Modeling and analysis of laminated shape memory alloy composite plates

Abstract

In the present work, static deformation, dynamic characteristic, and time response of laminated composite plates with surface-bonded shape-memory alloy (SMA) sheets are modeled. A modified higher order shear deformation theory with Ritz method is utilized. The energy balance equations are used to formulate the nonlinear heat transfer governing equations. The time response of the SMA/elastomer plate and the deflection as well as dynamic characteristics before and after activation of the SMA is computed. A Mathematica code is developed to analyze different plate problems. Parametric studies are performed for proposed structure system to demonstrate the effect of thickness ratio, aspect ratio, material properties, thermal expansion coefficient, electric input power, and thickness of SMA sheet on the transverse deflections, natural frequencies, and response time of the SMA layer. The obtained results are compared to the available studies solved by different theories.

Introduction

The shape-memory alloy (SMA) has been a subject of intensive research in the last decades due to its unique properties of one- and two-way shape-memory effect (SME), pseudoelasticity, and high damping capacity. Ghomshei et al. ^1,2 proposed a nonlinear finite element model and experimental test for the time response of a SMA actuator composed of matrix material with SMA sheets or wires embedded in or bonded to the matrix part. The model is developed based on a higher order shear deformation beam theory together with the von-Karman strain field. A one-dimensional constitutive equation with nonconstant material functions and sinusoidal phase transformation kinetics is used to model the thermomechanical behavior of the SMA actuator. The constitutive and phase transformation kinetic equations make distinction between the stress-induced martensite (SIM) and temperature-induced martensite (TIM) fraction. Balapgol et al. ^3,4 studied the deflection, natural frequency, and time response of SMA-laminated composite plate using finite element model with first-order shear deformation theory. The composite plate consists of a thin layer of SMA bonded to elastomer core. They concluded that the input power, heat sink strength, thermal conductivity, and thickness of the elastomer layer play important roles for controlling the time response of the SMA-laminated actuator. Gordaninejad et al. ⁵ presented a two-dimensional finite element model based on classical lamination theory, energy balance equations, and two-dimensional transition model of SMA layer for the response of thermally driven SMA/elastomer actuator. Wu et al. ⁶ derived a closed form solutions for the stress–strain–temperature response of a thermally driven SMA composite actuator neglecting the heat conduction in axial direction. Rogers et al. ⁷ used the Rayleigh–Ritz method to perform a linear analysis for simply supported plate embedded with SMA fibers. They studied the plate deflection, free vibration, buckling, and acoustic control. Lin et al. ⁸ proposed a closed form solution for symmetric composite beams embedded with SMA fibers with various boundary conditions. The resultant actuation forces and normal stress distribution were calculated for the proposed beams. One of the earliest models is the one-dimensional Tanaka’s model, ⁹ a macroscopic model that is derived from thermodynamic concepts and through experimental observations. Martensitic transformation was considered progressive through an internal variable, the volume fraction of martensite (ξ). The evolutionary equation was determined by considering the transformation micro-mechanism and it is expressed using an exponential function in the form of $\mathit{\xi }=\mathit{\xi }\left(\mathit{\sigma },T\right)$ Liang and Rogers ¹⁰ improved Tanaka’s model by directly matching experimental results to get the evolutionary equation that is expressed using the cosine function. The constitutive equation remains the same while the equations’ parameters are determined through experiments. An improvement of the Tanaka’s model was made by Brinson, ^11,12 who recognized that not all martensite that are converted to austenite will produce the recovery stress. Only the SIM is responsible for the SME. The martensite fraction is divided into two components: SIM and TIM. This model also does not assume constant material functions in the constitutive relationship. Furthermore, Brinson made some amendment that the constitutive equation will be valid at any temperature. This model was found to give a better representation of the SMA behaviors than the Liang and Rogers’s model. ¹⁰

Paine and Rogers ¹³ developed a model for SMA wire actuator for use in intelligent material systems. They determined the effect of the composite manufacturing on the nitinol performance. The results of their study investigating the effects of a “high temperature” thermoplastic composite processing cycle on the nitinol actuator. Their proposed actuators were exposed to a number of thermoplastic processing cycle simulations at 400°C. The critical parameters, such as processing time, training of the actuators, and the type of the actuators, were varied to determine their effect on the performance of the nitinol actuator. The obtained results agreed well with the data from the simulation tests.

Kalamkarov et al. ¹⁴ proposed asymptotic homogenization models for smart composite plates with periodically arranged embedded actuators and rapidly varying thickness.

Their models determine both local fields and effective elastic, actuation, thermal expansion, and hygroscopic expansion coefficients from three-dimensional local unit cell problems. They applied their models to rib-reinforced smart composite plates, smart sandwich plates with rib-like filler, wafer-reinforced smart composite plates, and sandwich smart composite plates with honeycomb filler. Their results conform with the classical laminate theory. They found that the differences in the values of the effective coefficients like piezoelectric actuators were attributed to the geometries of the respective unit cells. They conclude that the importance of the effective coefficients lies in the fact that they are universal in nature, and once determined they can be used to study a wide variety of boundary value problems.

In the present study, a two-dimensional nonlinear transient analysis of thermally driven composite elastomer plate with SMA sheet bonded on its surface is presented. A modified higher order shear deformation theory (MFSDT) with Ritz method is utilized. The two SMA constitutive models are used: a linear model that provides a one-dimensional relation between the martensite fraction with temperature and stress during the phase transformation, and Brinson’s model that divides the martensitic volume fraction into two parts, one corresponds to the fraction of the SIM, and the second represent the fraction of the TIM. The energy balance equation is used to solve a two-dimensional temperature distributions of the SMA layer. An α-family method is used to transfer the heat equation from ordinary differential equation to a set of algebraic equations. The time response and the deflection of the SMA/elastomer plate are computed. The natural frequencies are expressed at certain time (t) of the phase transformation and before and after activation of the SMA plate. The obtained results are compared to the available literatures.

Displacement field equations

The layout of a SMA composite plate actuator is shown in Figure 1. The plate consists of a layer of SMA of thickness (t_s ), which is bonded to a composite plate of thickness $\left(h-t_s\right)$ . The in-plane plate dimensions are a and b, respectively. The plate is bonded at its bottom to a heat sink maintained at temperature T _sink. The plate is subjected to a uniform transverse load (q). The Cartesian coordinate system is taken so that the xy-plane coincides with the midplane of the plate, and the z-axis is upwards normal to the midplane. The displacement field equations of Lo’s MFSDT ^15,16 are refined as MFSDT by representing the total rotation of the normal to the midplane by two components, bending and shear rotations. Also the proposed MFSDT shows the additional terms in the thickness direction. The displacement-based single-layer theories are more refined than the third-order theory they are cumbersome and the increase in accuracy is obtained by the increase in the number of unknown. The theory with cubic terms of the thickness direction yields quadratic variation of the transverse shear strains within each layer and represents the true distribution of the strains through thickness, and it does not require shear correction factors. The displacement field equations are given as follows ^17,18 :

Figure 1.

SMA-laminated plate with applied loading. SMA: shape-memory alloy.

$u\left(x,y,z\right)=u_0+z\left({\mathit{\theta }}_x-\frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }x}\right)+z^2{\mathit{\psi }}_x+z^3{\mathit{\xi }}_x$

1

$v\left(x,y,z\right)=v_0+z\left({\mathit{\theta }}_y-\frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }y}\right)+z^2{\mathit{\psi }}_y+z^3{\mathit{\xi }}_y$

$w\left(x,y,z\right)=w_0+z{\mathit{\theta }}_z+z^2{\mathit{\psi }}_z$

where $u_0$ , $v_0$ , and $w_0$ are displacements at the midplane in the x-, y-, and z-directions respectively. ${\mathit{\theta }}_x$ and ${\mathit{\theta }}_y$ are the rotations of the normal to the midplane about the y-axis, and x-axis, respectively. ${\mathit{\theta }}_z$ is a first-order displacement representing the extension in the plain normal to the midplane. ${\mathit{\psi }}_x,$ ${\mathit{\psi }}_y,$ and ${\mathit{\psi }}_z$ are second-order displacements or warping functions. ${\mathit{\xi }}_x$ and ${\mathit{\xi }}_y$ are third-order displacements or warping functions, where all the previous symbols are function of $\left(x,y,t\right)$ . h is the total laminate thickness.

The basic assumptions are that the SMA layers and the host layers are perfectly bonded, both thermal stresses and temperature in the phase transformation of SMA are considered, and the plate undergoes a small displacement.

Strain–displacement relationships

The strain–displacement relationships can be expressed in a matrix form as follows:

$\left\{\mathit{\epsilon }\right\}=\left\{{\mathit{\epsilon }}^0\right\}+z\left\{{\mathit{\epsilon }}^1\right\}+z^2\left\{{\mathit{\epsilon }}^2\right\}+z^3\left\{{\mathit{\epsilon }}^3\right\}$

2

where

${\left\{\mathit{\epsilon }\right\}}^T=\left[\begin{array}{cccccc}{\mathit{\epsilon }}_{xx}& {\mathit{\epsilon }}_{yy}& {\mathit{\epsilon }}_{zz}& {\mathit{\gamma }}_{yz}& {\mathit{\gamma }}_{zx}& {\mathit{\gamma }}_{xy}\end{array}\right]$

3

$\begin{array}{rl}\hspace{0.17em}& {\left\{{\mathit{\epsilon }}^0\right\}}^T=\left[\begin{array}{cccccc}\frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }x}& \frac{\mathrm{\partial }{v}_0}{\mathrm{\partial }y}& {\mathit{\theta }}_z& {\mathit{\theta }}_y& {\mathit{\theta }}_x& \frac{\mathrm{\partial }{u}_0}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{v}_0}{\mathrm{\partial }x}\end{array}\right]\\ \hspace{0.17em}& {\left\{{\mathit{\epsilon }}^1\right\}}^T=\left[\begin{array}{cccccc}\frac{\mathrm{\partial }{\mathit{\theta }}_x}{\mathrm{\partial }x}-\frac{{\mathrm{\partial }}^2{w}_0}{\mathrm{\partial }{x}^2}& \frac{\mathrm{\partial }{\mathit{\theta }}_y}{\mathrm{\partial }y}-\frac{{\mathrm{\partial }}^2{w}_0}{\mathrm{\partial }{y}^2}& 2{\mathit{\psi }}_z& 2{\mathit{\psi }}_y+\frac{\mathrm{\partial }{\mathit{\theta }}_z}{\mathrm{\partial }y}& 2{\mathit{\psi }}_x+\frac{\mathrm{\partial }{\mathit{\theta }}_z}{\mathrm{\partial }x}& \frac{\mathrm{\partial }{\mathit{\theta }}_x}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\mathit{\theta }}_y}{\mathrm{\partial }x}-2\frac{{\mathrm{\partial }}^2{w}_0}{\mathrm{\partial }x\mathrm{\partial }y}\end{array}\right]\\ \hspace{0.17em}& {\left\{{\mathit{\epsilon }}^2\right\}}^T=\left[\begin{array}{cccccc}\frac{\mathrm{\partial }{\mathit{\psi }}_x}{\mathrm{\partial }x}& \frac{\mathrm{\partial }{\mathit{\psi }}_y}{\mathrm{\partial }y}& 0& 3{\mathit{\xi }}_y+\frac{\mathrm{\partial }{\mathit{\psi }}_z}{\mathrm{\partial }y}& 3{\mathit{\xi }}_x+\frac{\mathrm{\partial }{\mathit{\psi }}_z}{\mathrm{\partial }x}& \frac{\mathrm{\partial }{\mathit{\psi }}_x}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\mathit{\psi }}_y}{\mathrm{\partial }x}\end{array}\right]\\ \hspace{0.17em}& {\left\{{\mathit{\epsilon }}^3\right\}}^T=\left[\begin{array}{cccccc}\frac{\mathrm{\partial }{\mathit{\xi }}_x}{\mathrm{\partial }x}& \frac{\mathrm{\partial }{\mathit{\xi }}_y}{\mathrm{\partial }y}& 0& 0& 0& \frac{\mathrm{\partial }{\mathit{\xi }}_x}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\mathit{\xi }}_y}{\mathrm{\partial }x}\end{array}\right]\end{array}$

4

Stress–strain relationships

The generalized stress–strain relationships can be written in contracted notation as follows:

${\mathit{\sigma }}_i=Q_{ij}{\mathit{\epsilon }}_j\hspace{0.17em}i,j=1,2,\dots 6$

5

The transformed stress–strain relationships for an orthotropic lamina oriented by an angle θ can be written as:

$\left\{\begin{array}{c}{\mathit{\sigma }}_x\\ {\mathit{\sigma }}_y\\ {\mathit{\sigma }}_z\\ {\mathit{\tau }}_{yz}\\ {\mathit{\tau }}_{zx}\\ {\mathit{\tau }}_{xy}\end{array}\right\}=\left[\begin{array}{cccccc}{\bar{Q}}_{11}& {\bar{Q}}_{12}& {\bar{Q}}_{13}& 0& 0& {\bar{Q}}_{16}\\ {\bar{Q}}_{21}& {\bar{Q}}_{22}& {\bar{Q}}_{23}& 0& 0& {\bar{Q}}_{26}\\ {\bar{Q}}_{31}& {\bar{Q}}_{32}& {\bar{Q}}_{33}& 0& 0& {\bar{Q}}_{36}\\ 0& 0& 0& {\bar{Q}}_{44}& {\bar{Q}}_{45}& 0\\ 0& 0& 0& {\bar{Q}}_{54}& {\bar{Q}}_{55}& 0\\ {\bar{Q}}_{61}& {\bar{Q}}_{62}& {\bar{Q}}_{63}& 0& 0& {\bar{Q}}_{66}\end{array}\right]\left\{\begin{array}{c}{\mathit{\epsilon }}_x\\ {\mathit{\epsilon }}_y\\ {\mathit{\epsilon }}_z\\ {\mathit{\gamma }}_{yz}\\ {\mathit{\gamma }}_{zx}\\ {\mathit{\gamma }}_{xy}\end{array}\right\}$

6

The elements of the transformed symmetric stiffness matrix $\left[\bar{Q}\right]$ are given in in the work by Reddy. ¹⁹

During martensitic transformation of SMA the modulus of elasticity of the metal is uniformly blended with the martensite and austenite phases. ³ The SMA layers and the host layers are perfectly bonded. The constitutive relationship of the SMA layer is:

$\left\{\mathit{\sigma }\left(\mathit{\zeta }\right)\right\}=\left[Q_s\left(\mathit{\zeta }\right)\right]\left(\left\{\mathit{\epsilon }\right\}-\left\{{\mathit{\alpha }}_s\right\}\mathrm{\Delta }T\right)$

7

where $\left[Q_s\left(\mathit{\zeta }\right)\right]$ is the SMA layer stiffness matrix function of the martensite fraction. ^17,18

$\left[Q_s\left(\mathit{\zeta }\right)\right]$ can be calculated by the inverse of the compliance matrix of the SMA layer $\left[S_{SMA}\right]$ , which is given in Appendix 1. The SMA layer stiffness matrix $\left[Q_s\left(\mathit{\zeta }\right)\right]$ is given by:

$\left[Q_s\right]={\left[S_{SMA}\right]}^{-1}=\left[\begin{array}{cccccc}\left({\mathit{\lambda }}_s+2G_s\right)& \mathit{\lambda }& \mathit{\lambda }& 0& 0& 0\\ \mathit{\lambda }& \left({\mathit{\lambda }}_s+2G_s\right)& \mathit{\lambda }& 0& 0& 0\\ \mathit{\lambda }& \mathit{\lambda }& \left({\mathit{\lambda }}_s+2G_s\right)& 0& 0& 0\\ 0& 0& 0& G_s& 0& 0\\ 0& 0& 0& 0& G_s& 0\\ 0& 0& 0& 0& 0& G_s\end{array}\right]$

8

where λ_s and the shear modulus G _s are given by:

${\mathit{\lambda }}_s=\frac{{\mathit{\nu }}_s{E}_s\left(\mathit{\zeta }\right)}{\left(1+{\mathit{\nu }}_s\right)\left(1-2{\mathit{\nu }}_s\right)}\mathrm{a}\mathrm{n}\mathrm{d}{G}_s\left(\mathit{\zeta }\right)=\frac{E_s\left(\mathit{\zeta }\right)}{2\left(1+{\mathit{\nu }}_s\right)}$

9

And the Young’s modulus E_s of the SMA layer is given by:

$E_s\left(\mathit{\zeta }\right)=\left(1-\mathit{\zeta }\right)E_A+\mathit{\zeta }{E}_M$

10

where E_A and E_M are the Young’s moduli in austenite and martensite phases, respectively.

The $\left\{{\mathit{\alpha }}_s\right\}$ is the thermal expansion coefficient vector of the SMA layer given by:

${\left\{{\mathit{\alpha }}_s\right\}}^T=\left[\begin{array}{cccccc}\mathit{\alpha }& \mathit{\alpha }& \mathit{\alpha }& 0& 0& 0\end{array}\right]$

11

and α is the thermal expansion coefficient of the SMA material.

It is clear that equation (7) is nonlinear equation in nature since $\left[Q_s\left(\mathit{\zeta }\right)\right]$ is function of $E_s\left(\mathit{\zeta }\right)$ and $E_s\left(\mathit{\zeta }\right)$ is function of the Young’s modulus in austenite and martensite phases, E_A and E_M , respectively.

SMA constitutive models

SMA linear model

The linear model provides a one-dimensional relation between the martensite fraction with temperature and stress during the phase transformation. ⁸ In the present model, the heating and cooling transition process is given as:

$\mathit{\zeta }=1-\frac{T-A_s}{A_f-A_s}+\frac{\left|\mathit{\sigma }\right|}{C_A\left(A_f-A_s\right)}$

12

$\mathit{\zeta }=1-\frac{T-M_f}{M_s-M_f}+\frac{\left|\mathit{\sigma }\right|}{C_M\left(M_s-M_f\right)}$

13

where $C_A$ and $C_M$ are the material constants that indicate the influence of the stress on the phase transformation temperatures. A_s and A_f are the start and finish temperatures of the phase transformation from martensite to austenite, respectively. M_s and M_f are the start and finish temperatures of the phase transformation from austenite to martensite, respectively. σ is the magnitude of the stress in the SMA layer, which is assumed to be the hydrostatic stress $\left|\mathit{\sigma }\right|=\left|{\mathit{\sigma }}_x+{\mathit{\sigma }}_y+{\mathit{\sigma }}_z\right|/\phantom{\left|\mathit{\sigma }\right|=\left|{\mathit{\sigma }}_x+{\mathit{\sigma }}_y+{\mathit{\sigma }}_z\right|3}3$ .

SMA Brinson’s model

The Brinson’s model ¹² made a significant improvement over Tanaka’s model ^{9 –11} and the Liang and Rogers’s model. ¹⁰ It recognizes the SIM as the only martensite that gives the functional property of SME and pseudoelasticity rather than the total martensite that contains both the TIM and the SIM. Brinson’s model assumes that the transformation depends only on the temperature and the stress, and the amount of transformation that occurs is described using the volume fraction of the SIM (ξ _S). Brinson’s model is quite popular for engineering applications since it is simple, accurate, and easy to implement into numerical applications. Brinson made a modification so that this model can be used at low temperatures by dividing the martensitic volume fraction into two parts:

$\mathit{\xi }={\mathit{\xi }}_S+{\mathit{\xi }}_T$

14

where ξ _S corresponds to the fraction of the SIM, and ξ _T refers to the fraction of the TIM given in Appendix 1.

Energy formulation

Hamilton’s principle

The governing differential equations of the whole structure are derived using Hamilton’s principle ¹⁹ :

$0={\int }_0^T\left(\mathit{\delta }U+\mathit{\delta }V-\mathit{\delta }K\right)dt$

15

where $\mathit{\delta }U$ is the virtual strain energy, $\mathit{\delta }V$ is the virtual work done by the applied loads, and $\mathit{\delta }K$ is the virtual kinetic energy. The virtual strain energy $\mathit{\delta }U$ is given by ¹⁹ :

$\mathit{\delta }U={\int }_A\left\{{\int }_{-\frac{h}{2}}^{\frac{h}{2}}\mathit{\delta }{\left\{\mathit{\epsilon }\right\}}^T\left\{\mathit{\sigma }\right\}dz\right\}dxdy$

16

By using the stress components, equations (6) and (7) yield:

$\begin{array}{r}\mathit{\delta }U=\int {\int }_A\{\sum _{i=1}^{kc}\left({\int }_{z_{i-1}}^{z_i}\mathit{\delta }{\left\{\mathit{\epsilon }\right\}}^T{\left[\bar{Q}\right]}_i\left\{\mathit{\epsilon }\right\}dz\right)\\ \hspace{0.17em}+\sum _{i=1}^{ks}\left({\int }_{z_{i-1}}^{z_i}\mathit{\delta }{\left\{\mathit{\epsilon }\right\}}^T{\left[Q_s\left(\mathit{\zeta }\right)\right]}_i\left(\left\{\mathit{\epsilon }\right\}-\left\{{\mathit{\alpha }}_s\right\}\mathrm{\Delta }T\right)dz\right)\}dxdy\end{array}$

17

where z_i are z-coordinates in z-direction from the midplane. Knowing that ${\left[\bar{Q}\right]}_i$ (transformed stiffness matrix of layer (i)) is for the each core layers and ${\left[Q_s\left(\mathit{\zeta }\right)\right]}_i$ is for the SMA layers, kc and ks are the total number of core layers and SMA layers, respectively.

The strain energy can be divided into two parts as follows:

$\mathit{\delta }U=\mathit{\delta }{U}_m-\mathit{\delta }{U}_T$

18

where $\mathit{\delta }{U}_m$ and $\mathit{\delta }{U}_T$ are the mechanical and thermal components of the strain energy defined as follows:

$\mathit{\delta }{U}_m={\int }_A\left\{\begin{array}{c}\mathit{\delta }{\left\{{\mathit{\epsilon }}^0\right\}}^T\left[A\right]\left\{{\mathit{\epsilon }}^0\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^0\right\}}^T\left[B\right]\left\{{\mathit{\epsilon }}^1\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^0\right\}}^T\left[D\right]\left\{{\mathit{\epsilon }}^2\right\}\\ +\mathit{\delta }{\left\{{\mathit{\epsilon }}^0\right\}}^T\left[E\right]\left\{{\mathit{\epsilon }}^3\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^1\right\}}^T\left[B\right]\left\{{\mathit{\epsilon }}^0\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^1\right\}}^T\left[D\right]\left\{{\mathit{\epsilon }}^1\right\}\\ +\mathit{\delta }{\left\{{\mathit{\epsilon }}^1\right\}}^T\left[E\right]\left\{{\mathit{\epsilon }}^2\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^1\right\}}^T\left[F\right]\left\{{\mathit{\epsilon }}^3\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^2\right\}}^T\left[D\right]\left\{{\mathit{\epsilon }}^0\right\}\\ +\mathit{\delta }{\left\{{\mathit{\epsilon }}^2\right\}}^T\left[E\right]\left\{{\mathit{\epsilon }}^1\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^2\right\}}^T\left[F\right]\left\{{\mathit{\epsilon }}^2\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^2\right\}}^T\left[H\right]\left\{{\mathit{\epsilon }}^3\right\}\\ +\mathit{\delta }{\left\{{\mathit{\epsilon }}^3\right\}}^T\left[E\right]\left\{{\mathit{\epsilon }}^0\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^3\right\}}^T\left[F\right]\left\{{\mathit{\epsilon }}^1\right\}+\mathit{\delta }{\left\{{\mathit{\epsilon }}^3\right\}}^T\left[H\right]\left\{{\mathit{\epsilon }}^2\right\}\\ +\mathit{\delta }{\left\{{\mathit{\epsilon }}^3\right\}}^T\left[J\right]\left\{{\mathit{\epsilon }}^3\right\}\end{array}\right\}dxdy$

19

where the sub-matrices are given as:

$\begin{array}{r}\left[A\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}dz,\hspace{0.17em}\left[B\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}zdz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}zdz\\ \hspace{0.17em}& \left[D\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}{z}^{2}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}{z}^{2}dz,\hspace{0.17em}\left[E\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}{z}^{3}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}{z}^{3}dz\\ \hspace{0.17em}& \left[F\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}{z}^{4}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}{z}^{4}dz,\hspace{0.17em}\left[H\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}{z}^{5}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}{z}^{5}dz\\ \hspace{0.17em}& \left[J\right]=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i\hspace{0.17em}{z}^{6}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\left[Q_s\right]}_i\hspace{0.17em}{z}^{6}dz\end{array}$

20

$\mathit{\delta }{U}_T={\hspace{1em}}_A\left\{\mathit{\delta }{\left\{{\mathit{\epsilon }}^0\right\}}^T\left[A^{\mathit{\alpha }}\right]+\mathit{\delta }{\left\{{\mathit{\epsilon }}^1\right\}}^T\left[B^{\mathit{\alpha }}\right]+\mathit{\delta }{\left\{{\mathit{\epsilon }}^2\right\}}^T\left[D^{\mathit{\alpha }}\right]+\mathit{\delta }{\left\{{\mathit{\epsilon }}^3\right\}}^T\left[E^{\mathit{\alpha }}\right]\right\}dxdy$

21

where A, B, and D are extensional, coupling, and bending stiffness matrices, respectively, and E, F, and H are higher order matrices.

$\left(\left[A^{\mathit{\alpha }}\right],\left[B^{\mathit{\alpha }}\right],\left[D^{\mathit{\alpha }}\right],\left[E^{\mathit{\alpha }}\right]\right)=\sum _{i=1}^k{\int }_{z_{i-1}}^{z_i}{\left[\bar{Q}\right]}_i{\left\{\mathit{\alpha }\right\}}_i\mathrm{\Delta }T\left(1,\hspace{0.17em}z,\hspace{0.17em}{z}^2,\hspace{0.17em}{z}^3\right)dz$

22

The strain vectors $\left\{{\mathit{\epsilon }}^0\right\}$ , $\left\{{\mathit{\epsilon }}^1\right\}$ , $\left\{{\mathit{\epsilon }}^2\right\}$ , and $\left\{{\mathit{\epsilon }}^3\right\}$ are given in equation (4).

The virtual work $\mathit{\delta }V$ due to the applied forces can be written as:

$\mathit{\delta }V={\hspace{1em}}_A\left\{p_z\left(x,y\right)\mathit{\delta }w\left(x,y\right)\right\}dxdy+F_{zi}\mathit{\delta }w\left(x_i,y_i\right)$

23

where $p_z\left(x,y\right)$ is the transverse distributed load, and F_zi is the concentrated force in the z-direction at point i. The virtual kinetic energy δK can be written as ¹⁹ :

$\mathit{\delta }K={\hspace{1em}}_V\mathit{\rho }\left[\mathit{\delta }{\left\{\dot{U}\right\}}^T\left\{\dot{U}\right\}\right]dV$

24

where

${\left\{U\right\}}^T=\left[\begin{array}{ccc}u& v& w\end{array}\right].$

25

For kc is core layers and ks is SMA layers. Equation (24) is represented as

$\delta K={\displaystyle {\int }_A\left\{{\displaystyle \sum _{i=1}^{kc}{\displaystyle {\int }_{z_{i-1}}^{z_i}{\rho }_{c_i}\left[\delta {\left\{\dot{U}\right\}}^T\left\{\dot{U}\right\}\right]dz+{\displaystyle \sum _{i=1}^{ks}{\displaystyle {\int }_{z_{i-1}}^{z_i}{\rho }_{s_i}\left[\delta {\left\{\dot{U}\right\}}^T\left\{\dot{U}\right\}\right]dz}}}}\right\}}dxdy,$

26

where ${\mathit{\rho }}_{c_i}$ and ${\mathit{\rho }}_{s_i}$ are the density of core layers and SMA layers, respectively. The displacement field (equation (1)) can be written as

$\left\{U\right\}=\left\{U^0\right\}+z\left\{U^1\right\}+z^2\left\{U^2\right\}+z^3\left\{U^3\right\}$

27

where

$\begin{array}{rl}\hspace{0.17em}& {\left\{U^0\right\}}^T=\left[\begin{array}{ccc}{u}_0& v_0& w_0\end{array}\right],\hspace{0.17em}{\left\{U^1\right\}}^T=\left[\begin{array}{ccc}{\mathit{\theta }}_x-\frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }x}& {\mathit{\theta }}_y-\frac{\mathrm{\partial }{w}_0}{\mathrm{\partial }y}& {\mathit{\theta }}_z\end{array}\right]\\ \hspace{0.17em}& {\left\{U^2\right\}}^T=\left[\begin{array}{ccc}{\mathit{\psi }}_x& {\mathit{\psi }}_y& {\mathit{\psi }}_z\end{array}\right],\hspace{0.17em}{\left\{U^3\right\}}^T=\left[\begin{array}{ccc}{\mathit{\xi }}_x& {\mathit{\xi }}_y& 0\end{array}\right]\end{array}$

28

Substituting equation (27) into equation (26), we can write:

$\mathit{\delta }K={\int }_A\left(\begin{array}{c}{I}_0\mathit{\delta }{\left\{{\dot{U}}^0\right\}}^T\left\{{\dot{U}}^0\right\}+I_1\mathit{\delta }{\left\{{\dot{U}}^0\right\}}^T\left\{{\dot{U}}^1\right\}+I_2\mathit{\delta }{\left\{{\dot{U}}^0\right\}}^T\left\{{\dot{U}}^2\right\}+I_3\mathit{\delta }{\left\{{\dot{U}}^0\right\}}^T\left\{{\dot{U}}^3\right\}\\ +I_1\mathit{\delta }{\left\{{\dot{U}}^1\right\}}^T\left\{{\dot{U}}^0\right\}+I_2\mathit{\delta }{\left\{{\dot{U}}^1\right\}}^T\left\{{\dot{U}}^1\right\}+I_3\mathit{\delta }{\left\{{\dot{U}}^1\right\}}^T\left\{{\dot{U}}^2\right\}+I_4\mathit{\delta }{\left\{{\dot{U}}^1\right\}}^T\left\{{\dot{U}}^3\right\}\\ +I_2\mathit{\delta }{\left\{{\dot{U}}^2\right\}}^T\left\{{\dot{U}}^0\right\}+I_3\mathit{\delta }{\left\{{\dot{U}}^2\right\}}^T\left\{{\dot{U}}^1\right\}+I_4\mathit{\delta }{\left\{{\dot{U}}^2\right\}}^T\left\{{\dot{U}}^2\right\}+I_5\mathit{\delta }{\left\{{\dot{U}}^2\right\}}^T\left\{{\dot{U}}^3\right\}\\ +I_3\mathit{\delta }{\left\{{\dot{U}}^3\right\}}^T\left\{{\dot{U}}^0\right\}+I_4\mathit{\delta }{\left\{{\dot{U}}^3\right\}}^T\left\{{\dot{U}}^1\right\}+I_5\mathit{\delta }{\left\{{\dot{U}}^3\right\}}^T\left\{{\dot{U}}^2\right\}+I_6\mathit{\delta }{\left\{{\dot{U}}^3\right\}}^T\left\{{\dot{U}}^3\right\}\end{array}\right)dxdy$

29

$\left\{{\dot{U}}^0\right\}$ , $\left\{{\dot{U}}^1\right\}$ , $\left\{{\dot{U}}^2\right\}$ , and $\left\{{\dot{U}}^3\right\}$ are given in Appendix 2:

$\begin{array}{r}{I}_0=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}dz,\hspace{0.17em}{I}_1=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}zdz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}zdz\\ I_2=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}{z}^{2}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}{z}^{2}dz,\hspace{0.17em}{I}_3=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}{z}^{3}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}{z}^{3}dz\\ I_4=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}{z}^{4}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}{z}^{4}dz,\hspace{0.17em}{I}_5=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}{z}^{5}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}{z}^{5}dz\\ I_6=\sum _{i=1}^{kc}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{c_i}\hspace{0.17em}{z}^{6}dz+\sum _{i=1}^{ks}{\int }_{z_{i-1}}^{z_i}{\mathit{\rho }}_{s_i}\hspace{0.17em}{z}^{6}dz\end{array}$

30

Ritz solution technique

The unknown displacements vector $\left\{\mathit{\gamma }\right\}=\left\{u_0,\hspace{0.17em}{v}_0,\hspace{0.17em}{w}_0,\hspace{0.17em}{\mathit{\theta }}_x,\hspace{0.17em}{\mathit{\theta }}_y,\hspace{0.17em}{\mathit{\theta }}_z,\hspace{0.17em}{\mathit{\psi }}_x,\hspace{0.17em}{\mathit{\psi }}_y,\hspace{0.17em}{\mathit{\psi }}_z,\hspace{0.17em}{\mathit{\xi }}_x,{\mathit{\xi }}_y\right\}$ is approximated by x- and y-polynomial functions that satisfy the plate boundary conditions. Two types of boundary conditions are considered, simply supported and cantilevered. The Ritz functions used in the present study are listed in Appendix 2 ^17,18 :

Equations of motion

The equations of motion of the plate structure are derived using the Ritz approximation technique. ^17,18 The strain vectors ${\left\{{\mathit{\epsilon }}^0\right\}}^T$ , ${\left\{{\mathit{\epsilon }}^1\right\}}^T$ , ${\left\{{\mathit{\epsilon }}^2\right\}}^T$ , and ${\left\{{\mathit{\epsilon }}^3\right\}}^T$ can be expressed in terms of the generalized coordinates $\left\{q_i\right\}$ as follows:

$\begin{array}{rl}\hspace{0.17em}& \left\{{\mathit{\epsilon }}^0\right\}=\left[{\bar{\mathit{\epsilon }}}^0\left(x,y\right)\right]\left\{q\right\},\hspace{0.17em}\left\{{\mathit{\epsilon }}^1\right\}=\left[{\bar{\mathit{\epsilon }}}^1\left(x,y\right)\right]\left\{q\right\}\\ \hspace{0.17em}& \left\{{\mathit{\epsilon }}^2\right\}=\left[{\bar{\mathit{\epsilon }}}^2\left(x,y\right)\right]\left\{q\right\},\hspace{0.17em}\left\{{\mathit{\epsilon }}^3\right\}=\left[{\bar{\mathit{\epsilon }}}^3\left(x,y\right)\right]\left\{q\right\}\end{array}$

31

where {q} is the generalized column vector of Ritz coefficients:

${\left\{q\right\}}^T=\left[\begin{array}{c}{\left\{q_1\right\}}^T\hspace{0.17em}{\left\{q_2\right\}}^T\hspace{0.17em}{\left\{q_3\right\}}^T\hspace{0.17em}{\left\{q_4\right\}}^T\hspace{0.17em}{\left\{q_5\right\}}^T\hspace{0.17em}{\left\{q_6\right\}}^T\hspace{0.17em}{\left\{q_7\right\}}^T\hspace{0.17em}{\left\{q_8\right\}}^T\hspace{0.17em}{\left\{q_9\right\}}^T\hspace{0.17em}{\left\{q_{10}\right\}}^T\hspace{0.17em}{\left\{q_{11}\right\}}^T\end{array}\right]$

32

$\left[{\bar{\mathit{\epsilon }}}^0\left(x,y\right)\right]$ , $\left[{\bar{\mathit{\epsilon }}}^1\left(x,y\right)\right]$ , $\left[{\bar{\mathit{\epsilon }}}^2\left(x,y\right)\right]$ , and $\left[{\bar{\mathit{\epsilon }}}^3\left(x,y\right)\right]$ are given in Appendix 3.

By substituting the assumed displacement functions into the total virtual strain energy (equation (18)), the virtual work (equation 23), and the virtual kinetic energy (equation (26)), one can obtain the equations of motion for the structure system:

$\left[M\right]\left\{\stackrel{\dot{}\dot{}}{q}\right\}+\left[K\right]\left\{q\right\}=\left\{F\right\}+\left\{F_0\right\}+\left\{F_T\right\}$

33

where $\left[K\right]$ and $\left[M\right]$ are the laminated stiffness and mass matrices, $\left\{F\right\}$ and $\left\{F_0\right\}$ are the distributed and concentrated mechanical load vectors, and $\left\{F_T\right\}$ is the thermal load vector given by:

$\left[K\right]={\int }_A\left\{\begin{array}{c}{\left[{\bar{\mathit{\epsilon }}}^0\right]}^T\left[A\right]\left[{\bar{\mathit{\epsilon }}}^0\right]+{\left[{\bar{\mathit{\epsilon }}}^0\right]}^T\left[B\right]\left[{\bar{\mathit{\epsilon }}}^1\right]+{\left[{\bar{\mathit{\epsilon }}}^0\right]}^T\left[D\right]\left[{\bar{\mathit{\epsilon }}}^2\right]+{\left[{\bar{\mathit{\epsilon }}}^0\right]}^T\left[E\right]\left[{\bar{\mathit{\epsilon }}}^3\right]\\ +{\left[{\bar{\mathit{\epsilon }}}^1\right]}^T\left[B\right]\left[{\bar{\mathit{\epsilon }}}^0\right]+{\left[{\bar{\mathit{\epsilon }}}^1\right]}^T\left[D\right]\left[{\bar{\mathit{\epsilon }}}^1\right]+{\left[{\bar{\mathit{\epsilon }}}^1\right]}^T\left[E\right]\left[{\bar{\mathit{\epsilon }}}^2\right]+{\left[{\bar{\mathit{\epsilon }}}^1\right]}^T\left[F\right]\left[{\bar{\mathit{\epsilon }}}^3\right]\\ +{\left[{\bar{\mathit{\epsilon }}}^2\right]}^T\left[D\right]\left[{\bar{\mathit{\epsilon }}}^0\right]+{\left[{\bar{\mathit{\epsilon }}}^2\right]}^T\left[E\right]\left[{\bar{\mathit{\epsilon }}}^1\right]+{\left[{\bar{\mathit{\epsilon }}}^2\right]}^T\left[F\right]\left[{\bar{\mathit{\epsilon }}}^2\right]+{\left[{\bar{\mathit{\epsilon }}}^2\right]}^T\left[H\right]\left[{\bar{\mathit{\epsilon }}}^3\right]\\ +{\left[{\bar{\mathit{\epsilon }}}^3\right]}^T\left[E\right]\left[{\bar{\mathit{\epsilon }}}^0\right]+{\left[{\bar{\mathit{\epsilon }}}^3\right]}^T\left[F\right]\left[{\bar{\mathit{\epsilon }}}^1\right]+{\left[{\bar{\mathit{\epsilon }}}^3\right]}^T\left[H\right]\left[{\bar{\mathit{\epsilon }}}^2\right]+{\left[{\bar{\mathit{\epsilon }}}^3\right]}^T\left[J\right]\left[{\bar{\mathit{\epsilon }}}^3\right]\end{array}\right\}dxdy$

34

$\left[M\right]={\int }_A\left(\begin{array}{c}{I}_0\mathit{\delta }{\left[{\bar{U}}^0\right]}^T\left[{\bar{U}}^0\right]+I_1\mathit{\delta }{\left[{\bar{U}}^0\right]}^T\left[{\bar{U}}^1\right]+I_2\mathit{\delta }{\left[{\bar{U}}^0\right]}^T\left[{\bar{U}}^2\right]+I_3\mathit{\delta }{\left[{\bar{U}}^0\right]}^T\left[{\bar{U}}^3\right]\\ +I_1\mathit{\delta }{\left[{\bar{U}}^1\right]}^T\left[{\bar{U}}^0\right]+I_2\mathit{\delta }{\left[{\bar{U}}^1\right]}^T\left[{\bar{U}}^1\right]+I_3\mathit{\delta }{\left[{\bar{U}}^1\right]}^T\left[{\bar{U}}^2\right]+I_4\mathit{\delta }{\left[{\bar{U}}^1\right]}^T\left[{\bar{U}}^3\right]\\ +I_2\mathit{\delta }{\left[{\bar{U}}^2\right]}^T\left[{\bar{U}}^0\right]+I_3\mathit{\delta }{\left[{\bar{U}}^2\right]}^T\left[{\bar{U}}^1\right]+I_4\mathit{\delta }{\left[{\bar{U}}^2\right]}^T\left[{\bar{U}}^2\right]+I_5\mathit{\delta }{\left[{\bar{U}}^2\right]}^T\left[{\bar{U}}^3\right]\\ +I_3\mathit{\delta }{\left[{\bar{U}}^3\right]}^T\left[{\bar{U}}^0\right]+I_4\mathit{\delta }{\left[{\bar{U}}^3\right]}^T\left[{\bar{U}}^1\right]+I_5\mathit{\delta }{\left[{\bar{U}}^3\right]}^T\left[{\bar{U}}^2\right]+I_6\mathit{\delta }{\left[{\bar{U}}^3\right]}^T\left[{\bar{U}}^3\right]\end{array}\right)dxdy$

35

${\left\{F_T\right\}}^T={\int }_A\left\{{\left[{\bar{\mathit{\epsilon }}}^0\right]}^T\left[A^{\mathit{\alpha }}\right]+{\left[{\bar{\mathit{\epsilon }}}^1\right]}^T\left[B^{\mathit{\alpha }}\right]+{\left[{\bar{\mathit{\epsilon }}}^2\right]}^T\left[D^{\mathit{\alpha }}\right]+{\left[{\bar{\mathit{\epsilon }}}^3\right]}^T\left[E^{\mathit{\alpha }}\right]\right\}dxdy$

36

$\begin{array}{r}{\left\{F\right\}}^T{=}_{\left[\begin{array}{c}\mathrm{0\; 0}{p}_z\left(x,y\right){\left\{a_3\right\}}^T\mathrm{0\; 0\; 0\; 0\; 0\; 0\; 0\; 0}\end{array}\right]}dxdy\\ {\left\{F_0\right\}}^T=\left[\begin{array}{c}\mathrm{0\; 0}{F}_{zi}{\left\{a_3\left(x_i,y_i\right)\right\}}^T\mathrm{0\; 0\; 0\; 0\; 0\; 0\; 0\; 0}\end{array}\right]\end{array}$

37

The terms $\left[{\bar{\mathit{\epsilon }}}^0\right]$ , $\left[{\bar{\mathit{\epsilon }}}^1\right]$ , $\left[{\bar{\mathit{\epsilon }}}^2\right]$ , and $\left[{\bar{\mathit{\epsilon }}}^3\right]$ are given in Appendix 3.

In case of free vibration, the right-hand side of equation (33) is equal to zero $\left(\left\{F\right\}=\left\{F_0\right\}=\left\{F_T\right\}=0\right)$ , thus equation (33) becomes:

$\left[K\right]\left\{q\right\}-\mathit{\lambda }\left[M\right]\left\{q\right\}=0$

38

And the natural frequency is given by:

$\mathit{\omega }=\sqrt{\mathit{\lambda }}$

39

which is computed at certain time t during the phase transformation.

Heat transfer equation

Modeling of the heat equation of SMA layer

The SMA layer is activated when exposed to a source of heat. Heat power is considered as a homogeneous heat source in the SMA layer. The thermal energy of the activated SMA layer is assumed to be lost by conduction through the core structure. A two-dimensional temperature distribution is considered by taking element dx–dy of the SMA layer, the energy balance equation can be given as ⁵ :

$C_s\frac{\mathrm{\partial }T}{\mathrm{\partial }t}-Q_s\frac{\mathrm{\partial }\mathit{\zeta }}{\mathrm{\partial }t}-k_s\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{x}^2}-k_s\frac{{\mathrm{\partial }}^{2}T}{\mathrm{\partial }{y}^2}+K_e\left(T-T_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}}\right)-P=0$

40

where $C_s\left(\mathrm{\partial }T/\phantom{\mathrm{\partial }T\mathrm{\partial }t}\mathrm{\partial }t\right)$ is the rate of change in the internal energy of the SMA layer. $C_s={\mathit{\rho }}_s{c}_s$ , in which ${\mathit{\rho }}_s$ and $c_s$ are the density in $\left[{\mathrm{k}\mathrm{g}\mathrm{m}}^{-3}\right]$ and the specific heat in $\left[\mathrm{J}\hspace{0.17em}{\mathrm{k}\mathrm{g}}^{-1}{{\int }^{\circ }\mathrm{C}}^{-1}\right]$ , respectively. $-\hspace{0.17em}{Q}_s\left(\mathrm{\partial }\mathit{\zeta }/\phantom{\mathrm{\partial }\mathit{\zeta }\mathrm{\partial }t}\mathrm{\partial }t\right)$ is the rate of energy contributed to the phase transformation of the SMA layer. Here $Q_s={\mathit{\rho }}_s{q}_s$ and q_s are the heat of transitions in $\left[J\hspace{0.17em}k{g}^{-1}\right]$ . $-k_s\left({\mathrm{\partial }}^{2}T/\phantom{{\mathrm{\partial }}^{2}T\mathrm{\partial }{x}^2}\mathrm{\partial }{x}^2\right)-k_s\left({\mathrm{\partial }}^{2}T/\phantom{{\mathrm{\partial }}^{2}T\mathrm{\partial }{y}^2}\mathrm{\partial }{y}^2\right)$ is the rate of heat conduction in the x- and y-directions through the SMA layer, and k_s is the thermal conductivity in $\left[W{m}^{-1\circ }{C}^{-1}\right]$ . $K_E\left(T-T_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}}\right)$ is the quasi-steady model for the heat lost by conduction through the core layer, when the core heat capacity is neglected. ¹⁹ $K_E=k_e/\phantom{k_e\left[t_s\left(h-t_s\right)\right]}\left[t_s\left(h-t_s\right)\right]$ ⁵ where k_e is the thermal conductivity of the core layers in $\left[W{m}^{-1\circ }{C}^{-1}\right]$ . T and T _sink are the temperatures in $\left[{\int }^{\circ }\mathrm{C}\right]$ of the SMA layer and the heat sink, which has a constant temperature and bonded to the lower surface of the core.

P is the input heat power density in $\left[W/\phantom{W{m}^3}{m}^3\right]$ generated by the electric current passed though the SMA layer in the heating process. P is zero if equation (40) is applied to the cooling process. t is the time in $\left(s\right)$ . Equation (40) can be written in dimensionless form as:

$\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }\mathit{\tau }}-R_{MA}\frac{\mathrm{\partial }\mathit{\zeta }}{\mathrm{\partial }\mathit{\tau }}-\frac{{\mathrm{\partial }}^2\mathit{\theta }}{\mathrm{\partial }{X}^2}-\frac{{\mathrm{\partial }}^2\mathit{\theta }}{\mathrm{\partial }{Y}^2}+R_k\left(\mathit{\theta }-S\right)-R_P=0$

41

where θ is the nondimensional temperature of the SMA, τ is the dimensionless time, R_MA is the dimensionless heat of transition of the SMA, R_k is the dimensionless thermal conductivity of the core, S is the heat sink strength that characterizes the heat conduction loss from the SMA through the core to the heat sink, and R_P is the dimensionless input power to the SMA layer. The dimensionless symbols are defined in Appendix 4.

Solution of the heat equation

The variational formulation of equation (41) over the plate area Ω is obtained by multiplying the equation by a weight function N_i :

${\int }_{\mathrm{\Omega }}{N}_i\left\{\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }\mathit{\tau }}-R_{MA}\frac{\mathrm{\partial }\mathit{\zeta }}{\mathrm{\partial }\mathit{\tau }}-\frac{{\mathrm{\partial }}^2\mathit{\theta }}{\mathrm{\partial }{X}^2}-\frac{{\mathrm{\partial }}^2\mathit{\theta }}{\mathrm{\partial }{Y}^2}+R_k\left(\mathit{\theta }-S\right)-R_P\right\}dXdY=0$

42

Integrating by parts and rearranging the equation elements yields:

${\int }_{\mathrm{\Omega }}\left\{N_i\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }\mathit{\tau }}-N_i\hspace{0.17em}{R}_{MA}\frac{\mathrm{\partial }\mathit{\zeta }}{\mathrm{\partial }\mathit{\tau }}-\frac{\mathrm{\partial }{N}_i}{\mathrm{\partial }X}\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }X}-\frac{\mathrm{\partial }{N}_i}{\mathrm{\partial }Y}\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }Y}+N_i\hspace{0.17em}{R}_k\left(\mathit{\theta }-S\right)-N_i\hspace{0.17em}{R}_P\right\}dXdY={\int }_{\mathrm{\Gamma }}{N}_i\left(\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }X}{n}_x-\frac{\mathrm{\partial }\mathit{\theta }}{\mathrm{\partial }Y}{n}_y\right)ds$

43

where n_x and n_y are the direction cosines of the boundary surface in the x- and y-directions. The variables θ and $\mathit{\zeta }$ are approximated over the plate by the following interpolation functions:

$\begin{array}{rlll}\hspace{0.17em}& \mathit{\theta }=\sum _{i=1}^4{N}_i\left(X,Y\right){\bar{\mathit{\theta }}}_i\left(\mathit{\tau }\right)& \hspace{0.17em}& \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}& \mathit{\zeta }=\sum _{i=1}^4{N}_i\left(X,Y\right){\bar{\mathit{\zeta }}}_i\left(\mathit{\tau }\right)\end{array}$

44

where $N_i\left(i=1,\hspace{0.17em}2,\hspace{0.17em}3,\hspace{0.17em}4\right)$ are the linear interpolation functions of the plate in x- and y-directions. ${\bar{\mathit{\theta }}}_i$ and ${\bar{\mathit{\zeta }}}_i$ are the unknown temperature and martensite fraction coefficients, respectively. Substituting equation (44) in equation (43) yields:

$\left[C^1\right]\left\{\frac{\mathrm{\partial }\bar{\mathit{\theta }}}{\mathrm{\partial }\mathit{\tau }}\right\}+\left[C^2\right]\left\{\bar{\mathit{\theta }}\right\}+\left[C^3\right]\left\{\frac{\mathrm{\partial }\bar{\mathit{\zeta }}}{\mathrm{\partial }\mathit{\tau }}\right\}=\left\{F^{th}\right\}$

45

where

$\begin{array}{r}{C}_{ij}^1{=}_{\mathrm{\Omega }}{N}_i{N}_{j}dXdY\\ C_{ij}^2{=}_{\mathrm{\Omega }}\left\{\frac{\mathrm{\partial }{N}_i}{\mathrm{\partial }X}\frac{\mathrm{\partial }{N}_j}{\mathrm{\partial }X}+\frac{\mathrm{\partial }{N}_i}{\mathrm{\partial }Y}\frac{\mathrm{\partial }{N}_j}{\mathrm{\partial }Y}+\hspace{0.17em}{R}_k{N}_i{N}_j\right\}dXdY\\ C_{ij}^3={-}_{\mathrm{\Omega }}\hspace{0.17em}{R}_{MA}{N}_i{N}_{j}dXdY\\ F_i^{th}{=}_{\mathrm{\Omega }}\left(R_P-R_{k}S\right)N_{i}dXdY{+}_{\mathrm{\Gamma }}{N}_i\left(\frac{\mathrm{\partial }\bar{\mathit{\theta }}}{\mathrm{\partial }X}{n}_x-\frac{\mathrm{\partial }\bar{\mathit{\theta }}}{\mathrm{\partial }Y}{n}_y\right)ds\end{array}$

The α-family method is used to transfer equation (45) from ordinary differential equations to a set of algebraic equations, in which a weighted average of the time derivative of a dependent variable is approximated at two consecutive time steps by linear interpolation of the values of the variable at the two steps.

For different values of α (0 < α < 1), the numerical integration schemes can be obtained. ²⁰

Assume constant time step Δt gives:

$\mathit{\alpha }{\left\{\frac{\mathrm{\partial }\bar{\mathit{\theta }}}{\mathrm{\partial }\mathit{\tau }}\right\}}^{n+1}+\left(1-\mathit{\alpha }\right){\left\{\frac{\mathrm{\partial }\bar{\mathit{\theta }}}{\mathrm{\partial }\mathit{\tau }}\right\}}^n=\frac{{\left\{\bar{\mathit{\theta }}\right\}}^{n+1}-{\left\{\bar{\mathit{\theta }}\right\}}^n}{\mathrm{\Delta }\mathit{\tau }}$

46

By rearranging terms of equation (45) gives:

$\begin{array}{rl}\left(\left[C^1\right]+\mathrm{\Delta }\mathit{\tau }\hspace{0.17em}\mathit{\alpha }\left[C^2\right]\right){\left\{\bar{\mathit{\theta }}\right\}}^{n+1}& =\left(\left[C^1\right]-\left(1-\mathit{\alpha }\right)\mathrm{\Delta }\mathit{\tau }\left[C^2\right]\right){\left\{\bar{\mathit{\theta }}\right\}}^n\\ \hspace{0.17em}& \hspace{0.17em}+\mathrm{\Delta }\mathit{\tau }\hspace{0.17em}\left(\mathit{\alpha }{\left\{F^{th}\right\}}^{n+1}+\left(1-\mathit{\alpha }\right){\left\{F^{th}\right\}}^n\right)\\ \hspace{0.17em}& \hspace{0.17em}-\mathrm{\Delta }\mathit{\tau }\left[C^3\right]\left(\mathit{\alpha }{\left\{\frac{\mathrm{\partial }\bar{\mathit{\zeta }}}{\mathrm{\partial }\mathit{\tau }}\right\}}^{n+1}+\left(1-\mathit{\alpha }\right){\left\{\frac{\mathrm{\partial }\bar{\mathit{\zeta }}}{\mathrm{\partial }\mathit{\tau }}\right\}}^n\right)\end{array}$

47

Knowing that

$\mathrm{\Delta }\mathit{\tau }\left(\mathit{\alpha }{\left\{\frac{\mathrm{\partial }\bar{\mathit{\zeta }}}{\mathrm{\partial }\mathit{\tau }}\right\}}^{n+1}+\left(1-\mathit{\alpha }\right){\left\{\frac{\mathrm{\partial }\bar{\mathit{\zeta }}}{\mathrm{\partial }\mathit{\tau }}\right\}}^n\right)={\left\{\bar{\mathit{\zeta }}\right\}}^{n+1}-{\left\{\bar{\mathit{\zeta }}\right\}}^n$

48

Thus, equation (47) can be written in a set of algebraic equations as follows:

$\left[\hat{K}\right]{\left\{\bar{\mathit{\theta }}\right\}}^{n+1}=\left\{\hat{F}\right\}$

49

where $\left[\hat{K}\right]=\left(\left[C^1\right]+\mathrm{\Delta }\mathit{\tau }\hspace{0.17em}\mathit{\alpha }\left[C^2\right]\right)$

$\left\{\hat{F}\right\}=\left(\left[C^1\right]-\left(1-\mathit{\alpha }\right)\mathrm{\Delta }\mathit{\tau }\left[C^2\right]\right){\left\{\bar{\mathit{\theta }}\right\}}^n+\mathrm{\Delta }\mathit{\tau }\hspace{0.17em}\left\{F^{th}\right\}-\left[C^3\right]\left({\left\{\bar{\mathit{\zeta }}\right\}}^{n+1}-{\left\{\bar{\mathit{\zeta }}\right\}}^n\right)$

Solution procedure

It is obvious that the equations of motion equation (33) and the heat equations equation (49) are coupled nonlinear equations. They can be decoupled if $\mathit{\zeta }$ is known. Since the solution of these two equations is a time-marching initial-value problem, a good estimate of $\mathit{\zeta }$ can be made based on the previous value at each time step. Once $\mathit{\zeta }$ is known, equation (49) is solved for $\left\{\mathit{\theta }\right\}$ and equation (33) is solved for $\left\{q\right\}$ . The step-by-step procedure for solving equations, equations (33) and (49) are given as follows:

Select initial values for ${\mathit{\theta }}^0$ and ${\mathit{\zeta }}^0$ and solve equation (28) for the initial deformation ${\left\{q\right\}}^0$ of the plate due to mechanical loading. To find natural frequency, Eigenvalue problem of equation (28) is solved based on ${\mathit{\zeta }}^0$ .

Calculate the new time $t_{n+1}=t_n+\mathrm{\Delta }t$ and estimate the martensite fraction ${\mathit{\zeta }}_{est}^{n+1}$ at $t_{n+1}$ by the free response model ${\mathit{\zeta }}_{est}^{n+1}=1-{\mathit{\theta }}_{est}^{n+1}$ , where ${\mathit{\theta }}_{est}^{n+1}$ is obtained by replacing the term $\left\{\frac{\mathrm{\partial }\bar{\mathit{\zeta }}}{\mathrm{\partial }\mathit{\tau }}\right\}$ by $\left\{-\frac{\mathrm{\partial }\bar{\mathit{\theta }}}{\mathrm{\partial }\mathit{\tau }}\right\}$ in equation (45).

Solve equation (33) for ${\left\{q\right\}}^{n+1}$ based on ${\mathit{\zeta }}_{est}^{n+1}$ and ${\mathit{\theta }}_{est}^{n+1}$ .

Solve equation (7) for ${\left\{\mathit{\sigma }\right\}}^{n+1}$ based on ${\left\{q\right\}}^{n+1}$ , ${\mathit{\theta }}_{est}^{n+1}$ , and ${\mathit{\zeta }}_{est}^{n+1}$ .

Calculate ${\mathit{\zeta }}^{n+1}$ using either linear or Brinson’s SMA constitutive models (equations 12 and 13) or equation (14) based on ${\left\{\mathit{\sigma }\right\}}^{n+1}$ .

Solve equation (49) for ${\mathit{\theta }}^{n+1}$ based on ${\mathit{\zeta }}^n$ and ${\mathit{\zeta }}^{n+1}$ .

Check convergence by testing the value $\left({\mathit{\zeta }}^{n+1}-{\mathit{\zeta }}_{est}^{n+1}\right)$ . If the convergence criterion is satisfied, repeat the above steps with a time increment $\mathrm{\Delta }t$ . If the convergence criterion is not satisfied, calculate the new estimated ${\mathit{\zeta }}_{est}^{n+1}$ by using relaxation. ³

Numerical results and discussion

• 1. The convergence and the validation of the present model is checked first for a laminated composite cantilever square plate (a = b = L) with side-to-thickness ratio $\mathit{\lambda }=L/h=\mathrm{1}00$ . The material properties of the individual layers are given in Table 1. The plate has symmetric lamination sequence [0/90/0]. A transverse uniform load (q ₀) is applied.

Table 1.

Material constants of the individual layers.

E 11/E 22	E 33/E 22	G 12/E 22 = G 13/E 22 = G 23/E 22	${\mathit{\nu }}_{12}={\mathit{\nu }}_{13}={\mathit{\nu }}_{23}$
15.4	1	0.5	0.3

The convergence of the normalized maximum deflection $\left(\bar{w}\right)$ and normalized stress resultants $\left({\bar{M}}_x\hspace{0.17em}and\hspace{0.17em}{\bar{M}}_y\right)$ are given in Figure 2. The results are given in normalized quantities as $\left({\bar{M}}_x,{\bar{M}}_y,{\bar{M}}_{xy}\right)={10}^4/q_0{a}^2\left(M_x,M_y,M_{xy}\right)$ and $\bar{w}=\left({10}^4{E}_{11}{h}^3/\phantom{{10}^4{E}_{11}{h}^3{a}^4{q}_0}{a}^4{q}_0\right)w_0$ .

In order to find suitable shape functions for the cantilever plate, the unknown displacements $u_0,\hspace{0.17em}{v}_0$ , ${\mathit{\theta }}_x,\hspace{0.17em}{\mathit{\theta }}_y,\hspace{0.17em}{\mathit{\theta }}_z$ , ${\mathit{\psi }}_x,\hspace{0.17em}{\mathit{\psi }}_y,\hspace{0.17em}{\mathit{\psi }}_z$ , ${\mathit{\xi }}_x,\hspace{0.17em}{\mathit{\xi }}_y$ are approximated by the x- and y-dependent functions, listed in. The transverse deflection $w_0$ approximate functions are alternated such that the number of terms of $w_0$ are increased from 1 to 49, and the total degrees of freedom are raised from 41 to 89. These approximate functions are the result of multiplication of two polynomials in x- and y-directions that satisfy the geometric boundary conditions using Pascal triangle.

Using three-term polynomials gives acceptable accuracy, as seen in Figure 2. Moreover, after the three-term polynomials, the stiffness matrix becomes ill conditioned, and the numerical solution begins to have a significant error. Thus, three-term polynomials are used in x- and y-directions to represent the transverse deflection $w_0\left(x,y\right)$ , which gives 49 total degrees of freedom.

Figure 2.

Convergence of the normalized maximum deflection $\bar{w}$ and stress resultants, ${\bar{M}}_x,\hspace{0.17em}{\bar{M}}_y\mathrm{a}\mathrm{t}x=0,y=b/2$ .

Table 2 shows the values of normalized deflection $\bar{w}$ and stress resultants ${\bar{M}}_x$ , ${\bar{M}}_y$ , and ${\bar{M}}_{xy}$ calculated by the present model in comparison with the available published results. The difference (Δ%) mentioned in Table 1 is the percentage difference in $\bar{w}$ , and, ${\bar{M}}_x$ , ${\bar{M}}_y$ relative to the reference values calculated by the finite element method using 2205 degrees of freedom, ²¹ listed in Table 2 with code FEM5. The comparison shows good agreement with the exact solutions.

Table 2.

Normalized maximum deflection $\bar{w}$ and stress resultants ${\bar{M}}_x$ , ${\bar{M}}_y$ , and ${\bar{M}}_{xy}$ of a laminated composite cantilever plate.

Reference	Code	Total degrees of freedom	x = 0, y = b/2		x = 0, y = b		x = 0, y = b/2
			$-\bar{w}$	Δ%	$-\bar{w}$	Δ%	${\bar{M}}_x$	Δ%	${\bar{M}}_y$	Δ%	${\bar{M}}_{xy}$
Present model	PM	49	15,502.8	0.348396	15,377.2	0.413186	5058.5	−0.70675	109.732	−8.64554	0
Ritz method 21	RM1	108	15,501	0.359967	15,382	0.3821	4967	1.114872	100	0.990099	0
	RM2	147	15,501	0.359967	15,384	0.369147	5068	−0.89588	102	−0.9901	0
	RM3	192	15,501	0.359967	15,384	0.369147	5078	−1.09496	102	−0.9901	0
Finite element method 21	FEM1	245	15,614	−0.36639	15,482	−0.26553	4261	15.17022	86	14.85149	0
	FEM2	605	15,582	−0.1607	15,459	−0.11657	4552	9.376866	92	8.910891	0
	FEM3	2205	15,567	−0.06428	15,449	−0.05181	4786	4.718296	97	3.960396	0
	FEM4	845	15,556	0.006428	15,441	0	5006	0.338443	101	0	0
	FEM5	2205	15,557	0	15,441	0	5023	0	101	0	0

Table 2 shows good agreement between the results calculated from the present model and those calculated using Ritz method and finite element method published in the work by Qatu and Algothani. ²¹

• 2. To verify the present thermal model, two cases are presented. Case (I) studies the stress-free thermal response of SMA-laminated plate, while the phase transformation response of a cantilever plate subjected to mechanical and thermal loads is considered in case (II). Parametric studies are then presented in case (III) to investigate the effect of plate thickness, aspect ratio, material properties, thermal expansion coefficient, and SMA sheet thickness on the transverse deflections, natural frequencies, and SMA’s response time. A set of computer programs are developed for this investigation using Mathematica 7.

Case (I): Free thermal response of SMA-laminated plate

The plate consists of 55-Nitinol for the SMA layer glued on the top of Dow Corning SYLGARD core. The SMA layer thickness $t_s=0.25\times {10}^{-3}m$ and a core layer thickness $(h-t_s)=2.25\times {10}^{-3}m$ . The total plate thickness is $h=2.5\times {10}^{-3}m$ . The reason for chosen such dimensions for the thickness of the SMA layers and the core layers is to compare the obtained results with the reference Wirtz. ²² The plate dimensions are taken as $a=b=0.25\hspace{0.17em}m$ . The material properties are given in Table 3. ^{3 –22}

Table 3.

Material properties for SMA and plate core. ^{3 –22}

Property	SMA (55-Nitinol)	Elastomer Sylgard
Density $\mathit{\rho }\hspace{0.17em}\left(\mathrm{k}\mathrm{g}/\phantom{\mathrm{k}\mathrm{g}{\mathrm{m}}^{\mathrm{3}}}{\mathrm{m}}^{\mathrm{3}}\right)$	6500	1050
Specific heat $c\hspace{0.17em}(\mathrm{J}/\phantom{\mathrm{J}(\mathrm{k}\mathrm{g}\hspace{0.17em}{\int }^{\circ }\mathrm{C})}(\mathrm{k}\mathrm{g}\hspace{0.17em}{\int }^{\circ }\mathrm{C}))$	883	1422
Thermal conductivity $k\hspace{0.17em}(W/\phantom{W(m\hspace{0.17em}{\int }^{\circ }\mathrm{C})}(m\hspace{0.17em}{\int }^{\circ }\mathrm{C}))$	17	0.146
Thermal expansion coefficient $\mathit{\alpha }\hspace{0.17em}({{\int }^{\circ }\mathrm{C}}^{-1})$	23×10−6
Heat of transition $q_s\hspace{0.17em}(\mathrm{J}/\phantom{\mathrm{J}\mathrm{k}\mathrm{g}}\mathrm{k}\mathrm{g})$	12,600
Material constant $C_A\hspace{0.17em}(\mathrm{M}\mathrm{P}\mathrm{a}/\phantom{\mathrm{M}\mathrm{P}\mathrm{a}\hspace{0.17em}{\int }^{\circ }\mathrm{C}}\hspace{0.17em}{\int }^{\circ }\mathrm{C})$	10.3
Material constant $C_M\hspace{0.17em}(\mathrm{M}\mathrm{P}\mathrm{a}/\phantom{\mathrm{M}\mathrm{P}\mathrm{a}\hspace{0.17em}{\int }^{\circ }\mathrm{C}}\hspace{0.17em}{\int }^{\circ }\mathrm{C})$	10.3
Young’s modulus of core $E_c\hspace{0.17em}(\mathrm{M}\mathrm{P}\mathrm{a})$		1300
Young’s modulus in martensite phase $E_M\hspace{0.17em}(\mathrm{M}\mathrm{P}\mathrm{a})$	26,300
Young’s modulus in austenite phase $E_A\hspace{0.17em}(\mathrm{M}\mathrm{P}\mathrm{a})$	67,000
Poisson’s ratio $\mathit{\nu }$	0.3	0.3
Martensite start temperature $M_s\hspace{0.17em}({\int }^{\circ }\mathrm{C})$	18.4
Martensite finish temperature $M_f\hspace{0.17em}({\int }^{\circ }\mathrm{C})$	9
Austenite start temperature $A_s\hspace{0.17em}({\int }^{\circ }\mathrm{C})$	34.5
Austenite finish temperature $A_f\hspace{0.17em}({\int }^{\circ }\mathrm{C})$	49

SMA: shape-memory alloy.

If we assume for validation purpose that $A_f-A_s=M_s-M_f=10{\int }^{\circ }\mathrm{C}$ , ²² then for 55-Nitinol $R_{ma}=1.4$ . The response time of the actuator of the SMA layer for martensite-to-austenite heating transition for an applied power $P=7.787\times {10}^{7}W/m^3$ $\left(R_p=0.2863\right)$ , and for austenite-to-martensite cooling transition for a sink temperature $T_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}}=-21{\int }^{\circ }\mathrm{C}$ (S = 3) are shown in Table 4 for both linear and Brinson’s SMA model and compared with Wirtz. ²² The response time is also calculated from the conservation of thermal energy in the SMA layer from basic heat transfer equation. ²³

$Q_{generated}=Q_{body}+Q_{dissipated}$

50

Table 4.

Response time of the actuator.

Model		Present (linear SMA model)	Present (Brinson’s SMA model)	Basic heat transfer equations24	Wirtz 22
Heating response	Time (s)	17.83	18.99	17.89	20.9
	τ	8.45	9	8.48	9.9
Cooling response	Time (s)	15.5	15.5	15.46	15.5
	τ	7.35	7.35	7.32	7.36

SMA: shape-memory alloy.

Case (II): Phase transformation of SMA-laminated plate

The phase transformation response of SMA layer located at the top, subjected to a uniformly distributed load, $q=-1000N/\phantom{N{m}^2}{m}^2$ and input heating power $P=5.6\times {10}^8\hspace{0.17em}W/\phantom{W{m}^3}{m}^3$ is studied. The plate dimensions are $a=0.25\hspace{0.17em}m$ , $b=0.14\hspace{0.17em}m$ with five layers for the core of thickness $1\times {10}^{-3}m$ each. The thickness of the SMA layer is $0.05\times {10}^{-3}m$ . The time step $\mathrm{\Delta }t=0.01\hspace{0.17em}\mathrm{s}$ is used. The material properties are given in Table 1. ³ The variations in martensite fraction in the activated SMA layer using the linear and the Brinson’s models are shown in Figure 3. In comparison with the results obtained by Balapgol ³ and Ghomshei. ² A good agreement is found between the proposed model and the mentioned references results. Known that in such a case the obtained results are obtained at certain small incremental time Δt, and it is assumed that there is no changes in the phase state of the SMA layer due to the uniform distributed mechanical load q, and subsequently due to the resultant stresses distribution through the thickness.

Figure 3.

Variation of martensite fraction $\left(\mathit{\zeta }\right)$ in activated SMA layer. SMA: shape-memory alloy.

Case (III): Parametric study of SMA composite plate

The plate used in case (I) is used for such parametric studies. The plate thickness is divided into 10 layers, the top layer is SMA, and the remaining 9 layers form the plate core. The material properties are given in Table 3 and the plate dimensions are shown in Figure 4. The plate span is $a=100\hspace{0.17em}\mathrm{m}\mathrm{m}$ , width $b=25\hspace{0.17em}\mathrm{m}\mathrm{m}$ , and thickness $h=3\hspace{0.17em}\mathrm{m}\mathrm{m}$ . The thickness of the SMA layer is $t_s=0.3\hspace{0.17em}\mathrm{m}\mathrm{m}$ . The plate is assumed to be thermally insulated on all four edges. The plate carries a downward load $F_z=\mathrm{2}0\hspace{0.17em}\mathrm{N}$ at the free end. The value of dimensionless time interval $\mathrm{\Delta }\mathit{\tau }=0.1$ is selected. An input heat power $P=2\times {10}^{7}W/m^3$ is used in the heating phase, and the sink temperature is $T_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}}=-20{\int }^{\circ }\mathrm{C}$ through the whole heating–cooling cycle.

Figure 4.

SMA-laminated cantilever plate. SMA: shape-memory alloy.

As the first step, the deflection response is calculated neglecting the effect of thermal expansion of the SMA layer. Then it is calculated considering the thermal expansion effect on the SMA layer only because the core layer is assumed at a constant temperature equal to the sink temperature, and the heating is only on the SMA layer.

Figure 5 shows the temperature response of the given plate in the heating–cooling cycle with input heat power $P=2\times {10}^{7}W/m^3$ applied until reaching the austenite-finish temperature (A_f ). Figure 6 shows the variation of the martensite fraction with the response time of the plate in the heating–cooling cycle. While heating from A_s to A_f , the SMA layer is transformed from fully martensite phase to fully austenite phase. This is an endothermic process which absorbs part of the input heat causing decrease in the heating rate of the SMA layer as seen in Figure 5. This transformation also causes reduction of the martensite fraction from 1 at A_s to 0 at A_f as shown in Figure 6. While cooling from M_s to M_f the SMA layer is transformed from fully austenite phase to fully martensite phase. This is an exothermic process, which emits heat causing decrease in the cooling rate of the SMA layer as seen in Figure 5. This transformation then increases the martensite fraction from 0 at M_s to 1 at M_f as seen in Figure 6.

Figure 5.

Temperature response of the SMA-laminated cantilever plate in the heating–cooling cycle with input power P = 2 × 10⁷ W/m³. SMA: shape-memory alloy.

Figure 6.

Variation of the martensite fraction with time response of the plate in the heating–cooling cycle with input power P = 2 × 10⁷ W/m³.

Figure 7 shows the lateral deflection of the plate at y = b/2 before activation (T < M_f) and after activation (T > A_f), the core Young’s modulus is 13 GPa. It is clear that the lateral deflection of the plate is decreased with the activation of the SMA layer due to the increase of its Young’s modulus from E_M in the martensite phase to E_A in the austenite phase as given in Table 3.

Figure 7.

Lateral deflection of the plate before activation (T < M_f ) and after activation (T > A_f ).

Effect of Young’s modulus of plate core

To study the effect of Young’s modulus of plate core (E_c ) on the lateral deflection of SMA-laminated cantilever plate, E_e is changed from 10 GPa to 80 GPa. Figures 8 and 9 show the lateral deflection of the plate before activation (T < M_f ) and after activation (T > A_f ), respectively, at different values of E_c . Figure 10 shows the effect of Young’s modulus of core (E_c ) on the dimensionless response time and maximum deflection of SMA-laminated cantilever plate at the middle point of the free end. Figure 11 shows the effect of Young’s modulus of core (E_c ) on the response time and the natural frequency of the plate.

Figure 8.

Effect of Young’s modulus of plate core (E_c ) on the lateral deflection of the plate before activation (T < M_f ).

Figure 9.

Effect of Young’s modulus of plate core (E_c ) on the lateral deflection after activation (T > A_f ).

Figure 10.

Effect of Young’s modulus of plate core (E_c ) on the plate dimensionless response time and maximum deflection $\left(\bar{w}=w/h\right)$ at the middle of the free end (x = a and y = b/2).

Figure 11.

Effect of Young’s modulus of plate core (E_c ) on the response time and natural frequency.

The natural frequency is calculated during the transformation between martensite and austenite phases. The coupled of non-linear governing equations is decoupled by substituting the initial value of ${\mathit{\theta }}^0$ and ${\mathit{\zeta }}^0$ , then solving the as quazi-linear equations of motion, Equation (33), for each time step $t_{n+1}=t_n+\mathrm{\Delta }t$ .

Effect of SMA thermal expansion

The lateral deflection is decreased in the heating stage and increased in the cooling stage. The temperature of the SMA layer is varied in the heating–cooling cycle as illustrated in Figure 5, while the temperature of the core layer is considered constant. Figure 12 shows the lateral deflection of SMA-laminated cantilever plate before and after with and without thermal expansion effect. It is clear that the activation decreases the deflection, and taking the thermal expansion effect into consideration adds an additional decrease of the deflection. While Figure 13 shows the effect of thermal expansion on the response time and the deflection of the cantilever plate at the free end. It is clear that the thermal expansion decreases the deflection without affecting the time response. The reason of that is that at the staring temperature (M_f ) the SMA layer is at martensite phase. Applying the transverse load in the positive z-direction causes the plate to be deformed upwards. While heating from M_f to A_s , the temperature of the SMA layer increases as seen in Figure 5, which causes the SMA layer to expand. Since this layer is positioned at a distance above the midplane, its expansion will produce an axial force above the neutral axis, which acts as a moment that forces the plate to deform downward to decrease the deformation gradually with heating under the same applied transverse load.

Figure 12.

Lateral deflection of SMA-laminated plate before activation (T < M_f ) and after activation (T > A_f ) with and without thermal expansion effect. SMA: shape-memory alloy.

Figure 13.

Effect of thermal expansion on the dimensionless response time and the maximum dimensionless deflection $\left(\bar{w}=w/h\right)$ of SMA-laminated plate at free end (x = a and y = b/2). SMA: shape-memory alloy.

Thus, heating from M_f to A_s has no effect on the SMA phase, it is still at martensite phase. The decrease in deformation is due to thermal expansion only. Once the temperature reaches A_s , the SMA layer starts to transform from martensite phase to austenite phase, which means gradual increase in the modulus of elasticity of the SMA layer and the total stiffness of the plate and subsequently causes a gradual decrease in the plate deflection until the temperature reaches A_f . After the plate is completely transformed to the austenite phase, it has the minimum deflection reached in the heating cycle. At A_f , no more heating power is added and the heat sink decreases the temperature gradually. Cooling from A_f to M_s will not affect the SMA layer phase but will cause thermal contraction. This contraction produces an axial compression force positioned above the neutral axis, which acts as a moment that forces the plate to deflect upwards, which means that the deflection increases gradually while cooling.

The cooling after M_s causes the SMA layer to be transformed to austenite phase, this gradually decreases the modulus of elasticity from E_A to E_M . Thus, the total stiffness of the plate also decreases causing more deflection under the same applied transverse load. The heat cycle is finished at M_f , while the deflection has the maximum value.

It is obvious that if the transverse load is downwards or the SMA layer is positioned at the bottom of the plate, the thermal expansion effect will be inverted. This means that heating increases deflection and cooling decreases deflection.

Figure 14 shows the effect of Young’s modulus of core (E_c ) on the lateral deflection of the plate after activation (T > A_f ) with thermal expansion effect. Figure 15 shows the effect of Young’s modulus of core (E_c ) on the response time and the maximum deflection of the plate at the middle of the plate free end with thermal expansion effect.

Figure 14.

Effect of Young’s modulus of plate core (E_c ) on the lateral deflection after activation (T > A_f ) with thermal expansion effect.

Figure 15.

Effect of Young’s modulus of plate core (E_c ) on the dimensionless response time and the maximum dimensionless deflection $\left(\bar{w}=w/h\right)$ at free end (x = a and y = b/2) with thermal expansion effect.

Effect of the SMA layer thickness

The effect of the SMA layer thickness on the response time, plate deflection, and natural frequency, a plate with the same value of the total thickness h = 3 mm, four values of SMA layer thickness (t_s ) are chosen such that (t_s /h = 0.1, 0.2, 0.3, and 0.4). Figure 16 shows SMA thickness effect on the temperature response of the plate in the heating–cooling cycle. Figure 17 shows the SMA thickness effect on the variation of the martensite fraction with response time of the SMA-laminated cantilever plate in the heating–cooling cycle. Increasing the SMA layer thickness will decrease and increase the response time during heating and cooling, respectively. Actually increasing the SMA thickness (t_s ) with the same total thickness (h) means decreasing the core thickness (t_e ). Both (t_s ) and (t_e ) affect the time response. As seen in the energy balance equation (40), the only term affected with the change of (t_s ) and (t_e ) is $K_E\left(T-T_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{k}}\right)$ , which represents the heat lost by conduction through the core layer such that $K_E=k_e/\phantom{k_e\left(t_s{t}_e\right)}\left(t_s{t}_e\right)$ , where k_e is the thermal conductivity of the core in $\left[W{m}^{-1\circ }{C}^{-1}\right]$ . It is clear that the cooling rate is inversely proportional to $\left(t_s{t}_e\right)$ . Figure 18 shows the relation between K_E and the SMA thickness ratio $(t_s/h)$ , it is clear that K_E decreases until the SMA thickness becomes equal to the core thickness $\left(t_s/h\right)=0.5$ then K_E increases again. This explains the increase in the heating rate and the decrease in the cooling rate with increasing the SMA thickness ratio $\left(t_s/\phantom{t_{s}h}h\right)$ as shown in Figure 16.

Figure 16.

SMA thickness effect on the temperature response of the SMA layer in the heating–cooling cycle with input power P = 2 × 10⁷ W/m³. SMA: shape-memory alloy.

Figure 17.

SMA thickness effect on the variation of the martensite fraction with time response of the SMA layer in the heating–cooling cycle with input power P = 2 × 10⁷ W/m³. SMA: shape-memory alloy.

Figure 18.

SMA thickness ratio (t_s /h). SMA: shape-memory alloy.

Figures 19 and 20 show the SMA thickness effect on the lateral deflection of SMA-laminated cantilever plate before and after activation, respectively. Figures 21 and 22 show SMA thickness effect on the response time and the maximum deflection of SMA-laminated cantilever plate at free end with dimensions and dimensionless values, respectively. It is obvious that increasing the SMA layer thickness decreases the plate deflection; this is because of increasing the stiffness of the plate with increasing the SMA thickness ratio. Figure 23 shows the SMA thickness effect on the response time and natural frequency of the proposed cantilever plate. The natural frequency decreases with increasing the SMA thickness ratio, this is because increasing the SMA thickness increases both stiffness and mass of the plate. However, the mass of the plate increases more than its stiffness does, which decrease the natural frequency of the plate.

Figure 19.

SMA thickness effect on the lateral deflection of the plate before activation (T < M_f ). SMA: shape-memory alloy.

Figure 20.

SMA thickness effect on the lateral deflection of the plate after activation (T > A_f ). SMA: shape-memory alloy.

Figure 21.

SMA thickness effect on the response time and the maximum deflection of the plate at free end (x = a and y = b/2). SMA: shape-memory alloy.

Figure 22.

SMA thickness effect on both dimensionless response time and the maximum deflection $\left(\bar{w}=w/\phantom{wh}h\right)$ of the plate at free end (x = a and y = b/2). SMA: shape-memory alloy.

Figure 23.

SMA thickness effect on the plate response time and natural frequency. SMA: shape-memory alloy.

Effect of plate aspect and thickness ratios

To study the effect of the plate aspect ratio (a/b) and the thickness ratio (a/h) on the plate deflection and natural frequency, four values of (a/b = 1, 2, 3, and 4) are chosen and a four values of (a/h = 10, 20, 50, and 100) are chosen. The plate length is taken constant (a = 0.1 m) and the SMA thickness ratio is taken constant (t_s /h = 0.1; Figure 4). The plate is loaded with a distributed load (q = 500 N/m²). The maximum lateral deflection $w\hspace{0.17em}\left(mm\right)$ and its normalized value $\left(\bar{w}\right)$ equation (51), before and after activation of SMA layer are calculated, and the activation ratios w after activation/w before activation are plotted in Figure 24. The natural frequency, Equation (51), before and after activation of SMA layer is calculated, and the activation ratios ω after activation/ω before activation are plotted in Figure 25. It is clear from Figures 24 and 25 that increasing the plate thickness ratio (a/h) causes decrease of the deflection activation ratio and increase of the natural frequency activation ratio which means that as the plate becomes thinner, a more response to the SMA activation is obtained. As the plate aspect ratio (a/b) increases, a large response to the SMA activation is obtained.

$\bar{w}=\frac{{10}^4{E}_e{h}^3}{a^4{q}_0}w,\hspace{0.17em}\hspace{0.17em}\bar{\mathit{\omega }}=\frac{\mathit{\omega }{b}^2}{h}\sqrt{\frac{{\mathit{\rho }}_e}{E_e}}$

51

Figure 24.

The effect of plate thickness ratio (a/h) and aspect ratio (a/b) on the deflection activation ratio (w after activation/w before activation).

Figure 25.

The effect of plate thickness ratio (a/h) and aspect ratio (a/b) on the natural frequency activation ratio (ω after activation/ω before activation).

Conclusion

The static response, the dynamic characteristics, and the time response of the SMA-laminated composite plates are obtained. A two-dimensional nonlinear transient analysis of thermally driven composite elastomer plate with SMA sheet bonded on its surface was presented.

The energy balance equation was utilized to solve the temperature distributions of the SMA layer. The time response of the proposed plate and the deflection as well as natural frequencies before and after activation of the SMA plate were computed.

Parametric studies have been performed and the following conclusions have been drawn:

The lateral deflection of the plate is decreased with the activation of the SMA layer due to the increase of the Young’s modulus of the SMA layer in the austenite phase.

Increasing the core material stiffness causes decreasing in deflection without affecting the time response.

Increasing the SMA layer thickness obviously decreases the response time during heating, and increases it during cooling until the SMA thickness becomes equal to the core thickness $\left(t_s/\phantom{t_{s}h}h\right)=0.5$ then the relation will be inverted.

Increasing the SMA layer thickness decreases both the plate deflection and natural frequency which depend on the ratios of stiffness and mass properties of SMA and core materials.

Increasing the plate thickness ratio (a/h) decreases the deflection activation ratio and increases the natural frequency and activation ratio which means that as the plate becomes thinner, it is more response to the SMA activation.

As the plate aspect ratio (a/b) increases a larger response to the SMA activation is obtained.

The plate deflection is affected by thermal expansion coefficient without affecting the time response according to the location of the SMA sheet away from the neutral axis and the transverse force direction.