## 3 Mode II Delamination Tests
### Relative Displacement between the Two Sub-Beams  
Utilizing Equation (24), the relative vertical displacement between the two sub-beams can be expressed as:  
\[ \begin{split}\Delta w=& w^{(2^{\prime})}-w^{(1^{\prime} )}=\\ &\big{[}l^{(2)}(P_{2}-P_{c})-l^{(1)}(P_{c}+P_{1})\big{]}(\alpha-x^{ \prime})\\ &-\mu P_{c}\bigg{(}b^{(2)}-\frac{h^{(2)}}{2}d^{(2)}-b^{(1)}+ \frac{h^{(1)}}{2}d^{(1)}\bigg{)}\bigg{(}-\frac{(x^{\prime})^{2}}{2}+\alpha x^{ \prime}-\frac{\alpha^{2}}{2}\bigg{)}\\ &+\big{[}d^{(2)}(P_{2}-P_{c})-d^{(1)}(P_{c}+P_{1})\big{]}\bigg{(} \frac{(x^{\prime})^{3}}{6}-\frac{\alpha^{2}x^{\prime}}{2}+\frac{\alpha^{3}}{3 }\bigg{)}\\ &+\Delta\phi|_{x^{\prime}=\alpha}(\alpha-x^{\prime})+\Delta w|_{x ^{\prime}=\alpha}+\frac{1}{2}(\mathsf{e}_{2}^{(1)}-\mathsf{e}_{2}^{(2)})( \alpha-x^{\prime})^{2}\end{split}  \tag{26}\]  
or in a matrix form:  
\[ \begin{split}\Delta w=&\begin{bmatrix}\Omega_{1}&\Omega_{2}&\Omega_{3}\end{bmatrix} \begin{bmatrix}P_{c}\\ P_{1}\\ P_{2}\end{bmatrix}+\Delta\phi|_{x^{\prime}=\alpha}(\alpha-x^{\prime})+\Delta w|_{x ^{\prime}=\alpha}+\frac{1}{2}(\mathsf{e}_{2}^{(1)}-\mathsf{e}_{2}^{(2)})( \alpha-x^{\prime})^{2}\end{split}  \tag{27}\]  
where  
\[ \begin{split}\Omega_{1}&=(x^{\prime}-\alpha)(l^{(1)}+l^{(2)})- \mu\bigg{(}-b^{(1)}+\frac{h^{(1)}}{2}d^{(1)}+b^{(2)}-\frac{h^{(2)}}{2}d^{(2)} \bigg{)}\bigg{(}-\frac{(x^{\prime})^{2}}{2}+\alpha x^{\prime}-\frac{\alpha^{2} }{2}\bigg{)}\\ &+\big{(}d^{(1)}+d^{(2)}\big{)}\bigg{(}-\frac{(x^{\prime})^{3}}{6 }+\frac{\alpha^{2}x^{\prime}}{2}-\frac{\alpha^{3}}{3}\bigg{)}\\ \Omega_{2}&=(x^{\prime}-\alpha)l^{(1)}+d^{(1)}\bigg{(}- \frac{(x^{\prime})^{3}}{6}+\frac{\alpha^{2}x^{\prime}}{2}-\frac{\alpha^{3}}{3 }\bigg{)}\\ \Omega_{3}&=-(x^{\prime}-\alpha)l^{(2)}-d^{(2)} \bigg{(}-\frac{(x^{\prime})^{3}}{6}+\frac{\alpha^{2}x^{\prime}}{2}-\frac{ \alpha^{3}}{3}\bigg{)}\end{split}  \tag{28}\]  
with \(\Delta w|_{x^{\prime}=0}=0\) since the two ends must be in contact in order for a contact force to be generated. A special solution of Equation (26) was presented in the work of [PERSON] and [PERSON] [14], where the relative deflections \(\Delta w|_{x^{\prime}=\alpha}\) at the crack tip, the relative rotations \(\Delta\phi|_{x^{\prime}=\alpha}\) at the crack tip, and the friction force between the two sub-beams \(\mu P_{c}\) were neglected. Equation (26) expressing the relative behavior between the two sub-beams at the cracked part of a composite laminate is provided for the first time in the literature. The relative deflections \(\Delta w|_{x^{\prime}=\alpha}\) and the relative rotations \(\Delta\phi|_{x^{\prime}=\alpha}\) at the crack tip are evaluated considering the displacement and rotation continuity conditions at the crack tip, where \(x^{\prime}=a\) for the left-hand side of the crack tip (i.e., the cracked part), and \(x=0\) at the right-hand side of the crack tip (i.e., the uncracked part):  
\[ \begin{bmatrix}\Delta u(m)\\ \Delta\phi(m)\\ \Delta w(m)\end{bmatrix} _{x^{\prime}=\alpha^{-}}= \begin{bmatrix}\Delta u(m)\\ \Delta\phi(m)\\ \Delta w(m)\end{bmatrix} _{x^{\prime}=\alpha}= \begin{bmatrix}\Delta u(m)\\ \Delta\phi(m)\\ \Delta w(m)\end{bmatrix} _{x=0^{+}}= \begin{bmatrix}\Delta u(m)\\ \Delta\phi(m)\\ \Delta w(m)\end{bmatrix} _{x=0} \tag{29}\]  
where the superscript \(m\) denotes one of the joint models. The solutions of \(\Delta u(m)\), \(\Delta\phi(m)\), and \(\Delta w(m)\) can be evaluated from the solutions of the interface compliance of each model according to Equations (A4), (A11) and (A24). The relative axial displacements \(\Delta u(m)|_{x=0}\), the relative deflections \(\Delta w(m)|_{x=0}\), and the relative rotations \(\Delta\phi^{(m)}|_{x=0}\) for each model are given by:\[ \begin{bmatrix}\Delta u^{(m)}\\ \Delta\phi^{(m)}\\ \Delta w^{(m)}\end{bmatrix} _{x=0} = \Delta[\mathcal{S}^{(m)}] \begin{bmatrix}N\\ M\\ Q\end{bmatrix}  \tag{30}\]  
where \(\Delta[\mathcal{S}^{(m)}]=[\mathcal{S}^{(m2)}]-[\mathcal{S}^{(m1)}]\) expresses the relative compliance between the two sub-beams for each model \(m\).