## 2 Theoretical Model  
Since most engineering theories of beams, plates and shells cannot recover all stresses accurately through their constitutive equations [6; 7], the modeling strategy proposed recently by present authors [12] is employed. A classical sandwich panel shown in Figure 1 is firstly divided into three parts, as shown in Figure 2. The top and bottom face sheets are modeled by Euler-Bernoulli beam theory and the core is directly modeled by two-dimensional (2D) elasticity theory without a priori assumption. After applying the continuity and equilibrium conditions at the interfaces (\(z=\pm c\)), the resultant differential equations of the top and bottom face sheets are used as the boundary conditions of the 2D core and then only the core needs to be analyzed. In this way, both displacement and stress in the soft-core can be accurately predicted. The Euler-Bernoulli beam assumption is usually adopted in the face sheets of sandwich panel/beam theories, e.g., the extended high-order sandwich panel theory (EHSAPT) [3], which has been validated numerically by present authors [12]. Denote \(f_{t}\), \(f_{b}\) and \(2c\) as the thickness of top face, bottom face and the core, respectively. The length of sandwich is \(L\) and a unit width is considered. For simplicity in presentation, the displacement components in \(x\) and \(z\) directions are denoted by \(u(x,z)\) and \(w(x,z)\). The material of the core is homogeneous and orthotropic. Therefore, the stress-strain relation is given by  
Figure 1: Sketch of a sandwich panel with a soft core. Figure 2: Sketch of the top face, bottom face and the core of a sandwich panel. (**a**) top face; (**b**) core; (c) bottom face. \[\left\{ \begin{array}{c}\sigma_{xx}\\ \sigma_{zz}\\ \tau_{xz}\end{array} \right\}=\left[Q\right]\left\{ \begin{array}{c}\epsilon_{xx} \\ \epsilon_{zz}\\ \gamma_{xz}\end{array} \right\}=\left[ \begin{array}{ccc}Q_{11}&Q_{13}&0\\ Q_{13}&Q_{33}&0\\ 0&0&Q_{66}\end{array} \right]\left\{ \begin{array}{c}\epsilon_{xx}\\ \epsilon_{zz}\\ \gamma_{xz}\end{array} \right\}=\left[ \begin{array}{ccc}\frac{E_{xx}}{1- \
u_{xz}\
u_{xx}}&\frac{\
u_{xx}E_{zz}}{1-\
u_{xz}\
u_{xx}}&0\\ \frac{\
u_{xx}E_{xx}}{1-\
u_{xz}\
u_{xx}}&\frac{E_{zz}}{1-\
u_{xz}\
u_{xx}}&0\\ 0&0&G_{xz}\end{array} \right]\left\{ \begin{array}{c}\epsilon_{xx}\\ \epsilon_{zz}\\ \gamma_{xx}\end{array} \right\} \tag{1}\]  
where symbols \(E\), \(G\), and \(\
u\) are elasticity modulus, shear modulus, and Poisson's ratio, respectively. The differential equations of the core, shown in Figure 2b, are given by  
\[\left\{ \begin{array}{c}Q_{11}\frac{\partial^{2}u}{\partial x^{2}}+Q_{66} \frac{\partial^{2}u}{\partial z^{2}}+\left(Q_{13}+Q_{66}\right)\frac{\partial ^{2}w}{\partial x\partial z}=0\\ Q_{66}\frac{\partial^{2}w}{\partial x^{2}}+Q_{33}\frac{\partial^{2}w}{\partial z ^{2}}+\left(Q_{13}+Q_{66}\right)\frac{\partial^{2}u}{\partial x\partial z}=0 \end{array} \right. \tag{2}\]  
where body forces are not considered. After applying the continuity and equilibrium conditions at the interface (\(z=-c\)), the differential equations of the bottom face sheet, shown in Figure 2c, are given by  
\[f_{b}E_{1}^{b}\left(u_{,xx}^{-c}+0.5f_{b}w_{,xxx}^{-c}\right)+Q_{66}\left( \frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)_{z=-c}=0 \tag{3}\]  
\[\frac{f_{b}{}^{3}}{12}E_{1}^{b}w_{,xxxx}^{-c}-\frac{f_{b}Q_{66}}{2}\left( \frac{\partial^{2}w}{\partial x^{2}}+\frac{\partial^{2}u}{\partial x\partial z }\right)_{z=-c}-\left(Q_{13}\frac{\partial u}{\partial x}+Q_{33}\frac{ \partial w}{\partial z}\right)_{z=-c}=0 \tag{4}\]  
where \(u^{-c}=u(x,-c)\) and \(w^{-c}=w(x,-c)\) are displacement components in \(x\) and \(z\) directions, and \(E_{1}^{b}\) is the elastic modulus of the bottom face sheet.