## Appendix A Solutions of the Governing Equation
### Semi-Rigid Joint Model  
The solutions for the upper sub-beam (\(s=1\)) for the semi-rigid joint model (\(m=2\)) are:  
\[\mathcal{N}^{(21)} =c^{(2)}e^{-\lambda x}+\mathcal{N}^{(11)}, \tag{20}\] \[\mathcal{M}^{(21)} =\frac{\eta}{\xi}c^{(2)}e^{-\lambda x}+\mathcal{M}^{(11)},\] \[\mathcal{Q}^{(21)} =-\bigg{(}\frac{h^{(1)}}{2}+\frac{\eta}{\xi}\bigg{)}c^{(2)}\lambda e ^{-\lambda x}+\mathcal{Q}^{(11)}\]  
For the evaluation of \(N\), \(M\), and \(Q\), the equilibrium equations at the crack tip are considered, i.e.,  
\[\mathcal{N}^{(1^{\prime})}|_{x^{\prime}=\alpha^{-}} =\mathcal{N}^{(21)}|_{x=0^{+}}-N_{c}, \tag{21}\] \[\mathcal{Q}^{(1^{\prime})}|_{x^{\prime}=\alpha^{-}} =\mathcal{Q}^{(21)}|_{x=0^{+}}-Q_{c},\] \[\mathcal{M}^{(1^{\prime})}|_{x^{\prime}=\alpha^{-}} =\mathcal{M}^{(21)}|_{x=0^{+}}+\frac{h^{(1)}}{2}N_{c}\]And, \(N\), \(M\), and \(Q\) are estimated as:  
\[\begin{split}& N=\mathcal{N}^{(1^{\prime})}|_{x^{\prime}=\alpha^{-}}- \mathcal{N}^{(11)}|_{x=0^{+}},\\ & Q=\mathcal{Q}^{(1^{\prime})}|_{x^{\prime}=\alpha^{-}}- \mathcal{Q}^{(11)}|_{x=0^{+}},\\ & M=\mathcal{M}^{(1^{\prime})}|_{x^{\prime}=\alpha^{-}}- \mathcal{M}^{(11)}|_{x=0^{+}}\end{split} \tag{10}\]  
\(c^{(m)}\) (\(m=2\)) and \(N_{c}\), \(Q_{c}\) are evaluated using the boundary conditions at the crack tip (\(x=0^{+}\)) (see Equation (10)) [11]:  
\[c^{(2)}=-\frac{(2M+h^{(1)}N)\xi}{h^{(1)}\xi+2\eta},\;\;N_{c}=-\frac{2(M\xi-N \eta)}{h^{(1)}\xi+2\eta},\;\;Q_{c}=Q+\lambda\bigg{(}M+\frac{h^{(1)}N}{2}\bigg{)} \tag{11}\]  
\(\lambda\) in Equations (11)-(11) physically expresses the decay rate \(\lambda=\sqrt{-\texttt{a}_{2}/\texttt{a}_{3}}\) with \(\texttt{a}_{3}=\bigg{(}l^{(1)}+l^{(2)}\bigg{)}\bigg{(}\xi\frac{h^{(1)}}{2}+ \eta\bigg{)}+\frac{1}{\texttt{R}_{2}^{2}}\bigg{(}d^{(1)}+d^{(2)}\bigg{)}\). The displacements and rotations' relationships between the two sub-laminates are derived from Equation (11) and considering the constitutive equations (Equation (4)):  
\[\begin{bmatrix}u^{(2s)}\\ \phi^{(2s)}\\ w^{(2s)}\end{bmatrix}=\begin{bmatrix}u^{(1s)}\\ \phi^{(1s)}\\ w^{(1s)}\end{bmatrix}+e^{-\lambda x}\Big{[}\mathcal{S}^{(2s)}\Big{]}\begin{bmatrix} N\\ M\\ Q\end{bmatrix} \tag{12}\]  
where the local deformation compliance matrix of each sub-laminate \(s\) at the joint \([\mathcal{S}^{(2s)}]\) with elements \(\mathcal{S}^{(2s)}_{ij}\) (\(i=1,2,3,\;j=1,2,3\)) is:  
\[\begin{split}\Big{[}\mathcal{S}^{(2s)}\Big{]}=\begin{bmatrix} \big{[}\mathcal{S}^{(21)}\big{]}\\ \mathcal{S}^{(22)}\big{]}\end{bmatrix}=\\ &+\frac{1}{\lambda}\frac{\xi}{h^{(1)}\xi+2\eta}\begin{bmatrix}a^{(1)}+ \frac{\eta}{\xi}b^{(1)}\\ b^{(1)}+\frac{\eta}{\xi}d^{(1)}\\ -\lambda\bigg{(}\frac{h^{(1)}}{2}+\frac{\eta}{\xi}\bigg{)}l^{(1)}+\frac{1}{ \lambda}\bigg{(}b^{(1)}+\frac{\eta}{\xi}d^{(1)}\bigg{)}\\ -a^{(2)}-\bigg{(}\frac{\eta}{\xi}+\frac{h^{(1)}+h^{(2)}}{2}\bigg{)}b^{(2)}\\ -b^{(2)}-\bigg{(}\frac{\eta}{\xi}+\frac{h^{(1)}+h^{(2)}}{2}\bigg{)}d^{(2)}\\ \lambda\bigg{(}\frac{h^{(1)}}{2}+\frac{\eta}{\xi}\bigg{)}l^{(2)}-\frac{1}{ \lambda}\bigg{[}b^{(2)}+\bigg{(}\frac{\eta}{\xi}+\frac{h^{(1)}+h^{(2)}}{2} \bigg{)}d^{(2)}\bigg{]}\end{bmatrix}\begin{bmatrix}h^{(1)}\\ 2\\ 0\end{bmatrix}^{T}\end{split} \tag{13}\]  
For \(x=0\), Equation (12) takes the form:  
\[\begin{bmatrix}u^{(2s)}\\ \phi^{(2s)}\\ w^{(2s)}\end{bmatrix}_{x=0}=\begin{bmatrix}u^{(1s)}\\ \phi^{(1s)}\\ w^{(1s)}\end{bmatrix}_{x=0}+\Big{[}\mathcal{S}^{(2s)}\Big{]}\begin{bmatrix} N\\ M\\ Q\end{bmatrix} \tag{14}\]  
In Equation (12), only the difference between the rotations of the upper and the lower sub-beam (\(\phi^{(21)}|_{x=0}-\phi^{(22)}|_{x=0}\)) was presented in [PERSON] and [PERSON] [30]. Their solution coincides with the above formulas. To the best of the authors' knowledge, Equations (12)-(13) are presented for the first time in the literature.