# An Elastic Interface Model for the Delamination of Bending-Extension Coupled Laminates
## Appendix C \(\mathcal{J}\)-Integral Calculation for the Crack-Tip Element  
The path-independent \(\mathcal{J}\)-integral was introduced by [PERSON] [57] as:  
\[\mathcal{J}=\int_{\Gamma}\left[\omega\,\mathsf{d}z-\left(t_{x}\frac{\partial u }{\partial x}+t_{z}\frac{\partial w}{\partial x}\right)\,\mathsf{d}\Gamma \right], \tag{26}\]where \(\Gamma\) is an arbitrary integration path surrounding the crack tip, \(\omega\) is the _strain energy density_ (per unit volume), \(t_{x}\) and \(t_{z}\) are the _stress vector components_ (referred to the outer normal to \(\Gamma\)), and \(u\) and \(w\) are the _displacement vector components_ along the coordinates \(x\) and \(z\). For the CTE, we choose an integration path, \(\Gamma\), encircling the interface between sublaminates as depicted in Figure 11. The path is subdivided into three segments, \(\Gamma_{12}\), \(\Gamma_{23}\), and \(\Gamma_{34}\), along which quantities entering the \(\mathcal{J}\)-integral have the expressions listed in Table 11.  
Accordingly, the following expression results:  
\[ \begin{split}\mathcal{J}=&\int_{0}^{\ell}-\left[- \tau\frac{\mathrm{d}}{\mathrm{d}x}\left(u_{1}+h_{1}\varphi_{1}\right)-\sigma \frac{\mathrm{d}w_{1}}{\mathrm{d}x}\right]\,\mathrm{d}x+\\ &+\int_{0}^{t}\left[\frac{\Phi}{t}-\left(t_{x}\frac{\mathrm{d}u}{ \mathrm{d}x}+t_{z}\frac{\mathrm{d}w}{\mathrm{d}x}\right)\right]\,\mathrm{d}z+ \\ &+\int_{\ell}^{0}-\left[\tau\frac{\mathrm{d}}{\mathrm{d}x}\left(u _{2}-h_{2}\varphi_{2}\right)+\sigma\frac{\mathrm{d}w_{2}}{\mathrm{d}x}\right] \left(-\mathrm{d}x\right),\end{split}  \tag{103}\]  
where \(t\) is the interface thickness and \(\Phi=\Phi(\Delta w,\Delta u)\) is the _interface potential energy_[27], such that:  
\[\sigma=\frac{\partial\Phi}{\partial\Delta w}\quad\text{and}\quad\tau=\frac{ \partial\Phi}{\partial\Delta u}. \tag{104}\]  
By recalling the definition of relative displacements, Equation (6), and assuming a vanishing interface thickness (\(t\to 0\)), we obtain:  
\[\mathcal{J}=\Phi\left[\Delta w(\ell),\Delta u\left(\ell\right)\right]-\int_{0} ^{\ell}\left[\sigma(x)\frac{\mathrm{d}\Delta w(x)}{\mathrm{d}x}+\tau(x)\frac{ \mathrm{d}\Delta u(x)}{\mathrm{d}x}\right]\,\mathrm{d}x. \tag{105}\]  
Next, by substituting Equation (104) into (105) and recalling that \(\,\mathrm{d}\Phi\) is an exact differential, we deduce:\[\mathcal{J} =\Phi\left[\Delta w(\ell),\Delta u\left(\ell\right)\right]-\int_{0}^{ \ell}\left[\frac{\partial\Phi}{\partial\Delta w}\frac{\mathrm{d}\Delta w(x)}{ \mathrm{d}x}+\frac{\partial\Phi}{\partial\Delta u}\frac{\mathrm{d}\Delta u(x)} {\mathrm{d}x}\right]\,\mathrm{d}x=\] \[=\Phi\left[\Delta w(\ell),\Delta u\left(\ell\right)\right]-\int_{ \left[\Delta w(0),\Delta u(0)\right]}^{\left[\Delta w(\ell),\Delta u(\ell) \right]}\frac{\partial\Phi}{\partial\Delta w}\,\mathrm{d}\Delta w+\frac{ \partial\Phi}{\partial\Delta u}\,\mathrm{d}\Delta u= \tag{111}\] \[=\Phi\left[\Delta w(\ell),\Delta u\left(\ell\right)\right]-\int_{ \left[\Delta w(0),\Delta u(\ell)\right]}^{\left[\Delta w(\ell),\Delta u(\ell )\right]}\mathrm{d}\Phi.\]  
Hence,  
\[\mathcal{J}=\Phi\left[\Delta w(0),\Delta u(0)\right], \tag{112}\]  
which holds for a general cohesive interface with potential-based traction-separation laws. For a brittle-elastic interface, the potential energy function specializes to:  
\[\Phi(\Delta w,\Delta u)=\frac{1}{2}k_{z}\Delta w^{2}+\frac{1}{2}k_{x}\Delta u^ {2}. \tag{113}\]  
By substituting Equation (113) into (112), we obtain:  
\[\mathcal{J}=\frac{1}{2}k_{z}\Delta w^{2}(0)+\frac{1}{2}k_{x}\Delta u^{2}(0), \tag{114}\]  
which is identical to Equation (42). For the present elastic interface model, the two addends in Equation (114) naturally identify the Mode I and II contributions to the energy release rate (see Section 4.1). This result holds for both the balanced and unbalanced cases, for which the analytical solution has been derived in Section 3. For a general cohesive interface model, Equation (112) yields the value of the \(\mathcal{J}\)-integral, but its decomposition into Mode I and II contributions is not straightforward.