## 2 Theoretical Problem  
Let us consider a straight composite laminated beam between two FG skins in a Cartesian coordinate system. The FG-FRCL beam has length \(L\), uniform width \(b\) and total thickness \(H\) in \(x,\ y\) and \(z\) direction, respectively, as shown in Figure 1. The governing equations are derived under the following assumptions: (a) external layers made of FGMs with uniform thickness, \(h_{f}\), which are perfectly bonded on the top and bottom side of the composite laminated beam; (b) fiber-reinforced laminated core, with thickness, \(h_{c}\), made of \(n=3\) layers with the same material and different lay-ups. This means that the FG-FRCL beam is made of \(N=n+2\) layers; (c) Euler-Bernoulli beam assumption, thus neglecting the effect of shear deformation and rotary inertia; (d) geometric von [PERSON] nonlinearity; (e) uniform temperature rise \(\Delta T\). According to the Euler-Bernoulli beam theory, the cross-sections of the beam remain orthogonal to the mid-plane after beam along \(x,\ y\) and \(z\) directions, denoted by \(u_{x}\), \(u_{y}\) and \(u_{z}\) respectively. \[u_{x}(x,z,t)=u(x,t)-z\frac{\partial w(x,t)}{\partial x} \tag{1}\]  
\[u_{y}(x,z,t)=0 \tag{2}\]  
\[u_{z}(x,z,t)=w(x,t) \tag{3}\]  
where \(u\) and \(w\) represent the displacement unknowns at the midplane along the \(x\) and \(z\) direction, respectively, \(\frac{\partial w(x,t)}{\partial x}\) is the rotation angle of cross section about the \(y\) axis and \(t\) is the time. As mentioned before, in order to consider the geometric nonlinearity, the von [PERSON] strain tensor is here adopted. Based on the von Karman strain tensor, the total nonlinear strain-displacement relationship can be defined in compact notation as follows [30; 31]  
\[\varepsilon_{ij}=\frac{1}{2}\Bigg{[}\frac{\partial u_{i}}{\partial x_{j}}+ \frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u_{k}}{\partial x_{i}} \frac{\partial u_{k}}{\partial x_{j}}\Bigg{]},\ (i,\ j,\ k)\ \epsilon\ [1,2,3] \tag{4}\]Using Equations (1)-(4), we obtain the following nonzero component of the strain tensor  
\[\varepsilon_{xx}=\frac{\partial u}{\partial x}-z\frac{\partial^{2}w}{\partial x^{ 2}}+\frac{1}{2}\bigg{(}\frac{\partial w}{\partial x}\bigg{)}^{2} \tag{5}\]  
To model the distribution of ceramic and metal phases along the thickness of both FG layers (i.e., top and bottom layers), the rule of mixture is applied. The beam thickness is, thus, numbered from the top surface at \(z=z_{0},\Big{(}z_{0}=-\frac{H}{2}\Big{)}\), to the bottom surface at \(z=z_{5},\Big{(}z_{5}=\frac{H}{2}\Big{)}\). Based on the rule of mixture, the arbitrary \(\Lambda\) material properties of the FGM layers, primarily, density, \(\rho\), modulus of elasticity, \(E\) and Poisson's ratio, vary from pure ceramic to pure metal as follows [18]:  
\[ \begin{array}{l}\Lambda_{1}=\Lambda_{c}\ +\ (\Lambda_{m}-\Lambda_{c})\Big{(} \frac{z-z_{5}}{z_{4}-z_{5}}\Big{)}^{p},z\ \in\ [z_{4},\ z_{5}]\\ \Lambda_{3}=\Lambda_{m}\ +\ (\Lambda_{c}-\Lambda_{m})\Big{(}\frac{z-z_{1}}{z_{0}-z _{1}}\Big{)}^{p},z\ \in\ [z_{0},\ z_{1}]\end{array}  \tag{6}\]  
where \(\Lambda_{1}\) and \(\Lambda_{3}\) are the material properties at the bottom and top layer, respectively, indices \(c\) and \(m\) refer to the ceramic and metal phase, respectively, and \(p\) is the volume fraction exponent that indicates the material variation profile through the thickness of FG layers. In presence of a thermal load, the stress-strain relationship for the \(k\)-th layer in the axial direction is defined as  
\[\sigma_{x}^{(k)}=Q_{11}^{(k)}\Big{(}\varepsilon_{xx}-\alpha_{x}^{(k)}\Delta T \Big{)} \tag{7}\]  
In which \(\Delta T\ =\ T\ -\ T_{0}\), is the temperature rise from the initial temperature \(T_{0}\), to temperature \(T\).