## Appendix C  
### General Elastic Analysis  
Below the formulas to estimate the distribution of stresses proposed by [PERSON] and [PERSON] [24] are described:  
Normal stress formula:  
\[\sigma_{z}=A_{1}\cdot cos(K_{5}y)\cdot cosh(K_{5}y)+A_{2}\cdot cos(K_{5}\cdot y )\cdot sinh(K_{5}y)+A_{3}\cdot sin(K_{5}y)\cdot cosh(K_{5}y)+A_{4}\cdot sin(K _{5}y)\cdot sinh(K_{5}y),\] (A16)  
where:  
\[D_{1}=\frac{E_{1}\cdot{h_{1}}^{3}}{12\cdot(1-{\mu_{1}}^{2})},\qquad D_{2}= \frac{E_{2}{h_{2}}^{3}}{12\cdot(1-{\mu_{2}}^{2})},\] (A17)  
\[K_{5}^{4}=\frac{E_{6}}{4\cdot t}\bigg{(}\frac{1}{D_{1}}+\frac{1}{D_{2}}\bigg{)},\] (A18)  
\[A_{1}=\frac{b_{3}\cdot R_{3}-2\cdot b_{2}\cdot sinh(K_{5}\cdot L)\cdot sin(K _{5}\cdot L)+b_{1}\cdot R_{6}+b_{4}\cdot R_{1}}{R_{5}},\] (A19a)  
\[A_{2}=\frac{b_{2}\cdot R_{2}-b_{3}\cdot sinh^{2}(K_{5}\cdot L)-b_{1}\cdot R_{ 4}-b_{4}\cdot sinh(K_{5}\cdot L)\cdot sin(K_{5}\cdot L)}{R_{5}},\] (A19b)  
\[A_{3}=\frac{b_{2}\cdot R_{2}-b_{3}\cdot sin^{2}(K_{5}\cdot L)-b_{1}\cdot R_{ 4}-b_{4}\cdot sinh(K_{5}\cdot L)\cdot sin(K_{5}\cdot L)}{R_{5}},\] (A19c)  
\[A_{4}=b_{1},\] (A19d)  
\[b_{1}=\frac{E_{4}}{2\cdot K_{5}2\cdot t}\bigg{(}\frac{M_{\mathcal{L}21}}{D_{2 }}-\frac{M_{\mathcal{L}11}}{D_{1}}\bigg{)},\qquad b_{2}=\frac{E_{4}}{2\cdot K_{ 5}2\cdot t}\bigg{(}\frac{M_{\mathcal{L}22}}{D_{2}}-\frac{M_{\mathcal{L}22}}{D_ {1}}\bigg{)},\] (A20a)  
\[b_{3}=\frac{E_{4}}{2\cdot K_{5}3\cdot t}\bigg{(}\frac{V_{\mathcal{L}21}}{D_{2 }}-\frac{V_{\mathcal{L}11}}{D_{1}}\bigg{)},\qquad b_{4}=\frac{E_{4}}{2\cdot K_{ 5}3\cdot t}\bigg{(}\frac{V_{\mathcal{L}22}}{D_{2}}-\frac{V_{\mathcal{L}12}}{ D_{1}}\bigg{)},\] (A20b)  
\[R_{1}=csht(K_{5}\cdot L)\cdot sin(K_{5}\cdot L)-sinh(K_{5}\cdot L)\cdot cos(K _{5}\cdot L),\] (A21a)  
\[R_{2}=csh(K_{5}\cdot L)\cdot sin(K_{5}\cdot L)+sinh(K_{5}\cdot L)\cdot cos(K_{5 }\cdot L),\] (A21b)  
\[R_{3}=csh(K_{5}\cdot L)\cdot sinh(K_{5}\cdot L)-cos(K_{5}\cdot L)\cdot sin(K_{ 5}\cdot L),\] (A21c)  
\[R_{4}=csh(K_{5}\cdot L)\cdot sin(K_{5}\cdot L)+cos(K_{5}\cdot L)\cdot sin(K_{ 5}\cdot L),\] (A21d)  
\[R_{5}=sinh^{2}(K_{5}\cdot L)-sin^{2}(K_{5}\cdot L),\] (A21e)  
\[R_{6}=csh^{2}(K_{5}\cdot L)-cos^{2}(K_{5}\cdot L).\] (A21f)Shear stress formula:  
\[\tau_{yz}=B_{1}\cdot cosh(K_{6}\cdot y)+B_{2}\cdot sinh(K_{6}\cdot y)+B_{3},\] (A22)  
where:  
\[K_{6}{}^{2}=\frac{4\cdot G_{a}}{t}\left(\frac{1-\mu_{1}{}^{2}}{E_{1}\cdot h_{1}}+ \frac{1-\mu_{2}{}^{2}}{E_{2}\cdot h_{2}}\right),\] (A23)  
\[B_{1}=\frac{c_{2}-c_{1}\cdot cosh(K_{6}\cdot L)}{K_{6}\cdot senh(K_{6}\cdot L)},\qquad B_{2}=\frac{c_{1}}{K_{6}},\qquad B_{3}=\frac{c_{3}}{L}-\frac{c_{2}-c_{1 }}{K_{6}{}^{2}\cdot L},\] (A24)  
\[c_{1}=\frac{G_{a}}{t}\left[\frac{1-\mu_{1}{}^{2}}{E_{1}\cdot h_{1}}\left(T_{c1 1}-\frac{6\cdot M_{c11}}{h_{1}}\right)-\frac{1-\mu_{2}{}^{2}}{E_{2}\cdot h_{2} }\left(T_{c21}+\frac{6\cdot M_{c21}}{h_{2}}\right)\right],\] (A25a)  
\[c_{2}=\frac{G_{a}}{t}\left[\frac{1-\mu_{1}{}^{2}}{E_{1}\cdot h_{1}}\left(T_{c1 2}-\frac{6\cdot M_{c12}}{h_{1}}\right)-\frac{1-\mu_{2}{}^{2}}{E_{2}\cdot h_{2 }}\left(T_{c22}+\frac{6\cdot M_{c22}}{h_{2}}\right)\right],\] (A25b)  
\[c_{3}=(T_{c12}-T_{c11})\left(1-\frac{y}{L}\right)+(T_{c21}-T_{c22})\left( \frac{y}{L}\right),\] (A25c)  
(\(D_{1}\), \(h_{1}\), \(\mu_{1}\), \(E_{1}\)) and (\(D_{2}\), \(h_{2}\), \(\mu_{2}\), \(E_{2}\)) correspond to adherent 1 and adherent 2, respectively.  
\(T_{c11}\), \(V_{c11}\), \(M_{c11}\), \(T_{c12}\), \(V_{c12}\), \(M_{c12}\), are the forces applied to the extremes of adherent 1.  
\(T_{c22}\), \(V_{c22}\), \(M_{c22}\), \(T_{c21}\), \(V_{c21}\), \(M_{c21}\), are the forces applied to the extremes of adherent 2.  
(**) For a better understanding, the nomenclature of the original variables \(b_{1}\), \(b_{2}\), and \(b_{3}\) of the shear stress formula in [24], are changed to \(c_{1}\), \(c_{2}\), and \(c_{3}\).