Stress relaxation properties of an epoxy-based shape-memory polymer considering temperature influence: experimental investigation and constitutive modeling

Abstract

Epoxy-based shape-memory polymer (ESMP), belonging to viscoelastic thermosets, exhibits strong temperature-dependent relaxation property caused by the glass transition process. Thoroughly experimental investigation and accurate numerical prediction are indispensable to be performed in the structures that demand for high precise structural controls. In this work, key parameters of an ESMP in stress relaxation process including initial modulus, elastic modulus, viscous modulus and relaxation time were experimentally investigated and constitutive-model-based analyzed to study the influence of the temperature, where the constitutive model was established by modifying the constant parameters of a neo-Hookean hyper-viscoelastic model to be functions of the temperature. A numerical simulation method was developed by coding UMAT in ABAQUS using the constitutive model and was verified by finite element simulations of stress relaxation experiments. The maximum deviation of 4.5% between simulations and experiments indicated the reasonability of the model in depicting stress relaxation process considering the influence of the temperature.

1 Introduction

Shape-memory polymers (SMPs) with shape storage and recovery abilities are drawing more and more interests in the recent researches of smart materials (Lin et al. 2018; Hanzon et al. 2018). Among them, epoxy-shape-memory polymer (ESMP) with high shape fixity and recovery is one kind of the most widely used materials due to the superior performance in thermal stability, chemical stability, machinability and low economic cost (Belmonte et al. 2017a, 2017b; Altuna et al. 2016). The extensive researches of ESMP in 4D printing (Monzón et al. 2017; Fan et al. 2018), actuators (Belmonte et al. 2017a, 2017b; Wang et al. 2016), morphing structures (Gong et al. 2017; Len et al. 2015), shape-memory composites (Chen et al. 2015; Beblo et al. 2015; Wang et al. 2014) and medical devices (Zainal et al. 2017; Santhosh Kumar et al. 2013) declare that ESMP is a popular shape-memory material. In this case, detailed investigations of ESMP on the material synthesis (Ji et al. 2017; Zheng et al. 2015), thermal-viscoelastic behaviors (Fan et al. 2016; Hu et al. 2012; Lewis and Dell 2016), mechanical property and its improvement (Shi et al. 2017; Beblo et al. 2015; Santiago et al. 2016; Shi et al. 2019) have drawn considerable attention. Among these research topics, investigations on the thermal-viscoelastic behaviors play a fundamental role in the targeted application designs of ESMP for they determine the structural behaviors directly.

ESMP composed of epoxy resin and curing additive is a kind of amorphous thermoset polymers (Chen et al. 2014), which would lead to significant viscoelastic property with rate-dependence and strain-dependence (Gutierrez-Lemini 2014; Belytschko et al. 2013). The viscoelastic behavior of ESMP would be sensitive to thermal conditions and present dramatic changes between rubbery and glassy states due to the glass transition process that triggers the shape-memory effect. As an essential viscoelastic behavior, the stress relaxation behavior is determined by the typical viscoelastic parameters including the initial modulus, elastic modulus, relaxation time, the ratio between elastic and viscoelastic modulus, etc. On the contrary, these parameters could also be obtained by investigating the stress relaxation process once a suitable constitutive model was applied to depict the stress–strain responses in the stress relaxation process (Gutierrez-Lemini 2014; Strobl 2007). As the stress relaxation phenomenon would have significant influence on the structure forming process and accurate thermal-controls of the ESMP structures. Therefore, it is crucial to comprehensively investigate the stress relaxation properties and their temperature-dependence before performing the analysis of ESMP structures. One should also take care that the viscoelastic behaviors would be quite different for various ESMPs. For instance, the material in Gu and Li (2013) presented significant viscoelastic behaviors near glassy transition region, but presented nearly elastic behaviors at both lower and higher temperatures. The material in Pieczyska et al. (2015) showed hyperelastic properties in rubbery phase and showed viscoelastic properties in glassy phase. One will see the ESMP investigated in this work is an epoxy resin system with aromatic-amine curing agent. This type of ESMP would show a behavior of viscoelastic properties at both rubbery and glassy states, which is distinctive for these ESMPs studied by others (Chen et al. 2014; Gu and Li 2013; Pieczyska et al. 2015). Due to this kind of material dissimilitude in mechanical properties caused by the chemical constituents or curing conditions, it is necessary to conduct thoroughly investigations before utilizing a novel ESMP.

Among the investigations on the temperature-dependence of the typical viscoelastic parameters and behaviors of ESMP, McClung et al. (2017) discussed the shape recovery behavior of an ESMP considering the loading rate and shape-memory cycles. They found that Poisson’s ratio at room temperature was independent of the strain rate, but was dependent on the strain magnitude. They also (McClung et al. 2012) explored the strain rate-dependence of ESMP in uniaxial tensile case at room temperature and high temperature, corresponding to glassy and rubbery states, respectively. Hu et al. (2017) characterized the static and the dynamic mechanical properties of an ESMP through dynamic mechanical analysis and uniaxial tensile experiments, where the elastic modulus, the transformation temperature and the Poisson ratio were evaluated around the glass transformation temperature. Ellson et al. (2015) developed a new thiol-epoxy SMP foam based on which they found that the glassy modulus varied from 19.1 MPa to 345 MPa and the rubbery modulus varied from 0.04 MPa to 2.2 MPa in static compression and dynamical experiments. Erel et al. (2015) examined the ESMP as a hybrid of hyperelastic and elastic material by conducting the parameter characterization experiments in simple tension and planar tension modes, which provided hyperelastic material parameters and failure strain in both rubbery and glassy states. Liu et al. (2018) have investigated the rate-dependence and strain-level-dependence of ESMP through a series of cyclic tensile tests, where they found that the loading rate or level had little effect on the shape recovery process. Gu and Li (2013) discussed the stress–strain hysteresis of an ESMP in the regime of viscoelasticity at various temperatures using uniaxial tensile experiments. They found that the viscoelastic response was very small at lower and higher temperatures, but became remarkable in the glassy transition region.

The above researches were mainly focused on the influence of strain (rate or magnitude) on viscoelastic and shape-memory properties, the stress–strain responses in rubbery and glassy states, as well as the influence of the temperature on elastic modulus and the Poisson ratio. However, the essential viscoelastic properties including initial modulus, elastic modulus, viscous modulus, relaxation time and their drastically temperature-dependence have not been adequately investigated. Especially in the glassy transition process, the viscous and elastic properties would present dramatic transition with the temperature, which is urgently to be considerably investigated before performing the analysis of ESMP structures. This work would carry out comprehensively investigations on the influence of the temperature on these parameters, where a constitutive model is also developed to depict the temperature influence.

The work is organized as follows. In Sect. 2, an ESMP was prepared by adding curing agent polyether amine to the epoxy resin, on which dynamic mechanical analysis (DMA) experiments and stress relaxation experiments are carried out to address the viscoelastic properties. In Sect. 3, a neo-Hookean hyper-viscoelastic model is modified to consider the temperature influence by assuming the material parameters are functions of the temperature. In Sect. 4, numerical method of the modified constitutive model is established and validated by simulations of stress relaxation experiments. At last, the conclusions are presented.

2 Material preparation and mechanical experiments

2.1 Material and specimen preparation

An ESMP composed of epoxy resin (E44) and the curing agent polyether amine (propylene glycol 2-aminopropyl ether) was developed through a classical curing process. The epoxy resin was firstly put into a container at room temperature and then heated to $60^{\circ}\mbox{C}$, where the polyether amine was added into the epoxy with molar mass ratio of 1:1. The two components were intensively mixed using stirrer and poured into a mold with dimensions of $1000~\mbox{mm} \times1000~\mbox{mm} \times2.5~\mbox{mm}$. The ESMP plate could be demolded after 2 h curing time at temperature $120^{\circ}\mbox{C}$ (Hu et al. 2017; Zheng et al. 2015). To avoid deckle edge in the specimen which would lead to crack propagation in tensile experiments, the raw plate was processed using high-smoothness milling machine to be $20~\mbox{mm} \times 5~\mbox{mm} \times1.5~\mbox{mm}$ for DMA experiments and $150~\mbox{mm} \times20~\mbox{mm} \times2.5~\mbox{mm}$ for stress relaxation experiments.

2.2 Dynamical mechanical experiments

Dynamic mechanical analysis (DMA) experiments were firstly conducted to address the glass transition process using a DMA 8000 machine produced by TA Instruments Inc. Three specimens were tested in single cantilever bending mode with an oscillation frequency of 1 Hz. Temperature sweep range was set to be $0\mbox{--}100^{\circ}\mbox{C}$ with heating rate of $3^{\circ}\mbox{C}/\mbox{min}$. Figure 1 plotted the modulus and tangent delta obtained in the DMA tests. Glass transition temperature ($T_{g}$), peak of loss modulus ($T_{l}$) and peak of tangent delta ($T_{t}$) were determined according to the standard of ASTME1640-13 (2018), during which the glass transition occurred. In the standard, $T_{g}$ was the intersect of the two tangents of storage modulus. One was constructed below transition temperature when the other was constructed at the inflection point approximately midway of the sigmoidal change. Experimental results of the three temperatures are listed in Table 1, where all the points lied among $30\mbox{--}50^{\circ}\mbox{C}$ with the rule of $T_{g} < T_{l} < T_{t}$. It meant the transition region would be located between $T_{g}$ and $T_{t}$.

Fig. 1

DMA experiment results: ($\mathbf{a}$) storage modulus, ($\mathbf{b}$) loss modulus and ($\mathbf{c}$) tan delta (Color figure online)

Table 1 Three temperature points defining the transition process ($^{\circ}\mbox{C}$)

As the DMA experiments revealed, loss modulus and tangent delta were higher in the transition region, which meant that the viscous properties of the material were more significant in the transition region. Besides, storage modulus, loss modulus and tangent delta presented dramatic variation in the transition region, which indicated the dramatic variation of viscoelastic properties. In order to address the variation and identify the temperature influence, stress relaxation experiments were carried out at temperature $20\mbox{--}60^{\circ}\mbox{C}$ as shown in Fig. 1 and Table 1.

2.3 Stress relaxation experiments

Stress relaxation experiments were conducted in an isothermal process using a universal material testing machine (UTM-4000) and a temperature chamber (WGDN series). Both two facilities were produced by Shenzhen Suns Technology Co., Ltd. Temperatures were detected by a thermal resistance (see Fig. 2). Nine specimens were tested at temperatures from $20^{\circ}\mbox{C}$ to $60^{\circ}\mbox{C}$ with the increment of $5^{\circ}\mbox{C}$. The clamped length of specimen was 25 mm in each end when the clear gauge length $L_{0}$ was 100 mm. The experiment was conducted as follows: (i) Fix the specimen on the jigs. (ii) Turn on the heat module of the tank room and hold about 40 minutes to make the thermal field steady. (iii) Apply the extension $\Delta_{i}$ to the specimen with a constant rate of $0.5~\mbox{mm}/\mbox{s}$. (iv) Hold on the extension for 400 s, where the stress relaxation occurred. As the deformation ability of the ESMP varied with temperature, the extension $\Delta_{i}$ was set to be 6 mm and 30 mm when the temperature was lower and higher than $T _{g}$, respectively.

Fig. 2

Facilities of stress relaxation experiments

Stress–strain and modulus–strain curves (here the modulus means the derivative of strain–stress curve) in the tensile loading process were depicted in Fig. 3. As could be seen, the modulus decreased with the increase of strain. Meanwhile, there was no intersection of the modulus curves, which meant the values of the modulus at low temperatures were always larger than that at high temperatures in the same strain state. As listed in Table 2, the attenuation ratio, which was equal to 100% minus the proportion of the remaining modulus (modulus at the last point of stretching) in the initial modulus, was over 87% at temperature blow $T _{g}$. When the temperature was higher than $T _{g}$, the attenuation ratio would decrease with the increase of the temperature.

Fig. 3

Stress–strain and modulus–strain curves in loading processes at various temperatures: (a) $20\mbox{--}30^{\circ}\mbox{C}$, (b) $35\mbox{--}45^{\circ}\mbox{C}$ and (c) $50\mbox{--}60^{\circ}\mbox{C}$ (Color figure online)

Table 2 Modulus values and their attention in tensile loading process

Figure 4 (the stress axis in $\log 10$ scale) gives the stress evolution in the relaxation process, from which we could see the stress values presented exponential decay as the time went by. According to Table 3, the attenuation ratio, which was equal to 100% minus the proportion of the remaining stress at 400 s in the initial stress, was more than 92% at temperature below $T _{g}$. Then it would decrease rapidly to 4.6% when the temperature was above $T _{g}$. From this one could conclude that the material behaved showing significant relaxation properties before $T _{g}$ and the relaxation phenomenon would be weakened with the increase of the temperature after $T _{g}$. To address this kind of drastic influence from the temperature, detailed investigations need to be carried out by analyzing the influence of the temperature on the material parameters, such as modulus, viscosity and relaxation time. In this case, a hyper-viscoelastic constitutive model was utilized to generate the parameters as follows.

Fig. 4

Stress relaxation process at various temperatures: (a) $20\mbox{--}30^{\circ}\mbox{C}$ and (b) $35\mbox{--}60^{\circ}\mbox{C}$ (Color figure online)

Table 3 Stress values and their attention in stress relaxation process

3 Model development and temperature influence analysis

3.1 Viscoelastic constitutive model considering temperature effect

ESMP involved in this work is a kind of thermoset, of which the mechanical responses could be sufficiently depicted by a neo-Hookean hyper-viscoelastic model (Tayeb et al. 2017). The neo-Hookean hyper-viscoelastic model could be written as Eq. (1) in the forms of left Cauchy–Green tensor $\underline{\mathbf{B}}$ and total volume strain $J$ (the details of the derivations are given in the Appendix). We write

$$\begin{aligned} \boldsymbol{\sigma} ( \underline{\mathbf{B}},t ) =& \frac{2}{J ( t )}C_{10} \int_{0}^{t} \Biggl( g_{\infty} + \sum _{i = 1}^{N} g_{i}e^{\frac{s - t}{\tau_{i}}} \Biggr) \biggl( \frac{d \underline{\mathbf{B}} (s)}{ds} - \frac{1}{3} \delta_{ij}\frac{d \underline{\mathbf{B}} (s)}{ds} \biggr)\,ds \\ &{}- \frac{2}{D_{1}} \bigl( J ( t ) - 1 \bigr)\delta_{ij} \mathbf{I} \end{aligned}$$

(1)

where $\boldsymbol{\sigma} ( t )$ is the total stress; $g_{\infty}$ and $g_{i}$ are normalized dimensionless parameters in the Prony series of $g ( t )$ and we have $g_{\infty} + \sum_{i = 1}^{N} g_{i} = 1$; $\tau_{i}$ is the relaxation time of the $i$th spring-slipper branch in a generalized Maxwell model; $C_{10}$ and $D_{1}$ are elastic material parameters; $s$ and $t$ are time; $\delta_{ij}$ is the Kronecker delta.

The values of $C_{10}$, $D_{1}$, $g_{\infty}$, $g_{i}$ and $\tau_{i}$ would be influenced by the temperature according to the entropy elasticity and the viscoelastic theory (Yang et al. 2018). Especially for the ESMP investigated in this work, the dramatic variation of viscoelastic properties during the transition process would cause significant change of parameter values. Therefore, a modified neo-Hookean hyper-viscoelastic constitutive model as Eq. (2) is proposed by assuming all the parameters to be functions of the temperature ($T$). We have

$$\begin{aligned} \boldsymbol{\sigma}(T,\underline{\mathbf{B}},t) =& \frac{2}{J ( t )}C_{10}(T) \int_{0}^{t} \Biggl( g_{\infty} ( T ) + \sum_{i = 1}^{N} g_{i}(T)e^{\frac{s - t}{\tau_{i} ( T )}} \Biggr) \biggl(\frac{d\underline{\mathbf{B}} (s)}{ds} - \frac{1}{3}\delta_{ij} \frac{d\underline{\mathbf{B}} (s)}{ds} \biggr)\,ds \\ &{}- \frac{2}{D_{1} ( T )} \bigl( J ( t ) - 1 \bigr) \delta_{ij} \mathbf{I} \end{aligned}$$

(2)

3.2 Material parameters calibration

To calibrate the parameters of the constitutive model Eq. (2), the stress evolution equations were derived in uniaxial tensile case and stress relaxation case (see the Appendix). A nonlinear fitting program was developed using the Levenberg–Marquardt algorithm based on ORIGIN. To reduce the number of the variables, the $\tau_{1}$, $\tau_{2}$ and $\tau_{3}$ were fixed to be constants as 5, 50 and 300 in the calibration process. Fitting results of the stress relaxation experiments data have been plotted in Fig. 4(a)–(b) while the material parameters are listed in Table 4. Up to now, the temperature influence could be analyzed by evaluating these material parameters.

Table 4 Material parameters of the constitutive model at various temperatures

3.3 Temperature influence analysis

Initial modulus and its viscous and elastic part were firstly analyzed. To avoid redundancy, we chose the shear modulus to analyze the viscous and elastic properties, as another kind of modulus would behave according to the same rules in view of continuum mechanics (Larson and Bengzon 2012). The initial shear modulus $G_{\mathrm{inital}}$ was equal to $2C_{10}$ in the neo-Hookean hyper-viscoelastic model and the shear modulus $G(t)$ would decay with time as Eq. (3) because of the viscous phenomenon (Ghoreishy 2012; Goh et al. 2004),

$$ G(t) = 2C_{10} \Biggl( g_{\infty} + \sum _{i = 1}^{N} g_{i}e^{\frac{ - t}{\tau_{i}}} \Biggr) = G_{\mathrm{inital}} \Biggl( g_{\infty} + \sum _{i = 1}^{N} g_{i}e^{\frac{ - t}{\tau_{i}}} \Biggr) $$

(3)

where the first and second terms on the right side correspond to elastic part modulus $G_{e}$ and viscous part modulus $G_{v} ( t )$. The initial viscous part modulus $G_{v}(0)$ could be derived by setting $t \to0$.

$G_{e}$ and $G_{v}(0)$ were calculated and plotted in Fig. 5(a) ($Y$-axis in log plot). As could be seen, $G_{e}$ and $G_{v}(0)$ reduced drastically with the increase of the temperature. $G_{v}(0)$ decreased to 0.27 MPa from 1554 MPa and was surpassed by $G_{e}$ at $55^{\circ}\mbox{C}$. As given in Fig. 5(b), the ratio of $G_{v}(0)$ to $G_{\mathrm{inital}}$ ($R _{v}$) and the ratio of $G_{e}$ to $G_{\mathrm{inital}}$ ($R _{e}$) would remain at nearly constant levels at temperature blow $T _{g}$, where the $R _{v}$ was more than 0.95. It meant over 95% of the initial modulus could be diminished in a relaxation process. When the temperature was above $T _{g}$, $R _{v}$ and $R _{e}$ started to present dramatic variation, where $R _{v}$ decreased and $R _{e}$ increased. The inflections of slope in $R _{v}$ and $R _{e}$ curves were located between $T _{g}$ and $T _{t}$ (transition region obtained from the DMA experiments), which meant the maximum slope was located in this area and the variation was most dramatic here. Besides, this kind of variation was not finished but became moderate after $T _{t}$.

Fig. 5

(a) Values of elastic and viscous modulus and (b) ratio of elastic and viscous modulus

Then the relaxation time of the material was investigated. The relaxation time $t_{c}$ is the time it costs for the re-equilibrium after loading step in a stress relaxation experiment. The re-equilibrium means that the stress value would decay to a constant value in the relaxation process (Gutierrez-Lemini 2014). A formula on the equilibrium state as Eq. (4) was set to determine $t_{c}$. The criterion of 1%, 0.1% and 0.01% were used to calculated $t_{c}$, respectively, which meant that the modulus attenuation value during one second was less than 1%, 0.1% and 0.01%. Calculation results were plotted in Fig. 6 using the data provided in Table 4. As could be seen, $t_{c}$ reduced with both the increase of the temperature and the increase of criterion value,

$$ \frac{C_{10}(t_{c} - 1) - C_{10}(t_{c})}{C_{10}(t_{c} - 1)} \times 100\% \le \mbox{Criterion} $$

(4)

Fig. 6

The time consumption to reach the equilibrium state under various criteria

4 Numerical simulations of stress relaxation considering temperature influence

As an essential problem in analyzing ESMP structure behaviors, the accurate predictions of stress relaxation response are indispensable to be performed in numerical simulations. This section would present the attempt to establish the numerical method in analyzing the temperature influence on stress relaxation of ESMP by coding the constitutive Eq. (2) and integrating it into ABAQUS.

4.1 Establishment of temperature functions of material parameters

The material parameters have been assumed to be continuous functions of the temperature, which could be obtained by data fitting on the discrete values listed in Table 4. After trial and error, the initial modulus $C_{10} ( T )$ could be commendably fitted by the Boltzmann function ($f ( T ) = A + B / \{ 1 + \exp [ ( T - C ) / D ] \}$, A, B, C and D were the targeted coefficients) due to its sharp decay with temperature. $g_{\infty} ( T )$ and $g_{i} ( T )$ were fitted by polynomial functions ($f ( T ) = A + BT + CT ^{2} + DT^{3}$). As the viscoelastic property of the ESMP would present dramatic variations, the difference of the magnitudes of the parameters was huge between glassy and rubbery states. To reduce the fitting error caused by it, the whole fitting was cut into two pieces at $40^{\circ}\mbox{C}$. The fitting results are plotted in Fig. 7 and listed in Table 5.

Fig. 7

Fitting results of the continuous temperature-functions: (a) initial modulus $C_{10}$, (b) $g_{\infty}$ and $g_{1}$, (c) $g_{2}$ and $g_{3}$

Table 5 Function coefficients of the temperature-functions of the material parameters

4.2 Numerical development of the constitutive model

A user subroutine to define specialized material properties (UMAT) in ABAQUS was developed according to the calculation process of Fig. 8, before which the incremental form of constitutive model Eq. (2) was derived to be Eq. (5) and Eq. (6) by stress decomposition (see the Appendix for derivation details).

$$\begin{aligned} \Delta\boldsymbol{\sigma} =& \Delta\mathbf{S} - \Delta\mathbf{p} = \Delta\mathbf{S}_{\infty} + \sum_{i = 1} ^{N} \Delta\mathbf{S}_{i} - \frac{2}{D_{1} ( T )} ( \Delta J - 1 ) \delta_{ij}\mathbf{I} \end{aligned}$$

(5)

$$\begin{aligned} \Delta\mathbf{S}_{i} =& \bigl( 1 - e^{\frac{ - \Delta t}{\tau _{i} ( T )}} \bigr) \biggl[ 2C_{10} ( T )g _{i} ( T )\tau_{i} ( T ) \biggl( \frac{\Delta\underline{\mathbf{B}}}{\Delta t} - \frac{1}{3}\delta_{ij} \frac{\Delta\underline{\mathbf{B}}}{\Delta t} \biggr) - \mathbf{S}_{i}^{n} \biggr] \end{aligned}$$

(6)

where $\Delta\boldsymbol{\sigma}$, $\Delta\mathbf{S}$ and $\Delta\mathbf{p}$ are the increment of total stress, deviatoric stress and hydrostatic stress, respectively. $\Delta\mathbf{S}_{\infty}$ is the elastic part of $\Delta\mathbf{S}$. $\mathbf{S}_{i}$ is the viscous part of $\mathbf{S}$ in the $i$th branch, where $\Delta\mathbf{S}_{i}$ is the increment form and $\mathbf{S}_{i}^{n}$ denotes the value of $\mathbf{S}_{i}$ in the end of last increment. $\Delta J$ and $\Delta\underline{\mathbf{B}}$ are increment of volume strain and left Cauchy–Green strain, respectively.

Fig. 8

Flow diagram of UMAT calculation process of the constitutive equation

Except for the material parameters $C_{10} ( T )$, $g_{\infty} ( T )$, $g_{i} ( T )$ and $\tau_{i} ( T )$ have been obtained in Sect. 4.1; the buck modulus-related parameter $D_{1}$ still needs to be calculated by Eq. (7) according to the laws of continuum mechanics (Gutierrez-Lemini 2014; Belytschko et al. 2013). We have

$$ D_{1} = \frac{3 ( 1 - 2\nu )}{2C_{10} ( 1 + \nu )} $$

(7)

where $\nu$ is Poisson’s ratio, the value form due to Diani et al. (2012) was used here.

4.3 Finite element simulations

The constitutive model of Eq. (2) was modified from a neo-Hookean hyper-viscoelastic model by assuming all the material parameters were continuous functions of the temperature. A neo-Hookean hyper-viscoelastic model has been widely verified to be accurate enough in depicting the mechanical responses of ESMP in the strain range of this work (Gutierrez-Lemini 2014; Belytschko et al. 2013; Dassault Systèmes 2018). Therefore, the accuracy of the modified model would be determined by the accuracy of the continuous functions. In this case, experiments at other temperatures than for that listed in Table 4 and the corresponding simulations need to be carried out to verify the accuracy of the continuous temperature functions, which could further reveal reasonability of the modified model.

Stress relaxation experiments at $27.5^{\circ}\mbox{C}$, $37.5^{\circ}\mbox{C}$ and $47.5^{\circ}\mbox{C}$ and the corresponding finite element simulations were carried out to check the prediction accuracy of the modified model. The procedures of this set of verification experiments were the same as the experiments in Sect. 2.3, where the extension were 6 mm for $27.5^{\circ}\mbox{C}$ and 30 mm for the other two. All the conditions in the simulations were akin to the experiments. The finite element model with the same dimensions as the specimen was built in ABAQUS and was meshed with 1 mm cubic hybrid element named C3D8H (Dassault Systèmes 2018). Each end was tied to a master node by the MPC beam constraint. One end was fully constrained when the other end was set to apply the extension. The analysis step of “static” type was used, where the geometric nonlinearity was considered. Fixed time increments were used in the calculations. 10 s, 1 s, 0.1 s, and 0.01 s were applied in various trial calculations to find the reasonable value of time increment. Almost the same results were found in the time increment of 0.1 s and 0.01 s in the trail calculations. Therefore, the increment time of 0.1 s was used in later calculations.

Simulation results comparing with experiment results were obtained as Fig. 9 after the calculations. Deviations between numerical simulations and experiments are listed in Table 6, which were evaluated at two points, including the initial stress and the remaining stress (stress at 400 s). The maximum deviation was found to be 4.5% at the remaining stress in $47.5^{\circ}\mbox{C}$. Generally, stress relaxation in the three experiments could all be commendably depicted, which meant the modification on the neo-Hookean hyper-viscoelastic model were reasonable in depicting the temperature influence. As the prediction accuracy of the modified model were decided by the accuracy of the parameter functions. Therefore, the accuracy of the model could be improved by adding a number of stress relaxation experiments in various temperatures before the establishment of the functions of parameters. Future work will focus on this.

Fig. 9

Simulation results of stress relaxation curves comparing with the data of the verification experiment

Table 6 Relative deviations of stress value between simulation results and experiment results

5 Conclusions

The temperature dependence of the viscoelastic properties of ESMP was successfully addressed by DMA experiments, stress relaxation experiments and material-parameter evaluations based on a neo-Hookean constitutive model. The ESMP involved in this work (an epoxy system with aromatic-amine curing agent) was detected to behave as regards viscoelastic properties in both rubbery and glassy states. The relaxation time was reduced as temperature increases. The variation of viscoelastic properties was not finished but became moderate after the peak of a tangent delta, while the most dramatic variation was located between the peak of the tangent delta and the glass transition temperature. Accurate numerical results of three verification experiments were obtained by utilizing the numerical method established by coding the modified neo-Hookean hyper-viscoelastic constitutive model into ABAQUS, which proved that modifying material parameters to be seen as continuous functions of the temperature could be reasonable in depicting the influence of the temperature.

Besides, only the constitutive equation in predicting the viscoelastic properties of ESMP was developed here, the constitutive model of fabric reinforced ESMP and the model in predicting the shape-memory process still remains to be done. Future work will focus on this.

Appendix

1.1 A.1 Development of Neo-Hookean hyper-viscoelastic model

As a kind of thermoset, the mechanical responses of ESMP including instantaneous (or elastic) part and time-dependent (or viscous) part (Gutierrez-Lemini 2014) could be described by the generalized Maxwell model (Tayeb et al. 2017; Dassault Systèmes 2018) as Eq. (A.1) and Eq. (A.2).

$$\begin{aligned} \boldsymbol{\sigma}(t) =& \int^{t}_{0} g(t-s)\frac{d\boldsymbol{\sigma}_{0}(s)}{ds}\,ds \end{aligned}$$

(A.1)

$$\begin{aligned} g(t) =& g_{\infty} + \sum^{N}_{i=1}g_{i}e^{\frac{-t}{\tau_{i}}} \end{aligned}$$

(A.2)

where the $\boldsymbol{\sigma}(t)$ is total stress, which is strain-dependent and time-dependent; the subscript in $\boldsymbol{\sigma}_{0}(t)$ represents the elastic response; the $g(t)$ is Prony series, where the $g_{\infty}$ and the $g_{i}$ are normalized dimensionless parameters, and have $g_{\infty} + \sum^{N}_{i=1}g_{i} = 1$. The $\tau_{i}$ is relaxation time of the $i$th spring-slipper branch in generalized Maxwell model.

In finite strain case, the Neo-Hookean hyperelastic model defined as Eq. (A.3) (Belytschko et al. 2013) could be employed to replace the linear Hooke’s law in calculating the elastic stress, which could describe the hyperelastic response of polymers.

$$\begin{aligned} U = C_{10}(I_{1} - 3) + \frac{1}{D_{1}}(J - 1)^{2} \end{aligned}$$

(A.3)

where $U$ represents the strain energy potential; $C_{10}$ and $D_{1}$ are elastic material parameters; $I_{1}$ and $J$ are strain invariants expressed in terms of the left Cauchy–Green tensor $\underline{\mathbf{B}}$: $I_{1} = \mathit{tr}(\underline{\mathbf{B}})$, $\underline{\mathbf{B}} = \underline{\mathbf{F}} \cdot \underline{\mathbf{F}}^{T}$, $\underline{\mathbf{F}} = J^{-1/3} \mathbf{F}$, $J = \det(\mathbf{F})$. The $\mathbf{F}$ is deformation gradient. The hyperelastic stress could be derived by partial differential of $U$ on $\underline{\mathbf{B}}$ and split as the deviatoric part (Eq. (A.4)) and the hydrostatic part (Eq. (A.5)).

$$\begin{aligned} \mathbf{S}_{0} =& \frac{2}{J}\operatorname{dev} \biggl(\frac{\partial U}{\partial I_{1}} \biggr)\underline{\mathbf{B}} = \frac{2}{J} C_{10} \biggl(\underline{\mathbf{B}} - \frac{1}{3}\delta_{ij} \underline{\mathbf{B}} \biggr) \end{aligned}$$

(A.4)

$$\begin{aligned} p_{0} =& -\frac{\partial U}{\partial J} = \frac{2}{D_{1}}(J - 1)\delta_{ij} \end{aligned}$$

(A.5)

where $\mathbf{S}_{0} = \boldsymbol{\sigma}_{0} + p_{0}\mathbf{I}$, $p_{0} = -\frac{1}{3}\mathbf{I}: \boldsymbol{\sigma}_{0}$, $\delta_{ij}$ is Kronecker delta. Then the Neo-Hookean hyper-viscoelastic model (Eq. (A.6)) to depict the mechanical response of ESMP could be obtained by inserting Eq. (A.4) and Eq. (A.5) into Eq. (A.1).

$$\begin{aligned} \boldsymbol{\sigma}(\underline{\mathbf{B}}, t) =& \frac{2}{J(t)}C_{10} \int^{t}_{0} \Biggl(g_{\infty} + \sum ^{N}_{i=1} g_{i} e^{\frac{s-t}{\tau_{i}}} \Biggr) \biggl(\frac{d\underline{\mathbf{B}}(s)}{ds} - \frac{1}{3} \delta_{ij}\frac{d\underline{\mathbf{B}}(s)}{ds} \biggr)\,ds - \frac{2}{D_{1}} \bigl(J(t) - 1 \bigr)\delta_{ij} \mathbf{I} \\ =& \mathbf{S}(\underline{\mathbf{B}},t) - \mathbf{p}(J,t) \end{aligned}$$

(A.6)

1.2 A.2 Calibration equations of the constitutive parameters

To identify the material parameters of the Neo-Hookean hyper-viscoelastic constitutive model, algebraic form of the constitutive model needs to be derived in the commonly used test modes such as uniaxial tension, plane tension and shear. In this work, the stress relaxation experiments in Sect. 2 are carried out in uniaxial tension state, therefore, the material-parameter calibration equations are derived in uniaxial mode. Assumption of incompressibility could be used (Diani et al. 2012), which means that $J(t)$ could be set as 1.0 in the constitutive model. Then the tensile stress could be given as Eq. (A.7).

$$\begin{aligned} \sigma_{11}(T,\underline{B}_{11},t) = \frac{4}{3} C_{10}(T) \int^{t}_{0} \Biggl(g_{\infty}(T) + \sum ^{N}_{i=1}g_{i}(T)e^{\frac{s-t}{f_{\tau}(T)}} \Biggr) \frac{d\underline{B}_{11}(s)}{ds}\,ds \end{aligned}$$

(A.7)

In the stress relaxation experiments, the strain evolutions in tensile direction could be given as Eq. (A.8).

$$\begin{aligned} \underline{B}_{11}(t) = \left \{ \textstyle\begin{array}{l@{\quad}l} \displaystyle \biggl(\frac{L_{0} + Vt}{L_{0}} \biggr)^{2} & t < t_{1} \\ \displaystyle \biggl(\frac{L_{0} + Vt_{1}}{L_{0}} \biggr)^{2} & t \geq t_{1} \end{array}\displaystyle \right . \end{aligned}$$

(A.8)

where $t_{1}$ is loading time; $L_{0}$ and $L$ are initial and current length of specimen; $V$ is the stretching rate.

Stress evolution in the relaxation process could be obtained by piecewise integration and inserting Eq. (A.8) into Eq. (A.7). Then, all the material parameters $C_{10}(T)$, $g_{\infty}(T)$, $g_{i}(T)$ and $\tau_{i}(T)$ at a certain temperature could be obtained using the stress formulation of Eq. (A.9) to fit the $\sigma - t$ data in stress relaxation experiments.

$$\begin{aligned} \sigma_{11}(T,t) =& \frac{4}{3} C_{10}(T) g_{\infty}(T) \underline{B}_{11}(t_{1}) + \frac{4}{3}C_{10}(T)\sum^{N}_{i=1}2 g_{i}(T)\frac{V\tau_{i}(T)}{L^{2}_{0}} \\ &{}\times \biggl[e^{(\frac{t_{1}-t}{\tau_{i}(T)})} \biggl(L_{0} + \frac{Vt_{1}}{\tau_{i}(T)} - V \biggr) + e^{(\frac{-t}{\tau_{i}(T)})} \bigl(V \tau_{i}(T) - L_{0} \bigr) \biggr] \end{aligned}$$

(A.9)

1.3 A.3 Incremental form of the constitutive equation

Incremental form of constitutive model needs to be derived when conducting the development of UMAT. Actually, the first and the second items in the right side of constitutive model Eq. (2) are the deviatoric part of stress and the hydrostatic part of stress, respectively. The hydrostatic stress $\mathbf{p}(T,J,t)$ can be directly written into incremental form as Eq. (A.10). For the deviatoric stress $\mathbf{S}(T,\underline{\mathbf{B}},t)$, the elastic part stress and viscous part stress need to be separately calculated as Eq. (A.11). For the elastic part stress, the incremental form could be directly given as Eq. (A.12).

$$\begin{aligned} \Delta \mathbf{p} =& \frac{2}{D_{1}(T)} (\Delta J - 1) \delta_{ij} \mathbf{I} \end{aligned}$$

(A.10)

$$\begin{aligned} \mathbf{S}(T,\underline{\mathbf{B}},t) =& 2 C_{10}(T) g_{\infty}(T) \biggl(\underline{\mathbf{B}} - \frac{1}{3} \delta_{ij} \underline{\mathbf{B}} \biggr) \\ &{}+ 2 C_{10}(T)\sum^{N}_{i=1}g_{i}(T) \int^{t}_{0} e^{\frac{s-t}{\tau_{i}(T)}} \biggl( \frac{d\underline{\mathbf{B}}(s)}{ds} - \frac{1}{3}\delta_{ij} \frac{d\underline{\mathbf{B}}(s)}{ds} \biggr)\,ds \\ =& \mathbf{S}_{\infty} + \sum^{N}_{i=1} \mathbf{S}_{i} \end{aligned}$$

(A.11)

$$\begin{aligned} \Delta \mathbf{S}_{\infty} =& 2 C_{10}(T) g_{\infty}(T) \biggl(\Delta \underline{\mathbf{B}} - \frac{1}{3}\delta_{ij}\Delta\underline{\mathbf{B}}\biggr) \end{aligned}$$

(A.12)

For the viscous stress, an assumption need to be introduced that the $\underline{\mathbf{B}}$ varies linearly with $t$ during the increment, which means that $d\underline{\mathbf{B}}(s)/ds = \Delta\underline{\mathbf{B}}/\Delta t$. Then, the integration of the second item in Eq. (A.11) could be written as Eq. (A.13). Equation (A.14) can be obtained after calculations on the integration equation.

$$\begin{aligned} \mathbf{S}_{i} \bigl(t^{n+1} \bigr) =& 2 C_{10}(T)g_{i}(T) \biggl[ \int^{t^{n}}_{0} e^{\frac{s-t^{n}-\Delta t}{\tau_{i}(T)}} \biggl( \frac{d\underline{\mathbf{B}}(s)}{ds} - \frac{1}{3}\delta_{ij} \frac{d\underline{\mathbf{B}}(s)}{ds} \biggr)\,ds \\ &{}+ \int^{t^{n} + \Delta t}_{t^{n}} e^{\frac{s-t^{n} - \Delta t}{\tau_{i}(T)}} \biggl( \frac{\Delta \underline{\mathbf{B}}}{\Delta t} - \frac{1}{3}\delta_{ij} \frac{\Delta \underline{\mathbf{B}}}{\Delta t} \biggr)\,ds \biggr] \end{aligned}$$

(A.13)

$$\begin{aligned} \mathbf{S}_{i} \bigl(t^{n+1} \bigr) =& e^{\frac{-\Delta t}{\tau_{i}(T)}}\mathbf{S}_{i}(t^{n}) + 2 C_{10}(T)g_{i}(T)\tau_{i}(T) \bigl(1 - e^{\frac{-\Delta t}{\tau_{i}(T)}}\bigr)\biggl(\frac{\Delta \underline{\mathbf{B}}}{\Delta t} - \frac{1}{3}\delta_{ij} \frac{\Delta \underline{\mathbf{B}}}{\Delta t}\biggr) \end{aligned}$$

(A.14)

where the superscript in $t^{n}$ denotes the $n$th increment. Then the incremental value of viscous stress $\Delta \mathbf{S}_{i}$ can be obtained by $\mathbf{S}_{i}(t^{n+1})$ minus $\mathbf{S}_{i}(t^{n})$ (given as Eq. (A.15)). Finally, the incremental form of total stress could be obtained as Eq. (A.16) by inserting Eq. (A.15), Eq. (A.12) and Eq. (A.10) into the constitutive model Eq. (2).

$$\begin{aligned} \Delta \mathbf{S}_{i} =& \bigl(1 - e^{\frac{-\Delta t}{\tau_{i}(T)}}\bigr) \biggl[2C_{10}(T)g_{i}(T)\tau_{i}(T) \biggl(\frac{\Delta \underline{\mathbf{B}}}{\Delta t} - \frac{1}{3}\delta_{ij} \frac{\Delta \underline{\mathbf{B}}}{\Delta t}\biggr) - \mathbf{S}^{n}_{i}\biggr] \end{aligned}$$

(A.15)

$$\begin{aligned} \Delta \boldsymbol{\sigma} =& \Delta \mathbf{S} - \Delta \mathbf{p} = \Delta \mathbf{S}_{\infty} + \sum^{N}_{i=1} \Delta \mathbf{S}_{i} - \frac{2}{D_{1}(T)}(\Delta J - 1) \delta_{ij}\mathbf{I} \end{aligned}$$

(A.16)