# An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates
## Appendix A Laminated Beam Stiffness
### Extension and Bending Stiffness  
For a laminated plate with a generic stacking sequence, the constitutive laws are:  
\[\left\{ \begin{array}{c}N_{x}\\ N_{y}\\ N_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}A_{xx}&A_{xy}&A_{xs}\\ A_{xy}&A_{yy}&A_{ys}\\ A_{xs}&A_{ys}&A_{ss}\end{array} \right]\left\{ \begin{array}{c}\epsilon_{x}\\ e_{y}\\ \gamma_{xy}\end{array} \right\}+\left[ \begin{array}{ccc}B_{xx}&B_{xy}&B_{xs} \\ B_{xy}&B_{yy}&B_{ys}\\ B_{xs}&B_{ys}&B_{ss}\end{array} \right]\left\{ \begin{array}{c}\kappa_{x}\\ \kappa_{y}\\ \kappa_{xy}\end{array} \right\} \tag{11}\]  
and:  
\[\left\{ \begin{array}{c}M_{x}\\ M_{y}\\ M_{xy}\end{array} \right\}=\left[ \begin{array}{ccc}B_{xx}&B_{xy}&B_{xs}\\ B_{xy}&B_{yy}&B_{ys}\\ B_{xs}&B_{ys}&B_{ss}\end{array} \right]\left\{ \begin{array}{c}\epsilon_{x}\\ \epsilon_{y}\\ \gamma_{xy}\end{array} \right\}+\left[ \begin{array}{ccc}D_{xx}&D_{xy}&D_{xs} \\ D_{xy}&D_{yy}&D_{ys}\\ D_{xs}&D_{ys}&D_{ss}\end{array} \right]\left\{ \begin{array}{c}\kappa_{x}\\ \kappa_{y}\\ \kappa_{xy}\end{array} \right\}, \tag{12}\]  
where \(N_{x}\), \(N_{y}\), and \(N_{xy}\) are the in-plane forces and \(M_{x}\), \(M_{y}\), and \(M_{xy}\) are the bending moments (per unit plate width); correspondingly, \(\epsilon_{x}\), \(\epsilon_{y}\), and \(\gamma_{xy}\) are the in-plane strains and \(\kappa_{x}\), \(\kappa_{y}\), and \(\kappa_{xy}\) are the bending curvatures; \(A_{ij}\), \(B_{ij}\), and \(D_{ij}\), with \(i,j\in\{x,y,s\}\), are the coefficients of the extensional, bending-extension coupling, and bending stiffness matrices, respectively, to be computed following classical laminated plate theory [49]:  
\[\left\{ \begin{array}{c}A_{ij}\\ B_{ij}\\ D_{ij}\end{array} \right\}=\sum_{k=1}^{n}Q_{ij}^{(k)}\left\{ \begin{array}{c}z_ {k}-z_{k-1}\\ \frac{1}{2}(z_{k}^{2}-z_{k-1}^{2})\\ \frac{1}{3}(z_{k}^{3}-z_{k-1}^{3})\end{array} \right\}, \tag{13}\]  
where \(Q_{ij}^{(k)}\), with \(i,j\in\{x,y,s\}\), are the in-plane elastic moduli of the \(k\)th lamina, included between the ordinates \(z=z_{k-1}\) and \(z=z_{k}\), in the global reference system, and \(n\) is the total number of laminae. For an orthotropic lamina, whose principal material axes, 1 and 2, are rotated by an angle \(\theta_{k}\) with respect to the global axes \(x\) and \(y\),  
\[Q_{xx}^{(k)} =Q_{11}^{(k)}\cos^{4}\theta_{k}+2(Q_{12}^{(k)}+2Q_{66}^{(k)})\cos ^{2}\theta_{k}\sin^{2}\theta_{k}+Q_{22}^{(k)}\sin^{4}\theta_{k},\] (14) \[Q_{yy}^{(k)} =Q_{yx}^{(k)}=(Q_{11}^{(k)}+Q_{22}^{(k)}-4Q_{66}^{(k)})\cos^{2} \theta_\[ \begin{split} Q_{11}^{(k)}&=\frac{E_{1}^{(k)}}{1-v_{12}^{( k)}v_{21}^{(k)}},\quad Q_{12}^{(k)}=Q_{21}^{(k)}=\frac{v_{12}^{(k)}E_{2}^{(k)}}{1-v_{12}^{( k)}v_{21}^{(k)}},\\ Q_{22}^{(k)}&=\frac{E_{2}^{(k)}}{1-v_{12}^{(k)}v_{21} ^{(k)}},\quad\text{and}\quad Q_{66}^{(k)}=G_{12}^{(k)}\end{split}  \tag{10}\]  
with \(E_{1}^{(k)},E_{2}^{(k)},v_{12}^{(k)},v_{21}^{(k)}\), and \(G_{12}^{(k)}\) being the in-plane elastic moduli of the lamina in engineering notation. For a plate to be modeled as a plane beam in the \(zx\)-plane, the out-of-plane internal forces and moments must be \(N_{y}=N_{xy}=0\) and \(M_{y}=M_{xy}=0\). Besides, the shear-extension and bend-twist coupling terms must vanish: \(A_{xs}=A_{ys}=0\) and \(D_{xs}=D_{ys}=0\). The bending-shear coupling terms must also be null: \(B_{xs}=B_{ys}=0\). Such conditions are fulfilled by laminates with single or multiple special orthotropic layers and antisymmetric cross-ply laminates. Furthermore, antisymmetric angle-ply and generic multi-directional laminates may satisfy the above-mentioned conditions for particular stacking sequences. As a consequence, Equations (11) and (12) reduce to:  
\[