## Abstract

A multi-mechanism-based phenomenological model is developed within the finite deformation kinematics framework for capturing the thermomechanical behaviour of shape memory polymers (SMPs) both during programming and in service. Particularly, the damage mechanisms in SMPs are studied within the continuum damage mechanics (CDMs) framework in which they are classified into *mechanical* or *physical* damage, induced during service condition, e.g. fatigue and *functional* damage induced during thermomechanical cycles, e.g. shape recovery loss. Statistical mechanics is incorporated to describe the initiation and saturation of these deformation mechanisms. The main advantage of the presented viscoplastic model, comparing to the existing counterparts, is its simplicity by minimizing the need for curve fitting, and capability in simulating the nonlinear stress–strain behaviour of amorphous, crystalline or semicrystalline SMPs. The developed viscoplastic CDM model takes into account several distinctive deformation mechanisms involved in the thermomechanical cycle of SMPs, including glass transition loss events, temperature-dependent material properties, stress relaxation, shape recovery transient events and damage effects. The established model correlates well with the experimental results and its computational capabilities provide material designers with a powerful design tool for future SMP applications.

## 1. Introduction

Shape memory polymers (SMPs) are capable of storing a prescribed shape indefinitely and recover them by specific external trigger, e.g. heat. SMPs have potential to be deployed in various engineering structures and devices [1,2]. Material designers in the automotive, aerospace and biomedical industries demand computationally efficient and mathematically simple constitutive laws for these materials. The developed models should not only be able to replicate the process of the programming and the shape recovery, *viz*. *thermomechanical* cycle, but also be able to describe the mechanical responses under mechanical loads, e.g. quasi-static or dynamic loads. The SMP programming and shape recovery process are well described by the thermomechanical (TM) cycle; see the electronic supplementary material, appendix A for a general four-step thermomechanical cycle of an amorphous SMP and its syntactic foam (pre-straining at temperature above the transition temperature, cooling to below the transition temperature while holding the prestrain/prestress constant, and unloading to fix a temporary shape, which completes programming; heating above the transition temperature leading to shape/stress recovery) [3]. It is noted that programming may not need the typical heating and cooling process. Programming can also be conducted at temperature below the transition temperature such as cold compression programming for amorphous thermosetting SMPs [4] or cold-drawing programming for semicrystalline thermoplastic SMPs [5–7].

Thermomechanical models for SMPs should not only be able to address the complex deformation mechanisms and the finite deformation kinematics in the thermomechanical responses of SMPs, but also be suited for the finite-element analysis (FEA) implementation which is an essential design tool for structure engineers. Thermomechanical modelling of SMPs has been extensively studied in the literature [4,5,8–24]. Most recently Qi and co-workers [25] reported the influence of programming conditions on shape fixity and free shape recovery and developed a unified model for shape memory behaviours. The underlying physical mechanisms of multi-shape memory were discussed by Yu *et al.* [26], where a multi-branch phenomenological model was proposed to describe such effects. Despite the fact that many of the developed models perform quite well in capturing specific mechanisms in thermomechanical responses of SMPs, lack of a multi-mechanism-based three-dimensional model including the continuum damage effects remains an open challenge in the research arena. This work aims at developing a multi-mechanism-based comprehensive model for the thermomechanical behaviour of SMPs and it constitutes the theoretical foundation for taking into account the damage effects on the SMP performance.

Mechanical components will experience different damage mechanisms during their manufacturing, transportation, installation and service life [27,28]. Continuum damage mechanics (CDM) provides a promising modelling approach to evaluate the role of these effects on the life and the performance of these products. In general, the damage mechanisms in SMPs can be categorized into two broad classes: (i) *mechanical damage* (MD) or physical damage: which is associated with the mechanical loads, e.g. impact or fatigue, during the service life of the SMP structures, represented by degradation in physical or mechanical properties such as modulus of elasticity [22,23,29–34]; and (ii) *functional damage* (FD) *or thermomechanical* cyclic damage: besides the service loads, SMPs experience thermomechanical loads during their TM cycles and the associated damage in these cycles give rise to loss of their functionality [4,13,14,22,23,35,36]. Under cyclic thermomechanical loadings, SMP may exhibit continuous degradation in functionality, such as reduction in shape recovery ratio (SRR) or stress recovery ratio. This type of damage degrades the functionality of SMP, but not necessarily deteriorates the elastic properties of SMP. Therefore, it is necessary to differentiate MD from FD. In addition to programming, TM cycles are needed in many applications such as in repeatedly healing wide-opened crack either by constrained expansion of SMP matrix [29,33,37] or constrained shrinkage of embedded SMP fibres [6,7,38]. FD damage has been widely observed experimentally [8,9,14,22,23,39], while the lack of a consistent theoretical framework for this effect is addressed in this work. The FD mechanisms can be ascribed to the slide and failure of the SMP molecular chains, which are in charge of the shape memory storage and the recovery processes. It is worth noting that both FD and MD should be considered to study the functionality and the performance of SMPs in real-world applications.

While many phenomenological viscoplastic models have been developed in the literature for modelling inelastic response of polymers, most of the up to date phenomenological models suffer from large numbers of material parameters with no clear calibration methodology [36]. In this work, a new viscoplastic model is also developed based upon the statistical mechanics. The model is mathematically simple and computationally efficient and may provide a powerful design tool to capture the rate- and temperature-dependent inelastic responses of SMPs. In addition, the developed model is tailored to obtain the material parameters directly from the macroscopic experimental data that minimizes the efforts for the model calibration (see the electronic supplementary material, appendix H). It is worth noting that most of the material parameters in the existing counterpart models need to be found from numerical curve fitting techniques, and yet there is no guarantee to get converging results. Furthermore, one of the major contributions of the presented viscoplasticity theory is that the rate- and temperature-dependent responses of a wide range of polymers, e.g. amorphous, crystalline or semicrystalline, can be simulated regardless of their morphology.

The developed framework is thoroughly verified with the available experimental data in the literature. In this work, no attempt is made to elaborate the SMPs chemical classifications, shape memory mechanisms (such as crystallization/melting transition, vitrification/glass transition, etc.) or structural origins of SMPs. Interested readers may find thorough reviews on these topics in [1,2,36,40–42]. Instead the main emphasis is given to providing the physical and molecular-level descriptions for different deformation mechanisms associated with the thermomechanical cycles as well as the mechanical loading of SMPs. The paper is organized as follows: in §2, the basic kinematics for a finite deformation mechanism is outlined. In §3, the temperature-dependent material properties are formulated; and in §4, the rate- and temperature-dependent viscoplastic models are developed. Section 5 is concerned with the shape memory modelling. Stress relaxation is investigated in §6, and a cyclic thermomechanical damage model is developed in §7. The computational aspects are briefly discussed in the electronic supplementary material, appendix E and the model performance is demonstrated in §8 by comparing with experimental data.

## 2. Kinematics and constitutive relation

Owing to the high strain levels associated with the thermomechanical cyclic loading of SMPs, the constitutive model should be able to take into account the finite deformation kinematics. In this framework, the deformation gradient, *F*_{ij}, correlating the Lagrangian *x*_{i}(*X*_{i}, *t*) and Eulerian *X*_{i}(*x*_{i}, *t*) material coordinates, is used
*F*_{ij}, into the elastic, plastic and damage gradients is an essential assumption for developing the coupled plasticity-damage return mapping algorithms [23,35,43,44]
*Eulerian* strain tensor, *ϵ*_{ij}, is then computed based on the deformation gradient
*δ*_{ij} is the Kronecker delta. The spatial velocity gradient is defined as *D*_{ij} and spin tensor, *ω*_{ij}, are, respectively, given by the following relationships:
*L*_{ijkl} is the fourth-order elastic stiffness tensor, ‘∇_{….}’ indicates the Jaumann rate and *σ*_{kl} is the Cauchy stress tensor. It is worthwhile noting that in the case of SMPs there are highly nonlinear coupling effects between large deformation, shape memory effect, rate- and temperature-dependent behaviours, inelastic deformation and damage mechanisms. This work provides interrelated constitutive models for each individual mechanism to account for these coupling effects.

## 3. Temperature-dependent shape memory polymer properties

It is well understood now that the shape memory effect in the SMP is an exhibition of temperature-dependent viscoelasticity [4,13]. In this work, two terms, *viz.* ‘Background’ and ‘Shape memory affected’ properties, are used to separate the thermomechanical behaviour of a *programmed* SMP from a *non*-*programmed* one. In other words, a non-programmed SMP behaves like conventional polymers; and the *background* properties are capable of capturing its thermomechanical properties. In the case of a programmed SMP, while the background properties are still effective in the viscoelasto-plastic responses of the SMP, the shape memory effect is modelled through *shape memory affected* properties. It should be noted while the ‘Background’ and ‘Shape memory affected’ properties are treated separately herein, they are implicitly coupled through interrelated constitutive relations.

### (a) Shape recovery and glass transition events

The statistical mechanics has been widely used in the literature to characterize the DMA test results [46]. In this context, two functions, which capture the *Gaussian distribution* and its *cumulative norm* over the recovery transition, i.e. *T*_{r} for a programmed SMP, and the glass transition, i.e. *T*_{g}, temperatures, are defined. The Gaussian (or normal) distributions for the recovery and the glass transition events are, respectively, indicted by ‘dnorm_{r}’ and ‘dnorm_{g}’ and their respective cumulative values are shown by ‘pnrom_{r}’ and ‘pnorm_{g}’ hereafter. These functions are mathematically prescribed as follows:
*T*(°C) is temperature, *T*_{#}(°C) indicates the mean value for the distribution, *σ*_{#} (°C) is the standard deviation and *σ*^{2}_{#} is its variance. The subscript ‘#’ will accordingly be replaced by ‘g’ or ‘r’ to indicate the ‘glass transition’ or the ‘recovery transition’ processes, respectively. The parameters in equation (3.1) are directly obtainable from the DMA test results in which *T*_{g} is the glass transition and *σ*_{g} is the bandwidth of the glass transition event. Furthermore, *T*_{r} indicates the recovery temperature and *σ*_{r} is the temperature bandwidth of the recovery process and both of them are readily obtained from the DMA and shape recovery experiments. To simplify the formulation, the ‘dnorm_{#}’ functions are normalized with respect to its maximum values
^{max}_{#} is the maximum value obtained from equation (3.1). Mathematical discussions and performance evaluations of equations (3.1)–(3.3) are given in the electronic supplementary material, appendix B.

These functions provide the mathematical formulation backbone for modelling the transient responses of SMPs over the glass transition and the recovery events. For example, the loss tangent, i.e. *δ*_{0} is the *initial loss* tangent value and it is obtained from the DMA tests results at low temperature, e.g. *T*=25°C for the polystyrene SMP. The ‘initial loss’ tangent can be correlated to the *low-temperature* energy levels in which the molecular mobility reaches a minimum value.

### (b) Background storage elastic moduli

The elastic tensile moduli *E* can be decomposed into an elastic storage modulus, i.e. *E*^{′}, and a loss modulus, i.e. *E*^{′′}, in which they configure a complex modulus, *viz*. *K*=*K*^{′}+*iK*^{′′}. The engineering moduli can be simply expressed by the magnitude of this complex expression, i.e. *E*^{′} and bulk, *B*^{′}, storage moduli are constitutively prescribed as follows:
*E*′_{0} and *B*′_{0} (MPa) are reference elastic tensile and bulk storage moduli, respectively, which are the low-temperature (*T*=0°C) values of these properties obtained from the DMA testing machine. Other mechanical properties are computable based upon calculated bulk and tensile moduli. For example, Poisson's ratio is calculated by *v*=−0.5 (*E*/3*B*−1).

## 4. Thermo-viscoplastic constitutive model

The plastic deformation in polymers is a shear-driven process. Thus, one may introduce the plastic flow rule as follows:
*C*_{1} is the rate sensitivity parameter, *n*_{1} is a hardening exponent, *μ* is the shear modulus and |*τ*| is the effective shear stress, to be defined in the following.

Polymers show a post-yield softening response that is attributed to overcoming the conformational rotation resistance of the segments. During the softening, the applied external energy is dissipated in rearrangement of the polymer network. Upon saturation of conformational rotations, the molecular network starts to stretch and the polymer shows the subsequent hardening effect. In the theory of plasticity, the *backstress* tensor is defined to capture the strain hardening effects. A physically consistent kinematic hardening model is developed herein for polymers that incorporate the statistical mechanics for describing the conformational rotation and chain stretching processes. The *mean inelastic stretch* *σ*_{α} is the standard deviation that indicates the bandwidth of the hardening region, and it will be saturated at *α*_{ij}, for the kinematic hardening effects in SMPs
_{1} (MPa) is a material parameter and it is determined from the numerical curve fitting techniques. The driving stress for producing the inelastic deformation rate is computed based on the following relation [47]:
*X*_{ij}=*α*_{ij}−(1/3)*α*_{kk}*δ*_{ij} is the deviatoric part of the backstress tensor, and *s*_{ij}=*σ*_{ij}−(1/3)*σ*_{kk}*δ*_{ij} is the deviatoric Cauchy stress. The effective shear stress, |*τ*|, is computed by [47]
*temperature- and rate-dependents* yield stress, and it is defined by
*τ*_{0} is the yield stress at the reference loading rate *C*_{2} is a material parameter which is available for polymers in the literature [48,49], and the last term in equation (4.8) takes into account the temperature effect on the yield stress. The material parameters *σ*_{α} are directly obtained from tensile or compression tests. The parametric study on *σ*_{α} is given in the electronic supplementary material, appendix C.

## 5. Shape memory modelling

In a thermomechanical cycle, there are three main factors which dominate the shape recovery in SMPs, these are: (i) the level of the programming strain, (ii) the holding time at the programming stage, i.e. stress relaxation effect and (iii) the heating rate [4,13,18,36,50,51]. The process of the programming and the recovery of SMPs includes several experimentally observed mechanisms including: (i) at higher holding times and at elevated temperature higher recovery strains are achieved, (ii) increasing the heating rates during the recovery process, only gives rise to higher transition temperature, without major effect on the magnitude of the recovered strain and (iii) the FD result in loss of shape memory functionality of SMPs. The FD effect is more significant at higher programming strain levels and at high cycle applications of SMPs [52]. These experimental observations need to be incorporated to develop a physically consistent framework for the shape recovery process.

### (a) Shape recovery modelling

The deformation mechanisms involved in thermal expansion and shape recovery processes of SMPs are totally different, although a simple phenomenological method to model the isobaric thermal deformation of SMPs is to neglect the physics of the deformation and phenomenologically model the whole thermal deformation process. In this case, the isobaric thermal deformation in SMPs are attributed to two basic mechanisms which are (i) the thermal expansion controlled by the background linear thermal expansion coefficient (LTEC), *β*_{b} and (ii) the shape recovery parameter, *β*_{rec}, to be defined in equation (5.2a). Hence, the isotropic thermal stretch vector, *I*_{i} is the unity vector, Δ*T*=(*T*−*T*_{0}) denotes the temperature increment and *T*_{0} is a reference temperature, *viz.* room temperature herein. In most of the cases, owing to the high level of strain recovery of the programmed SMP, the presence of the background LTEC is insignificant and *β*_{b} can be neglected. The recovered shape is then controlled by the shape recovery parameter *β*_{rec}, over the transition temperature *T*_{r}. The next step is to formulate *β*_{rec} based upon programming/recovery mechanisms. One may assume that the programming strain *ϵ*^{(prog)} has a direct effect on *β*_{rec}; let the ‘pnorm_{r}’ brings the recovery temperature effects in play and function *F*_{trel} takes into account the relaxation time *t*_{rel} effect. Then, recovery parameter *β*_{rec} reads
^{tminrel} is a reference value for the SRR under minimum relaxation (holding) time *t*^{min}_{rel} during the programming cycle. The recovery strain *ϵ*^{(rec)} is the measured recovered strain from a programmed SMP. The shape recovery parameter ℵ^{tminrel} then reads
*ϵ*^{(prog)}=0.30 with zero holding time, the recovery strain is *ϵ*^{(rec)}=0.219 which results in ℵ^{tminrel}≅0.73. One may note that for pure SMPs programmed by the classical heating, cooling and load-removing process, i.e. ‘hot’ programming, the SRR is close to 100% for the first few cycles, and the FD effects are insignificant; although SRR value will be decreased in higher cycles due to the FD mechanisms [8,22,52–54].

### (b) Stress relaxation effect on shape recovery

The function *F*_{trel} in equation (5.2a) accounts for the stress relaxation effect during programming. As depicted in figure 1, the holding time during the programming has a direct impact on the magnitude of the shape fixity ratio (SFR) obtained after the unloading process and also on the SRR measured through unconstrained heating process, i.e. free recovery [2,4,13,14,22,36,55]. The effect of the holding time (or structural relaxation time) on the SRR is captured through *F*_{trel} function, and an empirical constitutive relation is defined for this function as follows:
*n*_{2} is a material parameter to calibrate the effect of the relaxation time *t*_{rel} on the recovery strain *ϵ*^{(rec)}, and *t*^{max}_{rel} indicates the maximum relaxation time in a set of experiments and it can be prescribed by the plateau of the stress relaxation curve where the relaxation has its maximum effect on the shape recovery. The parameter ℵ^{tmaxrel} is a reference value for SRR at the maximum relaxation time *t*^{max}_{rel}, and it is obtained as follows:
^{tmaxrel}=0.96 is observed from figure 1 for *t*^{max}_{rel}=120 (min).

### (c) Heating rate effect on shape recovery

The major impact of the temperature rising rate is on the recovery temperature, i.e. *T*_{r}, while no significant change on the recovered strain level has been reported [4,13], as long as the dwelling time is sufficient. To take this effect into account, let the minimum temperature rising rate in a set of experiments be given by ^{−1}), which is 0.6°C/min in this work. The higher heating rates will elevate the recovery transition temperature, i.e. *T*_{r}. Then one may substitute *T*_{r} in equations (3.1) and (3.2) with *n*_{3} is a material parameter to capture the magnitude of shifts in the transition temperature *T*_{r} and

## 6. Stress relaxation response of shape memory polymers

Stress relaxation is a time- and temperature-dependent processes [56]. As already mentioned in §5*b*, the relaxation time, *t*_{rel}, has a direct impact on SRR and SFR of SMPs, in which higher relaxation times result in higher level of shape fixity and strain recovery. These effects are experimentally investigated for the polystyrene SMP programmed at the room (glassy) temperature and high (rubbery) temperature [3,4]. In the case of the room temperature programming of the SMP, as depicted in figure 1, the difference between SFR for zero relaxation time, i.e. Δ*t*=0 min, and maximum relaxation time (which leads to stabilization of stress relaxation), i.e. Δ*t*=120 min is notable [4]. On the other hand, the temperature level contributes to the polymer chain mobility where higher rates of stress relaxation are expected at elevated temperatures. The temperature effects have already been included into the *background* and *shape memory affected* properties and the temperature effect on the stress level is automatically accounted in the viscoplastic model. The time dependency is taken into account by introducing a relaxation function *F*^{trel}_{rel} that depends on the relaxation time *t*_{rel}. Thus, the relaxed stress *σ*_{rel} is defined by
*σ*_{0} is the reference stress at the respective temperature with zero relaxation time. The relaxation time function *F*^{trel}_{rel} is defined as follows:
*a*_{1} is a material parameter which controls the rate of the relaxation.

## 7. Continuum damage mechanics in shape memory polymers

The SMPs may experience two classes of damages which are: (i) physical damage or MD damages during their operational conditions and under their service loads and (ii) FD mainly during the thermomechanical cycles. In order to provide computational competency for FEA implementation of these damage effects, they will be characterized separately in the subsequent subsections. The ME damages can be computed incrementally at each material point in an FEA code by introducing a proper damage parameter which is updated during the course of deformation. On the other hand, the FD damage parameters are updated *cyclically* after each round of the thermomechanical cycle. The proposed framework aids the material designers to account for the full damage spectrum in SMPs and it provides a simple and effective damage computation tool. The proposed scheme is depicted in figure 2 in which the overall SMP damage is subdivided into MD and FD. Each of these categories includes specific type of damage mechanisms and they will be discussed in detail herein.

### (a) Mechanical damages

Regardless of the material type, almost all mechanical components experience specific type of damage mechanisms during their service life. In the case of polymeric materials, the MD mechanisms can be correlated to the molecular chain failures in the polymeric network. The accumulation of the failed chains may result in the loss of the mechanical properties of polymers such as a gradual reduction in the elastic tensile modulus [55,57]. In the case of ductile or low cycle fatigue, the MD damage parameter is constitutively developed by [22,23,54,58]
*σ*_{ij}, *D*_{ij} and *d* is the accumulated MD. Owing to the fact that the constitutive laws for the MDs in polymers have been extensively investigated in the literature, the details of the MD formulation are eliminated for the sake of brevity. For example, in [54,58] thermodynamic consistent damage models have been developed for glassy polymers and they can be readily applied for the case of ductile MDs in SMPs. The correlation between the damage deformation gradient, i.e. *et al*. [22,23].

### (b) Mechanical damage induced by thermomechanical cycles

The process of the programming and shape recovery of an SMP is associated with rearrangement of the polymeric chain and conformational changes in which the molecular network of SMPs is altered accordingly [54]. This process then affects the elastic moduli, e.g. elastic tensile modulus, due to these molecular-level rearrangement mechanisms. It is obvious that these changes are highly dependent on the thermomechanical cycle conditions. For example, changes in the elastic compressive modulus of an SMP system programmed at room temperature, see [4], notably differs from a system undergoes a high-temperature classical thermomechanical cycle, see [3,54]. Also the free or the constrained recovery process conditions will have a significant effect on the elastic moduli variations, because the rearrangement of the molecular network of SMP is affected by these constraint conditions. The following relation is proposed to capture the shape recovery affected variation of the elastic moduli
*E*_{0} is the elastic moduli of the intact bulk SMP (before undergoing the TM cycle) and *E*_{Nmax} is the experimentally measured elastic modulus after the *N*_{max}-th TM cycles. These values are directly obtained after each round of the TM cycle from the compression or tensile testing of the SMP sample. A scalar MD damage parameter is introduced as follows:
*N*_{i} indicates the number of the thermomechanical cycle, and *N*_{max} is the maximum number of the thermomechanical cycles in a set of experiments.

### (c) Functional damage

SMPs undergo a series of high- and low-temperature mechanical loads during TM cycles, see the electronic supplementary material, appendix A and figure 1. Furthermore, the programmed SMP sample can be recovered through either ‘free recovery’ process in which the SMP is heated without any external constraints and its SRR is measured. Or ‘constrained recovery’ where the SMP is confined during the recovery process and the magnitude of the recoverable stress is captured. The accumulation of the FD results in gradual degradation of the shape memory functionality of SMPs. In other words, FD may cause loss of memory capability but not necessarily load carrying capacity. Over several programming and recovering cycles, the FD mechanisms may lead to certain levels of inelastic deformation during the free shape recovery process. Contrary to the pseudoplastic deformation in SMPs, which is recoverable, this portion of the inelastic deformation is permanent, and cannot be recovered, leading to reduction in the shape recovery capability of SMP.

In §7*c*(i), the reduction in the SRR due to the accumulative FD is investigated for the case of free recovery of an SMP system programmed at room temperature (figure 1). The confined shape recovery is considered in §7*c*(ii) in which the effect of the FD on the stress recovery ratio is investigated.

#### (i) Cyclic degradation of the shape recovery ratio in thermomechanical cycles with free recovery

This effect is modelled herein by introducing a damage parameter into the formulation. The SRR of the damaged SMP for the maximum *t*^{min}_{rel} *or* *t*^{max}_{rel}, subscript ‘D’ indicates the SRR of the damaged SMP sample, and ℵ^{tminrel} and ℵ^{tmaxrel} are the SRRs of the undamaged SMP samples, defined in equations (5.2b) and (5.3b). The parameter *N*_{max}-th TM cycle, for SMP samples that are programmed with the minimum relaxation time *t*^{min}_{rel} and maximum relaxation time *t*^{max}_{rel}. Thus, the experimentally measurable parameter *N*_{max} indicates that the measurement is taken after the last TM cycle. The damage exponent 0.4 in equation (7.5a) appears to result in acceptable representation of the experimental data for our polystyrene-based SMP systems while it might need to be adjusted for different SMP systems.

#### (ii) Functional damage effect on stress recovery ratio in thermomechanical cycles with confined shape recovery

The thermomechanical behaviour of an SMP under one-dimensional confinement is investigated through strain-controlled programming and fully confined shape recovery tests (see [37,59] for the detailed test procedure). The SMP used in cyclic tests is MP 5510, purchased from Diaplex Company Ltd., a subsidiary of Mitsubishi Heavy Industries, Ltd. The TM cycle in this case includes three steps: step 1: stressing at elevated temperature; step 2: cooling and unloading the sample and step 3: confined shape recovery in which the recovery stress is measured. Figure 3 shows the stress recovery step of the SMP-based syntactic foam for the third, ninth and 15th TM cycles, where the cyclic TM loading results in gradual reduction in the stress recovery ratio. The recovery force in figure 3 consists of two components (i) thermal stress by thermal expansion and (ii) shape recovery effect in the *T*_{g} region. The magnitude of the recovered stress is considerably less than the applied programming stress. One may ascribe this reduction to the fact that the majority of the applied programming stress is consumed by (i) stress that leads to the molecular conformational changes (stress relaxation), which is the stored stress; (ii) stress that causes MD; (iii) stress that causes unrecoverable permanent deformation and (iv) stress that relates to elastic and viscoelastic deformation (springback). Clearly, the recovered stress is only a small portion of the applied programming stress. We aim at providing an empirical damage relation in the following to account for this effect in stress controlled TM cycles.

Let the stress recovery ratio be denoted by *Ω*, which is defined as
*σ*^{(rec)} and *σ*^{(prog)} are, respectively, the recovery and the programming stresses. The difference between the peak stress during programming and the stress corresponding to the springback in a TM cycle is assumed to be *σ*^{(prog)}. The cyclic loss of the stress recovery ratio is then prescribed by a mathematical relation in which the measured *Ω*_{0} and *Ω*_{Nmax} values are related to the damaged stress recovery ratio, *Ω*_{D}, as follows:
*Ω*_{0} and *Ω*_{Nmax} (<*Ω*_{0}) are, respectively, the values for the stress recovery ratio in the first and *N*_{max}-th cycles. One may note that the exponent, 0.4, is a material parameter that scales the nonlinearity in the behaviour of the damaged SMP.

## 8. Results and discussion

In the electronic supplementary material, appendix D, the model is outlined and various aspects of the developed scheme are summarized. The computational aspect for the developed framework is briefly addressed in the electronic supplementary material, appendix E. Electronic supplementary material, appendix H summarizes the parameters used in the model, and how each parameter is determined (experiment or curve fitting). In this section, the performance of the established model is studied step by step for different deformation mechanisms associated with the thermomechanical cycle of SMPs. The experimental data for a polystyrene-based SMP system backs up the simulations. The SMP chemical characteristics, sample dimensions or the details of the testing methods are elaborated elsewhere [3,4,16,29,37]. Table 1 summarizes the material properties that are used in predicting (i) the temperature-dependent properties, (ii) viscoplastic response, (iii) shape recovery, (iv) stress relaxation and (v) cyclic FD of the SMP system. It is emphasized that only bolded numbers in table 1 are obtained through curve fitting of the experimental data, and the rest of the material parameters are directly obtained from their respective experiments.

### (a) Modelling the loss tangent

The DMA experiments with heating rate of 5°C/min and frequency of 1 Hz were conducted to capture the glass transition loss tangent, i.e. *T*_{g}=67.7°C, while a modified glass transition *T*_{g}=75°C results in more accurate representation of the

### (b) Storage moduli

The DMA test reveals the response of tensile, *E*^{′}, and bulk, *B*^{′}, storage moduli over the glass transition event. The DMA test for the heating rate of 5°C/min and frequency of 1 Hz is depicted in figure 5. It is worth nothing that the bulk modulus measurement is complex and it is captured herein through correlating the DMA result for the shear and tensile moduli [60,61]. Two material parameters *E*′_{0} and *B*′_{0} in equations (3.5) and (3.6) are obtained from the DMA test data.

### (c) Tensile elastic modulus changes over glass transition loss event

The DMA test results provide an insight for the loss and storage dynamic modulus variation of SMP. The loading rate in the DMA testing has a significant effect on both the total tensile elastic modulus, *E* and the glass transition temperature, *T*_{g}. The quasi-static moduli are experimentally measured using our material testing system (MTS) which is empowered with a digital furnace and provides the isothermal testing condition. This task is accomplished through compression testing of SMP under three different temperature levels, (i) room temperature, (ii) near glass transition temperature and (iii) rubbery state temperature. The results are shown in figure 6 and the simulation uses equation (3.5) by replacing the *E*′_{0} with low-temperature static elastic tensile modulus, i.e. *E*=698 MPa. As depicted in figure 6, the captured glass transition of SMP by MTS is different from the DMA test result. As already mentioned, the glass transition temperature is a function of the loading rate and at higher strain rates *T*_{g} is shifted to the higher temperatures. Then a modified glass transition is prescribed in equations (3.1) and (3.2) for the case of quasi-static loading condition, i.e. *T*^{modified}_{g}=50°C, shown by the arrow in figure 6. The modified glass transition temperature is directly measured from the MTS data in figure 6.

### (d) Viscoplastic behaviour

The performance of equation (4.8) in capturing the rate- and temperature-dependent yield stress of SMP under compression is depicted in figure 7. Figure 7*a* shows the simulation and experimental results for the strain rate dependency of the yield stress, and figure 7*b* shows the simulation and experimental data for the temperature-dependent yield stresses. As discussed previously, owing to the nature of the applied quasi-static loading, a new glass transition needs to be introduced to the model, which is *T*^{modified}_{g}=50°C and shown in figure 7*b* with the arrow.

Figure 8*a* shows the experiments and simulations for the rate-dependent response of the SMP system, where the model is used to capture the strain rates of 0.003 and 0.005 s^{−1}. The material parameters for the viscoplastic model are summarized in table 1. The temperature-dependent stress–strain response of SMP system is obtainable by incorporating temperature-dependent elastic moduli in the constitutive relation (figure 6) and using the temperature-dependent yield stress (figure 7*b*). The experimental and simulation results for temperature-dependent viscoplastic responses are shown in figure 8*b*. The temperature-dependent responses are not quite matched for the mid-range temperature such as *T*=40°C but it perfectly captures room- and high-temperature responses. The deviation in the mid-range temperature data is sourced in the accumulative numerical errors comes from the modelled material properties. The deviations between the simulation and the experimental data for both the elastic tensile modulus in figure 6, and the yield stress in figure 7*b*, are accumulated when they are used in the viscoplastic model and they result in unmatched stress–strain curves for the mid-range temperatures.

### (e) Shape recovery

As discussed previously, the shape recovery process is affected by several factors and these factors are constitutively formulated in this work. The material parameters for the shape recovery modelling are summarized in table 1. At first step, the role of the programming strain level on the shape recovery process is investigated. Figure 9 depicts the experimental results for SMP programmed under classical high-temperature TM cycle with two tensile programming strain levels, i.e. *ϵ*^{(prog)}=+5% and +30%. The SMP is freely recovered with heating rate of *ϵ*^{(prog)}=+10% case.

The role of the heating rate effect is studied next. As shown in figure 10, the heating rate only affects the transition temperature, while no significant effect is observed for the magnitude of the recovered shape. Figure 11 represents the experimental and simulation results for the effect of the holding time (relaxation) during the programming cycle in which maximum SFR and consequently maximum SRR are obtained at longer holding time. Maximum SRR is obtained when the SMP sample is kept under applied tensile programming strain up to 120 min, while under zero relaxation time considerable differences in SFR and SRR are observed.

Electronic supplementary material, appendix F shows the performance of the established stress relaxation formula, equation (6.1), in capturing the experimental data.

### (f) Functional damage effect on shape recovery ratio

To study the effect of the FDs, the SMP sample is programmed at room temperature with compressive strain *ϵ*^{(prog)}=−0.5, under two different relaxation times *t*^{min}_{rel}=0 min and *t*^{max}_{rel}=120 min. Free shape recovery at a heating rate of *t*^{min}_{rel}=0 min and (ii) maximum holding *t*^{max}_{rel}=120 min times. Figure 12*a*,*b* represent the cyclic damage effect on SRR of the SMP system when the minimum and maximum relaxation times are used, respectively. Note *N*=0 indicates the completion of the first TM cycle and *N*_{max} is the final TM cycle number in the set of experiment after the first cycle. Table 1 summarizes the required material properties to capture the FD effects.

### (g) Mechanical damage effect during thermomechanical cycles

The bulk SMP in this study has an initial *E*_{0}=698 MPa elastic modulus. After six rounds of thermomechanical cycles, i.e. *N*_{max}=6, of the classical high-temperature compression programming (*ϵ*^{(prog)}=−0.3) and free recovery, the modulus is increased to *E*_{Nmax}=810 MPa. The performance of equation (7.3) in capturing the elastic compression modulus variations together with the experimental data are shown in figure 13. The increase in *E* is obviously correlated to the gradual alignment of the SMP molecular network with the externally applied compression force field. The process of the compression at the elevated temperature and the free recovery process are also schematically shown in figure 13.

### (h) Functional damage effect on the stress recovery ratio

Equation (7.6b) is used to capture the cyclic loss of the stress recovery ratio in figure 14. This result can be directly used as input parameters for investigating the performances of self-healing materials in high cycle applications, in which several damage-healing cycles are expected to be concurrently active.

To investigate the performance of the proposed framework in more general loading conditions, the thermomechanical cycle of a two-dimensional traditional programming process, reported by Li & Xu [62], is simulated in the electronic supplementary material, appendix G. To summarize the material parameters in this work, they are listed in the electronic supplementary material, appendix H, based upon their functionalities. It is shown in the electronic supplementary material, table H1 that most of the model parameters in this work are directly obtained from the experimental tests rather than numerical curve fitting techniques.

## 9. Concluding remarks

The computational solid mechanics framework is incorporated to establish the viscoplastic, shape recovery and cyclic thermomechanical damage constitutive models. Particularly, we have decoupled the damage in SMPs into physical damage or MD and FD. A consistent modelling approach is developed to capture the temperature-dependent properties, e.g. elastic moduli, yield stress and viscoplastic deformations in SMPs. The established framework provides a comprehensive three-dimensional model to analyse the mechanical and thermomechanical loading condition of SMPs. The model is also formulated within the finite deformation kinematics and is able to describe the SMP loading conditions, where the applied strain level may reach several tens of per cent. It is worthwhile to mention that most of the up to date viscoplatic models for polymers are designed for a specific type of polymer, such as amorphous or semicrystalline, and a large number of material parameters need to be found from curve fitting techniques [24,36]. One of the major contributions of the presented viscoplasticity theory is that the rate- and temperature-dependent responses of a wide range of polymers can be simulated regardless of their morphology. Furthermore, the complex nonlinear post-yielding response of polymers are simulated through a novel Gaussian description where only the centre and the band width of the nonlinearity is required to model the strain hardening effect. Many of the material parameters in the developed model are directly obtained from the experimental data, see the electronic supplementary material, appendix H, instead of the cumbersome numerical curve fitting techniques, such as root mean square error minimization. These features of the developed framework enhance not only the simplicity but also the generality of the model. The SMP experimental data are used to back up the simulations and it is shown that the model correlates well with the experimental results. In order to prove the applicability of the developed framework in complex problems, the performance of the developed thermomechanical model is examined in the case of two-dimensional thermomechanical cycles, and it is observed that the simulations agree well with the reported experimental data.

## Acknowledgements

This investigation was partially supported by Cooperative Agreement NNX11AM17A between NASA and the Louisiana Board of Regents under contract NASA/LEQSF (2011–14)-Phase3–05. This study was also partially supported by the NSF under grant no. CMMI1333997 and Army Research Office under grant no. W911NF-13-1-0145.

- Received March 13, 2014.
- Accepted July 31, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.