## 2 Theoretical Formulations
### Energy Functions of Lattice Sandwich Beams  
The analysis in the present paper is limited to linear conditions. Based on the classical sandwich beam theory, the Kirchhoff hypothesis holds, i.e., the straight material line normal to the beam midplane before deformation remains straight and normal to the midplane after deformation. Based on the above deformation hypothesis, the deformation process of the cross-section of the lattice sandwich beam under external force is given Figures 1c and 3. _HIJKL_ represents a straight line that is normal to the undeformed sandwich beam. If there is no shear strain, _HIJKL_ would rotate by an angle \(\partial w/\partial x\) to reach the position \(HT^{\prime}\)\(K\)'\(L\)', as shown in Figure 3b. If a shear strain \(\theta\) occurs in the core, _HIJKL_ moves to the position \(H^{\prime}\)\(T^{\prime}\)\(T^{\prime}\)\(K\)'\(L\)\". The lines \(H^{\prime}\)\(T^{\prime}\) and \(K\)\"\(L\)\" remain parallel with \(HT^{\prime}\)\(K\)'\(L\)', as the shear strains in the face sheets are assumed to be negligible. According to the geometric relationship in the figure, the deformation expression of any point at the cross-section of lattice sandwich beams with different truss cores under arbitrary boundary conditions are given as follows:  
\[ \begin{array}{l}u_{t}=-\frac{c}{2}\theta-z\frac{\partial w}{\partial x},w_{t }=w,(-\frac{h}{2}\leq z\leq-\frac{c}{2})\\ u_{c}=z(\theta-\frac{\partial w}{\partial x}),w_{c}=w,(-\frac{c}{2}\leq z\leq \frac{c}{2})\\ u_{b}=\frac{c}{2}\theta-z\frac{\partial w}{\partial x},w_{b}=w,(\frac{c}{2} \leq z\leq\frac{c}{2})\end{array}  \tag{1}\]  
where \(w\) and \(\theta\) represent the lateral displacement of lattice sandwich beams in the \(z\)-axis direction and the rotation angle in the \(y\)-axis direction, respectively. The thickness of the upper and lower panels and the height of the core layer are represented by \(t\) and \(c\), respectively. The horizontal displacements of the upper panel, the core layer and the lower panel in the \(x\)-axis direction are represented by \(u_{t}\), \(u_{c}\) and \(u_{\text{b}}\), respectively. The lateral displacements of the upper panel, the core layer and the lower panel in the \(z\)-axis direction are represented by \(w_{t}\), \(w_{c}\) and \(w_{\text{b}}\), respectively. Figure 3: Deformation of the equivalent lattice sandwich beam. (**a**) before deformation (**b**) after deformation. According to the above expressions and the definition of the material strain, the strain corresponding to the three layers of the equivalent lattice sandwich beam can be written as the expression of the displacement:  
\[ \begin{array}{l}\varepsilon_{t}=\frac{\partial u_{t}}{\partial x}=-\frac{ \zeta}{2}\frac{\partial\theta}{\partial x}-z\frac{\partial^{2}w}{\partial x^{2 }},(-\frac{h}{2}\leq z\leq-\frac{\zeta}{2})\\ \gamma_{c}=\frac{\partial u_{c}}{\partial x}+\frac{\partial w}{\partial x}= \theta,(-\frac{\zeta}{2}\leq z\leq\frac{\zeta}{2})\\ \varepsilon_{b}=\frac{\partial u_{b}}{\partial x}=\frac{\zeta}{2}\frac{ \partial\theta}{\partial x}-z\frac{\partial^{2}w}{\partial x^{2}},(\frac{ \zeta}{2}\leq z\leq\frac{h}{2})\end{array}  \tag{2}\]  
where \(\varepsilon_{t}\) and \(\varepsilon_{b}\) represent the normal strain of the upper panel and that of the lower panel, respectively. \(\gamma_{c}\) represents the shear strain of the core layer along the length of the beam. Based on Hooke's Law, the stress corresponding to the three layers of the equivalent lattice sandwich beams can be written as the expression of the displacements:  
\[ \begin{array}{l}\sigma_{t}=E_{x}\varepsilon_{t}=E_{x}(-\frac{\zeta}{2} \frac{\partial\theta}{\partial x}-z\frac{\partial^{2}w}{\partial x^{2}}),(- \frac{h}{2}\leq z\leq-\frac{\zeta}{2})\\ \tau_{c}=G_{\mathrm{zze}}\theta,(-\frac{\zeta}{2}\leq z\leq\frac{\zeta}{2})\\ \sigma_{b}=E_{x}\varepsilon_{b}=E_{x}(\frac{\zeta}{2}\frac{\partial\theta}{ \partial x}-z\frac{\partial^{2}w}{\partial x^{2}}),(\frac{\zeta}{2}\leq z\leq \frac{h}{2})\end{array}