Constitutive modeling and experimental validation of the thermo-mechanical response of a shape memory composite containing shape memory alloy fibers and shape memory polymer matrix

Abstract

Multifunctional shape memory composites composed of shape memory polymers and shape memory alloys exhibit superior shape memory properties; therefore, it is of great interest to researchers to model their thermo-mechanical behavior numerically. Although a number of constitutive models of shape memory alloys and shape memory polymers have been developed, very few models have been developed for shape memory composites and validated with experimental data. In this study, we first review separately constitutive models of shape memory alloys and shape memory polymers developed in previous studies. Both models were validated with thermo-mechanical tests conducted on a shape memory alloy fiber and shape memory polymer, respectively. A constitutive model for the shape memory composite was then developed utilizing the homogenization scheme. Shape memory composites containing 0.5% or 1% of the shape memory alloy fiber volume content embedded in the shape memory polymer matrix were fabricated. Thermo-mechanical tests were carried out on these shape memory composites to validate the proposed constitutive model. The experimental results showed that the proposed shape memory composite model was able to predict the general trend of the thermo-mechanical behavior of the shape memory composites.

Keywords

Shape memory composite shape memory alloy shape memory polymer constitutive modeling shape memory effect superelasticity analytical modeling thermo-mechanical response

Introduction

Shape memory materials are a group of adaptive materials that respond to external stimulus by changing their shape in predetermined manners. This phenomenon is known as the shape memory effect (SME) and can be triggered by various stimuli including heat, light, voltage, or magnetic field (Lendlein, 2010; Wei et al., 1998). Shape memory alloys (SMAs) and shape memory polymers (SMPs) are two classes of shape memory materials which have been studied extensively in the past two decades due to their wide application in a number of engineering fields including biomedical, aerospace, transportation, and textiles (Hu, 2007; Huang et al., 2010; Lendlein et al., 2010; Liu et al., 2014; Mather et al., 2009).

SMAs are thermally responsive and can be programed with thermal treatments. They exhibit the SME at relatively low temperatures and a behavior known as superelasticity at relatively high temperatures, above the austenite finish temperature (Brinson, 1993). The fundamental mechanism that lies behind these two phenomena is a reversible diffusionless phase transformation between the two phases: a high-temperature phase known as austenite and a low-temperature phase known as martensite. This transformation has been described as a thermoelastic process (Lexcellent, 2013). In general, these materials are capable of possessing great shape fixity at low temperatures, and they can recover up to 10% of its strain with a large recovery stress (Brinson, 1993). However, both the yield strength and elastic modulus of SMAs are low in the low-temperatures martensite phase, resulting in poor load bearing properties in the material (Lexcellent, 2013).

The mechanism responsible for the SME in SMPs differs significantly from that in SMAs due to the molecular network structure of polymers. SMPs can be considered to have two separate phases: the amorphous region containing soft segments and the crystalline region containing hard segments. The soft segments are able to accommodate a large amount of recoverable plastic strain, while the hard segments allow the fixation of the temporary shape when the material is cooled from a pre-strained state above the transition temperature (T_trans ) (Liu et al., 2006; Xie, 2011). Typically, this transition temperature is the glass transition temperature (T_g ) for amorphous polymers and it is the melting temperature (T_m ) for semicrystalline networks (Nguyen, 2013). Since the shape memory properties of SMPs can be tuned with processing parameters, polymer blending has been utilized as a method to create SMPs with improved overall shape memory and mechanical properties (Meng and Hu, 2009). In a recent research, polylactic acid (PLA) and thermoplastic polyurethane (TPU) have been melt-blended at different volume ratios, resulting in an observation of enhanced shape recovery and fixity rates (Lai and Lan, 2013). However, the elastic modulus of SMPs decreases with increasing temperatures as the polymer become rubbery above its glass transition temperature; the exact opposite trend is observed in SMAs (Nguyen, 2013). SMPs also exhibit relatively slow response time compared to SMAs; this is mainly caused by its relatively low thermal conductivity, and the lower stresses produced during recovery (Liu et al., 2007; Mather et al., 2009). Another interesting characteristic of SMPs is its viscoelastic behavior observed during thermo-mechanical cycles (Nguyen, 2013).

In order to combine and utilize the highly thermally dependent shape memory properties of SMAs and SMPs, shape memory composites (SMCs) containing SMAs and SMPs in various geometric configurations have been developed as intelligent structural systems with enhanced shape memory performance (Feng et al., 2013; Tobushi et al., 2006, 2010). Some distinct advantages of SMCs reported in the literature are (1) the SMP matrix can be designed to either passively accommodate the elastic strain imposed by SMAs or actively alter the effective properties of the SMC, (2) improved load bearing properties is observed at low temperatures since the SMP matrix reinforces the yield strength and the elastic modulus of SMA, (3) the recovery force of the SMC can be increased by approximately 70% compared to SMPs, and (4) close to 100% shape fixity and shape recovery rates can be achieved (Feng et al., 2013; Lexcellent, 2013; Tobushi et al., 2006).

Since the thermo-mechanical behavior of SMCs does not follow a linear trend, modeling efforts on the experimental performance of SMC systems and devices have lagged behind. In previous SMC constitutive models, various homogenization schemes have been applied to SMAs with static epoxy-based polymers, or SMPs with non-adaptive reinforcement fillers (Herzog and Jacquet, 2006; Jarali et al., 2008; Jiménez and Pellegrino, 2012; Tan et al., 2014). The shortcoming of these models is that both the static epoxy-based polymer and the non-adaptive reinforcement fillers weaken the overall SME of the system, and the inelastic strains of the SMA and SMP on the SMC systems were not adequately addressed. In another study, the rule of mixtures and Eshelby’s inclusion method have been used to homogenize the SMA and SMP to obtain the homogenized properties of SMC (Jarali et al., 2010). However, in this model, the temperatures of the SMA and SMP were modeled at extreme cases which are not achievable in practical experimental conditions. Therefore, further experimental validation of this model is necessary to prove its validity.

IIn this study, we present a constitutive model of SMC, describing its thermo-mechanical behavior by combining existing constitutive SMA and SMP models with the self-consistent homogenization scheme. The SMC was modeled in the geometry of a rectangular SMP matrix containing cylindrical SMA fibers uniformly distributed along the axial (or loading) direction. We also present a novel fabrication process for the SMC structure containing 0.5 and 1 vol% SMA fibers embedded in a biocompatible SMP matrix. Finally, the proposed constitutive SMC model was validated with the experimental data obtained from this study and good agreement was observed between the model and the experimental results. Prior to constructing the constitutive SMC model, we first review the existing constitutive models of SMA and SMP separately and validate them with the SMA and SMP materials chosen for the SMC structure in this study. The SMA model was validated with SMA fibers (Flexinol^¯) with a diameter of 0.25 mm, and the SMP model was validated with a biocompatible 50/50 TPU/PLA SMP blend developed in-house.

Review of existing SMA and SMP constitutive models

SMA constitutive models

The low-temperature martensite phase in NiTi-based SMAs is formed through the process of twining, which is associated with SME (Otsuka and Wayman, 1998a). Since martensite has a low degree of symmetry, a number of crystallographic variants can be formed depending on the direction of loading. SMAs utilized for practical applications are anisotropic polycrystalline materials which have been processed and trained in order to exhibit improved shape memory capacities. For instance, SMA fibers have been developed to exhibit a large recoverable elongation along the [111]_B2 fiber axis (Saburi, 1998). Due to its simple geometry, SMAs have been most widely studied in the wire (or fiber) geometry. One-dimensional (1D) macroscopic constitutive models have been developed to replicate the nonlinear mechanical response of SMA fibers under evaluated temperatures and loading conditions (Bernardini and Pence, 2002; Sittner et al., 2000; Zak et al., 2003). The difference between the existing models lies in the description of the phenomenon of hysteresis (such as theory of plasticity, phenomena of instability and rheology) or in the investigation of internal variables which are related to the fraction of martensite variants (Lexcellent, 2013).

The constitutive SMA model utilized in this study was first developed by Tanaka and Liang, and later modified by (Brinson, 1993; Brinson and Huang, 1996; Liang and Rogers, 1990; Tanaka and Iwasaki, 1985; Tanaka and Nagaki, 1982). It is formulated based on the constitutive law from thermodynamics and assumes that the stress in the SMA is correlated with three components: the transformational stress, the thermal stress, and the elastic stress. The model relies on eight basic material parameters: the elastic modulus of austenite (E_A ) and martensite (E_M ), the maximum transformation strain (ε_l), the martensite start (M_s ) and finish temperature (M_f ), the austenite start (A_s ) and finish temperature (A_f ), the transformation enthalpy of austenite (C_A ) and martensite (C_M ), the reorientation start stress ( $σ_{s}^{cr}$ ) and finish stress ( $σ_{f}^{cr}$ ), and the thermal expansion coefficient (α^f).

The thermo-mechanical behavior of SMA is described by the following constitutive equation which relates the stress (σ^f) to the strain (ε^f) and temperature (T) (Brinson, 1993)

σ^{f} - σ_{0}^{f} = (E (ξ) ε^{f} - E (ξ_{0}) ε_{0}^{f}) + (- E (ξ) ε_{l} ξ_{s} + E (ξ_{0}) ε_{l} ξ_{s 0}) + E (ξ) α^{f} (T - T_{0})

(1)

where E(ξ) is the elastic modulus of the material, and ξ is the martensite volume fraction, which is a function of the volume fraction of the thermally induced (twinned) martensite variant ξ_t and stress-induced martensite (detwinned) variant ξ_s. ξ_t and ξ_s are defined as

E (ξ) = E_{A} + ξ (E_{M} - E_{A})

(2)

ξ = ξ_{s} + ξ_{t}

(3)

In the above constitutive equations, the subscript “0” suggests the previous state prior to the current loading path. This means that the material properties and the loading conditions need to be solved iteratively during the computation. Thus, equation (1) can be further simplified to (Liang and Rogers, 1990)

σ^{f} = E (ξ) (ε^{f} - ε_{l} ξ_{s}) + E (ξ) α^{f} (T - T_{0})

(4)

The decision regarding the martensite fraction ξ can be made based on the transformation kinetics in a sinusoidal form, which was proposed by Liang and Rogers (1990). The conversion from twinned martensite or austenite to detwinned martensite is a series function of stress and temperature; these equations are reviewed as follows (Liang and Rogers, 1990)

For $T > M_{s}$ and $[σ_{s}^{cr} + C_{M} (T - M_{s})] < σ^{f} < [σ_{f}^{cr} + C_{M} (T - M_{s})]$

ξ_{s} = \frac{1 - ξ_{S 0}}{2} \cos {\frac{π}{σ_{s}^{cr} - σ_{f}^{cr}} [σ^{f} - σ_{f}^{cr} - C_{M} (T - M_{s})]} + (\frac{1 + ξ_{S 0}}{2})

(5a)

ξ_{T} = ξ_{T 0} - (\frac{ξ_{T 0}}{1 - ξ_{S 0}}) (ξ_{S} - ξ_{S 0})

(5b)

For $T < M_{s}$ and $σ_{S}^{cr} < σ^{f} < σ_{f}^{cr}$

ξ_{S} = \frac{1 - ξ_{S 0}}{2} \cos {\frac{π}{σ_{s}^{cr} - σ_{f}^{cr}} (σ^{f} - σ_{f}^{cr})} + \frac{1 + ξ_{S 0}}{2}

(5c)

ξ_{T} = ξ_{T 0} - \frac{ξ_{T 0}}{1 - ξ_{S 0}} (ξ_{S} - ξ_{S 0}) + Δ_{T ξ}

(5d)

where if $M_{f} < T < M_{s}$ and $T < T_{0}$

Δ_{T ξ} = \frac{1 - ξ_{T 0}}{2} {\cos [a_{M} (T - M_{f})] + 1}

(5e)

else, $Δ_{T ξ} = 0$

The reverse phase transformation kinetics equation is proposed as

For $T > A_{s}$ and $[C_{A} (T - A_{f})] < σ^{f} < [C_{A} (T - A_{s})]$

ξ = \frac{ξ_{S}}{2} {\cos [a_{A} (T - A_{s} - \frac{σ}{C_{A}})] + 1}

(6)

When SMA is subjected to a loading condition under M_f , twinned martensite starts to transform to detwinned martensite, equation (5) should be employed since the SME has primarily been utilized. In the high-temperature hysteretic response, the starting austenite phase is first transformed to detwinned martensite and the reverse transformation takes place in the unloading process. Thus, a combination of equations (5) and (6) should be used to determine the volume fraction of martensite.

SMP constitutive models

The current constitutive models of SMP can be organized into two categories in terms of the shape memory mechanisms: the glass transition mechanism of amorphous networks and the crystallization and melt transition mechanism of semicrystalline polymers (Nguyen, 2013). In this study, a thermally activated amorphous SMP blend was utilized as the matrix of the SMC. This material was chosen due to its relatively high stiffness and strength, its biocompatible properties, and its strain compatibility with the SMA fiber. Since this SMP is mostly amorphous, and the T_trans of this material is the same as its T_g , therefore, a model based on the glass transition mechanism of the amorphous networks was utilized for this study.

One of the most widely applied models using the glass transition mechanism is proposed by Liu et al. (2006). The model utilizes a phase transition approach to describe the phenomenon in which the imposed strain can be stored in a non-equilibrium state of glassy phase (also described as the “frozen phase”) during cooling as the mobility of the polymer chains decreases exponentially with decreasing temperature. The “frozen phase” starts to transform to the “active phase” when the temperature rises above T_g , to a point where the volume fraction of the “active phase” is 100%. In Liu’s model, the viscoelastic behavior of SMPs can be eliminated by fixing the heating/cooling rate when the strain imposed on the SMC is less than 10%. This model was utilized for this study since the strain involved in the SMCs in this study was 10% or less. However, it should be noted that once the imposed strain exceeds 10%, the predictive power of this model on the viscoelasticity behavior of SMPs becomes limited (Liu et al., 2006).

Four material parameters were essential in solving this model: the volume fraction of the frozen phase (ϕ_f), the modulus of the internal energetic deformation (E_i ), the modulus of the entropic deformation (E_e ), and the thermal expansion coefficient (α^m). Similar to the SMA model, the constitutive model for SMP divides the imposed stress into the elastic component, the stored component, and the thermal component and ties them together in terms of Hook’s law

σ^{m} = E^{m} (ε^{m} - ε_{S}^{m} - \int_{T_{h}}^{T} α^{m} dT) = \frac{ε^{m} - ε_{S}^{m} - \int_{T_{h}}^{T} α^{m} dT}{\frac{ϕ_{f}}{E_{i}} + \frac{1 - ϕ_{f}}{E_{e}}}

(7)

The stored strain $ε_{S}^{m}$ can be solved through the temperature derivative expression

\frac{d ε_{S}^{m}}{dT} = \frac{ε^{m} - ε_{S}^{m} \int_{T_{h}}^{T} α^{m} dT}{E_{e} (\frac{ϕ_{f}}{E_{i}} + \frac{1 - ϕ_{f}}{E_{e}})} (\frac{d ϕ_{f}}{dT})

(8)

where E_i was treated as a constant and can be measured at temperatures well below T_g .

Based on the rubbery elasticity of a network polymer, E_e was defined as a linear function of the temperature, the cross-linking density N, and Boltzmann’s constant k

E_{e} = 3 NkT

(9)

The frozen fraction ϕ^f can be expressed in terms of the temperature and two fitting parameters, c_f and n

ϕ_{f} = 1 - \frac{1}{1 + c_{f} {(T_{h} - T)}^{n}}

(10)

The temperature range utilized must be selected carefully based on a series of mechanical tests before applying the model; the range must cover the glass transition region between the glassy and the rubbery state.

Self-consistent homogenization technique of SMA and SMP

The idea of implementing a homogenization scheme to a composition structure containing heterogeneous components is to average the effective properties of the system as a whole by assuming each independent constituent to be idealized, homogeneous, and continuous at the microscopic level. In other words, the homogenized properties of a SMC, such as modulus, can be utilized to predict the response of the SMC in constitutive models. The goal of this study is to model the thermo-mechanical behavior of the SMC containing SMA wires embedded in a SMP matrix. The two-phase SMC is modeled as a transversely isotropic material with the plane of isotropy normal to the fiber direction (Figure 1). The self-consistent scheme developed by Pindera (2013) has been utilized; it is a modification of Eshelby’s equivalent inclusion method, specifically aimed at solving the problem of a two-phase composite with uniform dispersion of cylinders in a matrix. The elastic stiffness matrix in the Cartesian plane can be expressed in equation (11)

C = [\begin{matrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\ C_{12} & C_{23} & C_{22} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{2} (C_{22} - C_{23}) & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{66} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{matrix}]

(11)

Here the stiffness tensors are related to six engineering moduli, five of which are independent: the effective elastic modulus in the <100> direction ( $E_{100}^{C}$ ), Poisson’s ratio ( $ν_{100}^{C}$ ), plane strain bulk modulus (k), the transverse shear modulus ( $μ_{T}$ ), and the axial shear modulus ( $μ_{A}$ )

C_{11} = E_{100}^{C} + 4 K {(ν_{100}^{C})}^{2}

(12a)

C_{12} = 2 K ν_{100}^{C}

(12b)

C_{22} = k + μ_{T}

(12c)

C_{23} = k - μ_{T}

(12d)

C_{66} = μ_{A}

(12e)

The function of the effective elastic modulus is listed as follows

E_{100}^{C} = c^{m} E^{m} + c^{f} E^{f} + \frac{4 {(v^{m} - v^{f})}^{2}}{{(\frac{1}{k^{m}} + \frac{1}{k^{f}})}^{2}} [(\frac{c^{f}}{k^{m}} + \frac{c^{m}}{k^{f}}) - \frac{1}{k}]

(13)

where the superscripts c, m, and f denote the composite, matrix, and fiber, respectively. The variable c represents the volume fraction of each component in the composite. The plane strain bulk modulus of SMC is defined as

k = k^{m} + c^{f} (k^{f} - k^{m}) \frac{k + μ_{T}}{k^{f} + μ_{T}}

(14)

Using the homogenized compliance matrix in terms of the phase compliances and the stress concentration matrices, the homogenized transverse shear modulus in equation (15) yields

\frac{1}{μ_{T}} = \frac{1}{μ^{m}} + c^{f} (\frac{1}{μ^{f}} - \frac{1}{μ^{m}}) [\frac{1 + (\frac{1 - b}{b})}{1 + \frac{μ_{T}}{μ^{f}} (\frac{1 - b}{b})}]

(15)

where b is expressed as

b = \frac{k + 2 μ_{T}}{2 (k + μ_{T})}

(16)

In equations (13) and (15), the bulk modulus and shear modulus of the SMA and SMP phases can be described by considering each of them as homogeneous linear elastic materials. The SMP matrix is isotropic, and the SMA fibers are isotropic along the axis of deformation. These assumptions match the material properties of SMA and SMP in the model setup. Taking the matrix as an example, the expressions for bulk and shear moduli are

k^{m} = \frac{E^{m}}{3 (1 - 2 v^{m})}

(17a)

μ^{m} = \frac{E^{m}}{2 (1 + v^{m})}

(17b)

The homogenized elastic modulus of the composite in the [100] direction can be determined by substituting equations (14)–(18) into equation (13).

Proposed constitutive model of SMC

Assuming that the fiber-reinforced SMC with uniform dispersion of continuous SMA fibers is subjected to a uniaxial loading along the fiber direction similar to the SMA and SMP models, the general constitutive equation of SMC can be expressed through Hook’s law

σ^{C} = E_{100}^{C} ε_{elastic}^{C} = E_{100}^{C} (ε^{C} - ε_{inelastic}^{C})

(18)

The inelastic strain component $ε_{inelastic}^{C}$ of SMC consists of three terms: (1) the effect of the transformation strain in SMA on the composite strain ( $ε_{tr}^{C}$ ), (2) the effect of the stored strain in SMP on the composite strain ( $ε_{s}^{C}$ ), and (3) the effect of the thermal strain resulting from both SMA and SMP on the composite strain ( $ε_{th}^{C}$ ). Equation (18) can be further elaborated as

σ^{C} = E_{100}^{C} (ε^{C} - ε_{tr}^{C} - ε_{s}^{C} - ε_{th}^{C})

(19)

In addition, according to previous experimental data, the thermal strain of the SMA is several orders of magnitude smaller than its transformation strain; therefore, it can be safely removed from equation (18) without compromising the results (Brinson, 1993). Another study found that the thermal strain of the SMP is approximately 1/10 of the stored strain (Liu et al., 2006). Although the contribution of this term is not insignificant, its absence from the equation would still suffice to provide a good approximation of the thermo-mechanical behavior of the SMC; therefore, it has been removed from equation (18) for the purposes of simplifying the model. After eliminating the thermal strain component ( $ε_{th}^{C}$ ) in equation (18), the final constitutive equation is simplified to

σ^{C} = E_{100}^{C} (ε^{C} - ε_{tr}^{C} - ε_{s}^{C})

(20)

Based on equations (4) and (7), the stresses caused by the transformation strain and the stored strain on SMA and SMP are

σ_{tr}^{f} = E^{f} ε_{l} ξ_{s}

(21a)

σ_{s}^{m} = E^{m} ε_{s}

(21b)

As both SMP matrix and SMA fibers are considered to be active components in the system, the effective stress in the composite ( $σ_{eff}^{C}$ ) due to the transformation strain and the stored strain can be obtained as follows

σ_{eff}^{C} = σ_{tr}^{f} c^{f} + σ_{s}^{m} c^{m}

(22)

Substituting equations (21a) and (21b) into equation (22), the total effective inelastic strain ( $ε_{eff}^{C}$ ) in the composite due to the transformation strain and the stored strain yields

ε_{eff}^{C} = ε_{tr}^{C} + ε_{s}^{C} = \frac{E^{f} ε_{l} ξ_{s} c^{f} + E^{m} ε_{s} c^{m}}{E_{100}^{C}}

(23)

Subsequently, with the use of equations (13), (20), and (23), the mechanical properties of SMC can be determined at different temperatures under specific loading conditions. Thus, the uniaxial stress evolution of the SMC can be predicted. In the following sections, thermo-mechanical tests will be performed on fabricated SMCs and the results will be utilized to verify the predictive power of the proposed SMC constitutive model.

Experimental: materials and methods

SMA characterization

The SMA material utilized in this study was Flexinol NiTi wires that were 0.25 mm in diameter (Flexinol; Dynalloy Inc., CA, USA). They will be referred to as SMA fibers in the remainder of this article. Thermo-mechanical characterization and tensile tests were carried out to obtain the material properties of the SMA fibers, according to the ASTM F2516:2007e1 (2007). The tensile tests were performed on the SMA fibers with an AG-I universal tester with an outfitted TCE-N300 environmental chamber (Shimazu Corp., Tokyo, Japan).

The phase transformation temperatures at the stress-free state were first determined by constructing a stress–temperature phase diagram. Three isothermal tensile tests were then carried out using the SMA fibers at 82.5°C, 88°C, and 93°C in order to obtain the high-temperature hysteretic response. The isothermal tests were conducted because the critical stresses for phase transformation of the SMA wires were temperature dependent, and the wires exhibited superelastic behavior at temperatures above 80°C. In each test, the wires were strained to 8% followed by an unloading path. All the test data were recorded and plotted as stress–strain curves by a data acquisition computer. Finally, tensile tests were conducted on the SMA fibers at room temperature to investigate the shape memory properties of the SMA below its M_f temperature, and the results were plotted as stress–strain curves. A constant displacement rate of 0.8 mm/min was utilized to ensure complete phase transformations in the SMA samples.

SMP fabrication and characterization

The SMP matrix is made of a biocompatible SMP blend composed of TPU (Desmopan 385E; Bayer MaterialScience LLC, Pittsburgh, PA, USA) and PLA (3052D; NatureWorks LLC, Minnetonka, MN, USA) in the weight ratio of 50:50. The detailed fabrication procedure of the 50/50 TPU/PLA blend can be found in one of our previous studies (Song et al., 2014). Briefly, the neat TPU and PLA pellets were melt-blended with a micro-compounder (MICRO15; DSM, Geleen, Netherlands) for 10 min at 180°C. The extruded 50/50 TPU/PLA polymer strands were pelletized and compression molded (Hydraulic manual press Compression Molding, Carver 4386; Carver, Inc., Wabash, IN, USA) at 180°C for 10 min into a rectangular shape with the dimensions of length: 60 mm, width: 13 mm, and thickness: 1 mm. The edges of the samples were polished to ensure uniform stretching of the samples during the thermo-mechanical tensile tests.

The 50/50 TPU/PLA SMP blend samples were characterized as follows. The T_g of the 50/50 TPU/PLA blend was obtained with a differential scanning calorimetry (DSC) tester (Q2000; TA Instrument, New Castle, DE, USA). The elastic modulus of the SMP at room temperature (25°C) and 80°C was obtained with tensile tests. These two temperatures were chosen because they were below and above the T_g (60°C −62°C) of the SMP, respectively. The temperature-dependent thermal expansion coefficients were obtained with a thermo-mechanical analyzer (TMA) (Q400; TA Instrument).

The temperature range of 25°C–80°C was selected for the thermo-mechanical characterization of the SMP samples. This temperature range was selected due to the huge change in material’s properties over this temperature since it encompasses the T_g of the SMP. The SMP samples begin as a polymer in its glassy state at 25°C and gradually transform into the rubbery state as the temperature increases to its T_g at 60°C −62°C and continue to soften with further increase in temperature up to 80°C. The thermo-mechanical tests were conducted in the following steps: (1) the initial length (L₀ ) of the sample was measured prior to deformation, (2) the SMP sample was clamped to the tensile tester and the temperature was raised to 80°C and held for 5 min, (3) the sample was stretched to 10% strain, (4) the sample was cooled down to 25°C in the strained condition, (5) the strain was released and the stretched sample length (L₁ ) was measured, (6) the sample was heated up to 80°C in a water bath for 5 min, and (7) the recovered sample length (L₂ ) was measured. The shape fixity rate (R_f ) and shape recovery rate (R_r ) were calculated using the following equations (Xie, 2011)

R_{f} = \frac{(L_{1} - L_{0}) / L_{0}}{ε_{pre}} \times 100 %

(24a)

R_{r} = \frac{ε_{pre} - (L_{2} - L_{0}) / L_{0}}{ε_{pre}} \times 100 %

(24b)

where $ε_{pre}$ is the pre-stretched strain, which is 10% in this case.

The SMP shape recovery process was also carried out over the temperature range of 22°C–80°C in order to obtain the frozen fraction data for the constitutive model. The thermo-mechanical cycle was carried out until step 5, and then the SMP was recovered at 5°C increments in a hot water bath for 2 min, up to 80°C. The length of the sample at each recovery temperature was recorded.

SMC fabrication and characterization

A custom-made stainless steel mold was utilized for the fabrication of the fiber-reinforced SMC containing continuous SMA fibers and SMP matrix. The mold was designed to fabricate two rectangular SMC samples simultaneously with two different fiber volume contents. The dimensions of the samples were 66 mm length, 13 mm width, and 2 mm thickness. The detailed dimensions of the mold are shown in Figure 2.

When fabricating the SMC samples, two rectangular pieces of 50/50 TPU/PLA blend samples with half the thickness of the final sample were fabricated first (dimensions: 66 mm × 13 mm × 1 mm). One piece of the half-thickness SMP sample was placed into the SMC mold as a base layer first and then the SMA fibers were positioned in each groove of the mold. The SMA fibers were held in place by the notches on the mold and they were aligned along the axial (or loading) direction of the SMC samples (see Figure 3). Following that, the second layer of the half-thickness SMP sample was laid on top of the SMA fibers inside the mold. Some SMP powders, which were prepared by grinding the 50/50 TPU/PLA polymer pellets, were placed near the SMA wires in between the two layers of SMP sample to enhance the bonding between the SMP matrix and the SMA fibers. The mold containing the SMC samples was sandwiched between two Teflon sheets and two steel plates, and hot compressed for 10 min at 160°C (Hydraulic manual press Compression Molding, Carver 4386; Carver, Inc.). After the molding process, the SMC was taken out of the compression molder and cooled to room temperature by quenching it in water. The fabricated SMC samples were then taken out of the mold and polished to smooth out any rough edges. The advantage of fabricating two layers of SMP matrix first and sandwiching the SMA fibers in between is that straight wires could be obtained in the SMC samples. SMCs containing 1 and 2 vol% of SMA were fabricated; they correspond to 0.5% and 1% fiber volume content, respectively.

Isothermal tensile tests were carried out on the SMC samples containing 0.5 and 1 vol% SMA fibers at two temperatures: 25°C (which is also room temperature) and 80°C. These two temperatures were chosen in order to gain a better understanding of the mechanical behavior of the SMC in its active and inactive states. At the lower chosen testing temperature (25°C), both the SMA and SMP are in the inactive state, since it is below both the M_f of the SMA and the T_g of the SMP. At the higher chosen temperature (80°C), both the SMA and SMP are in their activated states, since it is above the A_f of the SMA and the T_g of the SMP. At this temperature and below a certain critical stress for slip, the SMA would display its characteristic superelastic behavior (Otsuka and Wayman, 1998b). During the tensile tests, the SMC samples were strained to 8% extension, followed by an unloading process. The corresponding stress–strain relationship was recorded.

Results and discussion

Simulation and experimental results of the SMA fibers

Experimental thermo-mechanical properties

For each isothermal thermo-mechanical test carried out at high temperatures (82.5°C, 88°C, and 93°C), a set of four critical transformation stresses was obtained from the corresponding stress–strain curve by finding the intersection of the linearized trends. The critical transformation stresses were the martensite start $(σ^{Ms})$ and finish $(σ^{Mf})$ stresses and the austenite start $(σ^{As})$ and finish $(σ^{Af})$ stresses. A typical stress–strain curve from these tests is presented in Figure 4. All the critical stresses for transformation obtained from the tensile tests at 82.5°C, 88°C, and 93°C are summarized in Table 1.

Table 1.

Experimental data of the temperature-dependent critical stresses for the phase transformations of the SMA fibers (Flexinol^¯) utilized in this study.

Temperature (°C)	Critical stress (MPa)
	σ^Ms	σ^Mf	σ^As	σ^Af
82.5	348	398	200	149
88	375	425	210	175
93	450	500	280	228

SMA: shape memory alloy.

By plotting the values of the temperature-dependent critical stresses for phase transformation presented in Table 1, a stress–temperature phase diagram can be established (shown in Figure 5). Although the experimental critical stresses do not yield precise linear results, one can still interpret the trend for the transformation stresses as linear functions of temperature (Brinson, 1993). Therefore, the values of the phase transformation temperatures at stress-free states M_s , M_f , A_s , and A_f can be obtained with simple linear extrapolation. The transformation enthalpy of austenite (C_A ) and martensite (C_M ) can be found as the slopes of the linear trend lines since they describe the relationship between temperature and critical stresses. Perfect agreement was found in our study between the C_M values, and a small discrepancy of 0.6% was observed between the C_A values. For the purposes of setting up the constitutive model, an average of the two C_A values was obtained.

In addition, the elastic modulus of austenite (E_A ) can be calculated using the austenite loading curve shown in Figure 4. Other material properties such as the martensite elastic modulus (E_M ), the reorientation start stress ( $σ_{s}^{cr}$ ) and finish stress ( $σ_{f}^{cr}$ ), and the maximum transformation strain (ε_l) can all be gathered from the stress–strain relationship obtained from the process of loading and unloading at room temperature by applying similar strategies. Consequently, all the thermo-mechanical properties of the SMA fibers can be obtained at this point. The results are summarized in Table 2.

Table 2.

Thermo-mechanical properties of the SMA fibers (Flexinol) utilized in this study.

Property	Symbol	Unit	Value
Elastic modulus of martensite	E_M	MPa	21,667
Elastic modulus of austenite	E_A	MPa	47,000
Thermal expansion coefficient of martensite (Song et al., 2014)	α_M	°C^-1	6.6E-06
Thermal expansion coefficient of austenite (Song et al., 2014)	α_A	°C^-1	1.1E-05
Poisson’s ratio (Song et al., 2014)	Γ	–	0.33
Maximum residual strain	ε_L	–	0.07
Martensite finish temperature	M_f	°C	42
Martensite start temperature	M_s	°C	47
Austenite finish temperature	A_f	°C	57
Austenite start temperature	A_s	°C	63
Transformation enthalpy of martensite	C_M	MPa/°C	9.63
Transformation enthalpy of austenite	C_A	MPa/°C	7.5
Reorientation start stress	$σ_{s}^{cr}$	MPa	65
Reorientation finish stress	$σ_{f}^{cr}$	MPa	98

SMA: shape memory alloy.

Data were obtained from experiments unless otherwise noted.

Numerical model validation

By implementing the relative material parameters in Table 2 into the constitutive law given by equation (4), the thermo-mechanical behavior of the SMA fibers can be simulated under different conditions. First, the operating temperature was set to 25°C (which is below M_f ), and equations (5c) to (5e) were used with the constitutive law to examine the low-temperature SME of the SMA fibers. The results obtained from the constitutive model and tensile tests are shown in Figure 6. Next, the superelasticity of the SMA fiber was investigated with the constitutive model by setting the operating temperature to 82.5°C (which is above A_f ). A comparison of the curves obtained from the modeled hysteric response and the experimental results is shown in Figure 7.

It can be seen in Figures 6 and 7 that the constitutive model was able to produce the general trend of the experimental nonlinear thermo-mechanical response of the SMA fibers. At both testing temperatures, the model was better at estimating the plateau stress than the loading regions of the stress–strain curve. A larger discrepancy was observed between the experimental and modeling results in these regions. The unloading curve is not observed in Figure 6 because the pseudoelastic behavior is dependent on the test temperature (Saburi, 1998). According to the literature, pseudoelastic behavior is not observed at temperatures below M_f but is fully apparent at temperatures above A_f (Saburi, 1998). In this study, the pseudoelastic behavior was not observed in samples tested at 25°C which is below the M_f of the SMA, see Figure 6. For samples tested at 80°C which is above the A_f of the SMA in this study, full pseudoelastic behavior was not observed; the SMA fibers recovered to approximately 1% strain, see Figure 7. This means that some permanent deformation occurred in the SMA fibers, and it did not achieve full recovery when strained to 8% extension. The authors chose to utilize 8% strain for the thermo-mechanical tests in this study because 10% strain is commonly utilized for SMA wires in the literature, and the manufacturer of the SMA fibers states that recovery upon 8% extension is achievable, although 4%−5% strain is recommended for a large number of repeated cycles (Brinson, 1993; Dynalloy, Inc., 2014; Liang and Rogers, 1990). Another reason was to increase the amount of observable deformation in the SMA fibers since the dimensions of the samples are very small.

The results obtained from the proposed constitutive model of SMA are closely dependent on the previously determined material parameters. The curves obtained from the experiments were not ideal since there is a noticeable amount of noise and fluctuation, which is a result of the limitations of the experimental setup. Due to their small diameter (0.25 mm), the SMA fibers were very sensitive to the change in temperature and air flow of the surrounding environment. During the high-temperature tests at 80°C, the temperature inside the environmental chamber was fluctuating due to the convection-based cooling and heating system. Despite the errors between the results obtained from the model and experimental results, the coefficients of determination (R²) of these results in Figures 6 and 7 were calculated as 0.81 and 0.73, respectively. These findings indicate fairly good fitting between the experimental and simulated data. In summary, the constitutive model of SMA was validated with experimental results, and it will be utilized for the derivation of the SMC model in this study.

Simulation and experimental results of the SMP matrix

Fitting parameters and material properties determination

In order to derive the temperature-dependent frozen fraction (Φ_f) of the SMP material, the fitting parameters c_f and n need to be calculated. Liu et al. (2006) suggested that the two major sources that contribute to the measured recovery strain during the heating process are the thermal strain ( $ε_{T}$ ) and the actual recovery strain which comes from the release of the stored strain. As a result, the thermal strain needs to be subtracted from the macroscopic recovery strain in order to obtain the actual recovery strain. The following equation was proposed

ε_{actual} = \frac{ε_{pre}}{ε_{measured} (25^{\circ} C) - ε_{T} (25^{\circ} C)} (ε_{measured} - ε_{T})

(25)

The frozen fraction in equation (10) can also be defined as

ϕ_{f} = \frac{ε_{actual}}{ε_{pre}}

(26)

Thus, the values of the frozen fraction at each recovery point can be determined with equation (10) or (26) using experimentally determined fitting parameters. The experimental values of the frozen fraction were obtained from shape recovery tests conducted in the temperature range of 22°C–80°C, at 5°C increments. The two sets of $ϕ_{f}$ values are graphed in Figure 8, and the values of c_f and n are adjusted to obtain the best fit between the data sets. A high degree of fitting (R² = 0.99) was obtained between the experimental frozen fraction values and the calculated values with c_f = 2.6 × 10^–10 and n = 6.16.

The elastic modulus of the 50/50 TPU/PLA blend was measured at both 25°C (glassy state) and 80°C (rubbery state). At 80°C, the elastic modulus is considered at the extreme case where there is no frozen phase inside the material. Since E_e is equal to E (at T = 80°C), the cross-linking density can be calculated with the use of equation (9), and the elastic modulus of the material can be obtained based on the given temperature. All the fitting parameters and the material properties of the 50/50 TPU/PLA sample were determined and are shown in Table 3.

Table 3.

Fitting parameters and material properties of the 50/50 TPU/PLA SMP blend utilized in this study.

Property	Symbol	Unit	Value
Glass transition temperature	T_g	°C	60–62
Elastic modulus at 25°C (RT)	E	MPa	600.05
Elastic modulus at 80°C	E	MPa	5.56
Thermal expansion coefficients below T_g	α^m _l	/°C	1.24E-04
Thermal expansion coefficients above T_g	α^m _h	/°C	2.16E-04
Fitting parameters	c_f	1/K⁴	2.62E-10
	n	–	6.16
Cross-linking density	N	mol/m³	614.1
Shape fixity rate at 25°C	R_f	%	19.06
Shape fixity rate at 80°C	R_f	%	99.42
Shape recovery rate at 25°C	R_r	%	100.00
Shape recovery rate at 80°C	R_r	%	98.03

TPU: thermoplastic polyurethane; PLA: polylactic acid; SMP: shape memory polymer; RT: room temperature.

Numerical model validation

The shape memory thermo-mechanical cycle of SMP can be described in four consecutive steps: (1) the tensile pre-strain in the rubbery state, (2) the strain storage, (3) the low-temperature unloading state, and (4) the free strain recovery. Using the constitutive law of SMP in equation (7) along with the 50/50 TPU/PLA blend material parameters, a MATLAB program was created to reproduce these four steps. The starting temperature was first set to 80°C, which was also used as the reference temperature, and then the stress evolution was determined iteratively as the strain was increased gradually to 10%. After that, equation (8) was applied to calculate the stored strain as the temperature dropped to 25°C while the material was kept under the 10% pre-strain constraint. The stored strain value calculated at each temperature point in the strain storage step was also used in the shape recovery step. The entire simulated thermo-mechanical cycle of the 50/50 TPU/PLA blend is presented in Figure 9.

The simulated stress–strain relationship and the experimental results in the pre-strain and strain storage stage are shown in Figure 10; a close fit of R² = 0.94 was obtained. However, in the experimental process, the time effect introduced by the viscoelasticity of the material could not be eliminated by simply establishing a constant heating/cooling rate as in the ideal case. Consequently, the experimental stress–strain response displays a parabolic shape rather than the modeled linear relationship. Since the material contracted with decreasing temperature in the pre-strain constraint, tensile thermal stresses were generated and built up exponentially once the temperature exceeded T_g . As a result, the thermal stresses created a positive contribution to the total stress during the process and the cooling curve appeared as a vertical line in the stress–strain space.

Additionally, the free strain recovery of the residual strain during heating of the sample by the model prediction was compared with experimental results, see Figure 11. The two sets of data were in close agreement, with a starting strain of 0.093 and 0.095, respectively. This implies that after unloading, the spring back effect took place as the mobility of the molecular chains was minimized and a quasi-equilibrium was reached in the material. At this point, most of the tension strain was locked in the glassy state. It should be noted that only a small change in strain was observed at the beginning and near the end of the process, suggesting that the selected testing temperature range was sufficient to describe the shape memory behavior of the sample matrix.

Experimentally, most of the shape recovery occurred at just below T_g over a temperature span of less than 10°C, where a relatively sharp transition can be observed (Figure 11). On the other hand, the model predicted a more gradual transition over the shape recovery temperature (Figure 11). This is most likely due to the difference in molecular mechanisms of the SME underlying the SMP experimentally and the model assumptions. Since R² for the two curves was 0.91, it can be concluded that the model is capable of reproducing the characteristic thermo-mechanical response of the 50/50 TPU/PLA blend within acceptable tolerance. From the modeled data, a slight compression state was observed after the material exceeded 100% recovery, indicating that inaccuracy exists in the thermal coefficient measurements in the rubbery state of the SMP.

Experimental and simulation results of the SMC

The fabricated SMCs containing 0.5% and 1% fiber volume contents are shown in Figure 12. It can be seen that the SMA fibers were embedded uniformly in the SMP matrix. Good bonding between the SMA and SMP was achieved with the fabrication technique, and no delamination was observed in samples in the first three thermo-mechanical cycles. In the “sandwich” fabrication process utilized in this study, the SMA fibers were able to contract freely between the two pieces of SMP polymer matrix when heated to 160°C during the compression molding process. Since 160°C is much higher than A_f , the SMA fibers would remain in its parent austenite phase, which is the contracted and activated state of the Flexinol wires. Upon cooling of the SMC, the ordered austenite phase in the SMA fibers would become the twinned martensite phase, and that would be the state of the SMA fibers in the SMCs post-fabrication. The SMA fibers were kept straight by the mold throughout the fabrication process, and no external stress was applied to the wires, in order to eliminate pre-deformation and improve the accuracy of the thermo-mechanical tests of the SMCs.

By implementing the material parameters of the SMA fiber and the SMP matrix from Tables 2 and 3 into equation (13), the homogenized elastic modulus in the fiber direction was computed. The calculated modulus was then utilized in the proposed constitutive law given by equation (20) to model the mechanical response of the SMC at different temperatures. A MATLAB program was written to simulate the stress–strain behavior of the SMC.

The shape memory properties of the SMC were first investigated at 25°C (Figure 13). This test temperature is below the T_g of the SMP and M_f of the SMA, the SMA fibers are in the twinned martensite phase, and the SMP matrix is in the passive state. A gradually increasing load was imposed on the SMC followed by an unloading process. The corresponding strain was computed and the stress–strain curve was obtained with the constitutive model. Experimentally, isothermal tensile tests were performed on the fabricated SMC samples containing 0.5% and 1% fiber volume fractions at 25°C. At this temperature, the effective properties of the SMC are dominated by the SMP since it is present in a much larger volume, and it is in the glassy state.

The constitutive model of SMC reproduces the range of the stress in the SMCs very well, within 3.3% error (Figure 13). The predictive power of the constitutive model was further validated with statistical analysis measures, the correlation coefficient (ρ), the root mean square error (RMSE), and the coefficient of determination (R²), using MATLAB (see Table 4). The ρ values were very high for all the loading and unloading conditions at this test temperature (exceeding 90%), demonstrating a very high degree of association between the experimental data and those predicted by the constitutive model. The R² values obtained were excellent for the unloading curves, but it was not ideal for the loading curves due to the large viscoelastic effects of the SMP. Since the constitutive model utilizes a linearized approach to simulate the mechanical response of the SMC, the sinusoidal shape in the experimental stress–strain curve was not reproduced by the model. The RMSE values were within the acceptable range, with less errors present in the unloading curve. Due to the complexity of the constitutive model presented in this study, the degree of error obtained is within the acceptable range of obtaining a reasonable model (Jarali et al., 2010; Liu et al., 2006; Nguyen, 2013). Taken together with the high correlation coefficient, it can be concluded that the proposed constitutive model provides a reasonable prediction of the experimental results at 25°C.

Table 4.

Correlation coefficient (ρ), the root mean square error (RMSE), and the coefficient of determination (R²) obtained from the constitutive model and experimental data from the loading and unloading curves of SMCs containing 1 and 0.5 vol% fiber content, at 25°C and 80°C.

	25°C				80°C
	1 vol% fiber		0.5 vol% fiber		1 vol% fiber		0.5 vol% fiber
	Loading	Unloading	Loading	Unloading	Loading	Unloading	Loading	Unloading
ρ	0.9478	0.9894	0.9198	0.9915	0.9299	0.8962	0.9587	0.9497
RMSE	4.1756	2.1036	2.0862	0.8243	0.5524	0.5344	0.2142	0.3803
R²	0.2957	0.8782	0.283	0.8809	0.8014	0.6784	0.8776	0.6977

SMC: shape memory composite.

The residual deformation observed in the unloading curve of the SMC could be due to the low fixity rate of the SMP at 25°C (see Table 3). These results agree well with a previous study which also observed a relatively large amount of residual deformation on a SMC with similar geometries tested at low temperatures (i.e. T_g = 25°C) (Tobushi et al., 2006). That same study also found that the residual strain decreased significantly when the deformation temperature was raised to T_g or higher (Tobushi et al., 2006); a similar trend is observed in our study where the residual strain decreased to less than 2%, see Figure 14. These results further confirm the validity of the constitutive SMC model developed in this study. In addition, a slope transition is found in the modeled stress–strain relationships, which implies the onset of the transformation of the martensite variants (i.e. from twinned to detwinned) in the SMA fibers (Saburi, 1998).

The load bearing capabilities of SMC at low temperatures were increased by more than 10-fold (from 9.02 to 95.12 N) with the addition of 0.5% volume fiber reinforcement. The force was further increased to 106.86 N in SMCs containing 1% volume fiber fraction. These results demonstrate the improved tensile properties of the SMC with the addition of SMA fibers.

The pseudoelastic behavior of the SMC at 80°C containing 0.5 and 1 vol% fiber is shown in Figure 14. Since this testing temperature is above the T_g of the SMP and A_f of the SMA, both the SMA fibers and SMP matrix are in the active phase during these tests. The interaction between the molecular chains was considered negligible in the SMC model due to the high conformational entropy of the polymer chains in the rubbery state. And an increase in the polymer chain mobility and a decrease in elastic modulus were observed. As a result, the SMA dominated the shape memory behavior at this temperature despite the small fiber content, and the elastic modulus of the SMA fiber was close to 4000 times that of the SMP. The stress–strain relationship exhibited by the SMC was similar to the SMA fibers obtained at high temperatures, especially at 1% fiber volume content, compare Figure 14 with Figure 7.

At 80°C, the high degree of association between the experimental and modeling data was again demonstrated by very high ρ values observed for the loading and unloading curves of the SMCs containing 1 and 0.5 vol% fiber content. The RMSE values obtained at this temperature were excellent since they were less than 0.6 for SMCs containing 1 vol% fiber and less than 0.4 for SMCs containing 0.5 vol% fiber. The error predicted with RMSE is much lower at this deformation temperature compared to 25°C. The R² values of the loading and unloading curves were higher than 0.8 and 0.67, respectively, demonstrating a good fit between the experimental and modeling data. Taken together, these statistical analysis results show that the proposed constitutive model predicted the loading responses of the SMC with fairly high accuracy, since a high correlation and small errors were observed. Furthermore, the SMC constitutive model proposed in this study yields a better prediction of the range of stress and strain of the experimental results compared to a previously developed constitutive SMC model, which was only able to predict the loading and unloading responses in the parent phase with relatively small errors but produced large errors in the product phase (Jarali et al., 2010).

Despite the limitations of the proposed constitutive SMC model, it can be concluded that it is able to simulate the general trend of the stress–strain behavior of the SMC containing 0.5% and 1% fiber volume content at both 25°C and 80°C. Future studies may expand on the proposed SMC model to take into account the nonlinear (or viscoelastic) behavior of the SMP component.

The response (or recovery) time of the SMCs developed in this study was approximately 4–7 s, which is similar to the recovery time of the SMP. Although the recovery time of the NiTi SMA fiber is less than 1 s for the tested length, it was not able to recover instantaneously as part of the SMC since it was embedded in a heat and electrically insulating SMP matrix. During the shape recovery process, the SMP is exposed to the heat source first and starts to recover. After a few seconds, the heat source reaches the SMA and it starts to recover. Therefore, the recovery time of the SMC is similar to the SMP.

The SMC actuators developed in this study are made with a biocompatible SMP matrix and can be utilized for biomedical applications. Although the NiTi SMA wires are not biocompatible by nature, they are completely embedded in the SMP matrix and not exposed to the surrounding environment. It should be noted that the T_g of the SMP matrix and the A_f of the SMA fibers were very close to each other (see Tables 2 and 3). This allows for the design of a SMC with enhanced shape recovery force, since both the SMA and SMP would contract at the same temperature. Another possible development of the SMC is to program the T_g of the SMP to be in the range of the M_f of the SMA fibers in order to achieve a two-way SME in the SMC. The 1D geometry of the SMC in this study can be developed into more complex structures in the future for applications such as surgical tools and artificial muscles.

Conclusion

The constitutive laws of SMA and SMP were studied separately and validated experimentally with Flexinol wires and 50/50 TPU/PLA SMP blend, respectively. A linear constitutive model of SMC was developed based on the individual SMA and SMP constitutive models using the self-consistent homogenization approach. SMC composites containing 0.5 and 1 vol% SMA fibers embedded in a SMP matrix along the loading direction were successfully fabricated with a custom-made compression mold. The proposed constitutive SMC model was then validated with the results from thermo-mechanical tests conducted on the SMCs at both high (80°C) and low (25°C) temperatures. The SMC model was able to predict with high accuracy the range of the stress of the SMC at low temperatures, and the plateau stress of the SMC at high temperatures. The SMA had the predominant effect on the shape memory properties of the SMC at high temperatures.

In conclusion, the proposed thermo-mechanical model was able to simulate the general trend of the thermo-mechanical responses of the SMC with reasonable tolerance. The proposed SMC model will be further developed in future studies to take into account the viscoelastic behavior of the SMP.

Footnotes

Acknowledgements

The authors would like to thank Farooq Al Jahwari and Vincent (Min Wen) Liu for their help with the modeling and design components, and Huntley H. Chang for his help with experiments involving SMP recovery.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the following agencies: Natural Sciences and Engineering Research Council (NSERC) of Canada, the Canada Research Chairs Program, and the Canada Foundation of Innovation.

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