Numerical Study on Shape Recovery Behaviors of Shape Memory Polymer Composite Hinges Considering Hysteresis Effect

Abstract

Shape memory polymer composite (SMPC) hinges have been researched as deployable structures in space missions due to their stable and controllable shape recovery behaviors. The elastic energy of the fabrics plays a dominant role in predicting the recovered shape of the hinges, as it strongly drives shape restoration. In this research, the shape recovery behaviors of SMPC hinges are numerically investigated by applying an equation that accounts for the hysteresis characteristics of the fabric reinforcement. The constitutive equation integrates the Mooney–Rivlin model, a viscoelastic, stored energy model, to characterize the hyperelastic properties varying with time, temperature, and shape recovery behaviors of the SMP matrix. Additionally, polynomial functions are introduced to represent the hysteresis effects and energy dissipation behavior of the fabrics. Since the elasticity of fabrics significantly affects the shape recovery of SMPCs, the developed constitutive equation enables accurate prediction of the recovered configuration. Finite element method analysis is performed based on this model and validated through comparison with experimental results. Finally, the constitutive equation is applied to investigate the shape memory response of SMPC hinges. The simulations present the significant design factors to increase the shape recovery ratio of the SMPC hinges.

1. Introduction

Deployable structures in space missions heavily rely on hinge mechanisms, which traditionally suffer from mechanical wear, lubricant evaporation in vacuum, and reduced functionality under extreme thermal cycling. These issues pose significant risks to mission longevity and reliability. Recent studies in shape memory polymer composite (SMPC) hinges offer a promising alternative [1,2,3]. SMPCs can be solid-state actuation without moving parts or lubricants, thereby eliminating traditional failure reasons. Their lightweight nature and high adaptability to space environments allow for simplified design, lower mass, and enhanced thermal stability. Furthermore, SMPC hinges exhibit excellent shape recovery and can be programmed for controlled deployment [4,5]. This technology represents a transformative approach to overcoming limitations of conventional hinge systems, with ongoing research focusing on improving recovery stress, deployment repeatability, and long-term performance in orbit.

Since SMPCs have temperature-dependent shape recovery characteristics, heating methods play a crucial role. Conventionally, heating SMPCs are achieved by putting functional particles into SMPs [6,7,8,9] or attaching a resistive Joule heating element to SMPCs [10,11,12]. However, those methods have limitations such as uneven temperature distribution, and reduction in recoverable deformation depends on particle content. Furthermore, the attachment of heating elements significantly reduces the deformable strain of the material. To overcome these limitations, SMPCs reinforced with conductive fabrics have been introduced as a viable solution. Several studies have demonstrated that such composites provide a more uniform temperature distribution while maintaining the inherent deformability of the structure [13,14,15]. Moreover, these composites exhibit tunable stiffness and shape recovery behavior, depending on both the temperature and the orientation of the reinforcing fibers.

For the practical application of SMPCs in deployable structures, it is crucial to accurately predict their thermomechanical responses, including their time- and temperature-dependent deformation and shape recovery characteristics. Some prior research has explored the advanced thermomechanical modeling of SMPCs, incorporating the hyper-viscoelastic nature of SMPs as well as the anisotropic behavior of the fabric reinforcements [16,17,18]. In particular, anisotropic, hyper-viscoelastic models have been used to evaluate the shape recovery of deployable SMPC antennas by accounting for thermal residual stresses [19]. In such systems, the shape recovery is driven by a combination of elastic energy from the fabrics and recovery energy stored in the SMPs [20,21]. Given that the elastic energy contribution from the fabrics generally exceeds that of the SMPs, it becomes essential to formulate an accurate constitutive model capable of capturing the anisotropic mechanical response of the fabric components.

Traditionally, elasto-plastic modeling approaches have been applied to characterize the plastic deformation [22,23], hysteresis [24], and cyclic loading behaviors [25,26] observed in SMPC systems. However, these models often require complex implementation within finite element frameworks and involve challenges in parameter identification. As a simplified alternative, the total strain formulation has been adopted, as it facilitates direct manipulation of the energy expressions. One such model based on continuum mechanics employs the total strain approach to describe the hyperelastic response of fiber-reinforced systems [27]. Many of the existing models in this category incorporate hyperelastic fiber formulations that reflect the influence of fiber orientation under large deformations [18,28,29,30,31]. Several models that incorporate the interaction mechanisms between the polymer matrix and fabrics have been shown to yield improved predictive accuracy [32,33]. Experimental investigations have further revealed that fabric reinforcements exhibit both energy dissipation and hysteretic characteristics [34,35], and that incorporating residual deformations can enhance the reliability of hysteresis predictions [36,37,38].

In this study, a numerical analysis is developed to investigate the shape recovery behavior of SMPC hinges by employing a constitutive model designed to represent the dissipative energy characteristics of fabric reinforcements. Due to the hysteretic nature of the shape recovery process, the proposed model enables more precise predictions of the final recovered shapes compared to conventional numerical approaches. The time- and temperature-dependent behaviors of SMPs are modeled using a combination of the Mooney–Rivlin framework, viscoelastic theory, and stored strain energy concepts. Additionally, polynomial expressions are utilized to represent the viscoelastic and hysteresis behaviors of the fabrics. The model parameters for SMPs are derived from tensile testing, dynamic mechanical analysis (DMA), and shape recovery experiments. For the fabric materials, parameters are obtained through tensile testing, stress relaxation, and hysteresis evaluation. The numerical analysis based on the finite element method is conducted, and its accuracy is evaluated through comparison with experimental results obtained from mechanical testing of SMPCs, such as uniaxial tensile, bias-extension, stress relaxation, and shape recovery tests. Finally, the shape recovery simulation of SMPC hinges under bending load are conducted using the constitutive equations. Assuming SMPC hinges are used in a solar panel array, the study numerically investigates the influence of bending load, cooling time, stowing time, and reheating time on the shape recovery behaviors.

2. Constitutive Equations

In this section, constitutive equations for shape recovery behaviors of SMPCs considering hysteresis effect are derived based on Helmholtz free energy. The constitutive equation for hyper-viscoelastic and shape memory characteristic of SMPs is explained in Section 2.1 and Section 2.2. The hyperelastic model for fabrics including hysteresis effect is expressed in Section 2.3.

In accordance with the Clausius–Planck inequality derived from the first and second laws of thermodynamics, a constitutive formulation can be established using the Helmholtz free energy function [39]. The representation of the Helmholtz free energy for composite materials is given by Equation (1):

$${\mathsf{\Psi }}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{e}}={\mathsf{\Psi }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}}\left(\mathbf{C},{\mathsf{\Gamma }}_1^{\mathrm{m}},\dots ,{\mathsf{\Gamma }}_{\mathrm{i}}^{\mathrm{m}}\right)+{\mathsf{\Psi }}_{\mathrm{f}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{c}}\left(\mathbf{C},{\mathsf{\Gamma }}_1^{\mathrm{f}},\dots ,{\mathsf{\Gamma }}_{\mathrm{j}}^{\mathrm{f}},{\mathbf{a}}_0,{\mathbf{b}}_0\right)$$

(1)

where ${\mathsf{\Psi }}_{\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{x}}$ and ${\mathsf{\Psi }}_{\mathrm{f}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{c}}$ denote the free energy contributions from the shape memory polymer matrix and the anisotropic fabric reinforcements, respectively. C represents the right Cauchy–Green strain tensor; ${\mathsf{\Gamma }}_{\mathrm{i}}^{\mathrm{m}}$ and ${\mathsf{\Gamma }}_{\mathrm{j}}^{\mathrm{f}}$ are internal variables describing the relaxation behavior of the matrix and fabric components. a₀ and b₀ correspond to the unit directions of weft and warp fibers in the reference configuration.

2.1. Constitutive Equation for the SMPs

The Helmholtz free energy function for the SMP matrix is expressed in Equation (2):

$${\mathsf{\Psi }}_{\mathrm{S}\mathrm{M}\mathrm{P}\mathrm{s}}={\mathsf{\Psi }}_{\mathrm{v}\mathrm{o}\mathrm{l}}\left(J\right)+{\mathsf{\Psi }}_{\mathrm{i}\mathrm{s}\mathrm{o}}\left(\overline{\mathbf{C}}\right)+\sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathsf{\gamma }}_{\mathrm{i}}^{\mathrm{m}}\left(\overline{\mathbf{C}},{\mathsf{\Gamma }}_{\mathrm{i}}^{\mathrm{m}}\right)$$

(2)

where $\overline{\mathbf{C}}=J^{-2/3}\mathbf{C}$, is the volume-preserving part of the right Cauchy–Green deformation tensor, and J is the determinant of the deformation gradient (volume ratio). ${\mathsf{\Psi }}_{\mathrm{i}\mathrm{s}\mathrm{o}}$ and ${\mathsf{\Psi }}_{\mathrm{v}\mathrm{o}\mathrm{l}}$ denote the volumetric and isochoric energy terms, respectively. The superscript m indicates association with the matrix, and ${\mathsf{{\rm Y}}}_{\mathrm{i}}^{\mathrm{m}}$ captures the viscoelastic dissipation potential. Assuming the material is nearly incompressible (J = 1), the isochoric part can be described using the Mooney–Rivlin formulation [40], as shown in Equation (3):

$${\mathsf{\Psi }}_{\mathrm{i}\mathrm{s}\mathrm{o}}=C_{10}\left(I_1-3\right)+C_{01}\left(I_2-3\right), {\mathsf{\Psi }}_{\mathrm{v}\mathrm{o}\mathrm{l}}=0$$

(3)

where C₁₀ and C₀₁ are material constants; I₁ and I₂ represent the first and second strain invariants.

The mechanical response of SMPs under deformation is modeled using a one-dimensional rheological representation (Figure 1), where spring and dashpot components represent equilibrium and non-equilibrium responses. Maxwell elements, consisting of a spring and dashpot in series, characterize the time-dependent viscoelastic behavior. The stored strain, ${\epsilon }_{\mathrm{s}}$, accounts for energy accumulation in the material. Based on the non-equilibrium stress formulation derived from this rheological model, the second Piola–Kirchhoff stress and corresponding elasticity tensors are given by the following relations:

$${\mathbf{S}}_{\mathrm{S}\mathrm{M}\mathrm{P}\mathrm{s}}=2\left[1+\sum _{\mathrm{i}=1}^{\mathrm{k}}\left\{{\beta }_{\mathrm{i}}^{\mathrm{m}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right)\right\}\right]\left\{C_{10}\mathbf{I}+C_{01}\left(I_1\mathbf{I}-\overline{\mathbf{C}}\right)\right\}+\sum _{\mathrm{i}=1}^{\mathrm{k}}{\mathbf{H}}_{\mathrm{i},\mathrm{n}}^{\mathrm{m}}$$

(4)

$${\mathbf{H}}_{\mathrm{i},\mathrm{n}}^{\mathrm{m}}=\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right)\left\{{\mathbf{Q}}_{\mathrm{i},\mathrm{n}-1}^{\mathrm{m}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right)-{\beta }_{\mathrm{i}}^{\mathrm{m}}{\mathbf{S}}_{\mathrm{i}\mathrm{s}\mathrm{o},\mathrm{n}-1}^{\mathrm{m}}\right\}, {\xi }_{\mathrm{i}}^{\mathrm{m}}=-0.5{\displaystyle \frac{∆t}{{\tau }_{\mathrm{i}}^{\mathrm{m}}}}$$

(5)

$$\begin{gathered} {\mathbf{Q}}_{\mathrm{i},\mathrm{n}-1}^{\mathrm{m}}= \\ {\beta }_{\mathrm{i}}^{\mathrm{m}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right){\mathbf{S}}_{\mathrm{i}\mathrm{s}\mathrm{o},\mathrm{n}-1}^{\mathrm{m}}+\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right)\left\{{\mathbf{Q}}_{\mathrm{i},\mathrm{n}-2}^{\mathrm{m}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right)-{\beta }_{\mathrm{i}}^{\mathrm{m}}{\mathbf{S}}_{\mathrm{i}\mathrm{s}\mathrm{o},\mathrm{n}-2}^{\mathrm{m}}\right\}, i=1,\dots , k \end{gathered}$$

(6)

$${\mathbb{C}}_{\mathrm{S}\mathrm{M}\mathrm{P}\mathrm{s}}=4C_{01}\left[1+\sum _{\mathrm{i}=1}^{\mathrm{k}}\left\{{\beta }_{\mathrm{i}}^{\mathrm{m}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{i}}^{\mathrm{m}}\right)\right\}\right]\left(\mathbf{I}\otimes \mathbf{I}-\mathbf{I}\mathbf{I}\right)$$

(7)

Here, the superscript m denotes that the quantities correspond to the matrix (SMPs), Δt refers to the increment of time, while ${\beta }_{\mathrm{i}}^{\mathrm{m}}$ and ${\xi }_{\mathrm{i}}^{\mathrm{m}}$ are the dimensionless variables associated with viscoelastic behavior, respectively; ${\tau }_{\mathrm{i}}^{\mathrm{m}}$ denotes the retardation time, ${\mathbf{Q}}_{\mathrm{i},\mathrm{n}-1}^{\mathrm{m}}$ represents the previous time-step stress, while ${\mathbf{H}}_{\mathrm{i},\mathrm{n}}^{\mathrm{m}}$ is the history-dependent stress component.

Based on the assumption that the SMPs are nearly incompressible materials, for the uniaxial tensile test, Equations (4) and (7) can be expressed as Equations (8) and (9).

$$S_u=2\left({\lambda }_u^2-{\displaystyle \frac{1}{{\lambda }_u}}\right)\left(C_{10}+C_{01}{\displaystyle \frac{2}{{\lambda }_u}}\right)$$

(8)

$$E=\left[1+\sum _{\mathrm{i}=1}^{\mathrm{k}}{\beta }_i^m\left\{\mathrm{e}\mathrm{x}\mathrm{p}\left({\zeta }_i^m\right)-1\right\}\right]E^{\mathrm{u}\mathrm{n}\mathrm{r}\mathrm{l}\mathrm{e}\mathrm{x}}$$

(9)

where ${\lambda }_u$ and E indicate the stretch ratio and elastic modulus in the loading direction, w is the frequency, and E^unrlex is the elastic modulus prior to stress relaxation. Equation (9) can be expressed in the frequency domain as follows [41]:

$$E=\left[1-\sum _{\mathrm{i}=1}^{\mathrm{k}}{\beta }_i^m\left\{{\displaystyle \frac{1}{w^2{\left({\tau }_i^{\mathrm{m}}\right)}^2+1}}\right\}\right]E^{\mathrm{u}\mathrm{n}\mathrm{r}\mathrm{l}\mathrm{e}\mathrm{x}}$$

(10)

Due to the matrix exhibiting time- and temperature-dependent behavior, the time–temperature superposition principle (TTSP) is utilized to identify the dimensionless parameters related to viscoelasticity based on their values at the glass transition temperature, T_g, as shown in Equation (11).

$${\zeta }_{\mathrm{i}}^{\mathrm{m}}\left(T\right)={\displaystyle \frac{{\zeta }_{\mathrm{i},T_{\mathrm{g}}}^{\mathrm{m}}}{{\alpha }_{\mathrm{T}}\left(T\right)}}$$

(11)

where T is the current temperature, ${\zeta }_{\mathrm{i},T_{\mathrm{g}}}^{\mathrm{m}}$ is the dimensionless parameter at T_g. The shift factor, α_T, which varies the temperature, is often expressed using the Williams–Landel–Ferry (WLF) equation [42].

$$\log {\mathsf{\alpha }}_{\mathrm{T}}={\displaystyle \frac{-C_1\left(T-T_{\mathrm{g}}\right)}{C_2+T-T_{\mathrm{g}}}}$$

(12)

where C₁ and C₂ are coefficients derived through interpolation of experimental shift factor data.

2.2. Constitutive Equation for Shape Memory Behaviors

SMPs possess the unique ability to retain a temporary shape and return to their original form when exposed to appropriate thermal conditions. When these materials mechanically deformed at temperatures above T_g, and subsequently cooled below T_g, the deformation is preserved in the form of internal strain energy. Even after the removal of the applied load, the deformed shape remains stable. Upon reheating the polymer above T_g, the stored energy is released, and the original shape is restored. To mechanically represent this phenomenon, the frozen fraction model is frequently utilized [43]. This model conceptualizes the polymer as comprising a blend of active and frozen molecular segments, where the frozen segments are responsible for energy storage. The proportion of the frozen phase is described using a logistic-type function as shown in Equation (13).

$${\varphi }_{\mathrm{f}}\left(T\right)={\left\{{\displaystyle \frac{1}{1+{\left(\frac{T_0}{T}\right)}^{\mathrm{h}}}}\right\}}^{\mathrm{s}}={\displaystyle \frac{{\epsilon }^{*}}{{\epsilon }_{\mathrm{p}\mathrm{r}\mathrm{e}}}}$$

(13)

In this equation, ${\varphi }_{\mathrm{f}}$ represents the frozen fraction; T₀, h, and S are empirically determined parameters, ${\epsilon }_{\mathrm{p}\mathrm{r}\mathrm{e}}$ is the pre-deformed strain, and ${\epsilon }^{*}$ is the recovered strain. Assuming that the stresses in the frozen and active phases are identical, the total strain can be expressed as the sum of the mechanical, thermal, and stored strains, as presented in Equation (14).

$$\mathbf{E}={\mathbf{E}}_{\mathrm{M}}+{\mathbf{E}}_{\mathrm{T}}+{\mathbf{E}}_{\mathrm{S}}$$

(14)

where E, E_M, E_T, and E_S are the total, mechanical, thermal, and stored strains, respectively. Those strains can be defined as Equations (15) to (17).

$${\mathbf{E}}_{\mathrm{M}}=\left\{{\varphi }_{\mathrm{f}}{\left({\mathbb{C}}^{\mathrm{l}}\right)}^{-1}+\left(1-{\varphi }_{\mathrm{f}}\right){\left({\mathbb{C}}^{\mathrm{h}}\right)}^{-1}\right\}:\mathbf{S}$$

(15)

$${\mathbf{E}}_{\mathbf{T}}=\left[{\int }_{T_0}^T\left\{{\varphi }_{\mathrm{f}}{\alpha }_{\mathrm{f}}+\left(1-{\varphi }_{\mathrm{a}}\right){\alpha }_{\mathrm{a}}\right\}dT\right]\mathbf{I}$$

(16)

$${\displaystyle \frac{d{\mathit{E}}_S}{dT}}=\left[{\left({\mathbb{C}}^{\mathrm{h}}\right)}^{-1}:{\left\{{\varphi }_{\mathrm{f}}{\left({\mathbb{C}}^{\mathrm{l}}\right)}^{-1}+\left(1-{\varphi }_{\mathrm{f}}\right){\left({\mathbb{C}}^{\mathrm{h}}\right)}^{-1}\right\}}^{-1}:\left(\mathbf{E}-{\mathbf{E}}_{\mathrm{S}}-{\mathbf{E}}_{\mathrm{T}}\right)\right]{\displaystyle \frac{d{\varphi }_{\mathrm{f}}}{dT}}$$

(17)

In the above, ${\mathbb{C}}^{\mathrm{l}}$ and ${\mathbb{C}}^{\mathrm{h}}$ represent the elastic stiffness tensors at low and high temperatures, respectively, while α_f and α_a denote the thermal expansion coefficients for the frozen and active phases.

2.3. Constitutive Equation for the Fabric

The constitutive behavior of fabrics under mechanical loads can be described using the Helmholtz free energy models expressed as Equation (18).

$${\mathsf{\Psi }}_{\mathrm{f}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{c}}={\mathsf{\Psi }}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e}}\left(\mathbf{C},{\mathbf{a}}_0,{\mathbf{b}}_0\right)+{\mathsf{\Psi }}_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}\left(\mathbf{C},{\mathbf{a}}_0,{\mathbf{b}}_0\right)+\sum _{\mathrm{j}=1}^{\mathrm{p}}{\mathsf{\gamma }}_{\mathrm{j}}^{\mathrm{f}}\left(\overline{\mathbf{C}},{\mathsf{\Gamma }}_{\mathrm{j}}^{\mathrm{f}},\mathbf{C},{\mathbf{a}}_0,{\mathbf{b}}_0\right)$$

(18)

Here, ${\mathsf{\Psi }}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e}}$ and ${\mathsf{\Psi }}_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}$ account for the fabrics’ response under uniaxial tension and shear, respectively. The third term ${\mathsf{{\rm Y}}}_{\mathrm{j}}^{\mathrm{f}}$ represents energy dissipation associated with viscoelastic effects. The unit vectors a₀ and b₀ define the orientations of the weft and warp fibers in the unreformed configuration. Assuming that the external load is applied in the y-direction (Figure 2), these vectors are given as Equation (19).

$${\mathbf{a}}_0=\left(\begin{array}{c}\cos {\theta }_a\\ \sin {\theta }_a\\ 0\end{array}\right), {\mathbf{b}}_0=\left(\begin{array}{c}\cos {\theta }_b\\ \sin {\theta }_b\\ 0\end{array}\right)$$

(19)

where ${\theta }_a$ and ${\theta }_b$ are the initial angle of the weft and warp fibers, respectively. The anisotropic response of the fabrics is captured using polynomial functions of pseudo-invariants Equations (20) and (21).

$${\mathsf{\Psi }}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e}}=\left\{\begin{array}{c}{k}_1{\left(I_4-1\right)}^4+k_2{\left(I_4-1\right)}^3+k_3{\left(I_4-1\right)}^2 \mathrm{for} \mathrm{weft} \\ k_4{\left(I_6-1\right)}^4+k_5{\left(I_6-1\right)}^3+k_6{\left(I_6-1\right)}^2 \mathrm{for} \mathrm{warp}\end{array}\right.$$

(20)

$${\mathsf{\Psi }}_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}=k_7{\left(I_{10}\right)}^4+k_8{\left(I_{10}\right)}^3+k_9{\left(I_{10}\right)}^2$$

(21)

Here, I₄, I₆, and I₁₀ are pseudo-invariants representing fiber stretch in the weft, warp directions, and the change in the angle between fibers, respectively. The coefficients ki (i = 1,⋯, 9) are obtained through curve-fitting to experimental data. To model the residual deformation due to hysteresis during unloading, the proposed invariants model is shown as Equation (22).

$$I_4\left(\mathbf{C},{\mathbf{a}}_0\right)={\mathbf{a}}_0·\mathbf{C}{\mathbf{a}}_0-{\delta }_{\mathrm{r}\mathrm{e},4}, I_6\left(\mathbf{C},{\mathbf{b}}_0\right)={\mathbf{b}}_0\mathbf{C}·{\mathbf{b}}_0-{\delta }_{\mathrm{r}\mathrm{e},6}$$

(22)

The residual deformations in the weft and warp direction ( ${\delta }_{\mathrm{r}\mathrm{e},4}$ and ${\delta }_{\mathrm{r}\mathrm{e},6}$) are defined based on loading unloading states as presented in Equation (23).

$$\begin{gathered} {\delta }_{\mathrm{r}\mathrm{e},4}=\left\{\begin{array}{c}0 \mathrm{for} \mathrm{loading}\\ {\left({\epsilon }_{\mathrm{r}\mathrm{e},4}+1\right)}^2-1 \mathrm{for} \mathrm{unloading}\end{array} \right. \\ {\delta }_{\mathrm{r}\mathrm{e},6}=\left\{\begin{array}{c}0 \mathrm{for} \mathrm{loading}\\ {\left({\epsilon }_{\mathrm{r}\mathrm{e},6}+1\right)}^2-1 \mathrm{for} \mathrm{unloading}\end{array}\right. \end{gathered}$$

(23)

Here, ${\epsilon }_{\mathrm{r}\mathrm{e},4}$ and ${\epsilon }_{\mathrm{r}\mathrm{e},6}$ denote the residual strains in the weft and warp directions, respectively.

The mechanical response of the fabrics is also modeled using a one-dimensional rheological system similar to that of SMPs as presented in Figure 2. ${\epsilon }_{\mathrm{r}\mathrm{e}}$ represents the remaining strain resulting from the hysteresis response of fabrics. From the definition of the non-equilibrium stress, the three-dimensional stress tensor and corresponding moduli can be given as Equations (24) to (28).

$$\begin{gathered} {\mathbf{S}}_{\mathrm{f}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{c}}=2\left[1+\sum _{\mathrm{j}=1}^{\mathrm{p}}{\beta }_{\mathrm{j}}^{\mathrm{f}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right)\right]\left[\left\{4k_1{\left(I_4-1\right)}^3+3k_2{\left(I_4-1\right)}^2+2k_3\left(I_4-1\right)\right\}{\mathbf{A}}_0\right. \\ +\left.\left\{4k_4{\left(I_6-1\right)}^3+3k_5{\left(I_6-1\right)}^2+2k_6\left(I_6-1\right)\right\}{\mathbf{B}}_0\right] \\ +2\left\{4k_7{\left(I_{10}\right)}^3+3k_8{\left(I_{10}\right)}^2+2k_9\left(I_{10}\right)\right\}{\mathbf{G}}_0+\sum _{\mathrm{j}=1}^{\mathrm{p}}{\mathbf{H}}_{\mathrm{j},\mathrm{n}}^{\mathrm{f}} \end{gathered}$$

(24)

$${\mathbf{H}}_{\mathrm{j},\mathrm{n}}^{\mathrm{f}}=\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right)\left\{{\mathbf{Q}}_{\mathrm{j},\mathrm{n}-1}^{\mathrm{f}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right)-{\beta }_{\mathrm{j}}^{\mathrm{f}}{\mathbf{S}}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e},\mathrm{n}-1}\right\}, {\xi }_{\mathrm{j}}^{\mathrm{f}}=-0.5{\displaystyle \frac{∆t}{{\tau }_{\mathrm{j}}^{\mathrm{f}}}}$$

(25)

$$\begin{gathered} {\mathbf{Q}}_{\mathrm{j},\mathrm{n}-1}^{\mathrm{f}}= \\ {\beta }_{\mathrm{j}}^{\mathrm{f}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right){\mathbf{S}}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e},\mathrm{n}-1}+\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right)\left\{{\mathbf{Q}}_{\mathrm{j},\mathrm{n}-2}^{\mathrm{f}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right)-{\beta }_{\mathrm{j}}^{\mathrm{f}}{\mathbf{S}}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e},\mathrm{n}-2}\right\}, j=1,\dots , p \end{gathered}$$

(26)

$$\begin{gathered} {\mathbf{G}}_0=-{\displaystyle \frac{1}{2}}\left(I_{10}+{\mathbf{a}}_0·{\mathbf{b}}_0\right)\left({\displaystyle \frac{1}{I_4}}{\mathbf{A}}_0+{\displaystyle \frac{1}{I_6}}{\mathbf{B}}_0\right) \\ +{\left(I_4{I}_6\right)}^{-{\displaystyle \frac{1}{2}}}{\mathbf{a}}_0·{\mathbf{b}}_0+\mathrm{I}\left({\mathbf{a}}_0·\mathbf{C}{\mathbf{b}}_0{\left(I_4{I}_6\right)}^{-{\displaystyle \frac{1}{2}}}-{\mathbf{a}}_0·{\mathbf{b}}_0\right) \end{gathered}$$

(27)

$$\begin{gathered} {\mathbb{C}}_{\mathrm{f}\mathrm{a}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{c}}=8\left\{1+\sum _{\mathrm{j}=1}^{\mathrm{p}}{\beta }_{\mathrm{j}}^{\mathrm{f}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\xi }_{\mathrm{j}}^{\mathrm{f}}\right)\right\}\left[\left\{6k_1{\left(I_4-1\right)}^2+2k_2\left(I_4-1\right)+k_3\right\}{\mathbf{A}}_0⨂{\mathbf{A}}_0\right. \\ +\left.\left\{6k_4{\left(I_6-1\right)}^2+3k_5\left(I_6-1\right)+k_6\right\}{\mathbf{B}}_0⨂{\mathbf{B}}_0\right]+8\left\{6k_7{I}_{10}^2+3k_8{I}_{10}+k_9\right\}{\mathbf{G}}_0⨂{\mathbf{G}}_0 \end{gathered}$$

(28)

where A₀ and B₀ are the cross-product of the unit vectors.

Assuming that the fabric is nearly incompressible, the pseudo-invariants can be transformed into a function of the stretch ratio such as $I_4={\lambda }_{\mathrm{w}\mathrm{e}\mathrm{f}\mathrm{t}}^2$, $I_6={\lambda }_{\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{p}}^2$, and $I_{10}=\mathsf{\Delta }\mathsf{\phi }$. The tensile and shear stresses can be rewritten as described in Equations (29) and (30).

$${\mathbf{S}}_{\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{l}\mathrm{e}}=\left\{\begin{array}{c}2\left[4k_1{\left({\lambda }_{\mathrm{w}\mathrm{e}\mathrm{f}\mathrm{t}}^2-1\right)}^3+3k_2{\left({\lambda }_{\mathrm{w}\mathrm{e}\mathrm{f}\mathrm{t}}^2-1\right)}^2+2k_3\left({\lambda }_{\mathrm{w}\mathrm{e}\mathrm{f}\mathrm{t}}^2-1\right)\right]\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{w}\mathrm{e}\mathrm{f}\mathrm{t}\\ 2\left[4k_4{\left({\lambda }_{\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{p}}^2-1\right)}^3+3k_5{\left({\lambda }_{\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{p}}^2-1\right)}^2+2k_6\left({\lambda }_{\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{p}}^2-1\right)\right] \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{w}\mathrm{a}\mathrm{r}\mathrm{p}\end{array}\right.$$

(29)

$${\mathbf{S}}_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{a}\mathrm{r}}=2\left\{4k_7{\left(\mathsf{\Delta }\mathsf{\phi }\right)}^3+3k_8{\left(\mathsf{\Delta }\mathsf{\phi }\right)}^2+2k_9\left(\mathsf{\Delta }\mathsf{\phi }\right)\right\}$$

(30)

3. Validation of the Constitutive Equations

In this section, the constitutive equtaions are verified by comparing with the experimental results of SMPCs. To obtain the coefficients of the constitutive equations in Section 3.1, the tests for SMPs and fabrics are conducted. Section 3.2 shows the comparison between simulation and experimental results for tensile and shape memory tests of SMPCs. The descriptions of the test method to validate models and related coefficients are summarized in

Table 1. Definition of the test methods for characterizing SMPs.

Characteristics	Test Methods	Coefficients
Hyperelasticity	Tensile	C10, C01
Viscoelasticity	DMA	βi, τi, c1, c2, Tg
Shape recovery	Thermomechanical cycle	T0, h, s
Thermal expansion	Increasing temperature	a, b, c

and

Table 2. Definition of the test methods for characterizing fabrics.

Characteristics	Test Methods	Coefficients
Tensile property	Tensile (0°, 90°)	k1, k2, k3, k4, k5, k6
Hysteresis effect	Loding-unloading	k1, k2, k3, k4, k5, k6
Viscoelasticity	Stress relaxation	βi, τi
Shear property	Tensile (±45°)	k7, k8, k9

.

3.1. Coefficients Acquisition Test

3.1.1. Model Coefficients Associated with SMPs

The model coefficients for hyperelasticity, viscoelasticity, stored strain energy, and thermal expansion are identified through uniaxial tension, DMA, shape recovery tests, and thermal strain data for SMPs. To prepare test specimens, except for DMA test, a thermoset polyurethane SMP (Diaplex Inc. MP4510) is placed into a dog-bone-shaped mold. Then it is cured at 75 °C for 2 h to obtain a thickness of 2 mm. The test is performed using a thermal chamber and materials testing system (MTS, MTS810), while specimen temperatures are monitored using a thermocouple (Figure 3). The dimensions of DMA test specimen are 20 mm (l) × 10 mm (w) × 2 mm (t).

The uniaxial tensile test is conducted in accordance with the standard method for evaluating the tensile properties of plastics [44]. During the test, the crosshead speed is maintained at 10 mm/min, and the temperature varies from 25 °C to 75 °C. As shown in Figure 4, the elastic modulus gradually declines when the specimen temperature exceeds 45 °C. Specimens tested at 25 °C and 35 °C exhibit failure at a stretch ratio near 1.05, because the SMP shows brittle behavior in this temperature range, where phase transition is minimal. The hyperealstic model coefficients are determined by interpolating the tensile data using Equation (4) as shown in

Table 3. Hyperelasticitiy model coefficients for shape memory polymers.

Characteristics	Temperature (°C)	C10 (MPa)	C01 (MPa)
Hyperelasticity	25	−4075	4356
	35	−1145	1230
	45	−3.017	5.263
	55	−1.136	2.123
	75	−0.379	1.030

.

The DMA test is performed using a DMA Q800 (TA Instrument) in accordance with the standard practice for the dynamic mechanical properties of plastics [45]. Using the fixed frequency with stress amplitude approach, the specimen is clamped at both ends, and the flexural modulus is measured over a temperature range of 25–75 °C at 5 °C intervals. The heating rate is set to 5 °C/min, and the test is conducted at three frequencies: 1 Hz, 10 Hz, and 30 Hz. The corresponding flexural moduli at different temperatures and frequencies are shown in Figure 5. Although the test results show that the flexural modulus transition appears around 55 °C, which is slightly different from T_g, this discrepany does not significantly affect the evaluation of the relaxation chracteristics. To construct the master curve covering the full frequency spectrum, the flexural modulus values at different temperatures are horizontally shifted to align with the reference temperature of T_g. The logarithmic shift factors applied in this study are presented in Figure 6, and the resulting time–temperature-shifted modulus data are illustrated in Figure 7. The shifted modulus data clearly demonstrates that the flextural modulus varies form 1.3 GPa to 10 MPa depending on the temperature and testing freqeucny. To derive the coefficients of the five-term viscoelastic model in the time domain from DMA data in the frequency domain, the model is transformed into the frequency domain [41]. These coefficients are extracted by fitting the master curve using the frequency domain expression for the elastic modulus. Additionally, the coefficients for the WFL equation are obtained by fitting the shif factor curve using Equation (12). A summary of the resulting coefficients for SMPs is provided in

Table 4. Viscoelasticity model coefficients for shape memory polymers.

Characteristics	i	βi (-)	τi (s)
Viscoelasticity	1	0.107	0.0154
	2	0.160	2.57
	3	0.235	213
	4	0.268	9580
	5	0.193	202,000

and

Table 5. WLF equation coefficients for shape memory polymers.

Characteristics	C1 (-)	c2 (°C)	Tg (°C)
Williams–Ladnel–Ferry (WLF) equation	52.433	166.429	45

.

The shape recovery test evaluates the frozen fraction as the temperature of SMPs increases. The frozen fraction is calculated by dividing the amount of strain recovered by the initially applied deformation. To ensure consistent thermal conditions for accurate numerical comparison with SMPCs, the SMPs are heated for a duration of 10 s. The measured frozen fractions at various temperatures (up to 70 °C) are illustrated in Figure 8. This figure indicates that little phase transition occurs below 35 °C, while continuous phase transition is observed as the temperature increases beyond this temperature. These findings provide a basis for explaining the brittle behavior of the SMP and SMPC at temperatures below 35 °C. The corresponding coefficients are derived by fitting the data using Equation (13), resulting in T₀ = 40.41 °C, s = 0.154, and h = –22.63.

Due to isotropic behavior of the SMPs, thermal strain is measured one-dimensionally as the specimen temperature increases from 25 °C to 75 °C. The thermal strain is represented as a second-order polynomial function of temperatures, that is, ${\mathsf{\epsilon }}_{\mathrm{T}}=a{T}^2+bT+c$, where temperature is expressed in Kelvin. The coefficients are determined by interpolating the experimental thermal strain as a = 7.75 × 10⁻⁷, b = −4.38 × 10−4, and c = 0.061.

3.1.2. Model Coefficients Associated with Fabrics

To identify the anisotropic and viscoelastic model coefficients, as well as the residual strains, uniaxial tensile, bias-extension, stress relaxation, and hysteresis tests are conducted on fabric specimens (Figure 9). The tested fabric (Soitex, HTL-1) is coated with silver powder and consists of nylon fibers in the warp direction and spandex fibers in the weft. The specimen is prepared in a rectangular geometry. Although nylon has temperature-dependent properties, it is experimentally observed that the mechanical properties of fabric in the warp direction are not significantly affected by temperature within the range of temperature encountered in this research (Figure 10). So, the tests for the fabric are conducted at room temperature. Strain measurements are obtained using a laser extensometer, while the angle between the weft and warp fibers is monitored through photogrammetry. The fabric specimen has a thickness of 0.31 mm. A crosshead speed of 10 mm/min is applied during both the loading and unloading phases.

In the uniaxial tensile and bias-extension tests, specimens are stretched along orientations initially aligned at 0/90° and ±45° with respect to the loading direction, respectively. The bias-extension test is specifically used to evaluate the shear behavior of woven fabrics. The corresponding results are shown in Figure 11 and Figure 12. Since the fabric consists of nylon fibers in the warp direction and spandex fibers in the weft direction, the stiffness and strength in the warp direction are much higher than those in the weft direction. Although the specimen can be stretched beyond a stretch ratio of 1.4 in both directions. In both tests, the tensile and shear stresses exhibit a sharp increase as the specimen is stretched. The coefficients associated with uniaxial and shear properties are extracted by interpolating the experimental data using Equations (29) and (30). The resulting parameters for the fabric materials are listed in

Table 6. Coefficients for the tensile properties of fabrics.

Characteristics	k1 (MPa)	k2 (MPa)	k3 (MPa)	k4 (MPa)	k5 (MPa)	k6 (MPa)
Tensile	0.0135	−0.0187	0.0545	−0.0782	0.2859	0.2618

and

Table 7. Coefficients for the shear properties of fabrics.

Characteristics	k7 (MPa)	k8 (MPa)	k9 (MPa)
Shear	0.7401	−0.5531	0.2107

.

The stress relaxation behavior of the fabrics is evaluated after the fabric is stretched to a ratio of 1.3, and the decreased stress is obtained over a period of 300 s while maintaining the constant stretch ratio (Figure 13). The stress in both weft and warp directions decrease by 0.023 MPa and 0.243 MPa, respectively. After 300 s, the stress variation is approximately 1%. The hyperelastic model coefficients are derived by interpolating the experimental relaxation data using a three-term viscoelastic model for fabrics, as presented in

Table 8. Coefficients for the viscoelasticity properties of fabrics.

Characteristics		Weft Direction		Warp Direction
	i	βi (-)	τi (s)	βi (-)	τi (s)
Viscoelasticity	1	0.03593	2.711	0.03257	16.31
	2	0.12848	20.98	0.00737	16.49
	3	0.05779	164.8	0.93712	3681

.

The hysteresis behavior, which reflects the stress–strain response during both loading and unloading cycles, is presented in Figure 14. The fabric specimens are stretched to a strain corresponding to a stretch ratio of 1.3 and subsequently unloaded until the stress returned to zero. The area ratio between the loading and unloading curves represents the ratio of dissipated energy. The energy dissipation ratio is measured as 0.35 in the weft direction and 0.59 in the warp directions. Additionally, the residual strains recorded in these directions are 0.057 and 0.039, respectively. The equation employed to interpolate the uniaxial test data is utilized to derive the coefficients of the hyperelastic model under loading conditions. The resulting coefficients for the fabrics are summarized in

Table 9. Coefficients for the hysteresis properties of fabrics.

Characteristics	k1 (MPa)	k2 (MPa)	k3 (MPa)	k4 (MPa)	k5 (MPa)	k6 (MPa)
Hysteresis (Unloading)	0.0537	0.0044	0.0174	1.1946	−0.7287	0.2250

.

3.2. Thermomechanical Characteristics of SMPCs

3.2.1. Fabrication and Experimental Setup for SMPCs

The SMPC specimen is produced using a single ply of fabric (Soitex, HTL-1) and the SMPs (Diaplex Inc. MP4510). The fabrication process consists of the following steps:

(1)

Place the fabric onto a flat mold surface.

(2)

Attach copper sheets to both fabric ends using an adhesive film (Loctite 9695) for consistent electrical conductivity across the SMPCs.

(3)

Cure the adhesive at 85 °C for 30 min followed by 125 °C for 2 h.

(4)

Pour the SMPs onto the fabric and layer it with a release film, caul plate, and breather.

(5)

Seal the assembly using vacuum film and sealant tape.

(6)

Cure the assembly at 80 °C for 2 h, followed by curing at 125 °C for 2 h.

The thickness of the fabricated specimen is approximately 0.32 mm, and its configuration is illustrated in Figure 15. The fiber volume fraction is measured to be 0.499 based on the density of the fabric and the SMPs. Due to the silver powder coating on the fabric, the specimen is capable of resistive heating. Temperature measurements using a thermocouple and thermal camera confirm that the specimen surface heats uniformly, reaching temperatures above 95 °C (Figure 16).

The results from the experiments are compared with those derived from numerical analysis. The analysis employs a total Lagrangian formulation with nonlinear geometry considerations, utilizing 8-node solid elements characterized by three translation degrees of freedom. The geometry of the analysis model is the same as actual dimensions of the SMPC specimen. The mesh consists of 40, 8, and 1 elements along the length, width, and thickness direction, respectively. Boundary conditions and loading scenarios are set identically to those used in the experiments. Due to the complexity of the model, convergence verification of the internal calculations is essential. Therefore, the analysis is carried out using a custom-developed MATLAB code (R2022b). In this analysis, SMPCs are assumed to be anisotropic materials, with their stress and elasticity tensors incorporated based on the equivalent model given in Equation (1).

3.2.2. Anisotropic Hyper-Viscoelastic Response

Time- and temperature-dependent mechanical responses are evaluated through uniaxial tensile and stress relaxation experiments. The tensile tests are conducted following the standard test method for the tensile properties of polymer matrix composite materials [46]. The tests are carried out at temperatures ranging from 25 °C to 75 °C, with a constant loading rate of 10 mm/min. To investigate the time-dependent behavior, the specimen is stretched to a stretch ratio of 1.3 at the same rate and then held in the deformed state for 7 min while monitoring the stress relaxation. Since viscoelastic effects are primarily exhibited above T_g, the relaxation tests are performed at 75 °C. Figure 17 presents the comparison between the measured data and the numerical analysis. Compared with the tensile response of SMPs, the SMPCs exhibit a higher modulus, which is 1.74 GPa. As the mechanical properties of the SMP degrade rapidly near T_g, the SMPC exhibits a drastic reduction in tensile modulus, decreasing from 1.74 GPa at 25 °C to 14.6 MPa at 75 °C. The SMPCs can undergo deformations exceeding 1.3 times their original length above T_g. During the stress relaxation phase, the stress decreases by approximately 13% within 1 min and by about 26% within a 7 min period. The consistency between the experimental tests and the analysis confirms that the proposed constitutive model reliably predicts the thermo-viscoelastic behavior of SMPCs.

3.2.3. Shape Memory Response

The thermomechanical cycle test is performed to investigate the shape recovery behavior of the SMCPs. The stress responses with respect to stretch ratio, time, and temperatures are illustrated in Figure 18. Initially, the specimen is stretched to a stretch ratio of 1.3 at 75 °C, applying a loading rate of 10 mm/min (isothermal loading). Subsequently, the temperature is lowered to 25 °C over a period of 100 s while maintaining the fixed deformed shape (cooling). Due to the temperature-dependent stiffness change, the stress increases by approximately 20% as the specimen cools. The specimen is unloaded at a constant rate of 10 mm/min while being held at a low temperature (isothermal unloading). During the reheating stage, the temperature is raised to 75 °C within 10 s, allowing the specimen to recover its original shape. The shape recovery response is analyzed by comparing the restored stretch ratios at various temperatures, as presented in Figure 19. As temperature rises, the release of stored strain energy enables the SMPC to return to its original configuration. Compared to neat SMPs, SMPCs demonstrated a higher recovery ratio, owing to the additional elastic energy stored in fabric reinforcement. However, approximately 6% of the original shape remained unrecovered, primarily due to hysteresis and residual deformation within the fabric layers. When the fabric is assumed to behave reversibly (as in conventional models), the recovery ratio falls below 3%. In contrast, the proposed model provided predictions that closely matched experimental observations and effectively represented the nonlinear effects associated with fabric behavior.

Since the stored strain is proportional to the rate of temperature change, the influence of cooling time is analytically examined (Figure 20). When the SMPCs are cooled to 25 °C within 2 s, the irrecoverable deformation is approximately 4.5% of the deformed shape. A similar residual deformation of 6% is observed when the cooling time is extended to 200 s. However, a significant increase in permanent deformation up to 17% is noted when the cooling time prolonged to 20,000 s. These results indicate that reducing the temperature within 200 s is sufficient to achieve effective shape fixitaion.

4. Numerical Study on SMPC Hinges

In this section, the shape recovery behaviors of SMPC hinges are numerically investigated using the developed constitutive equation. Section 4.1 shows the numerical model of a SMPC hinge under four-point bending load. Bending beahviors and recovered deformation with different temperature and time variables are presented in Section 4.2 and Section 4.3.

4.1. Numerical Model of SMPC Hinge

SMPC hinges, generally adapted to antenna or solar cell hinge mechanism in space applications, are exposed to bending load. Thus, the numerical model of SMPC hinge is conducted under a four-point bending condition as presented in Figure 21. It is assumed that the temperature of a SMPC hinge embedded with a layer of conductive fabric changes evenly. The fiber volume fraction is 0.01872 and the weft is arranged in the longitudinal direction. The geometric dimensions of the model are 160 mm (L), 40 mm (w), and 8 mm (t). One end of the SMPC hinge has a pin-supported boundary condition and the other end can move only in the longitudinal direction. The span length (L_S) is 80 mm and a distributed load (P) of 1.125 N/mm is exerted to apply bending moment. The SMPC hinge is bent until the maximum deflection becomes 48 mm.

4.2. Bending Beahviors with Different Temperatures

In order to verify the structural stability, the temperature-dependent bending characteristics of a SMPC hinge should be investigated. Bending behaviors are examined via bending analysis under a single loading–unloading condition as described in Figure 22. A SMPC hinge is deformed by applying bending moment (step 1). After unloading the bending moment (step 2), the elastic energy of the deformed beam is released. The analysis results at the temperatures of 25 °C and 75 °C are presented in Figure 23. The applied total force is 37.2 N and 7700 N at 75 °C and 25 °C, respectively. Because the stiffness of the SMPs is dominant at 25 °C, the deflection returns to zero while 24% of energy is dissipated after unloading. On the other hand, because the stiffness of the fabrics is prominently exhibited at 75 °C, irreversible deflection is shown while the energy dissipation ratio is 0.12. Under the assumption that the deformed hinge is stowed for a while at 75 °C, the effect of the stowing time on unloading behaviors is studied as presented in Figure 24. Due to the viscoelasticity of the SMPCs, the energy dissipation ratio increases to 0.119, 0.202, and 0.384 as the hinge is stored longer. The irreversible deflection is extended from 0.005 mm to 0.016 mm.

4.3. Shape Recovery Bhaviors with Different Time Variables

Figure 25 represents the loading conditions experienced by a SMPC hinge when the hinge is assembled on the ground and deployed in space, which are bending, stowed, residual stress-released, and shape recovery conditions. After the hinge is applied with bending moment at a temperature above T_g (step 1), the temperature decreases to fix the stowed configuration (step 2). During step 2, part of the elastic energy is stored. When the bending moment is eliminated (step 3), the remaining elastic energy of the hinge is released, and can maintain stowed shape without external force. By increasing the temperature, the hinge is deployed to initial shape as the stored energy is discharged (step 4).

The relationship between the deflection and temperature during shape recovery of SMPC hinge with respect to four time variables that determine recovered deflection is presented in Figure 26. The hinge is bent until the deflection becomes 48 mm. Considering the solar array stowing process, the hinge is assumed to bend within 220 s (approx. 4 min) to 10,220 s (approx. 3 h). As the bending time increased, unrecovered deflection increased from 1.3 mm to 14 mm. At constant bending time, the stored strain energy is regarded as it depends on the cooling time. Based on the numerical simulation in Section 3.2.3, the recovered deflection is simulated as the cooling time increased from 2 s to over 200 s. When the cooling time is 20,000 s (approx. 5 h), it is predicted that the hinge will restore the deformed shape until the deflection becomes 5.8 mm. And it is expected that the unrestored deflection will be reduced to 1.3 mm by decreasing cooling time to 2 s. In contrast to bending and cooling time, stowed and reheating time show negligible effect on recovered deflection. It is because shape recovery energy is determined before the hinge cools down. As bending time increases, the load required to bend the hinge decreases due to viscoelastic effect, leading to a reduction in applied energy. During phase transition, where the applied energy is converted to stored energy in the cooling stage, the energy loss increases with longer cooling times.

5. Discussion

A novel constitutive equation for SMPCs is proposed, incorporating both the viscoelastic behavior of the SMP matrix and the energy dissipation characteristics of the conductive fabric reinforcement. To determine the model coefficients, a comprehensive experimental approach is employed, including uniaxial tensile, bias-extension, stress relaxation, DMA, and shape recovery tests. The hysteresis test on the fabric layers enables the determination of coefficients related to energy dissipation, which directly contributes to the overall accuracy of the model. The proposed constitutive equation demonstrates excellent agreement with experimental results, effectively predicting mechanical response of SMPCs varying with time and temperature. Notably, the consideration of residual strains and hysteresis effect in the model significantly improves its ability to predict the shape recovery behaviors of SMPCs. The bending characteristics of SMPC hinge under four-point bending is numerically investigated using the constitutive equation. The results indicate that both bending and cooling times play a crucial role in achieving high recovery ratios. Specifically, it is observed that when the hinge is cooled down for over 200 s, shape recovery ratio is decreased from 97% to 88%, demonstrating that prolonged cooling times reduce its ability to fully recover its shape. This finding underscores the importance of controlling cooling time to optimize performance, especially in applications requiring fast shape recovery. In contrast, stowed and reheating time show minimal impact on shape recovery. These characteristics highlight two significant advantages of using SMPC in space applications. First, SMPC hinges offer high deployment reliability, even when the hinge remains stowed on the ground for several weeks or months. This makes SMPCs highly suitable for space missions where long storage times are common before deployment. Second, SMPs help minimize shape error after deployment, even when subjected to long reheating periods in deep space, a critical factor for precision in space systems. Given these characteristics, SMPC hinges can provide a simpler, more reliable alternative to traditional mechanical deployment mechanisms, reducing both the complexity and mass of the system. However, challenges still remain, such as outgassing issues under vacuum conditions, degradation of mechanical properties, and shape recovery performance in thermal cycle environments. In the future, the shape recovery behaviors of SMPC hinges will be investigated experimentally following thermal vacuum and thermal cycling tests. In addition, methods to prevent mechanical property degradation of SMPC hinges under radiation exposure will be researched.