Thermal Buckling and Vibration Analysis of SMA Hybrid Composite Sandwich Beams

Abstract

This work studies the buckling and free vibration behavior of Shape Memory Alloy Hybrid Composite (SMAHC) sandwich beams under a thermal environment. The sandwich beams consist of layers reinforced with SMAs and a FGM core, and they are simply supported at both ends. The higher order theory is combined with the Minimum Potential Energy principle or Hamilton principle to derive the governing equations of the thermal buckling and thermal vibration problems, respectively. The material properties of the beam are assumed as temperature-independent (TID) or temperature-dependent (TD). In the last case, two different types of thermal distribution are considered, namely a uniform and a linear distribution. The results based on the proposed formulation are verified against the reference literature, with a very good matching. A parametric study checks for the influence of different effective parameters such as thickness-to-length ratios, volume fraction powers, initial strain, volume fraction of SMA wires, and temperature distribution on the overall mechanical response of the selected structural member, with useful insights from a design standpoint.

1. Introduction

Today, sandwich structures are largely applied in many industrial applications, primarily in aerospace, marine, transportation, and civil applications due to their special properties, including lightweight, high stiffness, strength, and damping capability. Sandwich structures typically consist of two external layers and an internal core. The layer materials are usually made of metals [1], composites [2], or fiber-metal-laminate [3], whereas the core material includes foam [4], honeycomb [5], or Functionally-Graded materials (FGMs) [6]. The introduction of Shape Memory Alloys (SMAs) within the top/bottom layers is an efficient way to improve the sandwich properties. Intelligent materials such as SMAs, have some unique properties, such as a large recoverable strain field, together with a high flexibility shape memory effect and pseudo-elasticity [7]. On the other hand, a large recovery strain field and a certain flexibility of the SMAs for different load histories, temperatures, and stresses can be obtained for tunable thermal and mechanical properties [8].

In recent decades, different works from literature have shown an increased interest in the buckling and post-buckling behavior of composite plates, shells, and beams embedding SMAs [9,10,11,12,13]. Girish and Ramachandra [14] used a higher order shear deformation theory (HSDT) to study the buckling response of composite plates subjected to a uniform temperature distribution throughout the thickness. Kamarian and Shakeri [15] focused on an optimized buckling response of rectangular and skew composite plates in thermal environments by embedding different quantities of SMA wires. The authors applied the First Order Shear Deformation Theory (FSDT) and used a Generalized Differential Quadrature (GDQ) method to solve the governing equations of the problem. The Von Karman nonlinear displacement–strain relationships were used by Ho Roh et al. [16] to investigate the thermal post buckling of shape memory alloy hybrid composite shell panels. In such work, it was shown that the snapping phenomenon of shells can be prevented by embedding SMA wires in a composite structure.

Buckling and post-buckling phenomena are particularly important for structural beams in thermal conditions [17,18]. One well-known method to increase the thermal buckling resistance of sandwich beams relies on the introduction of SMAs into these structures [19]. Khdeir [20] studied the variation of the critical buckling temperature of cross-ply beams with various boundary conditions. The author also reported the effect of some parameters such as the length-to-thickness ratio and thermal expansion coefficients ratio on the critical buckling temperature. Bayat and Ekhteraei Toussi [21] applied a Layer-Wise Theory (LWT) to explore the thermal buckling and post-buckling behavior of laminated composite beams reinforced with SMA wires. The authors studied the effect of some parameters on the mechanical response, including the symmetric and antisymmetric layups, as well as the influence of the thickness/span ratio, SMA pre-strain, and volume fraction.

The free vibration of composite structures is another important aspect to investigate, as extensively found in literature [8,22,23]. Sang Park et al. [24] performed a vibration analysis of the composite plates embedded with shape memory alloy fibers by combining the FSDT basics with the Von Karman strain–displacement relations. Based on their results, the plates increased their stiffness for an increased volume fraction and initial strain of the SMA fibers; the authors also checked for the optimal design of fiber angle, stacking sequence, volume fraction, and initial strain of the SMAs.

Similarly, Parhi and Singh [25] studied the free vibration of SMA-embedded laminated composite shell panels, including spherical and cylindrical shell panels with TD material properties. The authors used a HSDT combined with the principle of virtual work and nonlinear von Karman strain displacement relations to handle the problem. Nekouei et al. [26] analyzed the free vibration of hybrid laminated composite cylindrical shells reinforced with SMA by applying the Brinson’s one-dimensional constitutive law to calculate the thermo-mechanical properties of SMAs. According to the study, the fundamental frequency was increased by adding a proper quantity of SMA wires to the cylindrical shells. At the same time, Roger and Barker [27] showed that the SMA wire heating induces an axial stress in the graphite–epoxy composite beam, where the introduction of 15% volume fraction of SMA wires is capable of increasing the overall natural frequency. Asadi et al. [28] studied the free vibration of thermally pre/post-buckled shear deformable SMA hybrid composite beams. The problem was tackled theoretically based on the FSDT, von Karman geometrical non-linearity, and extended Hamilton principle. The authors, in their work, verified a general improvement of the buckling temperature and post-buckling path of the shape memory alloy hybrid composite (SMAHC) beam by concentrating the SMA fibers in the center of the laminated beam. In addition, Alambeigi et al. [29] analyzed the free and forced vibration of sandwich beams made of a functionally-graded porous core and composite face layers embedding SMAs, immersed within an elastic foundation, modeled according to the Vlasov basics. Once again, it was found that the temperature, volume fraction of SMA, and porosity distribution are important parameters affecting the overall response of sandwich beams. Starting with the available literature on the topic, the present work is aimed at investigating the vibration and buckling response of sandwich beams with SMA reinforced external layers and a FGM core, under thermal conditions. More specifically, two types of thermal distribution are considered, namely, a linear and a uniform distribution. The governing equations for the thermal buckling and vibration rely on a high order theory, combined with the Minimum Potential Energy principle or Hamilton principle, respectively. To verify the accuracy of the proposed analytical method, our results are successfully compared to well-known references from literature. A large sensitivity analysis is, thus, performed to check for the effect of some important parameters such as types of thermal load, volume fraction, and initial pre-strained SMA wires, as well as different geometrical properties for both the face sheets and the core of the selected sandwich structure.

2. Materials Properties and Basic Equations of SMA

Let us consider a composite beam made of two layers and an FGM core. In more detail, the external layers are made of a graphite–epoxy composite with the addition of a SMA reinforcement. The selected structural member is here studied based on the Brinson’s model [30], due to its capability in considering the thermo-mechanical behavior at any temperature. According to this model, we introduce the following relation between the stress field, $\sigma$, strain field, $\epsilon$, thermal field, $T$, martensite fraction, $\mathsf{\xi }$, Young’s modulus of the SMA, $E\left(\epsilon ,\xi ,T\right)$, transformation tensor, $\Omega \left(\xi \right)$, and thermal tensor, Θ, i.e.,

$$d\sigma =E\left(\epsilon ,\xi ,T\right) d\epsilon +\Omega \left(\epsilon ,\xi ,T\right)d{\xi }_s+\Theta \left(\epsilon ,\xi ,T\right) dT$$

(1)

The Young’s modulus of the SMA is expressed as [31]:

$$E\left(\epsilon ,\xi ,T\right)=E\left(\mathsf{\xi }\right)=\frac{E_A}{1+\xi \left(\frac{E_A}{E_M}-1\right)}$$

(2)

where $E_A$ and $E_M$ are the Young’s moduli of a perfect austenite and martensite state, respectively. The martensite fraction is the summation of two parameters, i.e., the stress rise, ${\mathsf{\xi }}_s$, and temperature rise, ${\mathsf{\xi }}_T$. In addition, the transformation tensor is defined as:

$$\Omega \left(\xi \right)=-{\epsilon }_{L}E\left(\xi \right)$$

(3)

where ${\epsilon }_L$ is the maximum residual strain of the SMA wires obtained during a uniaxial tension test. There are two phases to measure the martensite fractions, depending on temperature and stress: (I) the phase transformation to M (martensite), (II) the phase transformation from M (martensite) to A (austenite). Phase (I) is described by the following equations:

If $T>M_s$ and ${\sigma }_s^{cr}+C_M\left(T-M_s\right)<\sigma <{\sigma }_f^{cr}+C_M\left(T-M_f\right)$

$$\begin{array}{c}{\xi }_s=\frac{1-{\xi }_{s0}}{2}\cos \left(\frac{\pi }{{\sigma }_s^{cr}-{\sigma }_f^{cr}}(\sigma -{\sigma }_f^{cr}-C_M(T-M_s))\right)+\frac{1+{\xi }_{s0}}{2}\\ {\xi }_T={\xi }_{T0}-\frac{{\xi }_{T0}}{1-{\xi }_{s0}}({\xi }_s-{\xi }_{s0})\end{array}$$

(4)

If $T<M_s$ and ${\sigma }_s^{cr}<\sigma <{\sigma }_f^{cr}$

$$\begin{array}{c}{\xi }_s=\frac{1-{\xi }_{s0}}{2}\cos \left(\frac{\pi }{{\sigma }_s^{cr}-{\sigma }_f^{cr}}(\sigma -{\sigma }_f^{cr})\right)+\frac{1+{\xi }_{s0}}{2}\\ {\xi }_T={\xi }_{T0}-\frac{{\xi }_{T0}}{1-{\xi }_{s0}}({\xi }_s-{\xi }_{s0})+{\Delta }_{T\xi }\end{array}$$

(5)

If $M_f<T<M_s$ and $T<T_0$

$${\Delta }_{T\xi }=\frac{1-{\xi }_{T0}}{2}\left(\cos (a_M(T-M_f))+1\right)$$

(6)

Otherwise, ${\Delta }_{T\xi }=0$. Phase (II), instead, accounts for the following relations:

If $T>A_s$ and $C_A\left(T-A_f\right)<\sigma <C_A\left(T-A_S\right)$

$$\begin{array}{ll}\xi =& \frac{{\xi }_0}{2}\left(\cos \left[\frac{\pi }{A_f-A_s}(T-A_s-\frac{\sigma }{C_A})\right]+1\right)\\ & {\xi }_s={\xi }_{s0}-\frac{{\xi }_{s0}}{{\xi }_0}({\xi }_0-\xi )\\ & {\xi }_T={\xi }_{T0}-\frac{{\xi }_{T0}}{{\xi }_0}({\xi }_0-\xi )\\ & C_A={\left(\frac{d{A}_s}{d\sigma }\right)}^{-1},C_M={\left(\frac{d{M}_s}{d\sigma }\right)}^{-1}\end{array}$$

(7)

where the subscript 0 refers to the initial state and $M_f$, $M_s$, $A_s$, and $A_f$ are the martensite finish temperature, martensite start temperature, austenite start temperature, and austenite finish temperature, respectively. $C_A$ and $C_M$ refer to the transformation constants for the austenite and martensite phases, respectively. More details regarding the mechanical properties of SMAs can be found in

Table 1. Material properties of SMA wires.

$\begin{array}{l}{E}_A=67\mathrm{GPa}\\ E_M=26.3\mathrm{GPa}\\ M_f=9°\mathrm{C}\\ M_s=18.4°\mathrm{C}\\ A_f=34.5°\mathrm{C}\\ A_s=49°\mathrm{C}\\ {\epsilon }_L=0.067\end{array}$

$\begin{array}{l}{C}_M=8\mathrm{MPa}/°\mathrm{C}\\ C_A=13.8\mathrm{MPa}/°\mathrm{C}\\ v_s=0.33\\ {\alpha }_s=10.26°\mathrm{C}\\ \theta =0.55\mathrm{MPa}/°\mathrm{C}\\ {\sigma }_s^{cr}=100\mathrm{MPa}\\ {\sigma }_f^{cr}=170\mathrm{MPa}\end{array}$

, accordingly to [30]. The relations defining the properties of a graphite–epoxy reinforced with SMAs (NiTi wires) with TD conditions are mentioned in Appendix A, whose properties are reported in

Table 2. Thermomechanical properties of the layers.

$\begin{array}{l}{E}_{1m}^{}=E_{1m}^0(1+E_{1m}^1\Delta T)\\ E_{2m}^{}=E_{2m}^0(1+E_{2m}^1\Delta T)\\ G_{12m}^{}=G_{12m}^0(1+G_{12m}^1\Delta T)\\ G_{23m}^{}=G_{23m}^0(1+G_{23m}^1\Delta T)\\ {\alpha }_{1m}^{}={\alpha }_{1m}^0(1+{\alpha }_{1m}^1\Delta T)\\ {\alpha }_{2m}^{}={\alpha }_{2m}^0(1+{\alpha }_{2m}^1\Delta T)\\ v_{12m}=0.22\\ {\rho }_m=1586{\mathrm{kg}/\mathrm{m}}^3\end{array}$

$\begin{array}{l}{E}_{1m}^0=155\mathrm{GPa}\\ E_{2m}^0=8.07\mathrm{GPa}\\ G_{12m}^0=4.55\mathrm{GPa}\\ G_{23m}^0=3.25\mathrm{GPa}\\ {\alpha }_{1m}^0=-0.07\times {10}^{-6}\\ {\alpha }_{2m}^0=30.1\times {10}^{-6}\end{array}$

$\begin{array}{l}{E}_{1m}^1=-3.53\times {10}^{-4}\\ E_{2m}^1=-4.27\times {10}^{-4}\\ G_{12m}^1=-6.06\times {10}^{-4}\\ G_{23m}^1=-6.06\times {10}^{-4}\\ {\alpha }_{1m}^1=-1.25\times {10}^{-3}\\ {\alpha }_{2m}^1=0.41\times {10}^{-4}\end{array}$

, in line with [31].

To define the variation of an arbitrary property throughout the thickness, such as the Young’s modulus ($E$), the coefficient of thermal expansion ($\alpha$), or the density ($\rho$), we introduce the following relation:

$$P(z)=(P_c-P_m){(z/h_c+1/2)}^n+P_m$$

(8)

where the subscripts c and m refer to a ceramic and metal phase, respectively, and $h_c$ and $n$ stand for the FGM thickness and volume fraction power, respectively. Similarly to face sheets, the mechanical properties of the FGM core are TD.

As FGMs are most commonly used in high temperature conditions with significant variations of material properties, it is reliable to assume TD mechanical properties for FGM cores to ensure an accurate prediction of the structural response. The material properties of the ceramic and metal phases are expressed as non-linear functions of the environment temperature; see [32,33]

$$P_i(T)=P_0(P_{-1}+1+P_{1}T+P_2{T}^2+P_3{T}^3)i=c,m$$

(9)

where $T$is the temperature in Kelvin and $P_i$ (i = −1, 0, 1, 2, 3) refers to the TD constants, as defined in

Table 3. TD constants of metal and ceramic phases.

Properties	Materials	P0	P−1	P1	P2	P3
Young’s elasticity modulus	Ti-6Al-4V	122.56 $\times$ 109	0	−4.586 $\times$ 10−4	0	0
	Zirconia	244.27 $\times$ 109	0	−1.371 $\times$ 10−3	1.214 $\times$ 10−6	−3.681 $\times$ 10−10
Coefficient of Thermal Expansion	Ti-6Al-4V	7.5788 $\times$ 10−6	0	6.638 $\times$ 10−4	−3.147 $\times$ 10−6	0
	Zirconia	12.766 $\times$ 10−6	0	−1.491 $\times$ 10−3	1.006 $\times$ 10−5	−6.778 $\times$ 10−11
density	Ti-6Al-4V	4429	0	0	0	0
	Zirconia	5700	0	0	0	0

[33].

3. Governing Equations Derivation

In this section, we describe the theoretical formulation here adopted to study the thermal buckling and vibration of a sandwich beam with an FGM core of thickness $h_c$ and SMA hybrid layers of thickness $h_a$, as depicted in Figure 1.

3.1. Equilibrium Equations

The analysis of the SMAHC sandwich beam is based on a HSDT, such that the displacement field, $\overline{u}(x,z,t),\overline{v}(x,z,t),\overline{w}(x,z,t)$, is defined as [34]:

$$\begin{array}{l}\overline{u}(x,z,t)=u_0(x,t)+\frac{5}{4}\left(z-\frac{4}{3h^2}{z}^3\right){\varphi }_x(x,t)+\left(\frac{1}{4}z-\frac{5}{3h^2}{z}^3\right)\frac{\partial w_0(x,t)}{\partial x}\\ \overline{v}(x,z,t)=0\\ \overline{w}(x,z,t)=w_0(x,t)\end{array}$$

(10)

where $u_0$ and $w_0$ are the displacement components along the $x$ and $z$ coordinate directions, respectively, of a point on the mid-surface. In addition, ${\varphi }_x$ refers to the rotation about the $x$ axis and $h$ is the total beam thickness. The kinematic relations at Cartesian coordinates (x-y-z) are defined as [34]:

$$\begin{array}{l}{\epsilon }_x=\frac{\partial \overline{u}}{\partial x}+\frac{1}{2}{\left(\frac{\partial \overline{w}}{\partial x}\right)}^2,{\epsilon }_y=\frac{\partial \overline{v}}{\partial y},{\epsilon }_z=\frac{\partial \overline{w}}{\partial z},{\gamma }_{xy}=\frac{\partial \overline{u}}{\partial y}+\frac{\partial \overline{v}}{\partial x},\\ {\gamma }_{xz}=\frac{\partial \overline{u}}{\partial z}+\frac{\partial \overline{w}}{\partial x}+\frac{\partial \overline{w}}{\partial x}\frac{\partial \overline{w}}{\partial z},{\gamma }_{yz}=\frac{\partial \overline{v}}{\partial z}+\frac{\partial \overline{w}}{\partial y}\end{array}$$

(11)

By substituting Equation (10) into Equation (11), we obtain the following equation:

$$\begin{array}{l}{\epsilon }_x={\epsilon }_x^{(0)}+z{\epsilon }_x^{(1)}+z^3{\epsilon }_x^{(3)}\\ {\gamma }_{xz}={\gamma }_{xz}^{(0)}+z^2{\gamma }_{xz}^{(2)}\end{array}$$

(12)

where ${\epsilon }_x^{(0)}$, ${\epsilon }_x^{(1)}$, ${\epsilon }_x^{(3)}$, ${\gamma }_{xz}^{(0)}$, and ${\gamma }_{xz}^{(2)}$ are defined as:

$$\left\{\begin{array}{l}{\epsilon }_x^{(0)}\\ {\epsilon }_x^{(1)}\\ {\epsilon }_x^{(3)}\end{array}\right\}=\left\{\begin{array}{l}\frac{du}{dx}+\frac{1}{2}{\left(\frac{d{w}_0}{dx}\right)}^2\\ \frac{5}{4}\frac{d{\varphi }_x}{dx}+\frac{1}{4}\frac{d^2{w}_0}{d{x}^2}\\ -\left(\frac{5}{3h^2}\frac{d{\varphi }_x}{dx}+\frac{5}{3h^2}\frac{d^2{w}_0}{d{x}^2}\right)\end{array}\right\},\left\{\begin{array}{l}{\gamma }_{xz}^{(0)}\\ {\gamma }_{xz}^{(2)}\end{array}\right\}=\left\{\begin{array}{l}\frac{5}{4}{\varphi }_x+\frac{5}{4}\frac{d{w}_0}{dx}\\ -\left(\frac{5}{h^2}{\varphi }_x+\frac{5}{h^2}\frac{d{w}_0}{dx}\right)\end{array}\right\}$$

(13)

The equilibrium equations are obtained according to the HSDT and the Principal of Minimum Potential Energy (PMPE), expressed as [34]:

$$\mathrm{\Pi }=U-W$$

(14)

where $U$ and $W$ refer to the strain energy and the external work, respectively, defined in variational form $\delta$ [35] as follows:

$$\delta U={\displaystyle \underset{\mathrm{\Omega }}{\int }({\sigma }_x^{}\delta {\epsilon }_x+{\sigma }_{xz}^{}\delta {\gamma }_{xz})}dxdz$$

(15)

or equivalently:

$$\begin{array}{cc}\delta U=& {\displaystyle \underset{0}{\overset{L}{\int }}{\displaystyle \underset{-\left(\frac{h_c}{2}+h_a\right)}{\overset{\frac{h_c}{2}+h_a}{\int }}{\sigma }_x\left\{\begin{array}{l}\frac{d\delta u}{dx}+\frac{d{w}_0}{dx}\frac{d\delta w_0}{dx}+\left(\frac{5}{4}\frac{d\delta {\varphi }_x}{dx}+\frac{1}{4}\frac{d^2\delta w_0}{d{x}^2}\right)z-\\ \left(\frac{5}{3h^2}\frac{d\delta {\varphi }_x}{dx}+\frac{5}{3h^2}\frac{d^2\delta w_0}{d{x}^2}\right)z^3\end{array}\right\}+}}\hfill \\ & {\sigma }_{xz}\left(\frac{5}{4}\delta {\varphi }_x+\frac{5}{4}\frac{d\delta w_0}{dx}-\left(\frac{5}{h^2}\delta {\varphi }_x+\frac{5}{h^2}\frac{d\delta w_0}{dx}\right)z^2\right)dzdx\hfill \end{array}$$

(16)

At the same time, Equation (16) can be redefined in terms of resultants ($N_x$, $M_x$, $N_{xz}$, $P_x$, $R_{xz}$), i.e.,

$$\begin{array}{ll}\delta U=& {\displaystyle \underset{0}{\overset{L}{\int }}\left(N_x\left(\frac{d\delta u}{dx}+\frac{d{w}_0}{dx}\frac{d\delta w_0}{dx}\right)+M_x\left(\frac{5}{4}\frac{d\delta {\varphi }_x}{dx}+\frac{1}{4}\frac{d^2\delta w_0}{d{x}^2}\right)+\right.}\\ & P_x\left(\frac{5}{3h^2}\frac{d\delta {\varphi }_x}{dx}+\frac{5}{3h^2}\frac{d^2\delta w_0}{d{x}^2}\right)+N_{xz}\left(\frac{5}{4}\delta {\varphi }_x+\frac{5}{4}\frac{d\delta w_0}{dx}\right)+\\ & \left.R_{xz}\left(-\left(\frac{5}{h^2}\delta {\varphi }_x+\frac{5}{h^2}\frac{d\delta w_0}{dx}\right)\right)\right)dx\end{array}$$

(17)

where $N_x$ is the in-plane force resultants; $M_x$ is the moment resultants; $N_{xz}$ is the final transverse force resultants; and $P_x$ and $R_{xz}$ are the stressed higher order resultants determined as a sum of the mechanical, thermal, and recovery resultants (denoted with superscript M, T, and r, respectively). Such parameters are defined as:

$$\begin{array}{l}{N}_x=N_x^M+N_x^T+N_x^r={\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{h_c/2+h_a}\left({\sigma }_x^M+{\sigma }_x^T\right)dz}+{\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{-h_c/2}{\sigma }_x^{r}dz}+{\displaystyle {\int }_{h_c/2}^{h_c/2+h_a}{\sigma }_x^{r}dz}\\ M_x=M_x^M+M_x^T+M_x^r={\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{h_c/2+h_a}z\left({\sigma }_x^M+{\sigma }_x^T\right)dz}+{\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{-h_c/2}z{\sigma }_x^{r}dz}+{\displaystyle {\int }_{h_c/2}^{h_c/2+h_a}z{\sigma }_x^{r}dz}\\ N_{xz}=N_{xz}^M+N_{xz}^T+N_{xz}^r={\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{h_c/2+h_a}\left({\sigma }_{xz}^M+{\sigma }_{xz}^T\right)dz}+{\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{-h_c/2}{\sigma }_{xz}^{r}dz}+{\displaystyle {\int }_{h_c/2}^{h_c/2+h_a}{\sigma }_{xz}^{r}dz}\\ P_x=P_x^M+P_x^T+P_x^r={\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{h_c/2+h_a}{z}^3\left({\sigma }_x^M+{\sigma }_x^T\right)dz}+{\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{-h_c/2}{z}^3{\sigma }_x^{r}dz}+{\displaystyle {\int }_{h_c/2}^{h_c/2+h_a}{z}^3{\sigma }_x^{r}dz}\\ R_{xz}=R_{xz}^M+R_{xz}^T+R_{xz}^r={\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{h_c/2+h_a}{z}^2\left({\sigma }_{xz}^M+{\sigma }_{xz}^T\right)dz}+{\displaystyle {\int }_{-\left(h_c/2+h_a\right)}^{-h_c/2}{z}^2{\sigma }_{xz}^{r}dz}+{\displaystyle {\int }_{h_c/2}^{h_c/2+h_a}{z}^2{\sigma }_{xz}^{r}dz}\\ \end{array}$$

(18)

Finally, from Equation (17), the equilibrium equations can be obtained as:

$$\begin{array}{l}\delta u_0=0\Rightarrow \frac{d{N}_x}{dx}=0\\ \delta {\varphi }_x=0\Rightarrow -\frac{5}{4}\frac{d{M}_x}{dx}+\frac{5}{3h^2}\frac{d{P}_x}{dx}+\frac{5}{4}{N}_{xz}-\frac{5}{h^2}{R}_{xz}=0\\ \delta w_0=0\Rightarrow -\frac{d}{dx}\left(N_x\frac{d{w}_0}{dx}\right)+\frac{1}{4}\frac{d^2{M}_x}{d{x}^2}-\frac{5}{3h^2}\frac{d^2{P}_x}{d{x}^2}-\frac{5}{4}\frac{d{N}_{xz}}{dx}+\frac{5}{h^2}\frac{d{R}_{xz}}{dx}=0\end{array}$$

(19)

In addition, the stress–strain ($\sigma -\epsilon$) relation involving the recovery stress and thermal strain at Cartesian coordinates are defined as:

$$\left\{\begin{array}{l}{\sigma }_x\\ {\sigma }_y\\ {\sigma }_z\\ {\sigma }_{yz}\\ {\sigma }_{xz}\\ {\sigma }_{xy}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{12}& C_{22}& C_{23}& 0& 0& 0\\ C_{13}& C_{23}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]\left(\left\{\begin{array}{l}{\epsilon }_x\\ {\epsilon }_y\\ {\epsilon }_z\\ {\gamma }_{yz}\\ {\gamma }_{xz}\\ {\gamma }_{xy}\end{array}\right\}-\left\{\begin{array}{l}{\alpha }_x\\ {\alpha }_y\\ {\alpha }_z\\ 0\\ 0\\ 0\end{array}\right\}\Delta T\right)+V_s\left\{\begin{array}{l}{\sigma }_1^r\\ {\sigma }_2^r\\ 0\\ 0\\ 0\\ {\sigma }_{12}^r\end{array}\right\}$$

(20)

where $C_{ij}$ (i,j = 1,2,..,6) are the stiffness matrix components. In addition, ${\alpha }_i$(i = x, y, z), $\Delta T$ stand for the thermal expansion and thermal variation, and $V_s$ and ${\sigma }_{ij}^r$ (i,j = 1, 2,3) refer to the volume fraction and recovery stress of the shape memory alloy, respectively. Thus, the mechanical and thermal stresses are defined as:

$$\left\{\begin{array}{l}{\sigma }_x^M\\ {\sigma }_{xz}^M\end{array}\right\}=\left\{\begin{array}{l}{C}_{11}{\epsilon }_x\\ C_{55}{\gamma }_{xz}\end{array}\right\},\left\{\begin{array}{l}{\sigma }_x^T\\ {\sigma }_{xz}^T\end{array}\right\}=\left\{\begin{array}{l}{C}_{11}{\alpha }_x\Delta T\\ 0\end{array}\right\}$$

(21)

Moreover, the relations between the stress resultants and the displacement field are obtained by combining Equation (18) and the stress-displacement relation, as briefly described in Appendix A (A5–A7). Finally, the equilibrium Equation (19) has been obtained according to the displacement field,$u_0,{\varphi }_x,w_0$.

3.2. Stability Equations

We now determine the stability equations of the beam under a thermal condition. The expansion of the total potential energy ($V$) based on the Taylor series reads as follows [36]:

$$\Delta V=\delta V+\frac{1}{2}{\delta }^{2}V+\frac{1}{6}{\delta }^{3}V+.......$$

(22)

where $\delta V$ is the first variation of the total potential energy, which is related to the equilibrium state, whereas the second variation, ${\delta }^{2}V$, is representative of the stability in the neighborhood of the equilibrium state. For all virtual displacements, if ${\delta }^{2}V>0$ the beam remains stable, otherwise (${\delta }^{2}V<0$) it becomes unstable. The last condition, ${\delta }^{2}V=0,$ is associated with the critical buckling condition [36]. In fact, the critical thermal buckling load disrupts the stability in equilibrium state. The main reason for obtaining the stability equations is related to the nonlinearity of the equilibrium equations. Let us assume that the displacement components ($u_0,{\varphi }_x,w_0$) undergo a small increment:

$$\begin{array}{l}{u}_0\to u_0^0+u_0^1\\ {\varphi }_x\to {\varphi }_x^0+{\varphi }_x^1\\ w_0\to w_0^0+w_0^1\end{array}$$

(23)

where $u_0^0,{\varphi }_x,w_0^0$ refer to the displacement field in the equilibrium state, whereas $u_0^1,{\varphi }_x^1,w_0^1$ are the associated components of a neighboring stable state in respect of the equilibrium state. The force resultants of a neighboring state are related to the equilibrium state as:

$$\begin{array}{l}{N}_x\to N_x^0+N_x^1\\ M_x\to M_x^0+M_x^1\\ P_x\to P_x^0+P_x^1\\ N_{xz}\to N_{xz}^0+N_{xz}^1\\ R_{xz}\to R_{xz}^0+R_{xz}^1\end{array}$$

(24)

Superscript ‘1’ refers to the stability state and superscript ‘0’ refers to the equilibrium state. By substituting Equations (23) and (24) into the equilibrium equation, Equation (19), a deviation of the force resultants from the initial equilibrium state can be obtained, as detailed in Appendix A, (A8). Finally, the stability equations can be obtained as follows:

$$\begin{array}{l}\delta u_0=0\Rightarrow \frac{d{N}_x^1}{dx}=0\\ \delta {\varphi }_x=0\Rightarrow -\frac{5}{4}\frac{d{M}_x^1}{dx}+\frac{5}{3h^2}\frac{d{P}_x^1}{dx}+\frac{5}{4}{N}_{xz}^1-\frac{5}{h^2}{R}_{xz}^1=0\\ \delta w_0=0\Rightarrow -N_{x0}\frac{d^2{w}_0^1}{d{x}^2}+\frac{1}{4}\frac{d^2{M}_x^1}{d{x}^2}-\frac{5}{3h^2}\frac{d^2{P}_x^1}{d{x}^2}-\frac{5}{4}\frac{d{N}_{xz}^1}{dx}+\frac{5}{h^2}\frac{d{R}_{xz}^1}{dx}=0\end{array}$$

(25)

where $N_{x0}$, for a uniform and linear temperature rise, is defined as:

$$N_{x0}={\displaystyle {\int }_{-(h_c/2+h_a)}^{(h_c/2+h_a)}{C}_{ij}(z)}{\alpha }_i(z)(T-T_0)dz,i,j=x,y,z$$

(26)

and $T_0$ is the reference temperature. When the beam buckles at $T=T_0+\Delta T$, such thermal value will be updated in Equation (26).

Let us now consider a sandwich beam whose temperature at the top and bottom layer surfaces is labeled as $T_t$ and $T_b$, respectively. The temperature distribution for fixed boundary conditions is obtained by solving the heat conduction equation along the beam thickness. If the beam thickness is thin enough, the temperature distribution is approximated as linear through the thickness direction z.

For a simply supported beam, the displacement components are assumed as:

$$\begin{array}{l}{u}_0^1=u_m\cos \left(\frac{m\pi x}{L}\right)\\ {\varphi }_x^1={\varphi }_m\cos \left(\frac{m\pi x}{L}\right)\\ w_0^1=w_m\sin \left(\frac{m\pi x}{L}\right)\end{array}$$

(27)

where m is the longitudinal wave number in the x-direction, and $u_m,{\varphi }_m,w_m$ are the unknown axial functions. Finally, by using Equations (17) and (25) and Appendix A (A8), we obtain the stability equations in terms of $u_m,{\varphi }_m,w_m$.

3.3. Free Vibration under Thermal Conditions

We can now determine the equations of motion for the vibration of SMAHC sandwich beams in a thermal environment, based on the Hamilton principle [37]:

$${\displaystyle \underset{t_1}{\overset{t_2}{\int }}\left(\delta KE-\delta PE\right)}dt=0$$

(28)

where KE is the kinetic energy of the beam and PE is the associated elastic potential energy. More specifically, the kinetic energy of the beam is defined as:

$$KE=\frac{1}{2}{\displaystyle {\int }_0^l{\displaystyle {\int }_{-(h_c/2+h_a)}^{h_c/2+h_a}\rho \left({\left(\frac{\partial \overline{u}}{\partial t}\right)}^2+{\left(\frac{\partial \overline{v}}{\partial t}\right)}^2+{\left(\frac{\partial \overline{w}}{\partial t}\right)}^2\right)}}dxdz$$

(29)

or, after some manipulation:

$$\delta KE={\displaystyle {\int }_0^L\left(\begin{array}{l}{I}_0\left(\frac{\partial u_0}{\partial t}\frac{\partial \delta u_0}{\partial t}+\frac{\partial w_0}{\partial t}\frac{\partial \delta w_0}{\partial t}\right)+I_1\left(\begin{array}{l}\frac{1}{4}\left(\frac{\partial \delta u_0}{\partial t}\frac{{\partial }^2{w}_0}{\partial x\partial t}+\frac{\partial u_0}{\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}\right)+\\ \frac{5}{4}\left(\frac{\partial \delta u_0}{\partial t}\frac{\partial {\varphi }_x}{\partial t}+\frac{\partial u_0}{\partial t}\frac{\partial \delta {\varphi }_x}{\partial t}\right)\end{array}\right)+\\ I_2\left(\begin{array}{l}\frac{25}{16}\frac{\partial {\varphi }_x}{\partial t}\frac{\partial \delta {\varphi }_x}{\partial t}+\frac{1}{16}\frac{{\partial }^2{w}_0}{\partial x\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}+\\ \frac{5}{16}\left(\frac{\partial \delta {\varphi }_x}{\partial t}\frac{{\partial }^2{w}_0}{\partial x\partial t}+\frac{\partial {\varphi }_x}{\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}\right)\end{array}\right)+\\ I_3\left(\begin{array}{l}-\frac{10}{6h^2}\left(\frac{\partial \delta u_0}{\partial t}\frac{\partial {\varphi }_x}{\partial t}+\frac{\partial u_0}{\partial t}\frac{\partial \delta {\varphi }_x}{\partial t}\right)-\\ \frac{10}{6h^2}\left(\frac{\partial \delta u_0}{\partial t}\frac{{\partial }^2{w}_0}{\partial x\partial t}+\frac{\partial u_0}{\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}\right)\end{array}\right)+\\ I_4\left(\begin{array}{l}-\frac{5}{2h^2}\left(\frac{\partial \delta {\varphi }_x}{\partial t}\frac{{\partial }^2{w}_0}{\partial x\partial t}+\frac{\partial {\varphi }_x}{\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}\right)-\frac{25}{6h^2}\frac{\partial {\varphi }_x}{\partial t}\frac{\partial \delta {\varphi }_x}{\partial t}-\\ \frac{5}{6h^2}\frac{{\partial }^2{w}_0}{\partial x\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}\end{array}\right)+\\ I_6\left(\begin{array}{l}\frac{50}{18h^4}\left(\frac{\partial {\varphi }_x}{\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}+\frac{\partial \delta {\varphi }_x}{\partial t}\frac{{\partial }^2{w}_0}{\partial x\partial t}\right)+\frac{25}{9h^4}\frac{\partial {\varphi }_x}{\partial t}\frac{\partial \delta {\varphi }_x}{\partial t}+\\ \frac{25}{9h^4}\frac{{\partial }^2{w}_0}{\partial x\partial t}\frac{{\partial }^2\delta w_0}{\partial x\partial t}\end{array}\right)\end{array}\right)}dx$$

(30)

where:

$$\begin{array}{l}(I_0,I_1,I_2,I_3,I_4,I_6)={\displaystyle {\int }_{-h_c/2}^{h_c/2}{\rho }^{FGM}(1,z,z^2,z^3,z^4,z^6)}dz+\\ {\displaystyle {\int }_{h_c/2}^{h_c/2+h_a}{\rho }^{Toplayer}(1,z,z^2,z^3,z^4,z^6)}dz+{\displaystyle {\int }_{-h_c/2}^{-(h_c/2+h_a)}{\rho }^{Bottomlayer}(1,z,z^2,z^3,z^4,z^6)}dz\end{array}$$

(31)

The elastic potential energy of the beam includes the strain energy due to vibration $\left(P{E}_p\right)$ and the primary stresses due to a thermal increase ($P{E}_T$), respectively, namely [37,38]:

$$PE=P{E}_p+P{E}_T$$

(32)

$$P{E}_p={\displaystyle {\int }_0^L{\displaystyle {\int }_{-(h_c/2+h_a)}^{h_c/2+h_a}({\sigma }_{xx}{\epsilon }_{xx}^L+{\sigma }_{xz}{\gamma }_{xz}^L)dxdz}}$$

(33)

$$P{E}_T={\displaystyle {\int }_0^L{\displaystyle {\int }_{-(h_c/2+h_a)}^{h_c/2+h_a}({\sigma }_{0xx}{\epsilon }_{xx}^{NL}+{\sigma }_{0xz}{\gamma }_{xz}^{NL})dxdz}}$$

(34)

where ${\sigma }_{ij}$σij and ${\sigma }_{0ij}$ $(i,j=x,z)$ refer to the stress tensor and pre-stress components due to the applied temperature field, respectively. In addition, ${\gamma }_{ij}^L$ and ${\epsilon }_{ij}^L$ correspond to the linear terms of the shear and normal strain tensor, respectively, whereas ${\gamma }_{ij}^{NL}$, ${\gamma }_{ij}^{NL}$, and ${\epsilon }_{ij}^{NL}$ are nonlinear terms of the shear and normal strain tensor, respectively. By considering Equations (13) and (33), the variational strain energy, $\delta P{E}_p$, is defined as:

$$\begin{array}{l}\delta P{E}_p={\displaystyle \underset{0}{\overset{L}{\int }}\left(N_x\frac{d\delta u}{dx}+M_x\left(\frac{5}{4}\frac{d\delta {\varphi }_x}{dx}+\frac{1}{4}\frac{d^2\delta w_0}{d{x}^2}\right)\right.+}\\ P_x\left(\frac{5}{3h^2}\frac{d\delta {\varphi }_x}{dx}+\frac{5}{3h^2}\frac{d^2\delta w_0}{d{x}^2}\right)+\\ \left.N_{xz}\left(\frac{5}{4}\delta {\varphi }_x+\frac{5}{4}\frac{d\delta w_0}{dx}\right)+R_{xz}\left(-\left(\frac{5}{h^2}\delta {\varphi }_x+\frac{5}{h^2}\frac{d\delta w_0}{dx}\right)\right)\right)dx\end{array}$$

(35)

In addition, the variational energy related to primary stresses (34) $\delta P{E}_T$ is defined as:

$$\delta P{E}_T={\displaystyle {\int }_0^l{N}_{0x}\frac{d{w}_0}{dx}\frac{d\delta w_0}{dx}}dx$$

(36)

where:

$$\begin{array}{l}{N}_{0x}=N_{0x}^T+N_{0x}^r\\ N_{0x}^T=-{\displaystyle {\int }_{-(h_c/2+h_a)}^{h_c/2+h_a}(C_{11}(z){\alpha }_x\Delta T)}dz\\ N_{0x}^r={\displaystyle {\int }_{-(h_c/2+h_a)}^{h_c/2+h_a}{V}_s{\sigma }_x^r}dz\end{array}$$

(37)

By substituting Equations (30), (35), and (36) into Equation (28), we obtain the following equations of motion:

$$\begin{array}{l}\delta u_0=0\Rightarrow \partial \frac{\partial N_x^{}}{dx}=I_0\frac{{\partial }^2{u}_0}{\partial t^2}+\frac{1}{4}{I}_1\frac{{\partial }^3{w}_0}{\partial x\partial t^2}+\frac{5}{4}{I}_1\frac{{\partial }^2{\varphi }_x}{\partial t^2}-\frac{10}{6h^2}{I}_3\left(\frac{{\partial }^2{\varphi }_x}{\partial t^2}+\frac{{\partial }^3{w}_0}{\partial x\partial t^2}\right)\\ \delta {\varphi }_x=0\Rightarrow \frac{5}{4}\frac{\partial M_x^{}}{\partial x}-\frac{5}{3h^2}\frac{\partial P_x^{}}{\partial x}-\frac{5}{4}{N}_{xz}^{}+\frac{5}{h^2}{R}_{xz}^{}=\frac{5}{4}{I}_1\frac{{\partial }^2{u}_0}{\partial t^2}+\frac{25}{16}{I}_2\frac{{\partial }^2{\varphi }_x}{\partial t^2}+\\ \frac{5}{16}{I}_2\frac{{\partial }^3{w}_0}{\partial x\partial t^2}-\frac{10}{6h^2}{I}_3\frac{{\partial }^2{u}_0}{\partial t^2}-\frac{5}{2h^2}{I}_4\frac{{\partial }^3{w}_0}{\partial x\partial t^2}-\frac{25}{6h^2}{I}_4\frac{{\partial }^2{\varphi }_x}{\partial t^2}+\frac{50}{18h^4}{I}_6\frac{{\partial }^3{w}_0}{\partial x\partial t^2}+\\ \frac{25}{9h^4}{I}_6\frac{{\partial }^2{\varphi }_x}{\partial t^2}\\ \delta w_0=0\Rightarrow N_{0x}\frac{{\partial }^2{w}_0^{}}{\partial x^2}-\frac{1}{4}\frac{{\partial }^2{M}_x^{}}{\partial x^2}+\frac{5}{3h^2}\frac{{\partial }^2{P}_x^{}}{\partial x^2}+\frac{5}{4}\frac{\partial N_{xz}^{}}{\partial x}-\frac{5}{h^2}\frac{\partial R_{xz}^{}}{\partial x}=I_0\frac{{\partial }^2{w}_0}{\partial t^2}\\ -\frac{1}{4}{I}_1\frac{{\partial }^3{u}_0}{\partial x\partial t^2}-\frac{1}{16}{I}_2\frac{{\partial }^4{w}_0}{\partial x^2\partial t^2}-\frac{5}{16}{I}_2\frac{{\partial }^3{\varphi }_x}{\partial x\partial t^2}+\frac{10}{6h^2}{I}_3\frac{{\partial }^3{u}_0}{\partial x\partial t^2}+\frac{5}{2h^2}{I}_4\frac{{\partial }^3{\varphi }_x}{\partial x\partial t^2}+\\ \frac{5}{6h^2}{I}_4\frac{{\partial }^4{w}_0}{\partial x^2\partial t^2}-\frac{50}{18h^4}{I}_6\frac{{\partial }^3{\varphi }_x}{\partial x\partial t^2}-\frac{25}{9h^4}{I}_6\frac{{\partial }^4{w}_0}{\partial x^2\partial t^2}\end{array}$$

(38)

where the displacement components are assumed as:

$$\begin{array}{l}{u}_0^{}(x,t)=u_m\cos \left(\frac{m\pi x}{L}\right)e^{i\omega t}\\ {\varphi }_x^{}(x,t)={\varphi }_m\cos \left(\frac{m\pi x}{L}\right)e^{i\omega t}\\ w_0^{}(x,t)=w_m\sin \left(\frac{m\pi x}{L}\right)e^{i\omega t}\end{array}$$

(39)

and $\omega$ refers to the natural frequency.

4. Results and Discussion

The numerical investigation starts with a validation of the proposed model against the literature and continues with a systematic analysis of the thermal buckling and thermal vibration of the sandwich beam for any kinds of thermal distribution, and varying input parameters $n$, $L/h$, ${\epsilon }_0$, and $V_{SMA}$. Thus, our results are first compared to predictions by Kiani and Eslami [39] based on the Euler–Bernoulli theory, as summarized in

Table 4. Critical bucking load of FGM beam with S-S supports for different value of n and L⁄h.

$\mathit{L}/\mathit{h}$	$\mathit{n}$
	0		1		2		10
	Current Study	[39]	Current Study	[39]	Current Study	[39]	Current Study	[39]
0.1	2365.8	2212.88	1030.1	959.01	854.7	797.03	901	851.48
0.05	596.40	545.72	255.09	232.72	211.17	192.66	225.20	206.22
0.025	142.40	128.93	57.03	51.14	46.45	41.56	50.08	44.91
0.013	33.40	29.51	9.53	7.84	6.93	5.52	7.93	6.43

in terms of critical bucking load of the FGM beam with simple supports (S-S) for different values of $L/h$ and volume fraction power, $n$, under a uniform temperature distribution. As is visible in this table, the good matching between the current formulation and [39] for each value of $L/h$ and volume fraction, $n$, confirms the accuracy of the proposed formulation, where the Euler–Bernoulli-based approach always yields more conservative results compared to higher-order assumptions, as employed in this work, with the main advantage of treating structural members with higher thicknesses. It is also worth observing that an increased value of $L/h$ and $n$ yields a progressive reduction in the critical buckling load, due to an overall increase in the structural flexibility.

After this validation step, the parametric study starts considering the variation of the critical temperature versus $h/L$ ratio, as shown in Figure 2, as provided by the present formulation and [39], with a reasonable matching among them, with some deviations due to the different theoretical assumptions, especially for higher thicknesses, where a higher order theory seems to be more accurate, accounting for any possible shear effect. Based on the plots in this figure, we note the monotone increase in the critical temperature for an increased $h/L$ ratio, for both a uniform and a linear thermal distribution. A linear distribution, however, always yields higher critical temperatures compared to a uniform distribution under the same geometrical assumptions for the structure.

The numerical investigation continues evaluating comparatively the free vibration of S-S-FGM beams, as estimated by the present model and other formulations from the literature. More specifically, we compare the first dimensionless frequency, $\Omega =\omega L^2/h\sqrt{{\rho }_m/E_m}$, as provided by our formulation against predictions from a classic beam theory [40] and an FSDT [41], as listed in

Table 5. First dimensionless frequency for FGM beam with an S-S boundary condition.

$\mathit{L}/\mathit{h}$	Result	$\mathit{n}$
		0	0.1	1	2	10
5	Nejati et al. [41]	6.8470	6.4990	4.8210	4.2510	3.7370
	Aydogdu et al. [40]	6.5632	6.2372	4.6533	4.1025	3.5610
	Present work	6.5134	6.1445	4.6234	4.0671	3.4966
20	Nejati et al. [41]	6.9510	6.5990	4.9070	4.3340	3.8040
	Aydogdu et al. [40]	6.9313	6.5808	4.8950	4.3234	3.7914
	Present work	6.9301	5.5251	4.8942	4.2330	3.7913

, for different values of $n$ and $L/h$, with a very good agreement among results, all proving the reliability and accuracy of the proposed formulation.

4.1. Thermal Buckling Analysis

In this section, we assess the sensitivity of the buckling response to the length ratio ($h/L$), volume fraction power ($n$), initial strain (${\epsilon }_0$), volume fraction ($V_{SMA}$), and thermal distribution. Figure 3 and Figure 4 show the monotone variation of the critical thermal buckling versus $h/L$ for different values of initial strain and volume fraction of SMA wires within the composite layers, respectively. In both figures, we account for both the TD- and TID-properties of the material, together with a uniform thermal distribution. According to the figures, by increasing ${\epsilon }_0$ and $V_{SMA}$, the critical thermal buckling increases. In addition, the value of the critical thermal buckling for TID-properties is higher than that one for TD properties. Based on the plots in both figures, the critical solutions based on TD and TID material properties assume the same value for low geometrical ratios (i.e., for $h/L=0.01$), while increasing their differences for higher values of $h/L$ ratios. An increased value of ${\epsilon }_0$ and $V_{SMA}$ enables a shift of the curves upwards, with an overall increase in the critical thermal buckling for both TD and TID material properties. In addition, Figure 5 depicts the variation of the critical thermal buckling versus $L/h$ for different values of $n$, under the double assumption of TD or TID material, together with a uniform thermal distribution and fixed values of $V_{SMA}=0.2$, ${\epsilon }_0=2\%$, and $h_c/h=0.8$. As shown in the plots of Figure 5, an enhanced value of $n$ yields an overall increase in the critical thermal buckling, for both TID and TD material properties. Additionally, for a fixed value of $n$ and $L/h$, the critical thermal buckling associated with TID material properties is always higher than the one related to TD material properties. The main differences among TD- and TID-based results rely on the different phenomenological models, where stress and strain distributions can be clearly affected by temperature together with any possible phase transformation.

Such sensitivity of the response to the input value of $n$ seems to reduce gradually, for higher geometrical values of $L/h$, until the critical thermal buckling becomes totally insensitive to TID or TD material properties.

The influence of linear and uniform thermal distributions on the critical thermal buckling is shown in Figure 6, while varying the geometrical ratio, $L/h$, from 10 up to 100. As clearly visible from the plots in this figure, the critical thermal buckling obtained for a linear thermal distribution is always higher than the one associated with a uniform distribution, under the same value of $V_{SMA}$. At the same time, for a fixed thermal distribution, an increased value of $V_{SMA}$ enables increased values of the critical thermal buckling, especially for higher $L/h$ ratios. The effect of ${\epsilon }_0$, $V_{SMA}$, and $n$ on the critical thermal buckling in linear thermal distribution is shown in Figure 7, Figure 8 and Figure 9, for $L/h$ ranging from 10 to 100. According to Figure 7 and Figure 8, the initial strain and the volume fraction both provide a similar effect on the critical thermal buckling. Indeed, in both figures the critical thermal buckling increases by increasing ${\epsilon }_0$ and $V_{SMA}$. The effect of ${\epsilon }_0$ and $V_{SMA}$ on the critical thermal buckling seems to increase for higher values of $L/h$. Another parameter affecting the critical thermal buckling response can be the SMA layers thickness ($h_a$), as depicted in Figure 9 for both linear and uniform thermal distributions, while assuming $n=1$, $h/L=0.04$, $V_{SMA}=0.2$, and ${\epsilon }_0=2\%$. More specifically, the critical thermal buckling increases by increasing $h_a$, whereas the critical thermal buckling for a linear distribution is always higher than the one associated with a uniform distribution.

4.2. Thermal Vibration Analysis

In this last section, we focus on the vibration response of the SMA hybrid sandwich beam in a thermal environment. Figure 10, Figure 11 and Figure 12 depict the variation of the first dimensionless frequency vs. the temperature for a uniform thermal distribution by considering the effect of ${\epsilon }_0$, $V_{SMA}$, and $n$, respectively. According to Figure 10 and Figure 11, the first dimensionless frequency tends to decrease for an increased temperature, except for an initial oscillating response for lower temperatures within the starting and ending point of the austenite phase. This effect becomes even more pronounced for higher values of ${\epsilon }_0$ and $V_{SMA}$. A different sensitivity is observed in Figure 12, because the first frequency can increase or decrease non-monotonically for different settings of the input parameters in terms of $n$and temperature. In these three figures, the curves intersect at one common point, at which the SMA wires activate. A recovery stress takes place in SMA wires between two distinct points of all the curves due to the enhanced temperature, with a recovery stress induced to the whole structure.

The sensitivity of the vibration response to the core thickness is plotted in Figure 13, under the following assumptions: $V_{SMA}=0.4$, $n=2$, ${\epsilon }_0=2\%$, and a uniform thermal distribution. As can be seen in this figure, the core thickness has a very pronounced effect on the first dimensionless frequency. An increased core thickness seems to provide an increased frequency response for lower temperatures, with a contrary variation of the response for temperatures higher than 200 °C. In Figure 14, we finally plot the variation of the first dimensionless frequency versus temperature by considering TD and TID conditions, under the double assumption of $V_{SMA}=0.4$ and $V_{SMA}=0.6,$ respectively. According to these plots, the first dimensionless frequency with TID properties is higher than that one for TD properties, whereas an increased value of $V_{SMA}$ yields a reduced value of the first frequency for lower temperatures, and an overall increase for temperatures higher than 100 °C.

5. Conclusions

The present work has focused on the buckling and free vibration response of a simply supported sandwich beam with an FGM core and external layers reinforced with SMA wires, immersed in a thermal environment. The proposed approach, based on a higher order theory and a variational energy formulation, has been verified in its reliability against predictions from literature, with an accurate response. After this preliminary validation, a systematic study followed for different input parameters, with interesting concluding remarks that could serve as theoretical benchmarks for a further computational treatment of the topic. More specifically, critical thermal buckling has been found to increase for both linear and uniform distributions, by increasing the initial strain field and the reinforcement volume fraction. At the same time, the critical thermal buckling for TD properties is always higher than the one associated with TID properties, especially for higher thicknesses of the structural member. An increased value of $n$ and layer thickness, together with a decreased length-to-thickness ratio, caused an overall enhancement of the critical thermal buckling, which is, in turn, affected by the different thermal distributions assumed in the problem. In detail, the critical thermal buckling, for a linear thermal distribution, always achieves higher values than a uniform distribution. As far as the vibration response is concerned, we have determined the first dimensionless frequency response of the sandwich beam, whose value has been revealed to increase for higher values of ${\epsilon }_0$, $V_{SMA}$, and $n$. Moreover, TD and TID properties can affect the first dimensionless frequency, so that the first dimensionless frequency of TID seems to be higher than TD.