## 2 Methods
### Stress-Strain Relations  
For the HCS plates, the stress-strain relations of the \(k\)th layer (\(k=1,2,3\)) in the thermal environment with respect to the global coordinate system are given as  
\[\mathbf{\sigma}^{(k)}=\mathbf{Q}_{in}^{(k)}\Big{(}\mathbf{\varepsilon}^{(k)}-\mathbf{ \varepsilon}_{0}^{(k)}\Big{)}\ \mathbf{\tau}^{(k)}=\mathbf{Q}_{tr}^{(k)}\mathbf{\gamma}^{(k)} \tag{6}\]  
where \(\mathbf{Q}_{in}^{(k)}\) and \(\mathbf{Q}_{tr}^{(k)}\) are the in-plane and transverse shear stiffness matrices. \(\mathbf{\varepsilon}_{0}^{(k)}\) is the initial strain vector caused by thermal expansion, given as  
\[\mathbf{\varepsilon}_{0}^{(k)}=\mathbf{\alpha}^{(k)}\Delta T(z) \tag{7}\]  
where \(\mathbf{\alpha}^{(k)}\) is the thermal expansion coefficient and \(\Delta T(z)\) is the temperature change through the plate thickness, which could have uniform, linear or parabolic distribution.  
For the bottom and top face sheets (\(k=1,\,3\)), the stiffness matrices and thermal expansion coefficient in the global coordinate system are obtained from those in the material coordinate system with transformation matrices between the global and material coordinates, which are expressed in detail in [38].  
For the foam-filled corrugated core (\(k=2\)), the effective stiffness matrices and thermal expansion coefficient are obtained from a homogenization method [19], in which the coupling effects between the corrugated structure and the filled foam are concerned. When the _MD_ of the foam-filled corrugated core coincides with the \(x\)-direction of the global coordinate system, the plane stress-reduced and transverse shear stiffness matrices the foam-filled corrugated core are given as  
\[\mathbf{Q}_{in}^{(2)}=\left[\begin{array}{ccc}\overline{Q}_{11}^{H}&\overline{Q}_ {12}^{H}&0\\ \overline{Q}_{12}^{H}&\overline{Q}_{22}^{H}&0\\ 0&0&Q_{66}^{H}\end{array}\right]\mathbf{Q}_{tr}^{(2)}=\left[\begin{array}{cc}Q_ {44}^{H}&0\\ 0&Q_{55}^{H}\end{array}\right] \tag{8}\]where \(\overline{Q_{ij}^{H}}=Q_{ij}^{H}-Q_{ij}^{H}Q_{j}^{H}/Q_{33}^{H}\left(i,j=1,2,3\right)\) and the effective stiffnesses \(Q_{ij}^{H}\) are comprised of contributions from the corrugated structure, the filled foam and the coupling effect between the corrugated structure and the foam, represented by \(Q_{ij}^{s}\), \(Q_{ij}^{fm}\) and \(Q_{ij}^{c}\), respectively, and \(Q_{ij}^{H}\) is given as  
\[Q_{ij}^{H}=Q_{ij}^{s}+Q_{ij}^{c}+(1-f)Q_{ij}^{fm} \tag{9}\]  
where the detailed \(Q_{ij}^{s}\), \(Q_{ij}^{c}\) and \(Q_{ij}^{fm}\) are given in Equations (A1)-(A3) (Appendix A).  
Moreover, the components of the effective thermal-stress tensor are given as  
\[\begin{array}{l}\beta_{1}^{H}=\frac{a_{s}E_{s}}{1-v_{s}^{2}}f+(1-f)\alpha_{ fm}Q_{11}^{fm}\\ \beta_{2}^{H}=\frac{a_{s}E_{s}}{1-v_{s}^{2}}\cos\theta+(1-f)\alpha_{fm}\Big{(} Q_{22}^{fm}+2Q_{23}^{fm}\Big{)},\\ \beta_{3}^{H}=\frac{a_{s}E_{s}}{1-v_{s}^{2}}\frac{t\sin^{2}\theta}{h_{2}}\frac{ t\sin^{2}\theta}{\cos\theta}+(1-f)\alpha_{fm}\Big{(}Q_{33}^{fm}+2Q_{23}^{fm} \Big{)}.\end{array} \tag{10}\]  
Therefore, the in-plane thermal expansion coefficient of the foam-filled corrugated core is obtained by  
\[\mathbf{\alpha}^{(2)}=\left\{\begin{array}{l}\alpha_{xx}^{(2)}\\ \alpha_{yy}^{(2)}\\ \alpha_{xy}^{(2)}\end{array}\right\}=\left(\mathbf{Q}_{in}^{(2)}\right)^{-1} \left\{\begin{array}{l}\beta_{1}^{H}\\ \beta_{2}^{H}\\ 0\end{array}\right\} \tag{11}\]  
The effective density of the foam-filled corrugated core is given as  
\[\rho^{(2)}=f\rho_{s}+(1-f)\rho_{fm} \tag{12}\]