# An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates
## Appendix A Laminated Beam Stiffness
### Shear Stiffness  
The shear stiffness of a laminated plate can be evaluated by imposing a strain energy equivalence, as usually done for homogeneous beams [56] and recently extended to laminated plates by [PERSON] et al. [25]. Here, we detail such a procedure for a laminated beam [68; 69; 70]. To this aim, first we recall the expression of the normal stress in the \(x\)-direction within the \(k\)th lamina [49]:  
\[\sigma_{x}^{(k)}(z)=Q_{xx}^{(k)}\left(\epsilon_{x}+zx_{x}\right)+Q_{xy}^{(k)} \left(\epsilon_{y}+zx_{y}\right)+Q_{xs}^{(k)}\left(\gamma_{xy}+zx_{xy}\right). \tag{10}\]  
Next, we substitute the expressions of the plate strain measures obtained in the previous section into Equation (10). Thus, we obtain:  
\[\sigma_{x}^{(k)}(z)=Q_{e0}^{(k)}\epsilon+Q_{x0}^{(k)}\kappa+z\left(Q_{e1}^{(k) }\epsilon+Q_{x1}^{(k)}\kappa\right), \tag{11}\]  
where:  
\[\begin{split} Q_{e0}^{(k)}&=Q_{xx}^{(k)}-\frac{A_{ xy}D_{yy}-B_{xy}B_{yy}}{A_{yy}D_{yy}-B_{yy}^{2}}Q_{xy}^{(k)},\quad Q_{x0}^{(k)} =-\frac{B_{xy}D_{yy}-B_{yy}D_{xy}}{A_{yy}D_{yy}-B_{yy}^{2}}Q_{xy}^{(k)},\\ Q_{e1}^{(k)}&=-\frac{A_{yy}B_{xy}-A_{xy}B_{yy}}{A_ {yy}D_{yy}-B_{yy}^{2}}Q_{xy}^{(k)},\quad\text{and}\quad Q_{x1}^{(k)}=Q_{xx}^{(k )}-\frac{A_{yy}D_{xy}-B_{xy}B_{yy}}{A_{yy}D_{yy}-B_{yy}^{2}}Q_{xy}^{(k)}.\end{split} \tag{12}\]  
Following the classical derivation of [PERSON]'s shear stress formula [56], we impose the static equilibrium in the \(x\)-direction of a short portion of beam defined by a given value of the \(z\)-coordinate. Thus, by recalling the inverse constitutive Equation (4) and static equilibrium Equation (8), with no distributed loads, the following expression is obtained for the shear stress in the \(k\)th lamina:  
\[\tau_{zx}^{(k)}(z)=Q\left(R_{k}-S_{k}z-T_{k}z^{2}\right), \tag{13}\]  
where \(Q\) is the transverse shear force (acting in the \(z\)-direction) and:  
\[\begin{split} R_{k}&=S_{k}z_{k}+T_{k}z_{k}^{2}+\sum _{j=\bar{k}+1}^{n}\left[S_{j}\left(z_{j}-z_{j-1}\right)+T_{j}\left(z_{j}^{2}- z_{j-1}^{2}\right)\right],\\ S_{k}&=\text{b}Q_{e0}^{(k)}+\text{d}Q_{x0}^{(k)}, \quad\text{and}\quad T_{k}=\frac{1}{2}\left(\text{b}Q_{e1}^{(k)}+\text{d}Q_{x1 }^{(k)}\right),\end{split} \tag{14}\]  
where b and d respectively are the bending-extension and bending compliances of the laminated beam, as given by Equation (3).  
The shear compliance, c, can now be calculated by imposing the following energy equivalence for a short beam segment:  
\[\frac{1}{2}\text{c}Q^{2}=\frac{1}{2}B\int_{-h}^{h}\frac{1}{G_{zx}^{(k)}}\left[ \tau_{zx}^{(k)}(z)\right]^{2}\,\text{d}z, \tag{15}\]  
where:  
\[G_{zx}^{(k)}=\frac{1}{\frac{\sin^{2}\theta_{k}}{G_{23}^{(k)}}+\frac{\cos^{2} \theta_{k}}{G_{31}^{(k)}}} \tag{16}\]  
is the transverse shear modulus of the \(k\)th lamina in the laminate reference system, depending on its out-of-plane shear moduli, \(G_{23}^{(k)}\) and \(G_{31}^{(k)}\). By substituting Equation (13) into (15), the shear compliance is obtained:\[\begin{split}\mathsf{c}=B\sum_{k=1}^{n}\frac{1}{G_{zz}^{(k)}}& \left[R_{k}^{2}\left(z_{k}-z_{k-1}\right)-R_{k}\mathsf{S}_{k}\left(z_{k}^{2}-z_ {k-1}^{2}\right)+\right.\\ &+\frac{1}{3}\left(S_{k}^{2}-2R_{k}T_{k}\right)\left(z_{k}^{3}-z_ {k-1}^{3}\right)+\\ &\left.+\frac{1}{2}\mathsf{S}_{k}T_{k}\left(z_{k}^{4}-z_{k-1}^{4 }\right)+\frac{1}{5}T_{k}^{2}\left(z_{k}^{5}-z_{k-1}^{5}\right)\right].\end{split} \tag{19}\]  
Lastly, the laminated beam shear stiffness is \(\mathsf{C}=1/\mathsf{c}\).