Active origami by 4D printing

We fabricate PAC hinges by creating computer aided design (CAD) files that specify the complete 3D architecture of the fibers and matrix then printing them using a multi-material polymer 3D printer (Objet 260 Connex, Stratasys, Edina, MN, USA). The layer-by-layer printing process works by depositing droplets of polymer ink onto the building platform, wiping them into a smooth film, and ultraviolet (UV) photopolymerizing the film. Once a layer is created, the platform moves down, and the next layer is printed. Several inkjet heads with separate material sources exist in the printing block, so multiple materials can be printed in each layer. In our work, each layer generally contains materials that constitute part of the matrix and part of the fibers. In addition, a hydrophilic gel is printed and used as a sacrificial material for the fabrication of complex geometries (Stiltner et al 2011).

Our strategy to create PAC hinges consisting of composites with a matrix that is elastomeric over our desired operating temperature range of between room temperature and about 100 °C and fibers that exhibit the shape memory effect (SME) over this temperature range. Thus we design our PACs to have a matrix with a glass transition temperature $\left( {{T}_{g}} \right)$ below 25 °C and fibers exhibiting SME in the range 25 °C–70 °C. To this end, we make use of the digital materials that are available with an Objet 3D printer. It provides two base materials: one is Tangoblack, a rubbery material at room temperature polymerized with a material ink containing urethane acrylate oligomer, Exo-1,7,7-trimethylbicyclo [2.2.1] hept-2-yl acrylate, methacrylate oligomer, polyurethane resin, and photo initiator; the other is Verowhite, a rigid plastic at room temperature polymerized with a material ink containing isobornyl acrylate, acrylic monomer, urethane acrylate, epoxy acrylate, acrylic monomer, acrylic oligomer, and photo initiator. The printer can also print digital materials that consist of varying compositions of these two materials that lead to different thermomechanical properties. In the current printing system, although users can tune thermomechanical properties of printed materials by choosing limited numbers of digital materials, we believe with the development of 3D printing technique, users will have more freedom of material choices. In this paper, we created PACs consisting of Tangoblack as the matrix (${{T}_{g}}$ ∼ −5 °C) and a digital material (termed Gray 60) with ${{T}_{g}}$ ∼ 47 °C.

We created PAC hinges to characterize experimentally by directly printing two-layer PAC laminates that are connected to inactive (rigid) panels. These panels can be used as end tabs to apply mechanical loads (figure 1(a)). The PAC laminates consist of two layers: one layer of matrix-only material and one layer of a PAC lamina with a prescribed fiber size and spacing (figure 1(a)). The composite architecture is characterized by the lamina thicknesses and volume fraction (determined from the size and spacing).

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Our hinges function via a mechanism of programmed strain mismatch (eigenstrain) between the two layers that leads to constant curvature bending over the hinge region, resulting in the plates on each side rotating an angle of $\theta$ with respect to each other (figure 1(b)). The strain mismatch is created by: i) stretching the hinge at an elevated programming temperature (${{T}_{H}}$, ${{T}_{H}}$ > ${{T}_{g}}$) to a prescribed strain (${{\varepsilon }_{0}}$), ii) cooling it to the usage temperature (${{T}_{L}}$, ${{T}_{L}}$ < ${{T}_{g}}$) while maintaining the strain ${{\varepsilon }_{0}}$, and then iii) releasing the load. Upon releasing the mechanical constraint, the hinge bends to an angle $\theta$ due to the combined effect of the entropic elasticity of the pure matrix material lamina and the shape memory effect of the PAC lamina (Ge Qi et al 2013). The hinge returns to its original flat shape after heating back to ${{T}_{H}}$.

We characterize the hinge performance by its bending angle $\theta$, which depends on the hinge materials (matrix and fiber thermomechanical constitutive behaviors), geometric parameters (hinge length L, and laminate/lamina configuration), and programming parameters (${{\varepsilon }_{0}}$, ${{T}_{H}}$, and ${{T}_{L}}$) (figure 1). In general our programming parameters are chosen so that rate effects do not play a role. In order to investigate the effects of these factors on hinge angle, we designed, fabricated, and carried out tests for a range of parameters including five different laminate/lamina configurations, three different programmed deformations (${{\varepsilon }_{0}}$ ∼ 10, 20, and 30%), and four different hinge lengths ($L$ = 2.5, 5.0, 7.5, and 10 mm). All of our tests were done with ${{T}_{H}}$ = 70 °C and ${{T}_{L}}$ = 25 °C. The laminate configuration is defined by the thickness of the hinge $t$ and the thickness of the PAC lamina $h$. The PAC lamina is characterized by the fiber volume fraction $\left( {{v}_{f}}=\pi {{R}^{2}}/\left( 2hw \right) \right)$, which we control by varying the fiber radius $R$ and the PAC lamina thickness $h$, while keeping the fiber pitch fixed at $2w$ = 1 mm. Table 1 shows the range of hinge parameters we used in our experiments. As described in section 3, we developed a theoretical model of the behavior of a PAC hinge, and this allows us to predict the result of hinge behavior beyond the range of parameters considered in our experiments.

Table 1.  Geometrical parameters that describe PAC lamina and laminates.

	Case I	Case II	Case III	Case IV	Case V
t (mm)	0.6	0.6	0.6	0.5	0.5
R (mm)	0.125	0.1	0.1	0.1	0.08
h (mm)	0.3	0.3	0.2	0.2	0.2
${{v}_{f}}$ (%)	16	10	16	16	10

We carried out experiments to determine the behavior of the PAC hinges. Figure 2 shows hinge angles (measured at ∼1 min after unloading) as a function of the programming stretch and hinge length, respectively, for various laminate/lamina configurations. The results in figure 2(a) are for hinges with $L$ = 5 mm and those in figure 2(b) for hinges with ${{\varepsilon }_{0}}$ = 20%. Figure 2 demonstrates behaviors that are important to understand for the design of PAC hinges. For a fixed material and geometric configuration, the hinge angle decreases (the bending increases) with increasing programming stretch. This is simply because the mismatch strain between the layers increases with applied stretch, and this drives increased bending. Furthermore, as the length $L$ increases, the hinge angle decreases. This is also easy to understand, as the strain mismatch results in approximately constant curvature over the length of the hinge, and so geometry dictates a larger curvature (or smaller hinge angle $\theta$) as $L$ increases. More details regarding the behavior of the PAC hinges are presented in the following section in the context of a theoretical model we develop to describe the behavior.

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In order to obtain parameters used for the models developed in the next section, we also conducted a series of fundamental thermomechanical tests, including dynamic mechanical analysis (DMA) tests, uniaxial tensile tests, thermal strain tests, and stress relaxation tests.

We measured the storage modulus and tanδ vs temperature for the matrix and fiber materials (figure 3(a)) in uniaxial tensile tests performing on a DMA machine (TA Q800) (frequency = 0.1 Hz; cooling rate = 2 °C min⁻¹; sample dimension 15 mm × 6 mm × 2 mm). In figure 3(a), within the temperature range from 100 °C − 50 °C, the storage modulus of the matrix soars from ∼0.7 MPa – ∼900 MPa, and that of the fiber soars from ∼6 MPa – ∼1.7 GPa. The peak of tanδ in the inset of figure 3(a) indicates the ${{T}_{g}}$ of the matrix is ∼−5 °C, and the ${{T}_{g}}$ of the fiber is ∼47 °C.

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The uniaxial tensile tests and thermal strain tests were also conducted on the DMA machine. The uniaxial tension tests were performed at 70 °C, where both the matrix and the fiber are at the rubbery state, and the samples with dimension 15 mm × 6 mm × 2 mm were stretched by 5% at a strain rate of 0.1%/s. In figure 3(b), the stress-strain behavior of the matrix and fibers shows a good linearity. Young's modulus of the fiber material is ∼6 MPa, while that of the matrix material is ∼0.7 MPa, which are consistent with those from the DMA tests. In the thermal strain tests, the termperature was descreased from 70 °C to 25 °C at a cooling rate of 2 °C min⁻¹, under a constant tensile loading of 0.001 N to prevent samples from buckling. In figure 3(c), both the fiber material and the matrix material contract linearly with coefficient of thermal expansions (CTEs) 2 × 10⁻⁴ °C⁻¹ and 2.3 × 10⁻⁴ °C⁻¹, respectively. In figure 3(d), we also tested the stress relaxations for fibers at 25 different temperatures from 0 °C – 60 °C with an inteval of 2.5 °C, which are used to construct the stress relaxation master curve and fit parameters in the multi-branch model.

For a material undergoing thermomechanical loading, the total deformation is due to both the thermal deformation (e.g., from thermal expansion/contraction, phase transformations, etc.) that if unconstrained does not give rise to stress and the mechanical deformation that gives rise to stress. The total deformation (or stretch, in the 1D case) of the material ${{\lambda }^{Total}}$ (${{\lambda }^{Total}}=l/L$, where $l$ is the deformed length in the current state and $L$ is the original length in the reference state) can be decomposed into the mechanical deformation $\lambda$ and the thermal deformation ${{\lambda }^{T}}$. That is, ${{\lambda }^{Total}}={{\lambda }^{T}}\lambda$. For the sake of convenience, we use subscripts M and F to differentiate the deformations of the matrix or the fiber.

The bending of our hinges results from the mismatch strain between layers of the PAC laminate that arise from the shape fixing of the deformed shape memory fibers. As such, we model it as a two-layer laminate; one layer is simply an elastomer with properties of the matrix, and the other layer is a unidirectional fiber-reinforced lamina. To facilitate the analysis, we set Cartesian coordinates at the geometric center of the PAC laminate with the z-axis in the fiber direction, the y-axis downward, and the x-axis within the plane of the laminate (figure 4(a)). We consider the following thermomechanical loading steps: we uniaxially stretch the hinge at ${{T}_{H}}$, maintain the strain, cool it to ${{T}_{L}}$, and then release the constraint.

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During the first step, the sample is uniaxially stretched to ${{\lambda }_{0}}$ $\left( {{\lambda }_{0}}=1+{{\varepsilon }_{0}} \right)$ in a time period of ${{t}_{1}}$ at a constant stretch rate. Here, it is reasonable to assume that both the matrix and the fibers have the same strain, i.e.:

$$\begin{eqnarray}&&{{e}_{M}}\left( t \right)={{e}_{t}}\left( t \right)={\rm ln} \left( \frac{t}{{{t}_{1}}}{{\varepsilon }_{0}}+1 \right),{\rm for}\;t\leqslant {{t}_{1}}.\end{eqnarray} \tag{ 1 }$$

At $t={{t}_{1}}$, ${{e}_{M}}\left( {{t}_{1}} \right)={{e}_{t}}\left( {{t}_{1}} \right)={\rm ln} {{\lambda }_{0}}$. In the cooling step, let the cooling rate to be $\dot{T}$; then the cooling time is ${{t}_{2}}=\left( {{T}_{H}}-{{T}_{L}} \right)/\dot{T}$. Both the matrix and fiber undergo thermal contraction, thus:

$$\begin{eqnarray}&&\lambda _{M}^{T}\left( t \right)=\lambda _{M}^{T}\left( \dot{T},{{T}_{H}} \right)\;{\rm and}\;\lambda _{F}^{T}\left( t \right)=\lambda _{F}^{T}\left( \dot{T},{{T}_{H}} \right).\end{eqnarray} \tag{ 2 }$$

The formulations of the thermal deformations $\lambda _{M/F}^{T}\left( \dot{T},{{T}_{H}} \right)$ will be introduced in equations (9) and (13), respectively.

Since the two ends of the hinge are fixed during the cooling, the total deformation (or stretch) does not change. This gives the mechanical deformations in the matrix and fibers as:

$$\begin{eqnarray}&&{{\lambda }_{M}}\left( t \right)={{\lambda }_{0}}/\lambda _{M}^{T}\left( t \right),{{\lambda }_{F}}\left( t \right)={{\lambda }_{0}}/\lambda _{F}^{T}\left( t \right),{\rm for}\;{{t}_{1}}\leqslant t\leqslant {{t}_{2}},\end{eqnarray} \tag{ 3a }$$

and the strains are

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {{e}_{M}}\left( t \right) & = & {\rm ln} \left[ {{\lambda }_{0}}/\lambda _{M}^{T}\left( t \right) \right],{{e}_{F}}\left( t \right) \\ {} & = & {\rm ln} \left( {{\lambda }_{0}}/\lambda _{F}^{T} \right),{\rm for}\;{{t}_{1}}\leqslant t\leqslant {{t}_{2}}. \\ \end{array}\end{eqnarray} \tag{ 3b }$$

After releasing the constraint, the hinge bends with a curvature $\kappa \left( t \right)$. Due to bending, the midplane at $y=0$ undergoes a change in deformation (or stretch) by $\Delta \lambda \left( 0,t \right)=1+{{\varepsilon }_{b}}\left( t \right)$ (figure 4(b)). Based on beam theory, other planes perpendicular to the y-axis deform by $\Delta \lambda \left( y,t \right)=1+{{\varepsilon }_{b}}\left( t \right)+\kappa \left( t \right)\cdot y$. Therefore, during bending, the total mechanical deformations (or stretches) in the matrix and fiber are:

$$\begin{eqnarray}&&\left\{ \begin{array}{ccccccccccccccc} {{\lambda }_{M}}\left( y,t \right)={{\lambda }_{0}}/\lambda _{M}^{T}\cdot \Delta \lambda \left( y,t \right) \\ {{\lambda }_{F}}\left( y,t \right)={{\lambda }_{0}}/\lambda _{F}^{T}\cdot \Delta \lambda \left( y,t \right) \\ \end{array} \right.,{\rm for}\;t\gt {{t}_{2}}.\end{eqnarray} \tag{ 4 }$$

Note that from the mechanics point of view, the hinge can be taken as a thin beam or plate. Therefore, although it can bend with a large rotation, the strain is usually small, and the Hencky strains are thus:

$$\begin{eqnarray}&&\left\{ \begin{array}{ccccccccccccccc} \begin{array}{lllllllllllllll} {{e}_{M}}\left( y,t,{{\lambda }_{0}},\lambda _{M}^{T},{{\varepsilon }_{b}},\kappa \right) \\ \quad ={\mkern 1mu} {\rm ln} \left( {{\lambda }_{0}}/\lambda _{M}^{T} \right)+{{\varepsilon }_{b}}\left( t \right)+\kappa \left( t \right)\cdot y \\ \end{array} \\ \begin{array}{lllllllllllllll} {{e}_{F}}\left( y,t,{{\lambda }_{0}},\lambda _{F}^{T},{{\varepsilon }_{b}},\kappa \right) \\ \quad ={\mkern 1mu} {\rm ln} \left( {{\lambda }_{0}}/\lambda _{F}^{T} \right)+{{\varepsilon }_{b}}\left( t \right)+\kappa \left( t \right)\cdot y \\ \end{array} \\ \end{array} \right.,{\rm for}\;t\gt {{t}_{2}}.\end{eqnarray} \tag{ 5 }$$

Since the bending occurs after releasing the external constraint, the total external force and total external moment applied to the hinge are zero (Dunn et al 2002, Ge Westbrook et al 2013, Westbrook Mather et al 2011):

$$\begin{eqnarray}\begin{array}{lllllllllllllll} \Sigma {\rm F} & = & \iint {{\sigma }_{M}}\left[ {{e}_{M}}\left( y,t,{{\lambda }_{0}},\lambda _{M}^{T},{{\varepsilon }_{b}},\kappa \right) \right]dydx \\ {} & {} & +\iint {{\sigma }_{F}}\left[ {{e}_{F}}\left( y,t,{{\lambda }_{0}},\lambda _{F}^{T},{{\varepsilon }_{b}},\kappa \right) \right]dydx \\ {} & = & 0, \\ \Sigma {\rm M} & = & \iint {{\sigma }_{M}}\left[ {{e}_{M}}\left( y,t,{{\lambda }_{0}},\lambda _{M}^{T},{{\varepsilon }_{b}},\kappa \right) \right]ydydx \\ {} & {} & +\iint {{\sigma }_{F}}\left[ {{e}_{F}}\left( y,t,{{\lambda }_{0}},\lambda _{F}^{T},{{\varepsilon }_{b}},\kappa \right) \right]ydydx \\ {} & = & 0. \\ \end{array}\end{eqnarray} \tag{ 6 }$$

In equation (6), the stresses on the matrix ${{\sigma }_{M}}$ and the fibers ${{\sigma }_{F}}$ can be calculated through constitutive equations (8) and (10) introduced in the following subsections, by inputting the corresponding mechanical Hencky strains in equation (5). In equation (5), the variables ${{\lambda }_{0}}$, $\lambda _{M/F}^{T}$, and $t$ are measurable or calculable, but ${{\varepsilon }_{b}}$ and $\kappa$ are two unknowns which can be calculated by solving equation (6).

Once the curvature $\kappa$ is calculated, the bending angle $\theta$ resulting from the geometrical relation (figure 4(b)) is:

$$\begin{eqnarray}&&\kappa \cdot L+\theta =\pi .\end{eqnarray} \tag{ 7 }$$

In summary, to compute the curvature $\kappa$ and the bending angle $\theta$, we first build up a set of two equations (equation (6)) to describe the total external force and total external moment. In the two equations, stresses on the fibers and matrix are calculated by their respective constitutive models, including measurable or calculable variables ${{\lambda }_{0}}$, $\lambda _{M/F}^{T}$, $t$, and two unknowns ${{\varepsilon }_{b}}$ and $\kappa$. The two unknowns can be computed by solving the two equations in (6). Once the curvature $\kappa$ is obtained, the bending angle $\theta$ can be calculated by equation (7). Using the developed model, we are able to describe the hinge bending angle as a function of the geometries, material constituents, and programming parameters.

Here, we note that the developed model is one dimensional, as it suffices to describe the operative deformation mode—bending of the hinges. The extension to 2D or 3D is important in general to extend the concepts here (tailored shape memory composite architectures in a laminated configuration) to more complex deformation modes, but it is not very important, in our opinion, to describe the deformation of the hinges in our work here. Indeed, we think this simplicity is somewhat elegant. The extension of the model to 2D or 3D is conceptually straightforward but practically challenging, as it will require nonlinear homogenization in 3D as well as characterization of the 3D constitutive behavior of the constituents. This would most likely be best done in a two-step approach (homogenization of the fibers and matrix within a layer followed by homogenization of the layers in the laminate) rather than in the combined manner we pursued here. As the goal of this model is to predict the macroscopic deformation of the laminate (curvature), we note a couple of treatments: (i) for simplicity we use a beam, rather than a plate, theory. We think this is warranted, given the simple deformation modes that our composite hinges exhibit; (ii) rather than homogenize each layer in the laminate (fibers and matrix) first and then proceed to model the laminate with its layers having effective homogenized properties, we homogenize within the layers (fibers and matrix) and throughout the laminate all in one step. This is tractable, given the 1D beam theory we adopt for simplicity, but in a full plate treatment it would probably be better to do each step separately. The end result is equivalent, though; (iii) in the homogenization of the fibers and matrix, we use a fairly sophisticated multi-branch constitutive model for the fibers that accounts for the nonlinear, time-dependent shape memory/fixing behavior as well as a simple hyperelastic model for the matrix. We realize that a more detailed model could be developed, at both the lamina level in terms of a fiber orientation distribution and inelastic behavior of the constituents (Dunn and Ledbetter 1997, Dunn et al 1996) and at the laminate level in terms of plate or even continuum behavior (Dunn et al 2002, Zhang and Dunn 2003, 2004), but we think the approach here is reasonable for understanding the basic behavior and designing components.

The matrix material shows entropic hyperelastic behavior over the operating temperature range of the hinge. Therefore, a simple hyperelastic model is adopted:

$$\begin{eqnarray}&&{{\sigma }_{M}}={{E}_{M}}\left( T \right){{e}_{M}},\end{eqnarray} \tag{ 8 }$$

where ${{e}_{M}}={\rm ln} {{\lambda }_{M}}$ is the Hencky strain, which can be readily incorporated into the bending theory, and ${{E}_{M}}\left( T \right)=3{{N}_{M}}kT$ is the temperature dependent Young's modulus due to the entropic elasticity where ${{N}_{M}}$ is the cross-link density, $k$ is Boltzmann's constant, and $T$ is the absolute temperature. Based on our experimental observations (figure 3(c)), we describe the thermal deformation by a linear relation with the CTE of the matrix material ${{\alpha }_{M}}$:

$$\begin{eqnarray}&&\lambda _{M}^{T}=1+{{\alpha }_{M}}\left( T-{{T}_{H}} \right).\end{eqnarray} \tag{ 9 }$$

The behavior of the fiber material over the temperature range is more complicated, as it exhibits the shape memory effect. To calculate the stress acting on the fiber material, we adopt a thermomechanical multi-branch model that decomposes the total deformation $\lambda _{F}^{Total}$ into a mechanical deformation ${{\lambda }_{F}}$ and a thermal deformation $\lambda _{F}^{T}$ (Westbrook, Kao et al 2011, Yu et al 2014). For the mechanical elements in the model, an equilibrium branch associated with elastic response and several nonequilibrium branches associated with viscoelastic response are arranged in parallel. Each nonequilibrium branch is taken to be a Maxwell element, where an elastic spring and a dashpot are arranged in series. The total stress (Castro et al 2010, Ferry 1961) acting on the fiber material ${{\sigma }_{F}}$ can be expressed as the sum of that in the equilibrium and nonequilibrium branches:

$$\begin{eqnarray}&&\begin{array}{lllllllllllllll} {{\sigma }_{F}}={\underbrace{{{E}_{F}}\left( T \right){{e}_{F}}}_{Equilibrium}} \\ \quad +{\underbrace{\sum \limits_{m=1}^{n} E_{non}^{m}\int \limits_{0}^{t} \frac{\partial {{e}_{F}}\left( T,t \right)}{\partial s}{\rm exp} \left[ -\int \limits_{s}^{t} \frac{dt\prime}{{{\tau }_{m}}\left( T \right)} \right]ds}_{Nonequilibrium}}. \\ \end{array}\end{eqnarray} \tag{ 10 }$$

In equation (10), the first term is the stress from the equilibrium branch, where ${{E}_{F}}\left( T \right)=3{{N}_{F}}kT$ is the temperature dependent Young's modulus and ${{e}_{F}}={\rm ln} {{\lambda }_{F}}$. The second term is the stress contribution from the nonequilibrium branches, where $E_{non}^{m}$ and ${{\tau }_{m}}\left( T \right)$ are the Young's modulus and the temperature dependent relaxation time for the mth branch. ${{\tau }_{m}}\left( T \right)$ can be expressed in terms of $\tau _{m}^{R}$ and a temperature dependent shifting factor ${{a}_{T}}\left( T \right)$:

$$\begin{eqnarray}&&{{\tau }_{m}}\left( T \right)={{a}_{T}}\left( T \right)\tau _{m}^{R},\end{eqnarray} \tag{ 11 }$$

where $\tau _{m}^{R}$ is the relaxation time for the mth branch at a reference temperature. Depending on whether the temperature is above, near, or below ${{T}_{g}}$, ${{a}_{T}}\left( T \right)$ is calculated by two different methods (O'Connell and McKenna 1999):

$$\begin{eqnarray}\begin{array}{lllllllllllllll} {\rm log} {{a}_{T}}\left( T \right) & = & -\frac{{{C}_{1}}\left( T-{{T}_{r}} \right)}{{{C}_{2}}+\left( T-{{T}_{r}} \right)},\left( T\geqslant {{T}_{r}} \right)\;{\rm and}{\rm ln} {{a}_{T}}\left( T \right) \\ {} & = & {\rm exp} \left[ -\frac{A{{F}_{C}}}{k}\left( \frac{1}{T}-\frac{1}{{{T}_{r}}} \right) \right],\left( T\lt {{T}_{r}} \right) \\ \end{array}\end{eqnarray} \tag{ 12 }$$

where ${{C}_{1}}$, ${{C}_{2}}$, and $A$ are material constants, ${{F}_{c}}$ is the configuration energy, $k$ is Boltzmann's constant, and ${{T}_{r}}$ is the reference temperature.

Generally as a polymer goes through the glass transition from the equilibrium rubbery state to the nonequilibrium glassy state, the thermal deformation is a function of temperature as well as time. In addition, the dependence on time can become very weak (as the time constant for this dependence can be very long) as the time in the nonequilibrium state increases (Yu et al 2014). In the past, multiple theories (Kovacs et al 1979, Moynihan et al 1976, Robertson et al 1984) have been developed to represent the evolution of the nonequilibrium volume change. In this paper, we take a simple empirical approach based on our experimental observations. We represent the thermal deformation during the temperature change as:

$$\begin{eqnarray}&&\lambda _{F}^{T}=1+{{\alpha }_{F}}\left( T-{{T}_{H}} \right),\end{eqnarray} \tag{ 13 }$$

where ${{\alpha }_{F}}$ is a linear thermal expansion coefficient of the fiber.

There are nine sets of parameters used in the thermomechanical constitutive models in total. They can be directly characterized by fitting the thermomechanical tests in section 2.

In the thermomechanical constitutive models for the matrix material, there are only two parameters, the crosslinking density ${{N}_{M}}$ and CTE ${{\alpha }_{M}}$. By simply using equation (8), ${{\sigma }_{M}}=3{{N}_{M}}kT{\rm ln} {{\lambda }_{M}}$, to fit the uniaxial tensile test in figure 3(b) for the matrix material at 70 °C, one can readily have $3{{N}_{M}}kT$ = 0.65 MPa and ${{N}_{M}}$ = 4.58 × 10²⁵ m⁻³. Using equation (9), $\lambda _{M}^{T}=1+{{\alpha }_{M}}\left( T-{{T}_{H}} \right)$, to fit the thermal strain test in figure 3(c), we have ${{\alpha }_{M}}$ = 2.3 × 10⁻⁴ °C⁻¹.

Among the parameters in the multi-branch model in equation (10), the crosslinking density, ${{N}_{F}}$, can be readily identified by simply using ${{\sigma }_{F}}=3{{N}_{F}}kT{\rm ln} {{\lambda }_{F}}$ to fit the stress-strain for the fiber material at 70 °C in figure 3(b), and ${{N}_{F}}$ = 4.23 × 10²⁶ m⁻³. Stress relaxation tests in figure 3(d) were used to identify parameters in nonequilibrium branches. A stress relaxation master curve at 35 °C in figure 5(a) was constructed by shifting relaxation curves using shift factors $\left( {{a}_{T}} \right)$ at different temperatures (figure 5(d)). The master curve can be described by Maxwell elements in parallel, and the stress relaxation modulus is:

$$\begin{eqnarray}\begin{array}{lllllllllllllll} E(t) & = & {{E}_{\infty }}+\sum \limits_{m=1}^{n} E_{non}^{m}{\rm exp} \left[ -\left( \frac{t}{\tau _{m}^{R}} \right) \right]\;{\rm with}{\mkern 1mu} \tau _{m}^{R} \\ {} & = & {{10}^{m-1}}\tau _{1}^{R},\;\left( m=2,...,n \right). \\ \end{array}\end{eqnarray} \tag{ 14 }$$

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In equation (14), ${{E}_{\infty }}$ is the relaxation modulus at time $t$ = $\infty$ (${{E}_{\infty }}$∼5 MPa in figure 5(a)); $\tau _{R}^{m}$ is the relaxation time for the mth branch at the reference temperature (35 °C). We assume that the relaxation time of the mth branch is a decade longer than the (m-1)-th branch. At time $t$ = 0, the relaxation modulus is $E\left( 0 \right)={{E}_{\infty }}+\mathop \sum \nolimits_{m=1}^{n} E_{non}^{m}$. Figures 6(a)–(c) present the model fitting for the stress relaxation. In figure 5(a), one nonequilibrium branch ($m$ = 1) was used to describe the stress relaxation modulus master curve. Based on equation (14), $\left( E\left( 0 \right)={{E}_{\infty }}+E_{non}^{1} \right)$, one has $E_{non}^{1}=700{\rm MPa}$ ($E\left( 0 \right)=705{\rm MPa}$ and ${{E}_{\infty }}=5{\rm MPa}$ in figure 5(a). Through the observation of the stress relaxation master curve in figure 5(a), the obvious stress relaxation occurs at ∼0.006 min Here, $\tau _{1}^{R}=0.36{\rm s}$ was taken for the relaxation time of the first nonequilibrium branch. It is shown that an increasing number of nonequilibrium branches is required to precisely describe the stress relaxation modulus master curve. Figures 5(b) and (c) show the model fitting for the stress relaxation with $m$ = 4, 8 nonequilibrium branches, and the model fitting improves dramatically by introducing more nonequilibrium branches. Here, we take $m$ = 2 as an example to demonstrate the fitting procedure. In figure 5(a), at time $t=\tau _{1}^{R}$, the discrepancy between the master curve and the model fitting is ∼250 MPa. This discrepancy can be corrected by introducing the second nonequilibrium branch with $E_{non}^{2}=250{\rm MPa}$. Based on equation (14), $\left( E\left( 0 \right)={{E}_{\infty }}+E_{non}^{1}+E_{non}^{2} \right)$, one has a new $E_{non}^{1}$ equal to 450 MPa. Assuming that the relaxation time of the second nonequilibrium branch is a decade longer than the first one, we have $\tau _{2}^{R}=10\tau _{1}^{R}$. Following the same fitting procedure, one has moduli for the all of the eight nonequilibrium branches ($E_{non}^{1}$ = 238 MPa, $E_{non}^{2}$ = 250 MPa, $E_{non}^{3}$ = 100 MPa, $E_{non}^{4}$ = 50 MPa, $E_{non}^{5}$ = 30 MPa, $E_{non}^{6}$ = 20 MPa, $E_{non}^{7}$ = 10 MPa, and $E_{non}^{8}$ = 2 MPa).

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The parameters ${{C}_{1}}$, ${{C}_{2}}$, and $A{{F}_{c}}/k$ in equation (12) can be obtained by fitting the shift factor-temperature curve (figure 5(d)). Based on the experimental observation in figure 3(c), the CTE of the fiber material $\left( {{\alpha }_{F}} \right)$ is 0.2 × 10⁻⁴ °C⁻¹. All values of parameters in the constitutive models are listed in table 2.

Table 2.  Lists of parameters for constitutive models.

	Parameter	Value
Matrix material
Crosslinking Density	${{N}_{M}}$	4.58 × 1025 m−3
CTE	${{\alpha }_{M}}$	2.3 × 10−4 °C−1
Fiber material
Crosslinking Density	${{N}_{F}}$	4.23 × 1026 m−3
Elastic Moduli on Nonequilibrium Branches	$E_{non}^{1}$, $E_{non}^{2}$, $E_{non}^{3}$, $E_{non}^{4}$ $E_{non}^{5}$, $E_{non}^{1}$, $E_{non}^{1}$, $E_{non}^{1}$	238, 250, 100, 50, 30, 20, 10, 2 MPa
Relaxation Time of the 1st Branch	$\tau _{1}^{R}$	0.36 s
WLF constant	${{C}_{1}}$	17.66
WLF constant	${{C}_{2}}$	51.6 °C
Pre-exponential parameter	$A{{F}_{c}}/k$	−20000 K
CTE	${{\alpha }_{F}}$	0.2 × 10−4 °C−1

As introduced in 3.1, the curvature $\kappa$ and the midplane strain ${{\varepsilon }_{b}}$ can be solved by incorporating the constitutive equations for the matrix and fibers from equations (8)–(13) with the corresponding mechanical Hencky strains in equation (5) into equation (6) (Details are straightforward, although tedious, and are shown in the appendix):

$$\begin{eqnarray}\left\{ \frac{{{\varepsilon }_{b}}\left( t \right)}{\kappa \left( t \right)} \right\}={{\left[ \begin{array}{ccccccccccccccc} A & B \\ B & D \\ \end{array} \right]}^{-1}}\left\{ \frac{{{N}_{t}}}{{{M}_{t}}} \right\}.\end{eqnarray} \tag{ 15 }$$

In composite laminate mechanics, $A$, $B$, and $D$ are termed the extensional stiffness, coupling stiffness, and bending stiffness, respectively. If the laminate is symmetric with respect to the geometric midplane, then $B$ = 0 (in our case it is not by design). ${{N}_{t}}$ and ${{M}_{t}}$ are called the thermal force and moment, respectively. Here they arise from the mismatch strain between layers in the laminate due to the shape fixing of the shape memory fibrous lamina; they are the driving force for the bending of the hinge. Equation (15) yields the curvature of the laminate:

$$\begin{eqnarray}&&\kappa =\frac{-B{{N}_{t}}+A{{M}_{t}}}{AD-{{B}^{2}}}.\end{eqnarray} \tag{ 16 }$$

Once the curvature $\kappa$ is calculated, the bending angle $\theta$ can be directly obtained by equation (7).

With the characterized parameters listed in table 2, we use our model to plot the predicted hinge angle vs applied strain at ${{T}_{L}}$ (figure 6(a)), and the hinge angle vs the hinge length (figure 6(b)) for the five different cross section profile cases of table 1. Table 3 shows the predicted laminate parameters (A, B, D, ${{N}_{t}}$, and ${{M}_{t}}$) for the laminate, with ${{N}_{t}}$ and ${{M}_{t}}$ computed for an applied strain of 30%. It is noted that as ${{\varepsilon }_{b}}\left( t \right)$ and the curvature $\kappa \left( t \right)$ are time dependent; all laminate parameters are computed at 1 min after unloading.

Table 3.  Laminate thermomechanical parameters that determine the hinge angle of a PAC laminate. (All parameters are computed at 1 min after unloading. ${{N}_{t}}$ and ${{M}_{t}}$ are computed at an applied strain of 30%).

	Case I	Case II	Case III	Case IV	Case V
$A$ $\left( {\rm N} \right)$	4.63	3.02	3.02	2.99	1.97
$B$ $\left( {\rm N}\cdot {\rm mm} \right)$	0.67	0.43	0.57	0.43	0.27
$D$ $\left( {\rm N}\cdot {\rm m}{{{\rm m}}^{2}} \right)$	0.12	0.076	0.13	0.074	0.047
${{N}_{t}}$ $\left( {\rm N} \right)$	−0.078	−0.066	−0.066	−0.059	−0.051
${{M}_{t}}$ $\left( {\rm N}\cdot {\rm mm} \right)$	−0.0048	−0.0031	−0.0059	−0.0031	−0.002

The hinge behavior as a function of applied strain and hinge length is easy to understand, but its behavior in terms of other microstructural parameters, e.g., cases I-V, is less straightforward to understand. Composite laminate mechanics provide a convenient way to understand the behavior of the PACs in terms of the microstructural parameters, specifically as it is expressed through the variation of A, B, D, ${{N}_{t}}$, and ${{M}_{t}}$. These are a function of the applied strain but reported for ${{\varepsilon }_{0}}$ = 30% in table 3. In essence, the five cases in table 3 systematically vary the volume fraction of the SMP fibers in the PAC lamina and the thicknesses of the two lamina that make up the laminate. Here we make a few observations important for the design of PAC hinges, based on our experiments and theory:

• Compared to Case I, the Case II hinge bends more because of the lower fiber volume fraction, with the lamina thicknesses equal.
• In Case III, the hinge bends more than that in Case II as the thickness of the lamina with the SMP fibers decreases, even though the volume fraction increases.
• The behavior of Case IV is comparable to that of Case II, and this arises because of the combination of the higher volume fraction for Case IV and the smaller thickness of lamina, and thus laminate.
• Case V, with the lowest fiber volume fraction and the thinnest lamina (and most compliant laminate) bends the most.

The hinge bending results from the interplay among the laminate parameters A, B, D, ${{N}_{t}}$, and ${{M}_{t}}$, which represent the effects of the microstructural parameters on the collective behavior of the shape memory fibers and the elastomeric matrix that contribute to produce the applied loading and laminate stiffness. Table 3 shows that the coupling between extension and bending (represented by the stiffness B and its role in equation (16)) significantly influences the bending of the laminate and the resulting hinge angle. Indeed it is through this coupling that our hinges operate as we apply a tensile strain. Via the internal workings of the lamina and laminate architecture, the laminate bends. Our results demonstrate that PAC laminate hinges can be designed to exhibit a controlled hinge angle, but because of the numerous design variables and their interacting influence, their design benefits greatly from a theory that can describe the observed behavior.