Constitutive Model for Shape Memory Polymer Based on The

Original Article

Journal of Intelligent Material Systems and Structures 2016, Vol. 27(3) 314–323 Constitutive model for shape memory Ó The Author(s) 2015 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav polymer based on the viscoelasticity DOI: 10.1177/1045389X15571380 and phase transition theories jim.sagepub.com

Xiaogang Guo1,LiwuLiu2,BoZhou3, Yanju Liu2 and Jinsong Leng4

Abstract As a kind of smart materials, shape memory polymer has covered broad applications for its shape memory effect. Precise description of the mechanical behaviors of shape memory polymer has been an urgent problem. Since the phase transition was a continuous time-dependent process, shape memory polymer was composed of several individual transition phases. Owing to the forming and dissolving of individual phases with the change in temperature, shape memory effect emerged. In this research, shape memory polymer was divided into different kinds of phases including frozen and active phases for its thermodynamics properties. Based on the viscoelasticity theory and phase transition theories, a new constitutive model for shape memory polymer which can explain the above assumptions was studied. In addition, a normal distribution model, which possesses less but physical meaning parameters, was proposed to describe the variation in the volume fraction in shape memory polymer during heating or cooling process.

Keywords shape memory polymer, constitutive model, phase transition, viscoelasticity and phase transition theories

Introduction filled with particles, carbon nanotubes, and short fibers (Basit et al., 2012; Tan et al., 2014; Yu et al., 2013, Since the shape memory polymer (SMP) was discovered 2014; Zhang and Li, 2013), were investigated. Now, in 1980s, it has drawn much attention and has been SMP composites and their products such as deployable vastly developed during the past decades (Ratna and and morphing structures have been researched and Karger-Kocsis, 2007; Wei et al., 1998). SMP possesses developed (Leng et al., 2011; Liu et al., 2014; Meng the ability of returning from deformed shape to original and Hu, 2009). shape under a reasonable stimulus such as temperature, In order to investigate the mechanical behaviors of light, electric field, magnetic field, pH, specific ions, or SMP, various constitutive models and a large tensile, enzyme (Behl and Lendlein, 2007; Fei et al., 2012; Hu et compression, torsion, and flexure experiments (Chen al., 2012; Leng et al., 2011; Meng and Li, 2013; Xu et et al., 2014; Diani et al., 2012; Guo et al., 2014; Liu al., 2010). And the thermal-actuated SMP has been widely studied as a typical one. With the variation in temperature, the thermal–mechanical cycle of shape 1College of Aerospace and Civil Engineering, Harbin Engineering memory effect (SME) is as shown in Figure 1. University, Harbin, China 2Department of Astronautical Science and Mechanics, Harbin Institute of Depending on the phase transition temperature Tg,the Technology, Harbin, China process can be classified into the following steps: (1) 3College of Pipeline and Civil Engineering, China University of Petroleum, heat and deform SMP at a high temperature above Tg; Qingdao, China (2) cool and remove the external force at a lower tem- 4Center for Composite Materials and Structures, Harbin Institute of Technology, Harbin, China perature; and (3) heat the pre-deformed SMP above Tg and then the polymer will recover to its original shape. Corresponding authors: Compared with other smart materials such as shape Yanju Liu, Department of Astronautical Science and Mechanics, Harbin memory alloy, SMP possesses the advantages of large Institute of Technology, Harbin 150001, China. deformation capacity, low density, and cost. However, Email: [email protected] the main drawbacks of SMP are their low actuation Jinsong Leng, Center for Composite Materials and Structures, Harbin forces and long recovery time. For the sake of their low Institute of Technology, Harbin 150080, China. stiffness and strength, SMP composites, which were Email: [email protected] Guo et al. 315

Applied Force Applied Force

Cooling under the Remove the force Heating constraint strain and heating

Original Shape Pre- deformed Shape Original Shape

Figure 1. Schematic of phase transition in shape memory effect during typical thermal–mechanical cycle. et al., 2006; Srivastava et al., 2010) were carried out. model for crystallization of SMP was proposed and the Tobushi et al. (1997) established a simple thermal– interaction between original amorphous and semi- mechanical model describing the mechanical behaviors crystalline phase was also investigated (Barot et al., of the thermal-actuated polymer using a rheological 2008). In order to explain the molecular mechanisms of model based on the viscoelasticity theory. Through tak- the SME in amorphous SMPs, Nguyen et al. (2008) ing nonlinear mechanical behaviors into account, a established a thermal-viscoelastic constitutive model nonlinear one-dimensional (1D) constitutive model for and assumed that structural relaxation and stress SMP was proposed in 2001 (Tobushi et al., 2001). relaxation were the key factors for SME. Taking the Considering the strain rate during the loading and non-equilibrium relaxation processes into account, unloading processes, Shim and Mohr (2011) built a Westbrook et al. (2011) established a 3D finite defor- new constitutive model which was composed of two mation constitutive model for amorphous SMP. Based Maxwell elements. Most of these modeling efforts have on micromechanics framework, Shojaei and Li (2013) adopted rheological models consisting of spring, dash- developed a multi-scale theory, which links micro-scale pot, and frictional elements in 1D constitutive models. and macro-scale constitutive behaviors of the polymers. Though these models are simple, they only agree quali- For the sake of the influence of material parameters on tatively with the experimental results. In the constitu- the response characteristics of the constitutive model, tive model of SMP, the concept of phase transition was Ghosh and Srinivasa (2013) proposed a 3D continuum also used. Although there is a slight difference between two-network for a thermal–elastic response model of physical nature of melting and glass transition, they SMP. And during the last few years, a new phenom- both can explain SME for certain macroscopic charac- enon for SMP—multi-SMEs—was discovered by which teristics. And the concept of phase transition is conve- SMP can memorize more than one temporary shape nient for analyzing the mechanical behaviors of SMP (Xie, 2010). And the research on the mechanisms of (Long et al., 2010). Based on modeling the crystalliza- multi-SMEs was investigated (Yu et al., 2012). tion and taking the phase transition into account, sev- The purpose of this article is to establish a constitu- eral constitutive models for SMP were also developed. tive model that can unify the linear elasticity and vis- In 2006, a new thermal–mechanical model that divided coelasticity into one equation. The concepts of phase the polymer into frozen and active phases was investi- transition and the theory of multi-SMEs were also con- gated, and a three-dimensional (3D) micromechanical sidered and described. Using the four-element model, constitutive model was built by Liu et al. (2006). In Liu the mechanical behaviors of the individual transition et al.’s (2006) article, the strain of the materials was phases were analyzed. The volume fractions of transi- decomposed into mechanical strain, thermal strain, and tion phases which can join them together were dis- storage strain which was the critical factor for the cussed, and a normal distributed equation that can SME. And the active and frozen phases were all treated simulate the change in these fractions in SMP during as elastic materials and could not describe the viscoe- heating or cooling process was built. Finally, in order lasticity of SMP at a constant loading condition and to confirm the parameters in this new constitutive temperature. However, the SMP in this article was model, the dynamic mechanical analysis (DMA) tests treated as the elastic materials, not time dependency and creep experiments were also accomplished. materials (Baghani et al., 2012). Based on Liu et al.’s (2006) work and the experience of investigating the The phase transition in SMP phase transition of shape memory alloys, Chen and Lagoudas (2008a, 2008b) studied the micromechanism The phase transition process in SMP during heating or of SME and created a 3D constitutive model for SMP cooling is complex. Obviously, SMP consists of a lot of under large deformation. Meanwhile, a constitutive molecule chains with different lengths and angles. 316 Journal of Intelligent Material Systems and Structures 27(3)

Taking the pre-stress and storage energy into account, and kinds of models such as trigonometric function the entropy of each chain also differs from the others, model and quadratic polynomial model have been built as well as their phase transition temperatures. During in order to indicate the relationship between the frac- cooling process from a high temperature, the phase tion and the temperature. But the cosine model con- transition will come out first in the molecule chain with sisted of two equations, so the slope of two equations higher entropy. That is, when cooled to a specific tem- at demarcation point Tg was not equal. The quadratic perature, phase transition in the molecule chain will polynomial model built in 2006 was an empirical equa- happen and the relevant parts will change from ‘‘active tion, whose two parameters were confirmed by fitting phase’’ to ‘‘frozen phase.’’ Thus, the specific tempera- the experiment results. ture is the phase transition temperature of those mole- In this article, a new volume fraction model that can cule chains. With the decrease in temperature, the indicate the phase transition accurately in a simple for- molecule chain will shrink while the stiffness of SMP mation was brought up and the volume fraction of the will increase. The shrinkage of the chain was due to the SMP during phase transition can be given by removal of residual stress formed in the polymer process step (shown in Figure 1). Based on the statistics, ðT (TT )2 1 g the molecule chains in SMP can be divided into several f = pffiffiffiffiffiffie 2S2 dT ð2Þ f S 2p classes by their phase transition temperatures. The vol- Ts ume fraction of each active class can be expressed by fi while the one of frozen phase is ff . In this model, the in which S is the variance of the normal distribution variation in volume fraction controls the strain release function and can be expressed as and storage in thermomechanical cycle. And they can T T be defined as S = f g ð3Þ n X Vf Vi f = f = f + f = 1 ð1Þ where n is a constant. T is the average value of the nor- f V i V f i g mal distribution and it is also the temperature at which in which ff and fi are the volume fraction of frozen the volume fraction is 0.5. phase and ith relevant active phase, respectively. V is As indicated in Figure 2, with the increase in tem- the total volume of the SMP. When the SMP is loaded perature, the volume fraction of frozen phase decreases during heating or cooling process, the strain energy will nonlinearly, the rate of which grows rapidly until to the almost be absorbed by active phase for conformation phase transition temperature Tg. The phase transition rotation of the polymer. When cooled down to phase temperature is about 320 K, while the starting tempera- transition temperature, the phase transition will happen ture is 273 K. When T=Tg reaches 1.04, at which the and the strain in ith active phase will also be stored as environment temperature is 358 K, the phase transition storage strain. Finally, the strain energy and deforma- ends and the volume fraction of frozen phase will tion are almost stored in SMP even after unloading at decrease to 0. Based on the comparison results, the nor- a low temperature. During the recovery process, the mal distribution function can describe the variation in strain energy and deformation stored in the ith active the volume fraction accurately. phase during loading and cooling process will release at its phase transformation temperature. Then, the deformed SMP will recover to its original shape. And experiment result 1.0 the SME will emerge. simulation result In addition, when a multi-load is applied on SMP at different temperatures, the mechanical strain caused by multi-stress will be stored in different active phases such as ith, jth, or kth. Thus, during the heating or recovery 0.5 process, the storage strain in certain active phase will be released when these phases experience the relevant temperature again, which is the reason for multi-SMEs. In addition, not only the storage strain but also the volume fraction of frozen phase viscoelasticity behaviors of SMP such as stress relaxa- 0.0 tion play an important role on SME. Based on the above discussions, the volume fraction 0.90 0.95 1.00 1.05 of frozen phase and active phase is a very important T/Tg parameter for SMP during phase transition process. Figure 2. Comparisons with simulation results using The researches about this issue have been carried out, experiment results (Liu et al., 2006) of volume fraction. Guo et al. 317

A new constitutive model for SME So, the strain in SMP which is the function of time and temperature can be defined as In the ith active phase, the strain is decomposed into T two parts: thermal expansion strain ei and mechanical Xn m S m m T strain ei . Thus, the total strain of active phase can be e = ff (e + ef )+ fiei + e = s(ff (T)Jf (T, t) expressed as 1 Xn S T m T + fi(T)Ji(T, t)+ff e + e ð10Þ ei = ei + ei ð4Þ 1 While the strain in frozen phase is constituted by S m For multi-load conditions, the strain in the SMP can storage strain e , mechanical strain ef and thermal T be given as expansion strain ef Xm S m T ef = e + ef + ef ð5Þ e = ff (T)Jf (T, t tj)sj 1 So, the strain of SMP during heating or cooling pro- Xm Xn ÀÁ S T cess can be finally shown as + fi(T)Ji T, t tj) + ff e + e ð11Þ 1 1

e = ff ef + Sfiei ð6Þ Here, tj is the time when sj applied during the loading And the total thermal expansion strain eT , which is or recovery process. divided into several parts, is given as The phase transition process in SMP is continuous definitely during cooling or heating process. In order to T T T simplify calculations, we assume that e = ff ef + fiei =(ff af + fiai)DT ð7Þ J (T, t)=J (T, t), a = a ð12Þ where DT is the change in temperature and ai and af i a i a are the thermal expansion coefficients of ith active and The thermal expansion strain eT can be finally frozen phases, respectively. expressed as In order to describe the mechanical behavior of SMP, the 1D rheological model for the new constitu- ðT 0 tive model is shown in Figure 3. Gf and Gf represent T T T e = ff ef + Sfiei = (ff af + faaa)dT ð13Þ the elastic modulus of frozen phase model and Gi and 0 0 T0 Gi are of the ith phase model. hf and hf are the viscosity coefficients of frozen phase model, while the coeffi- And the mechanical strain in active phase em can be 0 a cient of ith phase model can be expressed by hi and hi. simply expressed as Due to the viscoelasticity behaviors of active and frozen phases, the mechanical strains em and em are indi- m m i f ea = ei = sJa(T, t) ð14Þ cated as in which Ja(T, t) is the creep compliance of active phase 1 1 0 t m t=ti e = sJi(T, t)=s + 0 (1 e )+ i 1 1 t=t0 t Gi Gi hi J (T, t)= + (1 e a )+ 15 !a 0 ð Þ Ga Ga ha 1 1 0 t m t=tf ef = sJf (T, t)=s + 0 (1 e )+ Gf Gf hf Incorporating equations (1) and (14) into equation (10), we have In equation (8), the parameter t0 is the retardation 0 0 Xn time and can be expressed by G and h . That is S m m T e = ff (e + ef )+ fiea + e h0 ÀÁ1 t0 = ð9Þ S T G0 = sff (T)Jf (T, t)+fa(T)Ja(T, t) + ff e + e ð16Þ

' η ηn T G ' η G ' η1 G i G ' ε f η f 1 η i η n ηn f 1 i σ G ' ' ' G ' f G1 Gi n

Figure 3. 1D rheological representations for the developed model. 318 Journal of Intelligent Material Systems and Structures 27(3)

where G(T) is the elastic modulus of SMP during cool- ' η ε ' η G η f S G η a ε f f a a T ing process. In this model, the elastic modulus of SMP σ G' ' f Ga G(T) can be represented by the elastic modulus of frozen phase Gf and active phase Ga Figure 4. A simplified 1D rheological representation for the 1 f ff developed model. = a + ð19Þ G(T) Ga Gf

As Liu indicated, Gf is a constant and Ga possesses a linear relationship with temperature

η ' η G ' f εS G η a Gf = Constant, Ga = 3 NKT ð20Þ f η f a a ε T σ G' G' f a In order to explore the mechanical behaviors during recovery process, the total strain of SMP was decom- 1 ε 2 ε 3 ε 1 ε 2 ε 3 ε f f f a a a posed into several parts (as shown in Figure 5)

Figure 5. A simplified 1D rheological representation for the S T e = ef + ea + e + e developed model with several decomposed strains. 1 2 3 1 2 3 S T ð21Þ = ef + ef + ef + ea + ea + ea + e + e Thus, the storage strain eS will not undertake the 0 load so is the thermal expansion strain eT . Therefore, Here, fa, Ga, ta, and ha are the volume fraction, elastic modulus, retardation time, and viscosity coefficient of when the load is constant, these strains can be expressed 0 as follows active phase, while ff , Gf , tf , and hf are the correspondent parameters of frozen phase, respectively. 8 1 8 > s = Gf e 1 During cooling process with a constant stress or <> f <> s = Gaea de2 de2 strain, the phase transition comes out and the mechani- s = G0 e2 + h0 f s = G0 e2 + h0 a > f f f dt > a a a dt ð22Þ cal strain in ith active phase will be stored in frozen :> de3 : de3 f s = h a phase. So, the storage strain eS is given as s = hf dt a dt During the recovery process, the stress in the SMP is Vðf Vðf fðf 0 obviously and the boundary conditions would be pre- S 1 m V m v 1 m e = ea dv = ea d = ea df ð17Þ dicted that ds=dt = 0 and s = 0. So, the strain in SMP Vf Vf V ff 0 0 0 can be rewritten as 8 8 Equation (16), which combines viscoelasticity theory 1 > 0 = Gf e 1 <> f <> 0 = Gaea and phase transition model, is the constitutive model de2 de2 0 = G0 e2 + h0 f 0 = G0 e2 + h0 a for SMP during cooling or heating. In this theory, the > f f f dt > a a a dt ð23Þ :> de3 : de3 above viscoelasticity model shown in Figure 3 was f 0 = h a 0 = hf dt a dt replaced by a simplified developed model which con- sisted of only two phases—frozen phase and active And the total strain during recover process can be phase (Figure 4). Moreover, the two distinct phases finally indicated as possess the equal stress but different strains. 2 2 S T Generally, stress relaxation will happen during cool- erec = ef + ea + fa(T)e0 + e ð24Þ ing process at a constant strain e0. And with the t=tf t=ta S T = e0(f e + f e )+f (T)e + e decrease in temperature, the increase in elastic modulus f a a 0 will play an important role compared with stress Here, e0 is the strain that formed during cooling pro- relaxation. Meanwhile, the active and frozen phases cess due to the molecule relaxation. s0 and G are the during cooling process were also treated as elastomeric. original applied stress and elastic modulus, respectively. S And during cooling process, the mechanical strain will The parameter fa(T)e0 represents the released storage S be stored due to phase transition and becomes an irre- strain at temperature T, while e0 is the original storage coverable deformation which cannot undertake stress. strain before recovery process. When the deformed spe- Thus, the stress sm used for maintaining the deforma- cimen was heated at a low temperature, the stored tion which is the function of temperature can be finally strain was only released partly and the specimen could expressed as not recover to its original shape completely. And the residual strain will depend on the value of active and 0 S T sm = G(T)(e e e ) ð18Þ frozen phases at that temperature. Guo et al. 319

In the creep experiments at a constant temperature, 2.5 2100 storage modulus(MPa) the viscous flow rarely happens and the storage strain is 340K Tan Delta c 2.0 also 0. So, the strain in frozen phase ef and active phase 1800 ec can be, respectively, expressed as a 1500 1.5 s s Tan Delta c (t=tf ) 1200 ef = + 0 (1 e ) Gf Gf 1.0 s s 900 c (t=ta) ea = + (1 e )

0 Storage modulus(MPa) 600 0.5 Ga Ga

300 At the beginning of the creep, the power exponent 0.0 on the right side of equation (25) is nearly 1. So, the 0 c c 300 320 340 360 380 strains in frozen phase ef and active phase ea are all lin- Temperature (K) ear as time goes on Figure 6. Dynamic mechanical analysis test results of shape c c s s memory polymer. ec = ff ef + faea = ff + fa ð26Þ Gf Ga

When the time t is infinitely great, the power exponent is approximately 0 for the measurements of the Table 1. Parameters of styrene obtained through creep parameters tf and ta experiments at different temperatures. ! s s s s 21 e = f ec + f ec = f + + f + T (K) E (MPa) J (MPa ) t (s) c f f a a f 0 a 0 Gf Gf Ga Ga 348 420 1.67 180 ð27Þ 360 10 1.75 73 368 1 3.9 65

Experimental and simulated results In order to investigate the mechanical behaviors of Creep experiment at a constant temperature SMP and the correctness of the new constitutive model For the sake of determining the parameters in equa- built in this article, several experiments such as the tions (8) and (15), the creep experiments at several high creep experiments and DMA test were completed. In temperatures Tg were finished. the section of DMA tests, the phase transition tempera- The creep experiments were conducted using ture of SMP was obtained. And the experimental ZWICK/Z010 at several constant temperatures. The results are also shown in Figure 6. And the accuracy of SMP was first preheated for about 30 min at the target the phase transition model (equation (2)) was verified temperature before loading. The stress at 348 and in the section of comparison between the simulated and 360 K was 0.05 MPa, while the stress at 368 K was experimental results from Liu et al.’s (2006) research 0.01 MPa. The experiment results are shown in Figures work. Finally, the creep experiments under a constant 7 to 9. According to the DMA test and creep experi- stress and temperature (348 and 360 K) were conducted ments, several parameters of SMP at different tempera- to verify the precision of constitutive model built in this tures can be obtained and are listed in Table 1. article. In this article, all the specimens used for Though it is difficult to measure the retardation and mechanical experiments were made of styrene, one typi- relaxation time at a low temperature, these two para- cal kind of SMP. meters can be indirectly achieved following the Arrhenius-type behavior (O’Connell and McKenna, DMA test 1999) Figure 6 is the DMA test results of SMP. The experi- AFc 1 1 ln aT (T)= ð28Þ ment sample was cut into pieces of 19 mm 3 3mm kB T Tg 3 2 mm using laser cutting machine at the environment temperature of 300 K. The experimental fre- in which kB is the Boltzmann constant, A is a constant, quency and temperature range were 1 Hz and from 293 and Fc is the configurationally free energy which is the to 460 K, respectively. As indicated in Figure 6, the reciprocal of the configurationally entropy Sc. relationship between temperature and elastic modulus Significantly, Fc is a constant below the glass transition was obtained. The glass transition temperature Tg is temperature. aT is the time–temperature superposition about 340 K. shift factor 320 Journal of Intelligent Material Systems and Structures 27(3)

t 10 a = 29 348K, 0.05MPa T ð Þ t0

8 So, the retardation time in active phase at a room temperature can be indirectly obtained.

6 In this model, there were several parameters to be determined through the experiments at high or low temperature to describe the mechanical behaviors of

Strain(%) 4 SMP. Ga and Gf are the elastic moduli of active and frozen phases, which can be obtained from the tensile 2 experiments at high and low temperatures with a con- 0 stant strain rate. ta and ha are the retardation time and 0 0 viscosity coefficient of active phase, while tf and hf are 0 500 1000 1500 2000 2500 the correspondent parameters of frozen phase, respec- Time(S) 0 0 tively. Ga, ta, and ha canbemeasuredthroughthe 0 Figure 7. Creep behavior of shape memory polymer at 348 K creep experiments at a high temperature, while tf and and 0.05 MPa. hf were measured at a low temperature which will cost a lot of time. So, in this model, the Arrhenius- type equation was proposed to estimate the values of 0 tf and hf . Besides, the phase transition temperature 10 T would be obtained by DMA tests. And these para- 360K,0.05MPa g meters of SMP used in this constitutive model are listed in Table 2. 8 As shown in Table 2, the retardation time and viscosity coefficient of frozen phase are nearly 107–108 6 times than those of active phase. And the elasticity characters play a more important role than viscoelasticity characters in frozen phase. For simplification, fro- Strain(%) 4 zen phase is analyzed as elastomeric materials with an elastic modulus Gf (as shown in Figure 10). So frozen 2 phase also can be interpreted as the phase used to fix geometry of the materials. Based on the theories established in this article, the 0 0 50 100 150 200 250 300 350 simulated results of creep and recovery processes were Time(S) obtained. In addition, the comparison results between simulation and experiments at different temperatures Figure 8. Creep behavior of shape memory polymer at 360 K (348, 360, and 368 K) are indicated in Figures 11 to 13. and 0.05 MPa. As the creep results show, the mechanical property of SMP was elastic at the beginning of loading process. As time goes on, the viscoelasticity of SMP gradually emerges and the relationship between strain and time is 368K, 0.01MPa nonlinear. The creep compliance can be obtained if 4 plenty of time condition is allowed. As the comparison results show, the constitutive model built in this article 3 can describe the creep behaviors of styrene precisely, especially at 348 and 368 K.

2 Strain(%) Cooling experiment under a constant strain 1 In order to investigate the mechanical behavior of SMP during cooling process, the cooling experiments at a constant strain were carried out. The samples were first 0 0 200 400 600 800 1000 stretched to an elongation of 30% using a 50%/s strain Time(S) rate at the temperature of 381 K (Tg + 20 K). The respective temperature was kept constant for about Figure 9. Creep behavior of shape memory polymer at 368 K and 0.01 MPa. 60 s before cooling down to the room temperature (with a cooling rate of 0:8 K= min). Guo et al. 321

Table 2. Summary of shape memory polymer and simulation parameters.

0 0 Ga (MPa) Ga (MPa) Gf (MPa) ha (MPa s) hf (MPa s) ta (s) 0.85 0.26 1825 1.68 3 106 6.1 3 109 71

0 1 4 4 tf (s) Tg (K) AFckB af (10 =K) aa (10 =K)

4.75 3 106 340 18,000 0.7 1.4

Experimental Results ' η ε Simulation Results G a S εT G f a ηa 4 σ ' Ga 3

Figure 10. A simplified 1D rheological representation 2 considering frozen phase as elastomer. Strain(%)

10 Experiment Result Simulation Result 0 0 200 400 600 800 1000 8 Time(S)

Figure 13. Comparisons with simulation results used creep 6 experiment results at 368 K and 0.01 MPa.

Strain(%) 4 The comparisons between the simulation and experi- 2 ment results during cooling under a constant strain (30%) process are shown in Figure 14. At the begin- 0 ning of cooling process, the stress used for maintaining 0 500 1000 1500 2000 2500 3000 Time(S) pre-strain increased slowly and the stress was about 0.32 MPa at 381 K. When the temperature reached Figure 11. Comparisons with simulation results using creep 334 K, the stress grew rapidly and nonlinear. The ulti- experiment results at 348 K and 0.05 MPa. mate value of stress was 6.2 MPa at 302 K.

10 Recovery experiment under a constant temperature Simulation Results Experiment Results In Figure 15, the recovery process at 343 K of SMP 8 was tested. When the pre-stress was removed at 3580 s, the pre-strain deduced sharply and the strain energy was released. But due to low temperature, one part of 6 pre-strain and energy in the material were stored. So, the deformed SMP could not recover to its original Strain(%) 4 shape except at a higher temperature or with a plenty of time. Using equations (14) and (24), the simulated 2 results during the loading and recovery processes were obtained. In addition, the recovery process at 360 K was also 0 0 50 100 150 200 250 300 350 400 conducted and the experimental results are also shown Time(S) in Figure 16. As Figure 16 indicates, the deformed specimen can recover to its original shape completely at a Figure 12. Comparisons with simulation results using creep experiment results at 360 K and 0.05 MPa. high temperature (360 K) in a shorter period of time. Besides, the simulated results were also obtained based 322 Journal of Intelligent Material Systems and Structures 27(3)

7 on equation (24). Through the comparison between the Simulation Results Experiment Results simulated and experimental results, the constitutive 6 model built in this article can describe the recovery pro-

5 cess of SMP precisely.

4 Conclusion 3 Cooling Process

Stress(MPa) The key of this article is to capture the SME of 2 SMP. Equation (16) explains the mechanism of phase

1 transition and multi-SMEs. For simplicity, SMP was divided into two parts: frozen phase and active phase 0 while the strain was composed of several parts: the 300 320 340 360 380 Temperature(K) mechanical strain in frozen and active phases, storage strain, and thermal expansion strain. Here, the stor- Figure 14. Comparisons with simulation results using the age strain is the crucial element of SME. In addition, experiment results of cooling under a constant strain (30%). a normal distributed model was used to characterize the volume fraction which is a key factor of equations (16) and (18). Through creep and retardation experiments, the retardation and relaxation time of frozen and active phases were obtained directly or indirectly. 8 Experimental Results Through comparison with the experimental results, Simulated Results 7 the constitutive model established in this article can describe the SME and mechanical behaviors of SMP 6 accurately. 5

4 Acknowledgement Strain(%) 3 This work was supported by the National Natural Science Foundation of China (Grant No 11225211). 2

1 Declaration of Conflicting Interests 0 0 1000 2000 3000 4000 5000 6000 7000 The author(s) declared no potential conflicts of interest with Time(s) respect to the research, authorship, and/or publication of this article. Figure 15. Comparisons with simulation results using recovery experiment results at 343 K. Funding The author(s) received no financial support for the research, authorship, and/or publication of this article.

35 Experimental Results Simulated Results References 30 Baghani M, Naghdabadi R, Arghavani J, et al. (2012) A thermodynamically-consistent 3D constitutive model for 25 shape memory polymers. International Journal of Plasticity 20 35: 13–30. Barot G, Rao IJ and Rajagopal KR (2008) A thermodynamic

Strain (%) 15 framework for the modeling of crystallizable shape memory polymers. International Journal of Engineering Science 10 46: 325–351.

5 Basit A, L’Hostis G and Durand B (2012) Multi-shape memory effect in shape memory polymer composites. Materials 0 Letters 74: 220–222. 0 200 400 600 800 1000 1200 1400 1600 Behl M and Lendlein A (2007) Actively moving polymers. Time (s) Soft Matter 3: 58–67. Figure 16. Comparisons with simulation results using Chen J, Liu L, Liu Y, et al. (2014) Thermoviscoelastic shape recovery experiment results at 360 K. memory behavior for epoxy-shape memory polymer. Smart Materials and Structures 23: 055025. Guo et al. 323

Chen Y-C and Lagoudas DC (2008a) A constitutive theory Shim J and Mohr D (2011) Rate dependent finite strain con- for shape memory polymers. Part I. Journal of the stitutive model of polyurea. International Journal of Plasti- Mechanics and Physics of Solids 56: 1752–1765. city 27: 868–886. Chen Y-C and Lagoudas DC (2008b) A constitutive theory Shojaei A and Li G (2013) Viscoplasticity analysis of semi- for shape memory polymers. Part II. Journal of the crystalline polymers: a multiscale approach within micro- Mechanics and Physics of Solids 56: 1766–1778. mechanics framework. International Journal of Plasticity Diani J, Gilormini P, Fre´dy C, et al. (2012) Predicting ther- 42: 31–49. mal shape memory of crosslinked polymer networks from Srivastava V, Chester SA and Anand L (2010) Thermally linear viscoelasticity. International Journal of Solids and actuated shape-memory polymers: experiments, theory, Structures 49: 793–799. and numerical simulations. Journal of the Mechanics and Fei G, Li G, Wu L, et al. (2012) A spatially and temporally Physics of Solids 58: 1100–1124. controlled shape memory process for electrically conduc- Tan Q, Liu L, Liu Y, et al. (2014) Thermal mechanical consti- tive polymer–carbon nanotube composites. Soft Matter 8: tutive model of fiber reinforced shape memory polymer 5123–5126. composite: based on bridging model. Composites Part A: Ghosh P and Srinivasa AR (2013) A two-network thermome- Applied Science and Manufacturing 64: 132–138. chanical model and parametric study of the response of Tobushi H, Hashimoto T, Hayashi S, et al. (1997) Thermome- shape memory polymers. Mechanics of Materials 60: 1–17. chanical constitutive modeling in shape memory polymer Guo X, Liu L, Liu Y, et al. (2014) Constitutive model for a of polyurethane series. Journal of Intelligent Material Sys- stress- and thermal-induced phase transition in a shape tems and Structures 8: 711–718. memory polymer. Smart Materials and Structures 23: Tobushi H, Okumura K, Endo M, et al. (2001) Thermome- 105019. chanical properties of polyurethane-shape memory poly- Hu J, Zhu Y, Huang H, et al. (2012) Recent advances in mer foam. Journal of Intelligent Material Systems and shape-memory polymers: structure, mechanism, function- Structures 12: 283–287. ality, modeling and applications. Progress in Polymer Sci- Wei ZG, Sandstroro¨m R and Miyazaki M (1998) Shape-mem- ence 37: 1720–1763. ory materials and hybrid composites for smart systems. Leng J, Lan X, Liu Y, et al. (2011) Shape-memory polymers Part I: shape-memory materials. Journal of Materials Sci- and their composites: stimulus methods and applications. ence 33: 3743–3762. Progress in Materials Science 56: 1077–1135. Westbrook KK, Kao PH, Castro F, et al. (2011) A 3D finite Liu Y, Du H, Liu L, et al. (2014) Shape memory polymers deformation constitutive model for amorphous shape and their composites in aerospace applications: a review. memory polymers: a multi-branch modeling approach for Smart Materials and Structures 23: 023001. nonequilibrium relaxation processes. Mechanics of Materi- Liu Y, Gall K, Dunn ML, et al. (2006) Thermomechanics of als 43: 853–869. shape memory polymers: uniaxial experiments and consti- Xie T (2010) Tunable polymer multi-shape memory effect. tutive modeling. International Journal of Plasticity 22: Nature 464: 267–270. 279–313. Xu B, Fu YQ, Ahmad M, et al. (2010) Thermo-mechanical Long KN, Dunn ML and Jerry Qi H (2010) Mechanics of soft properties of polystyrene-based shape memory nanocom- active materials with phase evolution. International Journal posites. Journal of Materials Chemistry 20: 3442–3448. of Plasticity 26: 603–616. Yu K, Liu Y and Leng J (2014) Shape memory polymer/CNT Meng H and Li G (2013) A review of stimuli-responsive shape composites and their microwave induced shape memory memory polymer composites. Polymer 54: 2199–2221. behaviors. RSC Advances 4: 2961–2968. Meng Q and Hu J (2009) A review of shape memory polymer Yu K, Westbrook KK, Kao PH, et al. (2013) Design consid- Composites Part A: Applied Sci- composites and blends. erations for shape memory polymer composites with mag- ence and Manufacturing 40: 1661–1672. netic particles. Journal of Composite Materials 47: 51–63. Nguyen T, Jerryqi H, Castro F, et al. (2008) A thermoviscoe- Yu K, Xie T, Leng J, et al. (2012) Mechanisms of multi-shape lastic model for amorphous shape memory polymers: memory effects and associated energy release in shape incorporating structural and stress relaxation. Journal of memory polymers. Soft Matter 8: 5687–5695. the Mechanics and Physics of Solids 56: 2792–2814. Zhang P and Li G (2013) Structural relaxation behavior of O’Connell PA and McKenna GB (1999) Arrhenius-type tem- strain hardened shape memory polymer fibers for self- perature dependence of the segmental relaxation below T . g healing applications. Journal of Polymer Science Part B: The Journal of Chemical Physics 110: 11054. Polymer Physics 51: 966–977. Ratna D and Karger-Kocsis J (2007) Recent advances in shape memory polymers and composites: a review. Journal of Materials Science 43: 254–269.