#### Cyclic Thermomechanical Testing of Dual-Shape Polymers

Cyclic thermomechanical uniaxial tensile, bending, or compression tests are widely used to characterize SMPs, because a complete dataset of stress-temperature-strain over time can be generated. Such as modeling of stress–strain relationships is possible by fitting a theoretical model to existing datasets; in the same way, cyclic thermomechanical experiments serve as basis to evaluate and to fit theoretical models. As compared with a stress–strain relationship, where a constant strain rate is applied until a desired elongation is reached, which might be regarded as a single-step procedure, the thermomechanical evaluation of DSPs is a multistep procedure involving the creation of a temporary shape (called SMCP) and the recovery of the original shape under different conditions.

The first step during SMCP is the elongation of the sample to the programing strain ε_{m} at a temperature *T*_{deform}, which is above the transition temperature *T*_{trans} of the switching segment [see Fig. 1(a)]. At ε_{m}, the sample will be hold for a certain time to allow the sample to equilibrate. Subsequently, the sample will be cooled to *T*_{low}, which is below *T*_{trans}, either under constant strain (ε_{m}) or constant stress conditions (σ_{m} = const.) resulting in a maximum deformation ε(σ_{m}) = ε_{l}. During the cooling procedure, the formerly flexible switching segment will be fixed during the crystallization of the polymer chains, if *T*_{trans} = *T*_{m}, or during the transition from the rubbery into the glassy state, if *T*_{trans} = *T*_{g}. After *T*_{low} has been reached, the sample is unloaded and the temporary shape ε_{u} is obtained. The degree of how well the applied deformation ε_{m} or ε_{l} can be maintained is called the shape-fixity ratio *R*_{f}. *R*_{f} is calculated according to eq 1, while *N* is the number of repeated measurement cycles, if the sample is cooled under strain controlled conditions, or according to eq 2, if the sample is cooled under stress controlled conditions.

- (1)

- (2)

The recovery process can be investigated under stress-free (σ = 0 MPa) conditions to observe the shape recovery due to entropy elastic behavior of the switching segment above *T*_{trans}, when the polymer is heated to *T*_{high} ≥ *T*_{trans}. The temperature, at which the shape change occurs, is called the switching temperature *T*_{sw}. The recovered shape is determined as ε_{p} and the shape-recovery ratio *R*_{r}, which defines the degree of the shape recovery, can be obtained after one or more cycles, where each cycle is composed of the SMCP followed by the recovery procedure, and is defined in eq 3, if the programing was conducted under strain controlled conditions, or under stress controlled conditions (eq 4).

- (3)

- (4)

The stress-temperature-strain diagram in Figure 1 shows a simplified recovery process, where the sample fully recovers its original shape to ε_{p} = ε_{0}. Typically, ε_{p} remains higher than ε_{0} due to irrecoverable strain. Alternatively, the sample can be reheated under constant strain (ε = const.) conditions to measure the evolution of the stress at increasing temperatures. Here, the characteristic values that can be obtained from the stress–temperature curve for thermoplastic materials are the peak recovery stress σ_{max} and the respective temperature *T*_{σ}_{,max}. Above *T*_{σ}_{,max}, the recovery stress is reduced again due to softening of the polymer. In contrast to thermoplastic materials, the stress increases continuously for polymer networks and σ_{max} is maintained even at higher temperatures. The characteristic temperature *T*_{σ}_{,inf} is thus determined by the inflection point of the stress–temperature curve.26

#### Modeling of Dual-Shape Polymers

A schematic representation of a stress-temperature-strain diagram obtained in a cyclic thermomechanical tensile test of a thermoplastic DSP is shown in Figure 1(a), whereby in Figure 1(b) a simple viscoelastic model consisting of an elastic spring and a Maxwell model connected in parallel at different characteristic steps of the test cycle is illustrated. This model is called the standard linear solid (SLS).52 The elastic spring with a Young's modulus of *E*_{eq} represents the equilibrium response on stress by the permanent netpoints (hard segments). The nonequilibrium branch is presented by spring and dashpot in series. The dashpot has a viscosity η or a characteristic relaxation time τ = η /*E*_{r} , which depend strongly on temperature, representing the temperature dependent chain mobility of the switching segments (reversible crosslinks). At first, the viscosity of the dashpot is very low at *T*_{high}, as depicted in state 1 in Figure 1. At *T*_{high}, the deformation takes place so that the energy is stored mainly in the left spring of the equilibrium branch (see state 2 in Fig. 1), and the spring with modulus *E*_{r} of the Maxwell unit is almost undeformed compared with the initial state. During cooling to *T*_{low}, which is below *T*_{trans}, the viscosity or relaxation time of the dashpot increases by orders of magnitude. In state 3, the external load/stress has been removed at *T*_{low}. Because of the high viscosity, the Maxwell unit behaves like an elastic spring under constant stress. The stress free state (σ = 0) generates a redistribution of stored elastic energy in both springs combined with only a very small deformation ε_{m} − ε_{u}, because the modulus of the spring in the Maxwell unit of this model is in general much higher than the one in the equilibrium branch *E*_{r} ≫ *E*_{eq}. So, the deformation of the DSP at *T*_{high} is fixed. For the next step of the cycle, it is important to note that the deformation of the spring in the Maxwell unit is compressive, whereas the sample itself is still elongated. During heating, the sample from *T*_{low} to temperature above *T*_{trans}, the viscosity in the dashpot is dramatically decreased, and the force, created through the spring of the Maxwell unit moves at *T*_{sw} the viscous extension back to zero, and the original shape (state 1) is recovered.

For the understanding of the mechanism of the SME two features of the dashpot play an important role: (i) the viscous strain, which is frozen and locked at *T*_{low}, and (ii) the tremendous dependence of viscosity on temperature, which allows above *T*_{trans} to unlock viscous strain and to bring back the original shape.

The whole thermomechanical cycle has been at first theoretically described and modeled by Tobushi et al.34, 35 The model extended the SLS model with a slip element to account for dissipated energy due to internal friction, which results in an incomplete recovery. The model expressed the fact that during stress free recovery at a specific temperature, a certain part of the extension remained as irrecoverable strain. A comparison of experimental and calculated data35 for a segmented polyurethane composed of amorphous ether-based soft segments consisting with a *T*_{g} around 45 °C, which contained adipic acid and bisphenol A moieties extended with ethylene or propylene oxide and crystallizable hard segments synthesized from 4,4′-diphenylmethane diisocyanate (MDI) and 1,4-butanediol (BD) are shown in Figure 2. For a maximum extension of ε_{m} = 20%, all steps of the cycle: (1) stretching, (2) cooling, (3) unloading are qualitatively well described, as well as including the ε(*T*)-curve (not shown) during the stress-free recovery (4).

This model is a first example of a general approach to calculate thermomechanical properties of DSPs based on its viscoelastic properties. Other rheological models consisting of spring, dashpot, and frictional elements were applied to SMPs, for example, by Lin and Chen.31 These models were further developed and stepwise adapted to the specific nature of the thermal transition in the polymer, either a glass transition with *T*_{trans} = *T*_{g}40, 41, 43, 53, 54 among others or a melting transition *T*_{trans} = *T*_{m}.33, 37, 38 The models for SMPs with crystallizable switching segments followed a mechanical approach, in which the stress–strain behavior is described as a combination of spring or dashpot units,33 or the observed expansion and contraction is predicted based on so-called “constitutive equations” for a rubbery phase, a semicrystalline phase, the crystallization and melting process, which are adjusted to predict the conditions of a specific SMP.37

The current capability of viscoelastic models is demonstrated on the 3D finite deformation constitutive model for amorphous SMPs recently developed by Westbrook et al.,55 which is shown in Figure 3.

The authors investigated an acrylate-based covalently crosslinked polymer network (*T*_{g} = 42 °C). Isothermal uniaxial compression experiments were conducted at strain rates of 0.01/s and 0.1/s. To explore the rubbery and glassy behavior, the isothermal uniaxial compression tests were performed at various temperatures between 0 and 100 °C in 10 K increments. Further cyclic experiments, both under constrained and free conditions during the recovery step were carried out. For constrained recovery, the programing took place at *T*_{deform} = *T*_{high} = 60 °C, the unloading (fixing) took place at *T*_{low} = 10 °C, and the recovery again at *T*_{high} = 60 °C. During constrained recovery, a stress overshoot peak for an initial compression deformation is observed on the stress–temperature graph [Fig. 4(a)]. At stress-free recovery experiments of samples programed at *T*_{deform} = 40 °C and unloaded at *T*_{low} = 10 °C, the recovery temperature was varied from *T*_{high} = 30, 35, 40, to 50 °C. As it becomes obvious from Figure 4(b), the recovery behavior is strongly dependent on *T*_{high}.

These experiments were modeled with a viscoelastic model where every material point moves in space under conditions, which can be represented in 1D by the rheological model presented in Figure 3, where a thermal expansion element is arranged in series with mechanical elements. The mechanical elements consist of an equilibrium branch composed of an elastic spring with a Young's modulus of *E*_{eq}, a glassy nonequilibrium branch represented by a Maxwell element based on an elastic spring and a dashpot characterized by a relaxation time τ_{g} = η_{g}/*E*_{g}, which are placed in series, and several further Maxwell elements, presenting the rubbery nonequilibrium branches. This model represents an extension of the simple SLS model. The glassy branch describes the segmental relaxation of the polymer. This most fundamental relaxation process defines the monomer friction coefficient and it is on the bottom of all other and longer time scale chain relaxation processes. The model considers a finite number of *n* different relaxation processes in the rubbery (or molten) state. For the thermal expansion, a multiparameter theory has been used,56 which describes the volume departure of the polymer in the nonequilibrium situation below the glass transition. The elastic behavior of the DSP above *T*_{g} is modeled by a hyperelastic material model typically for rubbers, that is, the Arruda-Boyce eight chain model.57 For the nonequilibrium viscoelastic branches, one can assume that all branches follow the same viscous flow rules, but with different relaxation times. The relaxation times are temperature dependent. According to the thermo-rheological simplicity principle, a time-temperature superposition shift factor *a*_{T}(*T*) can be determined, which relates all *n* + 1 relaxation times

- (5)

with *i* = 0,…,*n* to a reference relaxation time τ, where the shift factor is equal to 1. For temperatures below *T*_{g}, the shift factor can be determined from an Arrhenius-type behavior,58 and at temperatures close to or above *T*_{g}, the WLF equation59 can be used.

The authors implemented the constitutive model into a FE software package and fitted at first model parameters for several key features: (i) the overall behavior of the isothermal uniaxial compression results at a strain rate of 0.01/s, (ii) the location and magnitude of the stress overshoot peak in the constrained recovery experiments, and (iii) the overall behavior in stress-free recovery experiments at all *T*_{high} values. It turned out that the number of nonequilibrium rubbery branches could be assessed to *n* = 2. The model was then capable to predict the strain rate dependency in the isothermal uniaxial compression experiments. For the constrained recovery conditions [see Fig. 4(a)], the model was able to capture magnitude and temperature location of the stress overshoot peak during heating. However, the overall behavior above 35 °C during cooling or heating and the location of the temperature at which the stress overshoot is still not well depicted. For the free recovery after a shape-memory cycle, the experimental results are well described [see Fig. 4(b)]. For *T*_{high} ≥ *T*_{g}, the stress-free recovery behavior (both the recovery rate ν_{rec} and the temperature interval Δ*T*_{rec}, where the transition from the temporary to the original shape starts (*T*_{start}) and ends (*T*_{end})) is equivalent excluding the thermal expansion for the case where *T*_{high} > *T*_{deform}.

The established model allows, for example, to study the influence of the heating rate on the recovery behavior up to the question of how fast the SMP can recover its memorized shape, if it could be heated instantaneously, or to simulate a magnetosensitive SMP composite to determine the effect of changing the particle size and concentration on the free recovery behavior.

The just discussed model represents implicitly also a tool to study the influence of the programing parameters *T*_{deform} and ε_{m} on the recovery behavior of the SMP due to the incorporated temperature dependency of the relaxation times. It should also be mentioned that this constitutive model needs 23 parameters. The model of Srivastava et al.60 is currently the most elaborated with 45 parameters. The effort in the parameter description, together with the often mathematically challenging formulation of these models, is at present a disadvantage for a wider application of these kind of models in the SMP research. From this point of view, simple modeling concepts, which only focus on the essential aspects of the shape-memory behavior, are of special interest.

An example is the approach of Bonner et al.61 to predict the recovery time of SMPs depending on programing parameters. A simple Kelvin–Voigt model (elastic spring in parallel with a dashpot) is used to analyze data of a transient stress dip test, where a sample is stretched at constant rate up to a certain maximum length, then the stretching direction is very quickly reversed for a short time, and finally, the strain development as function of time is observed and the recovery stress σ_{R} is measured. It was found that σ_{R} is a linear function of λ^{2} − 1/λ where λ is the draw ratio, further that σ_{R} decreases with increasing deformation temperature, and increases with increasing strain rate. It turned further out that the ratio of the recovery stress to the total stress is a good approximation invariant to the three programing variables *T*_{deform}, ε_{m}, and strain rate . The shape recovery of the SMP occurs when the spring component of the Kelvin–Voigt model or the stored stress in the material exerts a force capable of deforming the viscous component or the dashpot element of the model, which typically occurs at the temperature of shape recovery.

Into the same direction of a simplified model with a few characteristic parameters aims also the recently modified SLS model.36 The simple analytical solution allows to predict the nonisothermal free recovery of an amorphous DSP by only eight calibration parameters obtained by relatively quick dynamic mechanical thermal analysis (DMTA) measurements.

For the second set of general modeling approaches to calculate thermomechanical properties, the polymers' morphology is the starting point. Covalently crosslinked amorphous DSPs are described as a two-phase material composed of a glassy and rubbery phase. Liu et al.42 developed a constitutive model where the total strain ε can be represented as sum ε = ϕ_{f}ε_{f} + (1 − ϕ_{f})ε_{a} of two strain contributions from the frozen phase and the active phase. The strain in the frozen (glassy) phase ε_{f} can be determined from three contributions, that is, the (stored) entropic stain, the internal energetic strain, and the thermal strain. In the active phase, the strain deformation ε_{a} consists of two parts: the external stress induced entropic strain and the thermal strain.

The essential idea of this type of models is shown in Figure 5(a). In the glassy state at *T*_{low}, the major phase of a polymer is the frozen phase composed of frozen bonds, where conformational motions of chain segments are locked. By heating over a deformation temperature *T*_{deform}, the volume fraction of the frozen phase is reduced, and at the same time, the fraction of the active phase with free conformational motion is increasing till the polymer is at *T*_{high} in a practically full rubbery state. This process is assumed as fully reversible.

From a thermodynamic point of view, the frozen volume fraction ϕ_{f} is an internal state variable of the system and it is assumed that ϕ_{f} depends only on the temperature (*T*). Based on experimental results (a free strain recovery response during heating, where the sample has experienced predeformation ε_{pre} and strain-storage process), an analytical phenomenological function

- (6)

can be determined. The two numerical parameters *c*_{f} and *n* can be fitted from the experimental strain ratio data. Very recently, Kazakeviciute-Makovska et al.62 analyzed Liu's law (6), and other ϕ_{f}(*T*) relations obtained by different researchers43, 44, 47, 48, 53 and proposed a modified general function based on reduced temperature values

- (7)

with two parameters *a* and *b*, and following shape

- (8)

where *a* can be positive or negative, and *b* > 0. The influence of these parameters on the *T*-dependency of the frozen volume fraction is shown in Figure 5(b). Parameter *a* shifts the transition curve relative to *T*_{deform}, and *b* determines the width of the transition. The parameters *a* and *b* for the generalized evolution law for the frozen volume fraction may be obtained in a systematic way from stress–strain experiments. A simple approach has been used by Volk et al.,47 who assumed that the frozen volume fraction function [see Fig. 6(b)] is given by the normalized stress free recovery profile presented in Figure 6(a), that is, ϕ_{f}(*T*) = ε (*T*)/ε_{m}. Based on the generalized evolution law of the frozen volume fraction ϕ_{f}, it was possible to model several experimental sets of thermomechanical tests,63, 64 including the already discussed experiments of Tobushi et al.34

The two-phase approach was confirmed by Wang et al.48 Based on this theory, Chen and Lagoudas40, 53 reported an approach to describe the thermomechanical properties of SMPs under large deformations, where general constitutive functions of neo-Hookean type for nonlinear thermo-elastic materials are used for the active and frozen phases. This model was applied to analyze experimental cyclic tests for a polystyrene-based SMP network.47, 64 Figure 7 presents the comparison of experiment and modeling for different maximum extension values.

The model was calibrated by calculating values for model parameters (coefficient of thermal expansion, shear moduli, and the frozen volume fraction) using the data of the thermomechanical test with ε_{m} = 25%. Based on these data, the model allows to predict in qualitative agreement the full σ-*T*-ε tensile tests for small as well as large ε_{m} of 10%, 50%, and 100%. The model failed to capture the irrecoverable extension at the end of the recovery step. The model was also used to predict the stress free and constrained recovery curves for a polyurethane SMP, which were found to be in good agreement with experimental data.65

Despite its successes, the two-phase approach has a strong limitation. As Diani et al. discuss in ref.66, the volume fraction of each phase in such models must be given as a function of temperature ϕ_{f} = ϕ_{f}(*T*). This function is obtained from fitting experimental shape-memory data at a certain heating rate. Therefore, the resulting models can fit these experimental data but cannot predict the shape-memory behavior of the material under different heating rates or heating profiles.

Another important aspect was highlighted by Nguyen et al.43 with an own model development based on the nonequilibrium character of the glassy state. The authors pointed out that the concept of two mixed phases (frozen/active) is not in true agreement with the physical processes of the glass transition and thus results in “nonphysical” parameters, such as the volume fractions of the glassy and rubbery phases. Instead, it would be necessary to consider, as primary molecular mechanism, the time-dependent structural and stress relaxation of the glass forming polymer material. The model developed under this relaxation concept seems to reproduce the stress-free strain–temperature response, the temperature and strain-rate dependent stress–strain response, and important features of the temperature dependence of the shape memory response well.67 The modeling approach allowed recently to study the effect of physical aging (up to 180 days) on the stress free recovery of (meth)acrylate-based polymer networks.68 Another application of this stress-relaxation concept was the modeling of experiments where a SMP network was programed by cold compression.69

As extension of the discussed “two-phase” models, Kim et al.70 developed especially for shape-memory polyurethanes (SMPUs) a constitutive model consisting of three phases. SMPUs consist of hard and switching domains. The switching domains can form a variable state ranging from a rubbery (active) to a rigid (frozen) phase depending on temperature. Hard domains act as fixed (permanent) netpoints. The three-phase model divides the SMP into one hard segment phase that governs the viscoelastic behavior and two switching segmented phases (frozen and active phase) that undergo a reversible phase transformation as function of temperature. A three-element viscoelastic model (two springs and a dashpot) was used to present the hard segment phase. For the two switching segment phases, a Mooney-Rivlin solid model (hyperelastic spring) was used. The two hyperelastic springs for the frozen and active switching segment phases were linked in the model as series or parallel. The stresses in each phase were calculated using hyperelastic and viscoelastic equations, and combined into the total stress. The material properties for the constitutive equations were determined by differential scanning calorimetry and a series of uniaxial tensile stress–strain tests at three temperature conditions. Simulations of cyclic thermomechanical tests gave good agreement with experimental data and showed a certain dependence on the assumed linking model (parallel or series).