), the aterial is cooled to T L2 ( T T ), while keeping the external load causing the deforation at S4. At Step 6 (S6), after L2 unloading, the second teporary shape is fixed at back to trans II T L1. At Step 7 (S7), as the saple is heated, the aterial recovers into its first teporary shape. At Step 8 (S8), the peranent shape is reached by heating to. Recently, several ethods of achieving the t-sme were reported. Bellin and co-workers (Bellin, et al., 26, Bellin, et al., 27) developed two different polyer networks to achieve the t-sme. These two network systes are acroscopically hoogeneous with two icroscopic polyer segents. The two phase transitions are either glass transition of one segent and crystal-elt transition of the other one, or two crystal-elt transitions of two different segents. Xie (Xie, et al., 29) reported a different ethod of achieving the t-sme using a acroscopic bilayer crosslinked polyer structures with two well separated phase transitions. Recently, based on the fabrication of shape eory elastoeric coposites (SMECs) (Luo and Mather, 29), Luo and Mather (Luo and Mather, 21) introduced a new and broadly applicable ethod for designing and fabricating triple shape polyeric coposites (TSPCs) with well controlled properties. In the TSPC, an aorphous SMP (epoxy with T H 2 C 4 C, depending on the precise coposition) works as the atrix providing overall elasticity and provides one phase transition teperature and non-woven PCL (poly(ε-caprolactone)) icrofibers produced by electrospinning and incorporated into the atrix provides the other phase transition teperature T L2 T g

57 due to the crystal-elt transition of PCL ( 49 T 5 C). Copared with ethods reported previously, the approach for fabricating polyeric coposites is quite flexible, since one can tune the functional coponents separately to optiize aterial properties, opening up the potential to design for a variety of applications (Luo and Mather, 21). Constitutive odels were also developed in the past for SMPs. These odels were ainly for dual-shape SMPs, including the early odel by Tobushi et al. (Tobushi,Hayashi, et al., 1996), the odified standard linear solid odel with Kohlrausch-Willias-Watts (KWW) stretched exponential function by Castro et al. (O'Connell and McKenna, 1999), three diensional finite deforation odels for aorphous SMPs developed by Liu et al. (Liu, et al., 26), Qi et al. (Qi, et al., 28a), Nyugen et al. (Nguyen, et al., 28), Chen and Lagoudas (Chen and Lagoudas, 28a, b), Westbrook et al. (Ge,Luo, et al., 212); constitutive odels for crystallizable shape eory polyers were developed by Barot and Rao (Barot and Rao, 26), Westbrook et al. (Westbrook,Parakh, et al., 21), and Ge et al for SMEC (Ge,Luo, et al., 212). For ulti-shape SMP, Yu et al. (Westbrook, et al., 211) deonstrated that a ulti-branch odel can be used to capture the -SME in SMPs with a broad range of transition teperature. For the t-sme with dual transition teperatures, to the authors best knowledge, no odel has been reported to date. This paper investigates the triple-shape eory behavior of TSPC and presents a onediension (1D) odel to describe the tripe-shape eory behavior. In the odel, the acroscopic behavior of the coposite depends on two phases the coposite ade up of. Each phase of the coposite can undergo individual phase transition (the glass transition for the atrix and the crystal-elt transition for the fiber network) and contributes to the acroscopic behavior. The odel separately considers the glass transition and crystal-elt transition. This separate treatent allows the aterial paraeters used in the odel to be independently identified fro experients. The paraeter identification approaches are proposed and the necessary tests are

58 5 described. The paper is arranged in the following anner: In Section 2, the aterial is introduced briefly and experiental results including DMA, stress relaxation tests, uniaxial tension tests and triple-shape eory behaviors are presented. Section 3 introduces the 1D odel. In Section 4, paraeter identification is introduced and, with paraeters having been deterined by experients, siulations of stress and stored energy analysis for the triple-shape eory behavior are presented. 3.2 Materials and theroechanical behaviors Materials A TSPC was prepared using an epoxy-based copolyer theroset syste as the atrix and a poly(ε-caprolactone) (PCL) as fiber reinforceents. For the epoxy-based copolyer theroset syste, it consists of an aroatic diepoxide (diglycidyl ether of bisphenol-a or DGEBA), an aliphatic diepoxide (neopentyl glycol diglycidyl ether or NGDE) and a diaine curing agent (poly(propylene glycol) bis (2-ainopropyl) or Jeffaine D23) (Luo and Mather, 21). In this paper, the ole-% ratio DGEBA: NGDE=3:7 or D3N7 was chosen for all tests. The fabrication is siilar with previously reported shape eory elastoeric coposites (SMECs) (Luo and Mather, 29, 21) DMA experients DMA tests were conducted using a dynaic echanical analyzer (Q8 DMA, TA Instruents). The epoxy/pcl TSPC (a rectangular fil) was stretched by a dynaic tensile load at 1Hz. The teperature was first cooled at a rate of 1 C/in fro 1 C to -5 C, then after 1 in theral equilibrating at -5 C, it was heated up to 1 C at a rate of 1 C/in. Fig. 23 shows the tensile storage odulus with teperature. The heating trace and the cooling trace show different features. In the heating trace, it clearly shows two separated theral transitions, corresponding to the glass-rubber transition of epoxy and the

59 51 crystal-elt transition of PCL, respectively. As a result, the cascade of three storage odulus plateaus of decreasing agnitude with increasing teperature was observed. The first high odulus plateau (~1.5 GPa) exists below epoxy epoxy T g attributed to the glassy state of epoxy (Plateau I in Fig. 2). The second oderate odulus plateau (~1 MPa) lies between epoxy and PCL epoxy T g PCL T, where epoxy is in rubbery state but PCL is in seicrystalline state (Plateau II in Fig. 23). Above PCL PCL T, there is a low odulus plateau (~ 2MPa) (Plateau III, in Fig. 23). Different fro the heating trace, during cooling, the rubber-glass transition of epoxy and the elt-crystal transition of PCL are largely erged into one single transition and Plateau II disappears. This is priarily attributed to two close teperature ranges of two transitions, and the supercooling that is inherent to polyer crystallization. Fig. 23 DMA results for the epoxy/pcl TSPC: storage odulus with teperature during a cooling-heating cycle. During heating, the cascade of three storage odulus plateaus was observed. At Plateau I, the atrix (epoxy) is in the glassy state and the fiber network (PCL) is in the seicrystalline state. At Plateau II, the atrix is in the rubbery state, but the fiber network is still in the seicrystalline state. At Plateau III, the atrix is in the rubbery state and the fiber network is in the elting state. During cooling, Plateau II disappears Stress relaxation tests Stress relaxation tests were conducted using the DMA achine in order to yield the viscoelastic data required aterial paraeter estiations needed for our odeling. A rectangular

60 52 TSPC sheet with the diension of was used for tests. Stress relaxation tests were perfored at 16 different teperatures evenly distributed fro C to 3 C, a teperature safely below the elting point of the fiber network. The saple was preloaded by application of a sall, N force to aintain its straightness. After reaching the testing teperature, it was therally equilibrated for 3 in. Then a.1% strain was applied to the saple and the relaxation odulus was observed for 3 in. Fig. 24 shows the results of stress relaxation for 16 different teperatures. Base on the well-known tie teperature superposition principle (TTSP) (Brinson, 27, Rubinstein and Colby, 23), a relaxation aster curve was achieved by shifting relaxation curves to a reference teperature of T 16 C, with ref teperature-dependent shifting factors. The relaxation aster curve and shifting factors will be reported in Section 4. Fig. 24. Tensile relaxation odulus for teperatures varying fro C (indicate color or insert line) to 3 C (indicate color or insert line) with 2 C teperature intervals between experients Uniaxial tension tests The uniaxial tension tests for both the neat epoxy and the TSPC were conducted using the DMA achine. The diensions of the neat epoxy saple and the TSPC saple were and , respectively. Both saples were tested

61 53 at 4 C and 8 C respectively, at the loading rate of.5 MPa/in. The results of uniaxial tensions were used for paraeter identification, which will be introduced in Section Shape eory behavior Shape eory behaviors were tested on the DMA achine with a rectangular fil. The t-sme was achieved by following an eight-step-shape-fixing ethod proposed by Luo and Mather (Luo and Mather, 21): at Step 1 (S1), the aterial is initially stretched at a stress rate of.5 MPa/in to a constant load P 1 at a teperature T H epoxy PCL ( T T, T ). At Step 2 (S2), the aterial is cooled down at a rate of 2 C/in to a H g epoxy PCL teperature T ( T 2 T, T ) while aintaining the load P 1. At Step 3 (S3), it is heated up L2 L g epoxy PCL at a rate of 2 C/in to a teperature ( T T T ). At Step 4 (S4), after unloading, the first teporary shape is fixed at T L1 TL1 g L1. At Step 5 (S5), the aterial is stretched at the sae stress rate of S1 to a constant load P 2 (P 2 >P 1 ) and the aterial is at a new deforation state. At Step 6 (S6), it is cooled down to T L2 again, while keeping the constant load P 2. At Step 7 (S7), after unloading, the aterial achieves the second teporary shape at T L2. Finally, at Step 8 (S8), the aterial is heated up to T L1 and it recovers to its first teporary shape and peranent shape sequentially. In this paper, this approach is used where is 8 C, is 4 C and is C. TH TL 1 T L 2 It is ephasized that the cooling to at S2 and the heating to at S3 are necessary to TL2 T L 1 distinguish the two separated phase transitions: at S2, epoxy is in glassy state and PCL is in seicrystalline state and at S3, PCL is still in seicrystalline state but epoxy is in rubbery state. The strain-teperature plots are shown in Section 4 with three different noinal stress pairs (P 1 =.1 MPa, P 2 =.3 MPa; P 1 =.15 MPa, P 2 =.45 MPa; P 1 =.2 MPa, P 2 =.6 MPa). As reported by Luo and Mather (Luo and Mather, 21), the triple-shape behavior can be achieved by a so called one-step-fixing ethod. In order to better understand the echanis on

62 54 how to achieve two teporary shapes after a single fixing step, the aterial was tested by the one-step-fixing ethod. The aterial was stretched at a rate of.5 MPa/in to.2 MPa at 8 C and then cooled down to C at a rate of 2 C/in, while aintaining the load; after quickly unloading at C, alost 1% strain was fixed. During heating, it is quite interesting that there is a strain plateau at ~4 C-6 C, and at 8 C, the aterial was able to copletely recover into its peranent shape. In Section 4, we will show that three shapes can be observed: one at C, one at between 4 C and 6 C, and finally one at 8 C. 3.3 Model description Overall odel In this section, a one diensional (1D) odel is developed to capture the t-sme in the epoxy/pcl coposites and to clearly understand the echanis of shape fixing, stress and strain energy distribution during a shape eory cycle. In this odel, the atrix and fiber network are treated as a hoogenized syste. The acroscopic behavior of syste depends on the two phases (the atrix and the fiber network). The atrix as an aorphous SMP undergoes the glass transition and the fiber network as crystallizable polyer undergoes the crystal-elt transition. The fiber network is a liquid (elt) at high teperatures and an elastic solid at low teperature, where elts and crystals co-exist and their respective volue fractions depend on the teperature and their evolutions are described by the existing theory of crystallization. As the odel is priarily focusing on the shape fixing echanis, we do not seek a rigorous icroechanical description. In addition, the theral expansion is not introduced for this 1D odel, although it can be easily included. (Ge,Luo, et al., 212) For the TSPC syste, the siplest odel arranges stresses on the atrix and the fiber network in parallel (as shown in Fig. 25) and the total Cauchy stress total is given by: total M F V V, (5) M F

63 55 M F where and are the stresses acting on the atrix and the fiber network, V and V are the effective volue fractions of the atrix and the fiber network by considering the departure of stress distribution aong the phases, which are a ultiplication of stress concentration factors and volue fractions for the atrix and the fiber network, respectively, and V V 1. (Ge,Luo, et al., 212) Fig. 25 shows a rheological representation of the odel. In Fig. 4, the left red dashed box represents the stress acting on the atrix. A odified Standard Linear Solid (SLS) odel is adopted to consider both equilibriu and nonequilibriu behaviors of the atrix. Here, a spring is used to represent the equilibriu behavior and a KWW (Kohlrausch, Willias and Watts) stretched exponential function for stress relaxation is used to represent the nonequilibriu viscoelastic response (Willias and Watts, 197). The right blue dashed box represents the echanical eleents for the fiber network behavior. The fiber network changes its crystallinity with teperature: when T T, fiber elts do not carry load; when T T, fibers crystallize and can carry load. M M F F Fig. 25. Scheatic of overall odel: the total Cauchy stress consists of stresses on the atrix (red dashed box, left) and stress on fiber network (blue dashed box, right). For the stress on the atrix, a odified Standard Linear Solid (SLS) odel is adopted to consider the equilibriu and

64 56 nonequilibriu behaviors of the atrix. The ultiple springs of the fiber network odel represents fiber crystals fored at different ties. When fibers crystallize, the switch turns on and crystals appear with odulus, ; when fibers elt, the switch turns off and fibers lose their odulus Viscoelastic behavior for the atrix For the atrix, the odified SLS odel is used to consider both equilibriu and nonequilibriu viscoelastic behaviors. The total stress on the atrix consists of contributions fro both equilibriu and nonequilibriu behaviors: T, t T T, t M eq non. (51) The Cauchy stress fro the equilibriu response eq T follows the neo-hookean odel with entropic elasticity: 2 eq T NkT 1/, (52) where N is the crosslink density of the atrix, k is Boltzann s constant, T is absolute teperature and is the stretch along the uniaxial load. 1 and is the strain. For the nonequilibriu response, the stress should be: non te t, (53) where E is Young s Modulus and is the elastic strain of the nonequilibriu branch. Using non e KWW stretched exponential and Boltzann superposition principle, can be (Xie, et al., 29): e t t t dt exp ds, (54) s s T, t where is the total strain in the nonequilibriu, v is the viscous strain, T is the teperature dependent stress relaxation tie. is a aterial paraeter within a range between non e e v e and 1 characterizing the width of the stress relaxation distribution, i.e. a decrease of leads to an increase of the stress relaxation breadth. Once reaches 1, it is the narrowest stress relaxation

65 57 tie distribution and the nonequilibriu branch is reduced to the original Maxwell odel. The teperature dependent stress relaxation tie T is obtained using a shifting factor a : T a T, (55) T where is the stress relaxation tie at a reference teperature. It was found that depending on whether teperature is above and near or below T, the shifting factor a can be calculated by g T two different ethods (O'Connell and McKenna, 1999). For teperature above and near T g, the WLF (Willias-Landel-Ferry) equation (Willias, et al., 1955) is used: log a T 2 C1 T Tr C T T r, (56) where and C are aterial constants, and T is a reference teperature. For teperature below T g C1 2, an Arrhenius-type behavior developed by Di Marzio and Yang et al. (1997) (Di Marzio and Yang, 1997) is used: r lna T AF C 1 1, (57) k T Tr where A is aterial constant, F is the configuration energy and k is Boltzann s constant. c Mechanical behavior of the fiber network For the seicrystalline fiber network, crystallization and elting are functions of teperature and tie. We adopt the postulation that when a sall fraction of polyer crystals fors, it is in a stress-free state (Long, et al., 21, Westbrook,Parakh, et al., 21). However, in order to satisfy the boundary conditions, either overall or locally, this sall fraction will defor iediately. This stress-free state for new fored crystalline phases was referred as natural configuration by Rajagapol (Rajagopal and Srinivasa, 1998a, b), and has been used by any researchers to study crystallizing polyers (Ma and Negahban, 1995a, Ma and Negahban, 1995b,

66 58 Negahban, 1993, Negahban and Winean, 1992), rubbers with therally induced scissoring and reforation (Tobolsky, 196, Winean, 1999, 25, Winean and Min, 2, 23, Winean and Shaw, 28), shape eory polyers (Barot and Rao, 26, Barot, et al., 28, H. Jerry Qi, et al., 28), and light induced activated polyers (Long, et al., 29). Following this assuption, the stress in the fiber networks is (Long, et al., 21): where c tc t F FC FC v s t s ds, (58) t is the tie point when crystallization starts, FC t s is the stretch of the fiber crystals fored at t s and v s is the crystallinity of crystalline phases, which is a function of tie and teperature and is discussed below; FC is the stress function for fiber crystals. For the sake of siplicity, it siply assues the stress on fiber crystals follows neo-hookean behavior and the Cauchy stress is: FC 2 1 C, (59) where C is the shear odulus of fiber crystals. The increental description for stress on the fiber network during crystallization and elting was reported. (Ge,Luo, et al., 212) To describe crystallization kinetics (Avrai, 1941, 1939, 194), a siplified Avrai s phase transition theory is used. In polyer crystallization, the tie-teperature dependent crystallinity of polyer, vt, t, can be expressed as (Ge,Luo, et al., 212):, 1 exp (, ) v T t H V T t, (6) where H is the saturated crystallinity of a polyer under a certain condition, for PCL network H is taken as 25% (Ge,Luo, et al., 212) and V T, t is the volue fraction of crystalline phases without ipingeent. The calculation of V T, t during crystallization or elting was discussed in Ge et al. (Ge,Luo, et al., 212).

67 Strain energies Strain energies on the atrix and the fiber network were used to investigate the shape fixing echanis. For the equilibriu branch of the atrix and fiber crystals, as both of the adopt the Neo-Hookean constitution, the strain energy on the equilibriu branch of the atrix and individual fiber crystals is:, (61) where is the shear odulus and equal to NkT for the atrix and equal to C for fiber crystals; I 1 2 is the first deforation invariant, for the uniaxial loading case, I. The sign is used to distinguish tensile (+) or copressive (-) loading. For the strain energy of the fiber crystals, it follows: t F W v s W t s ds tc. (62) For the nonequilibriu branch of the atrix, the strain energy follows: 1 2 W E, (63) where E is Young s Modulus and is the elastic strain of the nonequilibriu branch non introduced in Eq. (54). W I 1 3 e 2 non e Results Paraeter identification In the odel, there are 11 paraeters in total (see Table 4). They were deterined by uniaxial tension and stress relaxation experients. Specifically, uniaxial tension tests were used to deterine the crosslinking density N of the atrix (epoxy), shear odulus of fiber crystals (PCL), and effective volue fractions V and V. Stress relaxation tests were used to identify the relaxation tie at 16 C, KWW stretching paraeter, the Young s odulus for nonequilibriu branch E non M, and constants in WLF equation and Arrhenius behavior. Fig. 26 shows the results fro uniaxial tensile experients of both the neat epoxy and TSPC at 8 C (Fig. 5a) and at 4 C (Fig. 26b). By fitting the experiental curve for the neat F

68 epoxy at 8 C, the crosslinking density N was identified as For the TSPC at F 8 C, as the fiber network elts,. Therefore, the total stress is: total M 2 V NkT ( 1 ). (64) By fitting the TSPC curve at 8 C (Fig. 5a), we obtained V =.47 and V 1V.53. For the TSPC at 4 C, as the crystallization of PCL finishes before the test, the Cauchy stress follows: total 2 2 V NkT 1 V H 1. (65) M F C M F M In Eq. (65), only the shear odulus C for PCL crystals is unknown and can be deterined easily by fitting the uniaxial tension for TSPC at 4 C. Fig. 26b shows the fitting result with C 9.2MPa. (a) (b) Fig. 26. Model fitting for uniaxial loadings: (a) at 8 C; (b) at 4 C. (Blue circles represent the odel fitting for the neat epoxy; red triangles represent the odel fitting for the TSPC). Based on the tie-teperature superposition principle, a relaxation aster curve at 16 C (Fig. 6a) can be achieved by shifting individual relaxation curves with shifting factors at different teperatures (Brinson, 27, Rubinstein and Colby, 23). The relaxation aster curve can be described by a odified standard linear solid odel with KWW stretch exponential, where the stress relaxation odulus should be:

69 61 E( t) E E exp 1 t. (66) In Eq. (66), at tie, E E E and when approaches, E E. Fro Fig. 27a, t 1 t we have ( ) and ( ). By considering the effective 8MPa E 1 148MPa E E E E volue fraction of the atrix, the Young s odulus of the nonequilibriu branch is E non E V 1 M and therefore Enon =315MPa. Using Eq. (66) to fit the stress relaxation aster curve at 16 C, paraeters and are deterined as 11s and.3, respectively. As introduced in Section 3, the shifting factors as functions of teperature can be described by WLF and Arrhenius equations at different teperature ranges. By fitting shifting factors with these two equations (Fig. 27b), paraeters in Eqs. (56) and (57) are deterined as C 1 =24, C 2 =5 ºC, Tr =16 ºC and AF k =-35K. c (a) Fig. 27 Model fitting for stress relaxation: (a) the stress relaxation aster curve at 16 C; (b) the shifting factors with teperature. (b) Table 4. List of aterial paraeters Paraeters Value Description Coposition

70 V.47 M V.53 F Epoxy Matrix MPa N 23 3 Enon 11s.3 Tr 16C 62 Effective volue fraction of the atrix (epoxy) Effective volue fraction of the fiber networks (PCL) Polyer crosslinking density Young s odulus for nonequilibriu branch Relaxation tie at C KWW stretching paraeter Reference teperature C1 24 WLF constant 5 C WLF constant C 2 AF k 35K Pre-exponential paraeter for Arrhenius-type behavior PCL Fiber Network 9.2MPa Shear odulus for PCL crystals C Coparison between odel siulations and experients Using paraeters in Table 4, the odel successfully predicts the triple-shape eory P1 P2 P1 P2 behavior under three different noinal stress pairs ( =.1 MPa, =.3 MPa; =.15 MPa, =.45 MPa; =.2 MPa, P =.6 MPa) (Fig. 28). It should be noted that these curves were not P1 2 used for paraeter identification. In Fig. 28, the odel predictions show good agreeent with experients and the largest discrepancy occurs at S4 unloading to achieve the first teporary shape and at S8 heating back to ~4 C to recover to the first teporary. We attribute this error priarily to the nonlinear deforation for the TSPC at 4 C, which the neo-hookean odel for PCL crystals is unable to satisfactorily represent. The faster relaxation odulus in the odel fitting (Fig. 27a) akes the odel predictions reach the first teporary at S8 earlier than experients.

71 63 (a) (b) (c) Fig. 28 Model predictions for triple-shape eory behavior under three different noinal stress pairs: (a) P1 =.1 MPa, P2 =.3 MPa; (b) P1 =.15 MPa, P2 =.45 MPa; (c) P1 =.2 MPa, P2 =.6 MPa. The odel also predicted the one-step-fixing shape eory behavior introduced in 2.4 with the.2 MPa iposed stress case (Fig. 29). Briefly, following high teperature deforation, then direct cooling to C and subsequent unloading, alost 1% strain was fixed. During continuous heating, recovery is interrupted as the strain reaches a plateau between PCL T, and then copletes as epoxy T g and PCL T is surpassed. In Fig. 29, the odel prediction shows good agreeent with the experient, indicating that the essential physics are captured.

72 64 Fig. 29 Coparisons between odel predictions and experiental result for the one-step-fixing shape eory behavior Stress and strain energy analyses for the triple-shape eory behavior In order to better understand the echanis of fixing teporary shapes, stress evolution and strain energy for the t-sme under the MPa stress pair were investigated. Fig. 3a provides the theroechanical loading history. Fig. 3b shows the variation of the strain as a function of tie. In Fig. 3c and e, the total stress is decoposed into the stress acting on the atrix and the stress acting on the fiber network. At S1, all stress coes fro the atrix since the fiber network (PCL) is in elt state. In addition, because the atrix is in its rubbery state, the stress on the atrix ainly coes fro the equilibriu branch; the stress on the nonequilibriu branch is alost zero due to large viscous strain developed, indicating the nonequilibriu branch is active for the subsequent shape eory effect. At S2 and S3, although crystals in the fiber network fored gradually, stress contribution fro the fiber network is negligible due to sall deforations on fiber crystals. At S4, after a quick unloading, the strain contracts ~5% (at S4 in Fig. 3b), which leads to a decrease of stress on the equilibriu branch of the atrix and the introduction of a copressive stress on the fiber network. Thus, the overall force balance is achieved and the total stress is zero. It is clear that the copressive stress on the fiber network

73 65 prevents the atrix fro returning to its stress-free (or initial) state. At S5 (the second loading), stresses on the atrix and the fiber network increase. At S6 (second cooling with holding a constant stress), stresses on the atrix and the fiber network stay constant. At S7 (unloading), stresses on the fiber network and on the equilibriu branch decrease slightly, but the stress on the nonequilibriu branch of the atrix changes quickly to a negative value, thus to prevent the aterial fro recovery and aintain the force balance. At S8, during heating, viscosity on the nonequilibriu branch decreases, causing the decrease of the copressive stress on this branch. When the stress on the nonequilibriu branch decreases to zero, the aterial recovers to the first teporary shape and the stresses on the equilibriu branch and the fiber network return to the sae values at S4. As the teperature is further increased to above the elting teperature, the stress on the fiber network starts to decrease; when fiber crystals copletely elt, the aterial recovers to its initial shape. Fig. 3 Stress and strain analyses for the triple-shape eory behavior: (a) the theroechanical loading history (stress and teperature vs tie). (b) Experiental output (strain). Total stress can be decoposed into the stress on the atrix (c) and the stress on the fiber network (e). The stress on the atrix (c) can be further decoposed into the one on the equilibriu branch ( E ) and the one on the nonequilibriu branch ( N ). (d) provides the strain energy on the atrix, which can be decoposed into the one on the equilibriu branch ( E ) and the one on the nonequilibriu branch ( N ). (f) provides the strain energy the fiber network. In (d) and (f), the strain energy is taken as negative if the deforation is copressive.

74 66 Fig. 3d and f show the variation of strain energy in the aterial. Coparing Fig. 3d and f, at S1, S2 and S3, the strain energy on the fiber network is nearly zero and ost strain energy is fro the equilibriu branch of the atrix. After the first unloading at S4, the strain energy on the equilibriu branch of the atrix drops and the strain energy on the fiber network is instantaneously generated to a negative value (due to copression). The energy release fro the atrix is stored in the fiber crystals and a strain energy loss of ~3 kj/ 3 (corresponding a 3% of loss) is observed. At S5 (loading at 4 C), the strain energies on both the atrix and the fiber network increase. At S6 (cooling to C), the strain energy on the atrix decreases slightly, but that of the fiber network reains nearly constant. Upon unloading at S7, the strain energies on the equilibriu branch and the fiber network change slightly by transferring the energy to the nonequilibriu branch in the for of copressive energy (-.2 kj/ 3 ). During heating at S8, the strain energy stored on both the fiber network and the atrix decreases and the nonequilibriu branch returns to zero when the aterial returns to the first teporary shape. Further heating the aterial to above its elting teperature, the energies stored on the fiber network and equilibriu branch decrease to zero and the aterial recovers to its peranent shape. It is also noted that since fiber crystals have a higher odulus than the equilibriu branch, the energy stored in the second teporary shape is uch higher than that stored in the first teporary shape. For the aterial studied here, the energy stored in the second teporary shape is 42.6 kj/ 3, which is 43% higher than that in the first teporary shape (8 kj/ 3 ) Stress and strain energy analyses for the one-step-fixing shape eory behavior Siilarly with the triple-shape eory behavior, stress and strain energy analyses were investigated for the one-step-fixing shape eory behavior. Fig. 31a provides the theroechanical loading history. Fig. 1b shows the variation of the strain as a function of tie. In Fig. 31c and e, the total stress is decoposed into the stress acting on the atrix and the stress

75 67 acting on the fiber network. During the loading step, all stress coes fro the atrix since the fiber network (PCL) is in elt state. In addition, because atrix is in its rubbery state, the stress on the atrix ainly coes fro the equilibriu branch; the stress in the nonequilibriu branch is alost zero due to large viscous strain developed, indicating the nonequilibriu branch is active for the subsequent shape eory effect. At the cooling step, although crystals in the fiber network fored gradually, stress contribution fro the fiber network is negligible due to sall deforations on the fiber network. During unloading, stresses on the fiber network and on the equilibriu branch decrease slightly, but the stress on the nonequilibriu branch of the atrix changes quickly to a negative value, thus to prevent the aterial fro recovery and aintain the force balance. During heating fro C to ~4 C, viscosity on the nonequilibriu branch decreases, causing the decrease of the copressive stress on this branch. When the stress on the nonequilibriu branch decreases to zero, the aterial arrives at a strain plateau fro ~4 C to 6 C (Fig. 31b) and the copressive stress on the fiber network aintains the force balance, which indicates part of the strain is fixed by copressed the fiber crystals. As the teperature is further increased to exceed the elting teperature, the stress on the fiber network starts to decrease; when fiber crystals copletely elt, the aterial recovers to its initial shape. Fig. 31d and f show the variation of strain energy in the aterial. Coparing Fig. 31d and f, at loading and cooling, the strain energy on fiber networks is nearly zero and ost strain energy is fro the equilibriu branch of the atrix. Upon unloading, the strain energies on the equilibriu branch and fibers networks change slightly by transferring the energy to the nonequilibriu branch in the for of copressive energy (-.3 kj/ 3 ). During heating fro C to ~4 C, the strain energies stored on the atrix decreases and the nonequilibriu branch returns to zero when the aterial arrives at a strain plateau fro ~4 C to 6 C. At the strain plateau, the energy of the fiber network is negative, indicating that the energy released fro the atrix is stored in the fiber crystals, and a strain energy release of ~5 kj/ 3 is observed. Further

76 68 heating the aterial to above its elting teperature, the energies stored on the fiber network and equilibriu branch decrease to zero and the aterial recovers to its peranent shape. Fig. 31. Stress and strain analyses for the one-step-fixing shape eory behavior: (a) the theroechanical loading history (stress and teperature vs tie). (b) Experiental output (strain). Total stress can be decoposes the stress on the atrix (c) and the stress on fiber networks (e). The stress on the atrix (c) can be decoposed into the one on the equilibriu branch ( E ) and the one on the nonequilibriu branch ( N ). (d) provides the strain energy on the atrix, which can be decoposed into the one on the equilibriu branch ( E ) and the one on the nonequilibriu branch ( N ). (f) provides the strain energy fiber networks. 3.5 Conclusion Triple-shape eory behaviors of TSPC were investigated in this paper. A 1D odel was presented to describe the tripe-shape eory behaviors. The odel separately considers the glass transition and elt-crystal transition. This separate treatent allows the aterial paraeters used in the odel to be independently identified fro experients. With paraeters identified fro stress relaxation tests and uniaxial isotheral tension tests, the odel is able to predict the tripleshape eory behaviors and the one-step-fixing shape eory behavior. The stress and strain energy analyses fro the odel revealed clearly the shape fixing echaniss for TSPC. For the triple-shape eory behavior, the first teporary shape is fixed by fiber crystals and the second teporary shape is fixed the nonequilibriu branch. For the one-step-fixing shape eory

77 69 behavior, the teporary shape is firstly fixed by the nonequilibriu branch at low teperature as the atrix in the glassy state. During heating, when the atrix transfers gradually fro the glassy state to the rubbery state, the fiber network becoes defored to fix a part of the teporary shape, which gives way to coplete recovery upon further heating.

78 7 CHAPTER IV PRINTED SHAPE MEMORY COMPOSITES 4.1 Introduction Three diensional (3D) printing or additive anufacturing is a process of aking 3D solid objects fro a digital odel. Distinct fro traditional achining techniques (subtractive processes relying on the reoval of aterial by ethods such as cutting and drilling), the 3D printing is realized by using additive processes and an objective is created by laying down successive layers of aterial. Since the start of the twenty-first century, the 3D printing technology has been developed rapidly and it is applied in a variety of the fields such as industrial design, footwear, jewelry, autootive, aerospace, dental and edical industries, etc. In this chapter, we pursue the general idea of control of acroscopic shape eory behavior via control of the coposite architecture by introducing a new paradig enabled by recent advances in digital anufacturing of ultiaterial architectures printed shape eory coposites (psmcs). We present a ethod based on 3D printing to fabricate coposite aterials with precisely-controlled fiber architectures that lead to coplex, anisotropic shape eory behavior. We deonstrate how anisotropic shape eory behavior derives in printed anisotropic shape eory elastoeric coposites (ASMECs) consisting of an elastoeric atrix and oriented poly (ε-caprolactone) (PCL) fibers that undergo a elt-crystal phase transforation during heating or cooling. The controllable anisotropy in theroechanical behaviors and shape eory behaviors are observed by tests, such as uniaxial tensions, the free theral strain tests, and shape eory cycles. A 3D anisotropic theroechanical constitutive odel is developed to capture the observed anisotropic behaviors and can be used to guide the design of psmc laina and lainates. This chapter is arranged in the following anner: In Section 2, the fabrication process of ASMECs is introduced. The theroechanical tests for ASMECs are presented, including DMA tests, uniaxial tensions, shape eory behaviors and theral strain tests. Section 3 introduces the

79 71 3D anisotropic theroechanical constitutive odel and the identification of the paraeters used in the odel. In section 4, the odel predictions of shape eory behaviors are copared with the experiental results. The paraetric studies such as the effect of volue fractions and the effect of the effective shear odulus of fibers are investigated. 4.2 Materials and experients PolyJet polyer jetting The layer-by-layer fabrication approach is used in the 3D printing process. In the 3D printing, the drop-on-deand inkjet printing is eployed to selectively deposit droplets of photopolyer directly onto a build platfor. This direct 3D printing technique is referred to as PolyJet. For the OBJet 3D printer (OBJet 26 Connex), as the inkjet print heads are connected with separate aterial sources (aterial cartridges with two photopolyer solutions (Verowhiteplus and Tangoblackplus) are installed in the OBJet 3D printer), ultiple aterial can deposited in a single layer and the aterials graded can be created. In addition, a hydrophobic gel is used as a sacrificial aterial or support aterial for the fabrication of coplex geoetries. During the printing process, once a layer of droplets is deposited (64 μ in standard ode; 32 μ in high-resolution ode), a roller evens out the horizontal surface of the layer. After that, two UV lights pass over the printed layer ultiple ties to cure the photopolyer. The PolyJet printing based on the layer-by-layer printing approach is well suitable for coponent ebedding, as the coponents can be directly ebedded in bulk, in powder or even in liquid for without destroying the previously printed layer. However, there are two inherent liitations in using the PolyJet to ebedded coponents. First, as the PolyJet software does not have the function to define the deposition of the sacrificial aterial, one has to anually reove the deposited sacrificial aterial fro the void where a coponent will be ebedded. Secondly, as the print heads is just over the printed part at a clearance of 4 μ, ebedded coponents cannot protrude fro previously deposited layer, or the print head ay be daaged.

80 Fabrication of printed shape eory coposites There are two approaches to reach printed shape eory coposites (PSMCs) using the 3D printer: the first one is straight printing of ultiple aterials; the second one is a ore coplex hybrid printing/ipregnation process with a sacrificial aterial and then a non-printed active aterial. For the first approach, as long as the coposite is designed using a CAD software, it can be printed directed by the 3D printer. Later, several applications fabricated by this approach will be deonstrated. Here, we priarily introduce the fabrication of the anisotropic shape eory elastoeric coposites (ASMECs), which were fabricated by the 3D printer using the second approach. The fabrication process follows six steps (Fig. 32): 1) Design the coposite in CAD. 2) Print the structure on the 3D printer. In the ASMEC case, the Tangoblackplus is chosen as rubbery atrix aterial. 3) Along with the coposite structure, an indicator is printed. Once the color or the shape of the indicator changes, it indicates that the channels copletely for, and the printing process should be paused iediately. 4) Manually reove the sacrificial aterial fro channels. 5) Fill the channels with PCL elt. This step needs to be operated at 8 C. 6) Resue the printing process to close up the channels. Coposites with seven fiber orientations (, 15, 3, 45, 6, 75 and 9 ) were ade. Fig. 32c takes the 9 fiber orientation to introduce the geoetry of the coposite. The saple has diension (length width thickness). The coordinate syste x- y-z is the globe coordinate syste, where the x-axis along the length direction and the y-axis along the width direction. The coordinate syste is the local coordinate syste. Originally, for the fiber orientation case, the 1-, 2-, 3-axes are coincident with the x-, y-, z-axes, respectively. In an arbitrary θ, using the 3-axis as a rotation axis, the 1-axis is rotated counterclockwise fro x-axis by θ. In the 9 fiber orientation case, the 1-axis is rotated counterclockwise by 9 and coincident with the y-axis. A zooed in figure for a repeating unit is used to introduce the size of fibers. The diension of the entire cross section is 2 L 3 L (here,

81 73 L=1 ). The fibers with square cross section b b. In the real saple, each end of the holes is sealed by a.25 thick cap to prevent PCL elt fro leaking. Regardless of the size of these caps, the volue fraction of fibers is: v b L 2 2 f 6. In following tests, b is taken to be L and v f becoes 1/6 (16.7%). b taken to be.7 L, where v f is 8.2% is used to investigate the effect of volue fraction. Fig. 32 The fabrication of ASMECs: (a) scheatics of the fabrication steps. (b) The corresponding photos of the fabrication steps in (a). (c) The scheatics of the geoetry of the coposites DMA tests DMA tests were conducted using the DMA achine for the seven fiber orientation cases and the neat rubbery atrix aterial (Tangoblackplus). For the coposite with seven fiber orientations, the diension of the saples is ~ For the neat rubbery atrix aterial, the diension is Before tests, saples were held at 3 C for 6 in to ake fibers (PCL) in the coposite crystallized. Then, it was heated fro 3 C to 1 C at a rate of 2 C/in. Fig. 33 shows storage odulus vs teperature for the eight cases. The elting

82 74 teperature (T ) of these coposites is ~63 C. The storage oduli of coposites with seven orientations follow the order: >15 >3 >9 >75 >45 >6.The storage odulus of the neat rubbery atrix aterial (RM) slightly increases as the teperature increasing, which indicates that in the teperature range the neat rubbery atrix aterial is in the rubbery state. The storage odulus of the neat rubbery atrix aterial is larger than the one of all coposites above T, and it is lower than all coposites below T as fibers in coposites crystallized. Fig. 33 The DMA results for the ASMECs with seven different fiber orientations and the neat rubbery atrix aterial Uniaxial tension tests Uniaxial tension tests were conducted using the DMA achine for the neat PCL at 3 C, and for the ASMECs with seven fiber orientations at 3 C and 8 C, respectively. For uniaxial tension tests at 3 C, before the tests were perfored, the saples (for the neat PCL with diension and for ASMECs with diension ~15 7 3) were held at 3 C for 6 in to ake sure PCL crystallized. For the neat PCL, it was stretched at a stress rate of 8MPa/in to 8MPa; For the ASMECs, they were stretched at a stress rate of.2mpa/in to.2mpa. Fig. 34a and b show the stress-strain behavior of the neat PCL and the ASMECs at 3 C. All of the exhibit nonlinear plastic deforation at large deforation. For

83 75 uniaxial tension tests for ASMECs at 8 C, the saples were held at 8 C for 6 in to ake PCL copletely elted. The saples were stretched at a stress rate of.5mpa/in to.5mpa. Fig. 34c shows the stress-strain behavior of the ASMECs at 8 C, which shows a good linearity for all coposites. Fig. 34d presents the coparison of Young s Modulus for ASMECs between 3 C and 8 C. Fig. 34e presents the coparison between DMA tests and uniaxial tensions for ASMECs. Circles represent Young s odulus at 3 C and triangles represent Young s odulus at 8 C. The DMA tests and uniaxial tensions show good agreeent. (a) (b) (c) (d) (e) (f) Fig. 34 Uniaxial tensions: (a) uniaxial tension for the neat fiber crystal (PCL) at 3 C. (b) Uniaxial tensions for the ASMECs at 3 C. (c) Uniaxial tensions for the neat rubbery atrix aterial at 8 C. (d) Uniaxial tensions for the ASMECs at 8 C. (e) Young s odulus vs fiber orientation for ASMECs at 3 C and 8 C, respectively. (f) Coparison between Young s odulus obtained fro the uniaxial tensions (triangles represent the results at 3 C and circles represent the results at 8 C.) and the storage odulus fro DMA tests of ASMECs.

84 Shape eory behaviors The SM behaviors of the ASMEC were investigated on the DMA achine. Fig. 35a shows the theroechanical cycle under which the shape eory behavior was perfored. In Fig. 35b, take the fiber orientation case for instance: at step 1 (S1), the saple was stretched at 8 C ( T H ) by gradually raping the tensile stress at a loading rate.5mpa/in to.5mpa. At step 2 (S2), the teperature was decreased to 3 C ( T L ) at a rate of 2 C/in while the noinal stress was fixed. It shows that when lowering the teperature, the strain initially increased. At step 3 (S3), it was isotherally held at 3 C for 6 in to crystallize PCL fibers. During the isotheral crystallization, an about 1% nonlinear strain drop was observed. At step 4 (S4), the stress was quickly released at 3 C and ost of the deforation was fixed. At step 5 (S5), the strain recovery was triggered by heating back to 8 C at a rate of 2 C/in. The strain was first linearly increased fro 3 C to ~55 C. Then, the strain suddenly dropped to ~-.4% at the teperature between ~55 C to 65 C. Fro 65 C to 8 C, the strain linearly recovered to zero. Fig. 35c shows the shape eory behavior for ASMECs with seven fiber orientations and for the neat rubbery atrix aterials. Eq. (1) was used to calculate the fixity R f. Table 5 lists the fixity for the eight cases. Fig. 35d shows the fixity vs fiber orientations. For ASMECs, between to 45, the fixity decreases draatically as increasing the degree of fiber orientation; as the fiber orientation higher than 45, the fixity stays around ~4% and the lowest case happens at 6. Interestingly, for the neat rubbery atrix aterial which does not have the shape eory effect, its fixity is supposed to be zero. However, the experient presented a negative value. (a) (b)

85 77 (c) Fig. 35 Shape eory behaviors for ASMECs and the neat rubbery atrix aterial: (a) theroechanical condition to achieve the shape eory behaviors. (b) the tie vs strain plot for the fiber orientation is used to deonstrate the shape eory behavior. (c) the tie vs strain plot for all seven fiber orientation cases of ASMECs and the neat rubbery atrix aterial. (d) the fixity vs fiber orientation for the ASMECs with seven fiber orientations and the neat rubbery atrix aterial. (d) Table 5. List of R f with different fiber orientations RM R f 85% 8% 67% 46% 34% 42% 5% -21% Theral strain tests The theral strain tests were conducted using the DMA achine. Fig. 36a presents the theroechanical condition for the theral strain test. During the entire test, a tiny static force (.1N) was applied to aintain the straightness of the saple. The teperature was firstly decreased fro 8 C to 3 C with the rate of 2 C/in. Then, it was held isotherally for 6in. After that, it was heated back to 8 C at the sae rate of cooling. Fig. 36b shows the theral strain vs tie for ASMECs and the neat rubbery atrix aterial. In order to better deonstrate the theral strain, the theral strain vs teperature was shown in Fig. 36c. Fig. 36d takes the fiber orientation case for exaple to describe the theral strain during the test. During cooling

86 78 fro 8 C to 3 C, the ASMEC with PCL elts contracts linearly with the coefficient of theral expansion (CTE). During isotheral crystallization at 3 C, it nonlinearly contracts by PT. During heating, it firstly expands linearly with CTE finally expands linearly with CTE. c, then expands nonlinearly by PT, and Fig. 36 Theral strain tests: (a) the theroechanical condition to achieve the theral strain under a stress free theral cycle. (b) the theral strain plot vs tie plot for the ASMECs with seven fiber orientations and the neat rubbery atrix aterial. (c) the theral strain vs teperature plot for the ASMECs with seven fiber orientations and the neat rubbery atrix aterial. (d) the theral strain vs teperature plot for the fiber orientation plot is used to describe the theral strain.

87 Model description Constitutive odel for anisotropic coposites In this paper, an anisotropic nonlinear elastic constitutive odels for fiber reinforced coposites proposed by Guo (Guo, et al., 26, 27) was adopted to describe the echanical response of the ASMEC syste. The odel developed by Guo treats both atrix and fibers are incopressible neo-hookean hyperelastic odel. The odel decoposes deforation gradient into two parts: the first part is coprised of uniaxial deforation along the fiber direction; the second part is coprised of all reaining deforation. The total strain energy can be constructed as: and the corresponding Cauchy stress is 1 1 W a I I I, (67) b pi B 1 I Fa Fa. (68) a iso b 3/2 4 Here, F is the deforation gradient and F=x X. Here we use X to represent the position of a aterial particle in the original (undefored) configuration, while x is the position of the corresponding particle in the current (defored) configuration. B is the left Cauchy-Green aniso deforation gradient and T B= FF. C is the right green Cauchy-Green deforation tensor and T C F F. 1 I and I 4 are the principal invariants of C : I1 trc and I a Ca, where a 2 4 F is a unit vector represents the fiber orientation of the reinforced fiber in the original (undefored) configuration and F is the fiber stretch. In Eqs. (67) and (68), a 1 v 1 v 1 v 1 v f f f f f f and b 2 f vv f 1 v 1 v f f f, where and f are shear oduli of the atrix and the fibers, respectively; v and v f are volue fractions of the atrix and the fibers and v vf 1.

88 8 In Eq. (68), p is an arbitrary hydrostatic pressure, the second ter is associated with the isotropic part of the Cauchy stress and the third ter is associated with the anisotropic part of the Cauchy stress Paraeter identification In Eq. (68), the Cauchy stress can be calculated using four paraeters (, v f ). The volue fractions v and identified, the Cauchy stress can be coputed. v f are deterined by design. As long as 2 In Fig. 37a, an incopressible Neo-Hookean odel NkT 1/ f, v and and f are is used to fit the stress-strain behavior of the neat rubbery atrix aterial (tangoblackplus) at 8 C, where N is the crosslinking density, k is the Boltzann s constant, and T is the teperature. Nk is identified as.822kpa/k. Thus, is.29mpa ( NkT, T 353K ). In Fig. 37b, an 2 incopressible Neo-Hookean odel c 1/ is used to fit the stress-strain behavior of the neat PCL at 3 C. The Neo-Hookean odel is able to fit the linear part. After strain ~2%, the saple starts to perfor nonlinear plastic deforation and the Neo-Hookean odel is unable to characterize. c is identified as 5MPa.

89 81 (a) Fig. 37 Uniaxial tensions are used to fit paraeters in the odel: (a) the uniaxial tension for the neat rubbery atrix aterial at 8 C is used to identify Nk and Nk is.822kpa/k. (b) The uniaxial tension for the neat fiber crystals (PCL) at 8 C is used to identify the shear odulus of fiber crystals and is 5MPa. c c (b) With NkT.25MPa ( T 33K), 5MPa, v.833 and v.167, f c f Eq. (68) was used to predict the uniaxial tension of the fiber orientation case at 3 C. Fig. 38a shows the coparison between experient and odel prediction. The odel prediction overestiates the stiffness of the coposite. The reason for this overestiation is that the odel assues the atrix and fibers are perfectly bonded, however, in the real coposite, the interaction between the atrix and fibers are not perfect and the load transfer between the atrix and fibers is not 1%. Considering the real interaction between the atrix and fiber, a paraeter is introduced and the effective shear odulus of seicrystalline PCL in the coposite becoes c c. In Eq. (68), f c c. The uniaxial tension of the fiber orientation case at 3 C was used to fit. In Fig. 38b, the odel is able to feature the stress-strain behavior at sall deforation and is identified as.18.

90 82 (a) Fig. 38 Model prediction and fitting for the uniaxial tension of the fiber orientation case: (a) the odel with 5MPa is used to predict the uniaxial tension of the fiber orientation f c case at 3 C. The odel prediction overestiates the stiffness of the coposite. (b) By fitting the uniaxial tension of the fiber orientation case at 3 C, the paraeter is identified as.18 and the effective shear odulus of the fiber in coposite at 3 C is 9MPa. (b) With NkT.25MPa ( T 33K ), 9MPa, v.833 and f c c v.167, Eq. (68) was used to predict the uniaxial tensions of the other six fiber orientation f cases at 3 C. Fig. 39 shows with.18 or 9MPa, the odel is able to predict stressstrain behavior at sall deforation. It is found that when the degree of the fiber orientation is high, the odel is less sensitive to. c (a) (b) (c)

91 83 Fig. 39 The odel with (d) (e) (f) 5MPa ( 1 ) and 9MPa (.18 ) predicts uniaxial f tensions of other six fiber orientations at 3 C: (a) the 15 fiber orientation case. (b) The 3 fiber orientation case. (c) The 45 fiber orientation case. (d) The 6 fiber orientation case. (e) The 75 fiber orientation case. (f) The 9 fiber orientation case. The odel with.18 predicts the experiental results well at sall deforation and the odel is less sensitive to at the high degree fiber orientation case. f In Fig. 4, with NkT.29MPa ( T 353K ), v.833 and v.167, the f uniaxial tensions of the fiber orientation case at 8 C was used to fit (or f c ) at 8 C using Eq. (68). It turns out that (or f c ) is nearly zero that iplies the PCL elts contributes negligible stiffness to the entire coposite at 8 C. With f at 8 C, Eq. (68) was used to predict the uniaxial tensions of the other six fiber orientation cases (Fig. 41). The odel predicts the stress-strain behavior well.

92 84 Fig. 4 The uniaxial tension of the fiber orientation case at 8 C is used to identify f at 8 C and it is nearly zero. (a) (b) (c) (d) (e) (f) Fig. 41 The odel with f predicts uniaxial tensions of other six fiber orientations at 3 C: (a) the 15 fiber orientation case. (b) The 3 fiber orientation case. (c) The 45 fiber orientation case. (d) The 6 fiber orientation case. (e) The 75 fiber orientation case. (f) The 9 fiber orientation case.

93 85 Figure 42 shows the prediction of Young s odulus of the coposites with different fiber orientations fro to 9, at 3 C and at 8 C, respectively. The predictions show good agreeent with experiental results. Considering the case where fibers and the atrix are perfectly bonded with 5MPa at 3 C, the prediction shows if the fiber orientation is lower f than 45, the stiffness of the coposite is draatically increased by changing fro 9MPa to f 5MPa. However, if the fiber orientation is higher than 45, the stiffness of the coposite is less sensitive to. f Fig. 42 The coparison of Young s odulus vs fiber orientation between experiental results and odel predictions Models for theral strain following for: During a heating-cooling cycle, the coposite has theral deforation gradient in T T Fx F xy T T T F Fyx Fy. (69) T Fz All of these coponents in theral deforation gradient are teperature and tie dependent. Thus, they are:

94 86 T x T y T z T xy T yx T x T y T z T xy T F T, t 1 T, t, F T, t 1 T, t, F T, t 1 T, t, F T, t T, t 2, F T, t T, t 2. yx (7) Fig. 43a shows a sall eleent. Assuing there is no tractions acting on any of its six faces. Therefore, as teperature changing, the strain is free theral strain. The free theral strains are: T 1 ( Tt, ), T 2 ( Tt, ) and T 3 ( Tt, ). In a ore general case shown in Fig. 43b, if the fibers are oriented θ degree counterclockwise fro the x-axis, the theral strains in the x-y-z global coordinate syste can be transfored fro the fiber local coordinate syste: T t 1 T t 2 T t T t 2 T t 1 T t T t 3 T t T t T t T t T t T t,, cos, sin,, T 2 T 2 T x,, cos, sin,, T 2 T 2 T y,,,,, =,, T T T T z yz xz T T T xy,, 2 1, 2, sin cos. (71) (a) (b) Fig. 43 Scheatics of a fiber reinforced coposite: (a) a sall eleent with three free theral strains. (b) A ore general case, where the fibers are orientated by θ degree counterclockwise fro the x-axis, the theral strains of the coposite in the x-y-z global coordinate syste can be transfored fro the fiber local coordinate syste.

95 87 In the ASMEC case, T T Tt is, Tt, T T Tt is 9, Tt, 2, x 1, for the fiber orientation case and x for the 9 fiber orientation case. As long as, we can describe T T, Tt, and 9, Tt, x, we can describe the theral strain between the based on Eq. (71). x In Fig. 44a, during cooling fro T H to T L ( T H =8 C, T L =3 C) at a cooling rate Q ( Q 2 C/in ), the theral strains are: T TH T TH T T Q T T Q x 1 1 x 2 2 (72), T, t T, t Qdt, 9, T, t T, t Qdt. During isotheral crystallization at T L, the theral strains are: T T, T, t, T, v t dt, t T T PT x x L TLTH 1 c Q Q L H T T 9, T, t 9, T, v t dt, t T T PT x x L TLTH 2 c Q Q L H (73) where vc theral strains are: t is the relative crystallinity changing fro to 1. During heating fro T L to T H, the TT T T T c PT x x L iso 1 c 1 c 1 c L T L H Q, T, t, T, t v t 1 v t v t dt, Q TT T T T c PT x x L iso 2 c 2 c 2 c L T L H Q 9, T, t 9, T, t v t 1 v t v t dt. Q (74) In fig. 44b, paraeters are identified by fitting the theral strains at and 9 (dash-dot c PT line representing odel fitting). They are 1.2% / C, 1.2% / C, 1 1.5%, c PT 2.2% / C, 2.2% / C, 2.15% five orientations are predicted in Fig. 44c. T. Based on Eq. (71),, Tt, with the other x

96 88 (a) (b) (c) Fig. 44 Model fitting and prediction for free theral strain: (a) free theral strain vs teperature for the and 9 fiber orientation cases. (b) Model fitting for free theral strain vs teperature for the and 9 fiber orientation cases. (c) Model prediction for free theral strain vs teperature for the other five fiber orientation cases Theroechanics during phase evolution During a theroechanical cycle, the total deforation gradient total F can be decoposed into the theral deforation gradient M F, which gives rise to stress: T F and the echanical deforation gradient total T M F F F. (75) During phase evolution, f gradually increases fro to a certain value. Typically, in this coposite syste, f gradually increases fro to c. a and b in Eq. (68) are also gradually changing fro a 1 v f 1 v f to 1 v 1 v 1 v 1 v f f c a c, f f c and fro b vv f 1 v f to b c 1 v 1 v 2 c vv f, f f c respectively. Before phase evolution, at tie t,, and the echanical deforation gradient M is F. Therefore, the Cauchy stress is: T b1 I M M 3/2 M M a 4 f pi F F F a F a. (76)

97 89 During phase evolution, at tie t t, as a crystalline phase with v c,1 fors, the effective shear odulus of the fiber in the coposite becoes c,1 vc,1c. a and ac,1 and b c,1, and the increents of a and b are: a,1 a c,1 a and b,1 b c,1 b b in Eq. (68) becoe. (77) As the newly fored crystalline phase foring at the configuration where deforation gradient is F and it is stress-free, the odulus increents a,1 and b,1 are not associate with the M M deforation gradient F. Meanwhile, accopanying with the creation of the new phase, an M increental deforation gradient F1 stress at tie t t is: ay be applied to the aterial. Therefore, the Cauchy b 1 I, M M T M 3 2 M M 1 pi a,f1 F1, 4 F1 a F1 a F1 a b I M M T M 3 2 M M a,1f1 1 F1 1,1 1 4 F1 1, a 1 F1 1a1 F1 1a1, (78) where, a, a, b, b M M M 1 1 F F F, M F M M 11 F 1, a1 F a M F a 1 1 and 4, I F a Fa Fa. Extending the algorith into the case at tie t n t, the Cauchy stress is: 1 I, M M T M 3 2 M M n pi a, Fn Fn b, 4 Fn a Fn a Fn a n i1 I M M T M 3 2 M M a, ifi n Fi n b, i 1 4 Fi n, a i Fi nai Fi nai, (79) where a, i a vc, ic a vc, i1c, b, i b vc, ic b vc, i1c, n M M M Fn Fi F, i1 n M M Fi n i F, ji a i M i1 M i1 F a I F a Fa Fa and vc, i vc, j. F a, 4, i j1

98 Phase evolution rule The odified Avrai s theory (M. Avrai, 1941, 1939, 194, Ozawa, 1971) was use to describe the crystallinity during the isotheral crystallization. At tie t : i t n vci, 1 exp k t, (8) where k is a constant related the growth rate of crystallization and n is a constant related to the diension of the crystals. 4.4 Results Shape eory behaviors With 12 paraeters listed in table 6, the odel was able to predict the shape eory behaviors for the seven fiber orientation cases and for the case of the neat rubbery atrix aterial. Fig. 45 shows the coparison between experients and odel predictions. The odel prediction shows good agreeent with experients. The ajor error occurs at three steps: at S2 and S3, the error ight be due to the theral expansion; at S4 (unloading), prediction shows better fixity than experient, due to the plastic deforation that the odel cannot characterize. Table 6. List of aterial paraeters Paraeter Value Description Coposition v.833 Volue fraction of Matrix v.167 Volue fraction of Fiber f Mechanical behavior N.822 kpa / K Polyer crosslinking density 9MPa The effective odulus of fibers in the coposite c Theral strains C 4 1 CTE at high teperature for the C case 2 1 C 4 1 CTE at high teperature for the 9 C case

99 91 c 1 c 2 PT 1 PT C 4 1 CTE at low teperature for the C case 2 1 C 4 1 CTE at low teperature for the 9 C case Theral strain due to crystallization for the C case Kinetics of crystallization k 8 1 Theral strain due to crystallization for the 9 C case 1 1 in Growth rate of the crystalline phases n 3 Diension of the crystalline phases (a) (b) (c) (d) (e) (f) (g) (h) Fig. 45 Model predictions for shape eory behaviors: (a) the fiber orientation case. (b) the 15 fiber orientation case. (c) The 3 fiber orientation case. (d) The 45 fiber orientation case. (e) The 6 fiber orientation case. (f) The 75 fiber orientation case. (g) The 9 fiber orientation case. (h) The rubbery atrix aterial Paraetric study Fixity can be iproved by increasing volue fraction of fibers ( v f ). In Fig. 46a, the fixity is enhanced by increasing v f. It is also found that as v f, where it is actually the neat

100 92 rubbery atrix aterial, the fixity turns into negative, which is consistent with the experiental observation. The effective shear odulus of fibers ight be changed by changing the interface between the atrix and fibers or choosing different aterial with different stiffness as fiber. The changing of the effective shear odulus of fibers affects the fixity of the coposite. Fig. 46b shows the fixity at arbitrary fiber orientation with different effective shear odulus ( ). As fiber orientation 45, apparently, when c is increased, the fixity is iproved. As fiber orientation 45, the effect is still observed, but it is liited as 2MPa. c c (a) Fig. 46 Paraetric studies: (a) effect of volue fractions of fibers. (b) Effect of effective shear odulus of fiber crystals. (b) Effect of theral strain The odel was used to investigate the effect of the theral strain on eory behaviors. Fig. 47 shows the coparison of a shape eory cycle between experient and odel predictions with/without theral strain for fiber orientation case. For the case without theral strain, during S2 (cooling), the strain goes up further than the one with theral strain. During S2, as the atrix in the rubbery state and fiber elted, the odulus of the coposite is decreased by

101 93 decreasing teperature due to entropic elasticity. In order to balance the constant load applied, the aterial defors ore at lower odulus. However, during cooling, theral strain always decreases. This is the reason that the slope of the strain without theral strain is larger than the one with theral strain. At S3 (isotheral crystallization), for the case without theral strain, the strain drop due the crystallization disappears. Based on Eq. (1), the fixity with theral strain is R f u u 1% and the one without theral strain is R f 1%. Fro Fig. 47, obviously, u u T and T, where T is the theral strain at 3 C and T. Therefore, we always have u u indicating that during unloading, only echanical strain changes. We also always have that R f Rf. Fig. 47b also shows the sae coparison for the rubbery atrix aterial. For the case without theral strain, at S4 (unloading), the strain iediately drops to zero and the fixity for u this case ( R f 1% ) turns to zero. This is reasonable for the aterial doesn t have shape T eory effect, whose fixity ust be zero. For the case with theral strain, after unloading turns u to be negative, since it includes the theral contraction. The negative u akes the fixity R f u 1% to be negative. In Fig. 47c, the odel predicts the fixity without the theral strain for all cases. It shows that the fixity without theral strain is always higher than the one with theral strain. For the neat rubbery atrix case, once the theral strain is not considered, the fixity turns to be zero.

102 94 (a) (b) (c) Fig. 47 Effect of theral strain: (a) coparison of the fiber orientation case with theral strain and without theral strain. (b) Coparison of the rubbery atrix aterial with theral strain and without theral strain. (c) Coparison of fixity with theral strain and without theral strain. 4.5 Conclusion In this chapter, the fabrication of ASMECs by 3D printer is introduced. Experients including DMA tests, uniaxial tensions, theral strain tests and shape eory behavior tests show the ASMECs exhibit isotropic behaviors. A 3D anisotropic theroechanical constitutive is developed to capture the anisotropic shape eory behaviors of the ASMEC. The paraetric studies such as the effect of volue fractions and the effect of the effective shear odulus of fibers are investigated using the odel.

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108 1 APPENDIX A PREDICTION OF TEMPERATURE-DEPENDENT FREE RECOVERY BEHAVIORS OF AMORPHOUS SHAPE MEMORY POLYMERS 1. Introduction Shape eory polyers (SMPs) are active aterials that can recover a pre-defored shape in response to an environental stiulus such as teperature (Di Marzio and Yang, 1997, Lendlein and Kelch, 22, 25, Liu, et al., 27, Luo and Mather, 29, Xie, 21), light (Jiang, et al., 26, Koerner, et al., 24, Lendlein, et al., 25, Li, et al., 23, Long, et al., 29, Ryu, et al., 212, Scott, et al., 26, Scott, et al., 25, Westbrook,Castro, et al., 21), oisture (Huang, et al., 25), and agnetic field (Mohr, et al., 26). The ost attractive feature of SMPs is their capability of undergoing very large deforation. For exaple, deforation excess of 11% was reported (Lendlein and Kelch, 22). This advantage allows SMPs to be used in any applications such as actuation coponents in icrosyste, bioedical devices, active surface patterning, aerospace deployable structure, and orphing structures (Lendlein and Kelch, 22, 25, Liu, et al., 26, Liu, et al., 24, Tobushi,Hara, et al., 1996, Willias, et al., 1955, Yakacki, et al., 27). Aong the SMPs developed recently, therally triggered SMPs are the priary focus in the SMP research (Yakacki, et al., 27). For therally triggered SMPs, they undergo a typical four-step theroechanical shape eory (SM) cycle to achieve the SM behavior (Castro, et al., 21, Di Marzio and Yang, 1997, Lendlein and Kelch, 22, Liu, et al., 26, Qi, et al., 28a). At Step 1, the polyer is heated to a teperature above its phase transition teperature T trans (the phase transition teperature can be either glass transition teperature T g for aorphous SMPs or elting teperature T for seicrystalline SMPs. Specifically, in this paper, the siplified odel is developed for aorphous SMPs, which include covalently cross-linked glassy theroset networks and

109 11 physically cross-linked glassy copolyers(liu, et al., 27), and Ttrans Tg ), then defored to a desired shape. At Step 2, the polyer is cooled down to a teperature below T trans, while aintaining the initial deforation constraint. At Step 3, the deforation constraint is reoved and the teporary shape is fixed. At Step 4, as heating back to a teperature above T trans polyer recovers into its peranent (initial) shape. Steps 1-3 are also referred to as the prograing step and Step 4 is the recovery step. For certain applications where the shape recovery is iportant, Step 4 is under a free recovery condition, where the initial deforation constraint is reoved during heating., the In the design of SMP applications, theoretical odels are desired to predict the behaviors of targeted design. In the past, several odels (Chen and Lagoudas, 28a, b, Ge,Luo, et al., 212, Liu, et al., 26, Nguyen, et al., 28, Qi, et al., 28a, Srivastava, et al., 21) were developed and ost of the were used to predict the coprehensive behaviors of SMPs under threediension finite deforation. For exaple, recently, Westbrook et al. (Ge,Luo, et al., 212) developed a odel that considers finite deforation stress-strain behaviors (including yielding, post-yielding, and rate dependency), constrained recovery behaviors and free recovery behaviors. Because of the coplicated thero-echanical properties of SMPs, it is not a surprise that these odels involve a large nuber of aterial paraeters (typically 2-3). In order to obtain these paraeters, one has to conduct a series of experients, soe of which are tie-consuing. For exaple, one has to conduct uniaxial experients at several distinctive teperatures and strain rates; free recovery experients have to be conducted and each of the could take about 4 to 8 hours, depending on the teperature rate used. After the experients, aterial paraeter fitting is also tie-consuing as ost of these odels contain ~2-3 paraeters. Finally, these odels are only useful to design engineers when they are ipleented into finite eleent codes. Certainly, once all these tie-consuing works are copleted, one can conduct high fidelity analysis to iprove designs. However, for any engineers, an estiation of key features of shape

110 12 eory behaviors, such as free recovery tie, can be very helpful, especially at the early stage of a aterial developent or a device design. Such an estiation should be based on a siple odel containing a liited aount of aterial paraeters that can be easily identified experientally. To our best knowledge, no such a odel is currently available. This paper takes a first step toward this direction and provides a siple odel to predict the teperature dependent free recovery behavior of SMPs. This odel is based on a odified standard linear solid (SLS) odel with Kohlrausch-Willias-Watts (KWW) stretched exponential function and requires only eight paraeters that can be deterined by stress relaxation tests. This paper is arranged in the following order: section 2 introduces aterials synthesis, experients including stress relaxation tests, and free recovery experients; section 3 presents the odified standard linear solid odel incorporating KWW stretched relaxation; in section 4, paraeter identification, odel predictions and paraetric studies are introduced; finally, conclusions are ade in section Materials and experients 2.1 Materials SMPs were synthesized following Yakacki et al. (27) (Yakacki, et al., 27). Briefly, tert-butyl acrylate (tba) onoer and crosslinker poly (ethylene glycol) diethacrylate (PEGDMA) were ixed according to a pre-calculated ratio, and 2,2-diethoxy-2- phenylacetophenone was added as photoinitiator. The polyer solution was injected between two glass slides or into a glass tube, which then was placed under the UV lap for polyerization. After 1 in, the SMP aterial was reoved fro the glass slides or tube and put into an oven for 1 hour at 8 C. The SMP aterials were achined into the proper shapes for different experients, which include stress relaxation test and free recovery test. 2.2 Stress relaxations

111 13 Stress relaxations were conducted by using a Dynaic Mechanical Analysis (DMA) tester (TA Instruents, DMA Q8). A rectangular sheet with a diension of was used for tests. Stress relaxation tests were perfored at 13 different teperatures evenly distributed fro 15 C to 45 C with 2.5 C interval. The saple was preloaded by N force to aintain the straightness. After reaching the testing teperature, it was allowed 3 in for the theral equilibriu. Then a 1% strain was applied to the saple and the relaxation odulus was easured within 3 inutes. Fig. 48 shows the results of stress relaxation under 13 different teperatures. Base on the well-known tie teperature superposition principle (TTSP) (Brinson, 27, Rubinstein and Colby, 23), a relaxation aster curve at 27.5 C can be constructed by shifting relaxation curves with shifting factors at different teperatures. The relaxation aster curve and shifting factors are reported in section 4. Fig. 48 Stress relaxations at teperatures varying fro 15 C to 45 C with 2.5 C interval. 2.3 Free recovery tests with raping teperature Free recovery tests were designed to validate the odel and were not used for odel paraeter identification. The cylindrical speciens (1 diaeter and 1 height) were tested using a MTS Universal Materials Testing Machine (Model Insight 1) equipped with a

112 14 custoized theral chaber (Thercraft, Model LBO). The tests were conducted following the four-step thero-echanical shape eory cycle described above and were presented in detail in Westbrook et al. 25. At the prograing teperature T H1 (4 C), the specien was initially copressed by 2% at a strain rate of.1/s. The specien was then allowed 1 in to relax before being cooled to the shape fixing teperature T L (2 C) at a linear rate of 2.5 C/in. Since the teperature rate was low, it was estiated that the teperature distribution in the saple was unifor. Once T L was reached, the specien was held for one hour then the copressive constraint was reoved. During the free recovery, the teperature was increased to a high teperature T H 2 at the sae rate of cooling and once the teperature reached T H 2, the specien recovered into its peranent shape. The shape (or strain) recovery in the specien was easured using a laser extensoeter (EIR, Model LE-5). Four high teperatures ( T H2 ) for free recovery were chosen as 3 C, 35 C, 4 C and 5 C. The results of free recovery tests were plotted in section 4 for the coparison between odel predictions and experients. 3. Model description As studied by Castro et al. (Castro, et al., 21), a standard linear solid (SLS) odel odified with Kohlrausch-Willias-Watts (KWW) stretched exponential relaxation can capture the SM behavior of an SMP. Below we use SLS-KWW odel and seek an analytical solution for free recovery. We start with odeling the isotheral free recovery with the SLS odel, then extend it to the nonisotheral case, and finally incorporate the KWW stretched exponential. 3.1 Isotheral free recovery A standard linear solid odel is used to describe the equilibriu and nonequilibriu behaviors of an SMP. As shown in Fig. 49, a linear spring with Young s odulus E is used to represent the equilibriu behavior; for the nonequilibriu viscoelastic response, the spring with

113 15 Young s odulus E 1 is used to represent the elastic response and the dashpot with viscosity represents the viscous behavior. The cobination of spring and dashpot gives rise to a corresponding relaxation tie E1. The nonequilibriu branch is also referred to as Maxwell odel with a stress relaxation odulus: exp t E1 E t. (A1) The total deforation is e and the deforation in the nonequilibriu branch can be decoposed into the elastic deforation e e and the viscous deforation e v. Fig. 49 A scheatics of the standard linear solid odel: a linear spring on the left with Young s odulus E represents the equilibriu behavior; on the right, the nonequilibriu branch consists of a spring with Young s odulus E 1 representing the elastic response and a dashpot with viscosity representing the viscous behavior. During the free recovery, the total stress should be zero, i.e.. (A2) total Ee E1e e Equilibriu Nonequilibriu

114 16 Fro Eq. (A2), the relation between the total deforation e and the elastic deforation in the nonequilibriu branch e e can be obtained as: e E e e. (A3) E1 In addition, the deforations and deforation rates in each branch have the following relationship: e ee ev, (A4a) e ee ev. (A4b) In the nonequilibriu branch, the stress in the dashpot should be equal to the stress in the spring, i.e. Ee E e. (A5) 1 v 1 e Dashpot Spring Cobing Eq. (A3) and Eq. (A5), one finds, e E v e. (A6) E 1 Considering Eq. (A3), Eq. (A4b) and Eq. (A6), one obtains, de E E 1 e E1 E1 1 dt. (A7) We denote the total deforation at the beginning of the free recovery as e and the deforation during the free recovery as and is defined as e, where et e 1 r t. 1 ax r t is the strain recovery ratio, r t e t e, (A8)

115 17 where e ax is the strain easured at the beginning of the free recovery step and et is the instantaneous strain during the free recovery. rt was originally proposed by Lendlein and Langer (Lendlein and Langer, 22) to quantify the aount of the initial fixed deforation that is recovered during a free recovery test. Integrating of Eq. (A7) on both sides, it follows: e 1 de t E E 1 dt e e E1 E. (A9) 1 Solving Eq. (9), one has the relation between t and : t E E 1 ln 1 E1 E 1. (A1) 3.2 Nonisotheral free recovery During the nonisotheral free recovery, the SMP is heated at a rate of Q C/in fro a low teperature T L. After t in, the teperature is T TL Qt. Note here, we assuing the heating rate to be low and therefore the teperature distribution is unifor within the saple. Under the nonisotheral condition, the relaxation tie (or viscosity) of the SMP strongly depends on teperatures. For a rheologically siple polyer, this variation should follow the tie teperature superposition principle (TTSP) with a shifting factor: T a T T, (A11) where is a reference relaxation tie. It was found that depending on whether teperature is above and near or below T g, the shifting factor a T can be calculated by two different ethods (O'Connell and McKenna, 1999). For teperature above and near T g, the WLF (Willias- Landel-Ferry) equation (Willias, et al., 1955) is used:

116 18 loga T 2 C1 T Tr C T T r, (A12) where C 1 and C2are aterial constants, and T r is a reference teperature. For the teperatures below T g, an Arrhenius-type behavior developed by Di Marzio and Yang et al. (1997) (Di Marzio and Yang, 1997) is used: ln a T AF C 1 1, (A13) k T Tr where A is aterial constant, F c is the configuration energy and k is Boltzann s constant. Considering Eq. (A9) and Eqs. (A11)-(A13), for the teperatures below T r (or T Qt T ), one have: L r e e 1 t c exp 1 1 b L r de E E AF dt e E E k T Qt T. (A14a) As the teperature is raised above T r (or TL Qt Tr), one have 1 1 L r r C T QtT L de 1 1 t E E AF c C2 TL Qt Tr 1 Q exp dt T 1. rt dt L e E1 E1 kb TL Qt Tr Q e T T e (A14b) Therefore, the relation between t and becoes, for the teperatures below T r, 1 t ln 1 exp 1 1 E E AFc 1 1 dt E E kb TL Qt Tr, (A15a) and for the teperatures above T r,

117 19 1 T 1 L r rt C T QtT L E E AF 1 1 t c C2 TL Qt T ln 1 Q exp dt T 1 rtl E 1 E1 kb TL Qt Tr Q r dt. (A15b) 3.3 KWW stretched exponential In ost polyers, the relaxation behavior is non-syetrical, non-exponential, non- Debye process, which cannot be captured by a single exponential function described by Eq. (A1) (Alvarez, et al., 1991, Lindsey and Patterson, 198, Svanberg, 23). The well-known Kohlrausch, Willias and Watts (KWW) (Willias and Watts, 197) stretch exponential function is ost popular and widely used to describe stress relaxation in polyers. In the KWW odel, the paraeter ( 1) is used to stretch the stress relaxation tie. The exact origin of this stretched exponential relaxation behaviors in glass-foring systes reain unclear. It is often viewed as a consequence of the dynaic heterogeneity of the syste, eaning that the stretching behavior is caused by the distribution of a series single exponential relaxation processes. However, studies have also shown that such a stretching behavior could be intrinsic to certain relaxation processes. (Angell, et al., 2, Phillips, 1996) rewritten as: Regardless of the physical origin, the stress relaxation odulus in Eq. (A1) can be 1 exp t E E t. (A16a) Eq. (A16a) can also be interpreted as a regular relaxation in the stretched tie space t with the tie apping of t t. This new tie space is also called aterial clock (Caruthers, et al., 24,

118 11 McKenna, 1989). Eq. (A16a) then recovers into a siple exponential relaxation behavior in the aterial clock t : 1 exp t E E t, (A16b) where. It can be shown that Eq. (A1) is also valid in the aterial clock: de E E 1 dt. (A17) e E E Since t t, dt with 1 dt t dt. Eq. (A17) becoes: de E E 1 e E1 E1 1 t 1 dt. (A18) When a SMP is under the isotheral free recovery, the relation between the recovery ratio and tie is: 1 E E t 1 E 1 E1 1 ln 1 t dt. (A19) When a SMP is under the nonisotheral free recovery, for teperatures below T r, the relation between the recovery ratio and tie is: 1 t c t dt 1 1 b L r ; (A2a) E E AF ln 1 exp E E k T Qt T For teperatures above T r, the relation between t and becoes: 1 T 1 L r rt C T QtT L E E AF 1 1 t c 1 C2 TL Qt Tr 1 ln 1 Q exp t dt T 1. r T t dt L E 1 E1 kb TL Qt T r Q (A2b)

119 Results and discussion 4.1 Paraeter identification In Eq. (A2a) and Eq. (A2b), which are used to predict free recovery behaviors, there are eight paraeters in total and they are listed in Table 7. All of the are deterined by stress relaxation tests. Base on the tie-teperature superposition principle, a relaxation aster curve at 27.5 C (Fig. 5a) can be constructed by shifting individual relaxation curves in Fig. 48 with shifting factors at different teperatures (Fig. 5b) (Brinson, 27, Rubinstein and Colby, 23). The ater relaxation curve can be described by a odified standard linear solid odel with KWW stretch exponential, where the stress relaxation odulus is: E( t) E E exp 1 t. (A21) In Eq. (A21), at tie we have E 3.5MPa ( t, E E E1, and as t approaches, E E. Fro Fig. 5a, E ) and E1 142MPa ( E E ). Using Eq. (A21) to fit the stress relaxation aster curve at 27.5 C, paraeters and are deterined as 2.5sec and.6, respectively. The odel fitting of the stress relaxation can be iproved better by introducing one siple exponential branch (or Maxwell branch). However, by introducing this additional branch, it is ipossible to ake the relatively siple solution in the for Eq. (A2b). As introduced in Section 3.2, the shifting factor as a function of teperature can be described by WLF and Arrhenius equations at different teperature ranges. In Fig. 5b, by fitting shifting factors with these two equations, paraeters in Eq. (A12) and (A13) are deterined as C 1 =51.6, C 2 =17.44, T r =27.5ºC and AFc k =-4. Copared with a reported siilar odel (Castro, et al., 21), paraeters identified by stress relaxation tests are in the range of those in the reported paper.

120 112 (a) Fig. 5 Paraeter fitting for stress relaxation. (a) The stress relaxation aster curve at 27.5 C. (b) The shifting factors with teperature. (b) Table 7 Paraeters for odel prediction Paraeter Value Description E 3.5MPa Elastic odulus in equilibriu branch E 1 142MPa Elastic odulus in nonequilibriu branch.6 KWW stretching paraeter 2.5s Relaxation tie at reference teperature T r 27.5 C Reference teperature C Paraeter for WLF equation C 1 2 AF k c b 51.6 C Paraeter for WLF equation 4K Pre-exponential paraeter for the Arrhenius-Type behavior 4.2 Model predictions for free recovery behaviors Fig. 51 shows the nonisotheral free recovery behaviors fro experients introduced in Section 2. The strain recovery ratio rt defined in Eq. (A8) was used to characterize free recovery behaviors. Clearly, the recovery rate and degree of recovery are faster and larger at higher recovery teperatures. Using paraeters in Table 7, the odel is able to predict free recovery behaviors. Fig. 51 shows the coparison between odel predictions and experients for free recovery behaviors. Overall, the odel is able to predict the recovery tie under different recovery teperatures and

121 113 odel predictions show good agreeent with experients. The recovery rate in predictions is slightly higher than the one in experients, especially at T H 2 =3ºC and 35ºC. The ain reason for this error is that in the stress relaxation fitting in Fig. 5a, the odulus in the odel fitting is relaxed faster than the experiental data. Fig. 51 Coparison between experients for free recovery. The odel also predicts free recovery behaviors under infinite heating rate, where teperatures instantaneously reach recovery teperatures. This represents the fastest recovery a SMP can achieve. Here, in order to copare the recovery rates, we define the recovery tie as the tie needed for 95% of recovery. In Fig. 52, for TH 2 5 C, the recovery tie is 5 seconds. For TH 2 4 C, it takes ~2.5 in to recover. For TH 2 35 C, the recovery tie is extended to ~36 in. For TH 2 3 C, it needs ~98 in (15.1 hours) to recover, as the recovery teperature is close to T g.

122 114 Fig. 52 Model predictions for free recovery behaviors under infinite heating rate. 4.3 Paraetric study In soe SMP applications, such as cardiovascular stents (Yakacki, et al., 27), recovery tie is an iportant feature of SMPs, as the faster it recovers, the shorter tie an operation takes and the less pain patients suffer. Here, paraetric studies are explored using the odel to investigate how to iprove the recovery tie by changing aterial properties. All of these predictions are under infinite heating rate with the recovery teperature of TH 2 35 C. In Fig. 53, the recovery tie is decreased by increasing the equilibriu odulus ( E ), or KWW stretching paraeter ( ), or decreasing the nonequilibriu odulus ( E 1 ), or the relaxation tie ( ). Table 2 gives the coparison of recovery tie (the tie needed for the 95% recovery) by changing aterial paraeters by the sae percentages. As indicated in the table, the recovery is ore sensitive to rather than other paraeters. When.6, the recovery tie is 36in. If is reduced by 4% to.36, the recovery tie is 83.8 days. If is increased by 6% to.96, the recovery tie is only 2.5sec.

123 115 (a) (b) (c) (d) Fig. 53 Predictions for free recovery at 35 C with different aterial paraeters. (a) By changing E. (b) By changing E 1. (c) By changing. (d) By changing. Table 8 Free recovery tie with different aterial paraeters. In each case, only one paraeter was varied to the percentage indicated in the table whilst the rest of the paraeters were the sae as those in Table 7. Paraeters 6% 8% 1% 12% 14% 16% E 83.6in 52in 36in 26.7in 2.7in 16.6in E 15.5in 24.8in 36in 48.6in 62.7in 78.2in in 28.7in 36in 42.9in 5.1in 57.2in 83.8days 78in 36in 5in 1.2in 2.5s

124 116 Above findings provide valuable insights on controlling the free recovery of a given SMP by tuning aterials properties. For exaple, both E and E 1 can be controlled via the choices of onoer and crosslinkers as well as their olar ratios. As shown by the WLF and Arrhenius relationships, the relaxation tie at a given teperature depends strongly on the T g of the SMP, which can be controlled via the forulations of the SMPs. Lastly, the KWW exponent can also be synthetically controlled. For polyers, it is known that characterizes the distribution of the relaxation processes, and is thus related to the heterogeneity (both structurally and dynaically) of the polyer. For exaple, the network structures of thiolene-based SMPs are uch ore hoogenous than that of the (eth) acrylate-based SMPs, because of their unique free-radical-ediated step-growth polyerization echanis (Hoyle and Bowan, 21). By ixing thiolene with (eth) acrylate onoers, systeatical variations in can be achieved. 5. Conclusion In this paper, a odified standard linear solid odel with KWW stretched relaxation exponential is used to obtain an analytical solution for capturing the nonisotheral free recovery behaviors of SMP. This solution requires only eight paraeters that can be obtained fro stress relaxation experients. The theoretical predictions of free recovery behavior show a good agreeent with experiental results for an acrylate-based SMP. Paraetric studies using this solution reveal that the free recovery tie can be reduced by increasing the equilibriu odulus ( E ) or KWW stretching paraeter ( ), or by decreasing the nonequilibriu odulus ( E 1 ) or the relaxation tie ( ) and is ost sensitive to the KWW stretching paraeter.

125 117 APPENDIX B THERMOMECHANICAL BEHAVIORS OF A TWO-WAY SHAPE MEMORY COMPOSITE ACTUATOR 1. Introduction Shape eory polyers (SMPs) are a class of eerging sart aterials, which are capable of fixing their teporary shape and recovering their peranent shape in response to environental stiulus such as teperature (Di Marzio and Yang, 1997, Ge,Luo, et al., 212, Ge,Yu, et al., 212, Lendlein and Kelch, 22, 25, Lendlein and Langer, 22, Liu, et al., 27, Nguyen, et al., 28, O'Connell and McKenna, 1999, Qi, et al., 28a, Westbrook, et al., 211, Willias, et al., 1955, Xie, 21, Xie, et al., 29), light (Jiang, et al., 26, Koerner, et al., 24, Lendlein, et al., 25, Li, et al., 23, Long, et al., 29, Rickert, et al., 22, Scott, et al., 26, Scott, et al., 25, Thoas and Lendlein, 22, Westbrook,Castro, et al., 21), oisture (Huang, et al., 25) or agnetic field (Mohr, et al., 26). SMPs can be further categorized as one-way SMPs (1W-SMPs) and two-way SMPs (2W-SMPs), based on whether the actuation is reversible or not. In the first kind, the transition fro the teporary shape to the peranent shape cannot be reversed by siply reversing the stiulus, or in order to achieve the teporary shape again after recovery, a new prograing step is necessary. In the second kind, the transition between the teporary shape and the peranent shape is reversible. Both one-way shape eory (1W-SM) effect and two-way shape eory (2W-SM) effect can find their applications. However, fro aterial processing and fabrication point of view, it is relatively ore difficult to achieve the 2W- SM effect. Several attepts were ade in the past to achieve a free-standing 2W-SM effect by developing a SMP coposite such as a polyeric lainate developed by Taagawa (Taagawa, 21); a bilayer polyeric lainate was introduced by Chen et al. (Chen, et al., 21, Chen, et al., 28). Recently, Chung reported a SMP, poly(cyclooctene) (PCO), which is capable of exhibiting both 1W- and 2W-SM effects (Chung, et al., 28). However, in order to achieve the 2W-SM

126 118 effect, a constant load (such as a dead weight) has to be applied to assist the reversible actuation. In typical SMP applications, it ight not be desirable to apply a constant external load to achieve the 2W-SM effect. To overcoe this disadvantage, we developed a PCO based coposite actuator that deonstrated a nonlinear reversible actuation under the teperature stiulus (Bellin, et al., 26). This coposite was fabricated by ebedding a PCO strip in its pre-stretched teporary shape into an elastoer atrix. Upon a heating and cooling cycle, the coposite actuator could be actuated fro a straight shape to a bent shape, and then be returned to its straight shape. Fig. 1a shows the actuation during a heating-cooling cycle and Fig. 54b shows the transverse displaceent of the end of the coposite actuator as a function of teperatures. In Fig. 54b, there are three key features of the free-standing 2W-SM effect: 1) the actuation is a highly nonlinear function of teperatures with a large hysteresis loop. 2) The training cycle of heatingcooling leads to an actuation different fro other cycles. After the training cycle, the aount of actuation is stabilized to a saller actuation. 3) During heating, the actuator first oves in a direction opposite to the actuation direction then oves back along the actuation direction. (a)

127 119 (b) Fig. 54 An actuation during a heating-cooling cycle: (a) the actuation shows the free-standing 2W-SM effects; (b) transverse displaceent of the actuator end for five theral cycles (Bellin, et al., 26). The goal of this paper is to theoretically investigate the theroechanical behaviors of this 2W-SMP coposite actuator and the echaniss for the observed phenoena during the actuation cycles, and to provide insight into how to iprove designs. The paper is arranged in the following anner. In section 2, aterials and fabrication for the actuator are briefly introduced, and the theroechanical behavior for the PCO SMP is presented. In section 3, the existing theroechanical constitutive odel for the PCO SMP is reviewed. The odel for the actuator is developed in section 4 by integrating the constitutive odels for the PCO SMP into a lainate coposite theory. In section 5, predictions fro the analytic odel are presented and the characterization of actuator is analyzed. Several paraetric studies such as effects of width ratio, odulus of the atrix and prograing stress are discussed. 2. Materials and experients 2.1 Materials The prograed 2W-SMP serving as a strip ebedded into the atrix. The 2W-SMP was synthesized by using Poly(cyclooctene) (PCO) (Evonik-Degussa Corporation, Vestenaer

128 with a trans content of 8% and dicuyl peroxide (DCP) (>98% purity, Aldrich). Details about the preparation of the saple were discussed by Chung et al (Chung, et al., 28). Fig. 55a shows the 1W-SM effect for PCO: the saple was stretched by an external load (7kPa) at T H (here TH 7 C ). Then, the teperature decreased fro T H to T L (here TL 15 C ), while the external load was aintained. After unloading at L T, PCO was capable of staying at the teporary shape. Finally, the saple recovered the peranent shape as being heated back fro T L to T H (Chung, et al., 28, Westbrook,Parakh, et al., 21). The 2W-SM effect can be achieved essentially by following the sae steps as those in 1W-SM effect, except that the external load was aintained during heating. The actuation strain used to characterize the 2W-SM effect, is defined as: R act T LTH LT H Ract T, which was L T, (B1) where LT H is the length of the saple at T H, L is the length as teperature varies. Fig. 55b shows Ract T teperature plot for 2W-SM behaviors of PCO under three different external loads (5 kpa, 6 kpa and 7 kpa). During cooling, at the teperature above ~35 C, the actuation strain increases alost linearly at a relatively sall slope. At the teperature between ~35 C to ~3 C, the actuation strain increases draatically during crystallization. At the teperature below ~3 C, the actuation strain saturates to ~25%. The agnitude of the actuation strain strongly depends on external load, i.e., higher external load results in higher actuation strain. During heating, at the teperature below ~5 C, the actuation strain increases slightly. At the teperature between ~5 C to ~55 C, the actuation strain decreases draatically due to elting of crystalline phase. Finally, at the teperature above ~55 C, the actuation strain decreases at a relatively sall slope siilar to that of cooling first region s slope (Chung, et al., 28, Westbrook,Parakh, et al., 21).

129 121 (a) (b) Fig. 55 PCO exhibits both 1W- and 2W-SM behaviors: (a) Stress-Teperature-Strain plot for 1W-SM behavior under an external load 7 kpa. (b) Actuation Strain-Teperature plot for 2W- SM behaviors under three different external loads (Chung, et al., 28, Westbrook,Parakh, et al., 21). An acrylate-based polyer was chosen as the atrix for the actuator. It was synthesized by 55 wt% poly(ethylene glycol) diethacrylate (PEGDMA), Mn 75 (PEGDMA 75, Aldrich) and 45 wt% tert-butyl acrylate (tba) (98% purity, Aldrich) and 2,2-diethoxy-2- phenylacetophenone (99% Purity, Aldrich) was added as the photoinitiator. The glass transition g teperature of this polyer is T 1 C, below the teperature range of the actuator heatingcooling cycle (15 C~7 C). Details were introduced by Ortega et al. (Ortega, et al., 28). 2.2 Actuator fabrication and characterization The polyer coposite actuator was fabricated by using PCO and PEGDMA. The details of fabrication were provided in Westbrook, et al. (Bellin, et al., 26). Briefly, the PCO strip was first prograed into a stretched shape under a stress of 7 kpa at 7 C, followed by cooling to T L =15 C then unloading. The prograed PCO strip was iediately placed into an aluinu old. After sealing the old by two glass slides, the PEGDMA solution was injected into the

130 122 old using a syringe. After polyerization and curing, the coposite aterial was reoved fro the old and sectioned into the required actuator diensions. The actuator characterization experients followed five heating-cooling cycles with each cycle as the following: first, the teperature was held constant at TL 15 C for 5 in and then heated to T H =7 C at a rate of 2 C/in. Once T H was reached, the teperature was held constant at T H for 1 in and then cooled to T L at a rate of 2 C/in. Lastly, the teperature was held constant at T L for an additional 5 in. The results of the characterization experients were presented in introduction (Fig. 54). 3. Constitutive odels for PCO and PEGDMA 3.1 Constitutive odel for the 2W-SMP The constitutive odel for the PCO SMP to capture both 1W- and 2W-SM behaviors was studied by Westbrook (Westbrook,Parakh, et al., 21). A brief review of this odel is provided below Mechanics for the PCO SMP PCO is a crystalizable crosslinking polyer. Therefore, it is assued that the PCO is a ixture of a rubbery phase and a crystalline phase. The volue fraction of each phase is deterined by the instantaneous teperature, deforation, and corresponding rate and history. A siple large deforation elasticity odel is adopted by assuing that both the rubbery phase and the crystalline phase follow the siilar for of stress-strain behavior: N kt ln, ln, (B2) R R R R C C C where the subscript R and C represent the rubbery phase and the crystalline phase, respectively. N R is the cross-link density, k is Boltzann s constant and T is the absolute

131 teperature. 123 NRkT gives the shear odulus for the rubbery phase of 2W-SMP and is the shear odulus for the crystalline phase. is the stretch ratio, and ln is the Hencky strain, which is used as it conveniently converts the ultiplication of stretches into additive strains and thus siplifies the work of tracking deforation in individual crystalline phases (Westbrook,Parakh, et al., 21). It should be noted that the stretch in the rubbery phase R generally is different fro the stretch in the crystalline phase C. Assuing that the theroechanical loading starts at a high teperature, at tie t, PCO consists of 1% rubbery phase. If a certain load total applied to 2W-SMP, the rubbery phase should defor by R, therefore: total R R. (B3) As the teperature is lowered to below crystallization teperature, crystallization starts. Since polyer crystallization is a relatively slow and continuous process, one iportant assuption is that when a sall fraction of polyer crystals is fored, it is in a stress-free state (Long, et al., 21). This stress-free state for newly fored crystalline phases was referred as natural configuration by Rajagapol (Rajagopal and Srinivasa, 1998a, b). In the case PCO, this evolution of polyer crystal phase also affects the echanical deforation and is attributed to underlying echanis for the 2W SM effect. More details about effects of crystal phase evolution on the echanics were discussed by Westbrook (Westbrook,Parakh, et al., 21). By considering the volue fractions of the rubbery phase and individual crystalline phase fored at different ties, the total stress at tie t becoes: 1 f fi i, (B4) total R R C i1 where f i1 f. Note that Eq. (B4) can also be applied to the case where there is no new i crystalline phase fors by siply setting f i.

132 124 During heating, the existing crystalline phases start to elt and recover to the rubbery state. At tie t t t, it is assued that the last fored crystalline phase with f e elts. 1 Concurrently to elting, a sall deforation elt is induced. Following the sae assuption for echanical deforation during crystallization, at tie t t n t,(t is the tie when elting starts) the crystalline phase with volue fraction elts and the deforation is f e n1 induced by n elt. The total stress is: n e n n n k e k e total 1 f e n R elt R fi C elt i k 1 i1 k1, (B5) where f e n f. e n i i Theral expansion coefficient t tc t : where Westbrook (21) presented the effective coefficient of theral expansion (CTE) at tie f i T 1 fir fic tran i1 i1 T R is the CTE of the rubbery phase,, (B6) C is the CTE of the crystalline phase, and tran is the volue expansion ratio during phase transition fro the rubbery phase to the crystalline phase; T is the teperature increent at the -th tie increent. Thus, the theral strain of 2W- SMP at tie t tc t is: T i i TT. (B7) i Evolution rule for the 2W-SMP

133 125 Westbrook (21) introduced the evolution rule for the 2W-SMP, based on Avrai s phase transition theory odified by Gent (Westbrook,Parakh, et al., 21). Crystallization occurs after the teperature is lower than the crystallization teperature T c and the crystallization rate is: f k f f T T, if ( T Tc and crit ), (B8) c c c crit where f is the volue fraction of the crystalline phase, k c is the crystallization efficiency factor and f is the saturated volue fraction. Siilar to the elting process, elting starts as the teperature above the elting teperature and the elting rate is: f k f T T, if ( T T ), (B9) where k is the elting efficiency factor. The total rate of crystalline foration is: f f f. (B1) c More details about the evolution rule were presented by Westbrook (Westbrook,Parakh, et al., 21). 3.2 Constitutive odel for the atrix aterial For the sake of siplicity, the atrix aterial follows the sae stress-strain behavior with the 2W-SMP rubbery phase: N kt ln (B11) Ma Ma Ma Ma where N is the cross-link density and NMakT is the shear odulus for PEGDMA. Ma The theral strain of the atrix aterial follows: where Ma is CTE for the atrix aterial. MaT Ma H T T (B12) 4. Model for the 2W-SMP coposite actuator

134 126 In this section, the odel for the actuator is developed by integrating the constitutive odels for the 2W-SMP into a lainate coposite theory. For the sake of siplicity, heat transfer is not considered in this case and we assue the teperature is uniforly distributed inside and outside the actuator. This is a close approxiation as the actuator is generally thin. 4.1 Geoetry The actuator is treated as a lainate. Fig. 56a shows the diensions of the cross section of the actuator (diensions listed in Table. 9). The xy plane passes through the iddle of the 2W- SMP strip where the z-coordinate defines the thickness. The top surface is z 1 and the botto is z 2, so z h h 2, z h h 2 and 1 T 2 2 B 2 z1 z2 h1. Based on the Euler-Bernoulli bea theory, taking the strain at z is o follows: ( o z o ), the strain due to bending along the z-axis z (B13) where is the curvature of the bea (Fig. 56b). It is ephasized here that z is not the neutral axis and o is not necessary to be zero. During actuation, the strain due to bending is generally sall ( 1.5% ), which is proved by the siulation shown later. Therefore, during actuation, we approxiate the Hencky strain equal to the engineering strain and in Eq. (B13), the engineering strain is replaced by the Hencky strain: z where o is the stretch in the xy plane and ln ln o z, (B14) z is the stretch along z-axis. Eq. (B15) helps to unify the strain description in the 2W-SMP and the increental strain during actuation. The transverse displaceent d is related the curvature of the actuator during bending through the relationship: d 1 cos L (B15)

135 127 where L is the length of the actuator and listed in Table 9. (a) Fig. 56 The geoetry of the actuator: (a) diensions of the cross section. (b) The scheatic of the bending of the actuator. (b) Table 9. Diensions of the cross section of the actuator Paraeter Value Description Actuator h Thickness w Width L Length Ebedded PCO Specien h 2.81 Thickness w Width h B.63 Offset 4.2 Analysis during actuation In this section, deforations during actuation including deforations on the 2W-SMP strip and the atrix are analyzed. We first investigate the aterial points on the xy plane ( z ). For portions out of the xy plane, the Hencky strain follows Eq. (B14). Therefore, in following, if it is not noted, all stretches and stresses discussed are those in the xy plane. Before actuation, since the 2W-SMP strip is ebedded under a load free condition, based on Eq. Error! Reference source not found., the total stress on the 2W-SMP strip is:

136 128 e A A f fi i 1, (B16) A total R R C i1 where f e A fi, R i1 A is the strain in rubbery phase and the strain in the crystalline phase i fored at t tc i t. During actuation, raising the teperature to above the elting teperature triggers the shape recovery of the 2W-SMP strip, which tends to contract. However, due to the existence of the atrix, the contraction of the 2W-SMP strip is constrained. Considering theral expansion, the deforation on the 2W-SMP strip can be decoposed into: I. The echanical deforation SM (Fig. 57), which gives rise to stress acting on the 2W- SMP strip. II. The theral expansion T, where T 1 T and T 1 at TL 15 C (Fig. 57). Therefore, at tie t t n t during actuation, for the portion of the 2W-SMP in the xy plane, A n n A the total deforation including prograing is and the total stress is: T SM R A e n A 1 f n fi i, (B17) An n n total e R SM R C SM i1 where f e n e n i i1 f. For the portion of the 2W-SMP out of the xy plane, the total stress is: e n A z 1 f z f z e, (B18) n where z follows Eq. (B14): SM An n n A total n R SM R i C SM i i1 n SM z ln ln z (B19) n SM In the atrix, as it constrains the recovery of the 2W-SMP strip, it is pulled back by the n 2W-SMP strip with Ma correspondingly and the stress on the atrix should be: A n n Ma Ma Ma. (B2)

137 129 For portions out of the xy plane, based on Eq. (B14), the Hencky strain of the atrix follows: n Ma z ln ln z. (B21) n n n By adding the theral expansion of the atrix, the total strain of the atrix is MaT MaT Ma (in n n n Fig. 57), and MaT 1 MaT, MaT 1 at TL 15 C. n Ma Fig. 57 Scheatic of deforations of the actuator: the brown bar represents the 2W-SMP strip and the blue bar represents the atrix. Before actuation, the 2W-SMP strip and the atrix have the sae length. During actuation, the deforation of the 2W-SMP strip can be decoposed into the echanical deforation SM due to the SM effect and the constraint fro the atrix, and the theral deforation T, and the deforation of the atrix could be decoposed into the echanical deforation Ma stretched by the 2W-SMP strip and the theral expansion MaT. Considering the geoetric copatibility, the 2W-SMP strip and the atrix should still have the sae length, that is TSM MaT Ma. Considering the geoetric copatibility, in the xy plane, the length of the 2W-SMP strip should be always equal to the length of atrix (Fig. 57). Therefore, during actuation, the deforation on the 2W-SMP strip should be equal to the deforation on the atrix (Fig. 57): TSM MaT Ma. (B22) According to Eq. (B22), the Hencky strain of the atrix in the xy plane can be calculated directly as following: Ma T SM MaT ln ln ln. (B23) Based on Eq. (B21) and Eq. (B23), any Hencky strain on the atrix along z-axis is:

138 13 z ln ln ln z. (B23) Ma T SM MaT 4.3 Solutions During actuation, there is no external load and oent applied to the actuator. Therefore, at t t n t, we have: A F F F, An An An total S Ma M M M, An An An total S Ma (B25) where F A n S and F A n Ma are the forces acting on the cross section of the 2W-SMP strip and the atrix, respectively: h2 2 An An n S w2 total SM h2 2 F,, z dz, h2 z1 An An n An n Ma 1 Ma Ma 2 Ma Ma z h F w,, z dz w,, z dz, (B26) M A n S and M A n Ma are oents, h2 2 An An n S w2 total SM h2 2 M,, z zdz, h2 z1 An An n An n Ma 1 Ma Ma 2 Ma Ma z h M w,, z zdz w,, z zdz. (B27) n Totally, there are three unknown quantities: SM and Eqs. (B25)-(B27). n, Ma and. They can be solved by Eq. (B21) 5. Results 5.1 Prediction for the actuator characterization experient

139 131 As is solved during heating-cooling actuation, based on Eq. (B15), the analytic odel presents the transverse displaceent varying with teperatures under five theral cycles. In the odel, there are 13 paraeters in total (listed in Table 1), including 5 paraeters for the theroechanical behavior of the 2W-SMP (PCO), 6 paraeters for the evolution rule of the 2W-SMP, and 2 paraeters for the theroechanical behavior of the atrix (PEGDMA). Paraeters for 2W-SMP (PCO) are directly fro Westbrook (Westbrook,Parakh, et al., 21) and paraeters for the atrix (PEGDMA) were identified by siple tests including uniaxial tension and stress free theral expansion test. Fig. 58 presents the odel prediction for the heating-cooling actuation cycles, which shows a good agreeent with the experiental results. Table 1 Paraeters for Analytic Model Description Paraeter Value Mechanical Behavior for the 2W-SMP (PCO) Polyer Crosslinking Density for Rubbery Phase N R k (Pa/K) Crystalline Phase Modulus μ c (MPa) 16 CTEs for the 2W-SMP (PCO) Rubbery Phase CTE α R ( C -1 ) Crystalline Phases CTE α C ( C -1 ) Phase transition volue expansion ration α tran ( C -1 ) Evolution Rule for the 2W-SMP (PCO) Crystallization Teperature T c ( C) 42 Melting Teperature T ( C) 47 Volue Fraction at Tie t= f (-).8 Critical Stretch for Crystalline Phase λ crit (-) 1.5 Crystallization Efficiency Factor k c (-) Melting Efficient Factor k (-) Theroechanical Behavior for the Matrix (PEGDMA) Polyer Crosslinking Density for the Matrix N Ma k (Pa/K) Matrix CTE α Ma ( C -1 ) 1 1-4

140 132 Fig. 58 Predictions for the heating-cooling actuation cycles. The odel successfully predicts three key features characterized in the Introduction. For the large hysteresis loop during actuation, this is attributed to the hysteresis of crystallization and elting for the 2W-SMP. For the actuation difference between the training cycle and others, it is essentially because of different loading histories on the 2W-SMP strip between the prograing cycle and later cycles. In Fig. 59, in the prograing step, a constant load (7kPa) was applied to achieve stretched shape ( 95%, in Fig. 8a) during cooling. After unloading at TL 15 C, the stretched 2W-SMP strip returned to 73.95% (Fig. 6b) and the strip was iediately placed into an aluinu old for the actuator fabrication. Therefore, at the beginning of the training cycle, the strain on the 2W-SMP strip was 73.95%. During heating in the training cycle, the constrained recovery of the 2W-SMP strip (fro 73.95% to 72.85% in Fig. 6b) led to the bending of the actuator. During cooling in the training cycle, the crystallization was triggered as the teperature below crystallization teperature. Differently fro the prograing cycle, however, the stress acted on the 2W-SMP strip was aintained by the atrix, therefore was a variable with teperature (Fig. 59) and dependent on the actuator bending. As a result, the strain of the 2W-SMP strip was 73.79% at the end of the train cycle (in Fig. 6). Therefore, the strain at the beginning of the first cycle is 73.79%, which is different fro the one at the beginning of the

141 133 training cycle (73.95%). For the opposite oving at the early stage during heat, where the elting of the 2W-SMP does not start, based on Eq. Error! Reference source not found., the effective CTE is / C, which is higher than the CTE of the atrix aterial (1 1-4 / C). Thus, PCO strip expands ore than the atrix, which leads the actuator to bend to the opposite direction to the actuation(fig. 61b). Once elting starts, the SM effect leads the 2W-SMP strip to contract and the actuator bends to the positive direction (Fig. 9c). Fig. 59 Stress vs teperature on the 2W-SMP strip during actuation including prograing (a) Fig. 6 Strain on the 2W-SMP strip during prograing and actuation: (a) including prograing; (b) during actuation. (b)

142 134 (a) (b) (c) Fig. 61 Scheatics of actuation (blue strip inside is the 2W-SMP strip and the white part around is the atrix): (a) original position of the actuator; (b) when the 2W-SMP strip expands ore than the atrix, it is defined that the actuator bends to the negative direction; (c) as the 2W-SMP strip contracts due to the SM effect, it is defined that the actuator bends to the positive direction. 5.2 Paraetric studies The odel provides also a good tool to explore the actuator design space which is constrained by the old geoetry, prograing and fabrication. In general, the axiu transverse displaceent can be iproved by reducing the actuator width, using softer atrix aterial or increasing the prograing stress. In Fig. 62, the 3D plots show the noralized transverse displaceent at TH 7 C varies with odulus of the atrix ( E fro 5MPa to 15MPa) and the prograing stress ( P fro Ma 5kPa to 2kPa) under three different width ratios ( R w =2, 2.82 and 4. Here, Rw w1 w2 ). It clearly shows that the transverse displaceent at 7 C is enhanced by decreasing the width ratio, choosing softer atrix and increasing the prograing stress. However, in Fig. 62a, when the odulus of the atrix is close to 5MPa and the prograing stress is larger than ~17kPa, the transverse displaceent is decreasing, instead of increasing, by increasing the prograing stress.

143 135 Based on Eq. (B15), the axiu transverse displaceent occurs at crit L.74, where crit.5 1. When is larger than crit, the tip of the actuator bends back and the transverse displaceent decreases when is still increasing. In other word, where the width ratio and the odulus of the atrix are low and the prograing stress is high such as the case in Fig. 62a, the axiu transverse displaceent does not occur at T H, although the axiu does. When the width ratio is high enough such as the case in Fig. 62b and c, both the axiu transverse displaceent and the axiu occur at T H. Fig. 63 shows the bending angle ( L ) of the actuator under extree conditions: low width ratio ( Rw 2), low odulus of the atrix ( EMa 1 5MPa ) and high prograing stress ( P 15 ~ 2kPa ). Siilar tendency with the noralized transverse displaceent, the increase of the prograing stress results in increase of the bending angle under constant width ratio and odulus of the atrix; the decrease of the odulus leads to the increase of the bending angle under constant width ratio and prograing stress. The green grid plane arks L 2, where the actuator curls a copletely closed circle.

144 136 (a) (b) (c) Fig. 62 3D plots showing the noralized transverse displaceent ( d L) at 7 C varying with odulus of the atrix ( E Ma ) and the prograing stress ( P ) under three different width ratios: (a) Rw 2; (b) Rw 2.82, the red star represents the current actuator; (c) Rw 4.

145 137 Figure 63. The bending angle ( L ) of the actuator under extree conditions ( Rw 2, EMa 1 5MPa and P 15 ~ 2kPa ), where the green grid plane represents L 2 and the actuator curls a copletely closed circle. 6. Conclusions An analytic odel for the free-standing polyer coposite actuator is developed. The actuator consists of a prograed 2W-SMP strip ebedded into a rubbery atrix. The actuation is triggered by the SM effect and the external load is provided by the bending of the atrix. In this paper, by taking advantage of existing constitutive for the 2W-SMP, an analytic odel is built to solve a bending proble for the coposite actuator during heating-cooling cycles. The odel successfully captures actuation behaviors and helps better understand phenoena during actuation. As a good design tool, the odel explores the actuator design space. The odel quantitatively presents the increase of the axiu transverse displaceent by reducing the actuator width, using softer atrix aterial or increasing the prograing stress.