## 2 Modeling and Governing Stress Equations
\begin{array}{c}\tau_{1}(x)=-q_{1}c_{1}exp^{q_{1}x}+q_{1}c_{2}exp^{-q_{1}x }-\sqrt{q_{2}+q_{3}}Ic_{3}exp^{x\sqrt{q_{2}+q_{3}i}}-\sqrt{q_{2}+q_{3}}Ic_{4} exp^{x\sqrt{q_{2}-q_{3}i}}+\\ \hskip 14.226378 pt\sqrt{q_{2}+q_{3}}Ic_{5}exp^{-x\sqrt{q_{2}+q_{3}i}}+\sqrt{q_ {2}-q_{3}}Ic_{6}exp^{-x\sqrt{q_{2}-q_{3}i}}\end{array}  \tag{25}\]  
Based on the third relation obtained from FBD of adherend No. 1, the changes of \(Q_{i}\) can be expressed in the following equation. \[N_{1}(x)=c_{1}exp^{q_{1}x}+c_{2}exp^{-q_{1}x}+c_{3}exp^{x\sqrt{q_{2}+q_{3}i}}+ c_{4}exp^{x\sqrt{q_{2}-q_{3}i}}+c_{5}exp^{-x\sqrt{q_{2}+q_{3}i}}+c_{6}exp^{-x \sqrt{q_{2}-q_{3}i}}+\frac{T\delta_{4}}{\delta_{3}} \tag{26}\]  
Appendix A expresses the coefficients of \(\omega\)i (_i_ = 1-6) in the above equation. Now, the formulas defined for \(N_{1}\), \(M_{1}\), and \(Q_{1}\) and governing boundary conditions can be employed for adherend 1 to specify differential equation constants. Assume that the coordinates origin is located on one left end of adherend 1 (Figure 5), so the boundary conditions are  
\[\boxed{x=0}\ :\ N_{1}=M_{1}=Q_{1}=0 \tag{27}\]The differential equation coefficients are calculated by substituting these boundary conditions into Equations (23), (24) and (26). Therefore, the peel stress equation can be determined by  
\[\sigma_{1}(x)=\omega_{1}q_{1}c_{1}exp^{q_{1}x}-\omega_{2}q_{1}c_{2}exp^{-q_{1}x}+ \omega_{3}\sqrt{q_{2}+q_{3}}ic_{3}exp^{x\sqrt{q_{2}+q_{3}}}+\omega_{4}\sqrt{q_ {2}-q_{3}}ic_{4}exp^{x\sqrt{q_{2}-q_{3}}l}\] \[-\omega_{5}\sqrt{q_{2}+q_{3}}ic_{5}exp^{-x\sqrt{q_{2}+q_{3}}l}- \omega_{6}\sqrt{q_{2}-q_{3}}ic_{6}exp^{-x\sqrt{q_{2}-q_{3}}l} \tag{29}\]  
A finite element model in ANSYS was used to validate the analytical solution and find the maximum peel stress location and magnitude in the adhesive layer [12]. It was assumed that the adherends are Graphite-Epoxy T5208/300 and adhesive layers are Method 408. The mechanical properties of these materials are mentioned in Table 1.  
Accordingly, the maximum peel stress occurred at both ends of the adhesive area, and the analytical solution results are perfectly acceptable, compared to the finite element model results [12]. The differences between analytical and FEM solutions of peel stress along the adhesive area are shown in Figure 6. The maximum peel stress calculated via analytical solution is 473.08 kPa, where the FEM gives a 501.65 kPa output in the same conditions. The differences between those two methods are 6.04 percent. The horizontal axis in Figure 6 is the length of the adhesive area, and the vertical axis illustrates the peel stress magnitude. As mentioned before, the highest peel stress amplitude occurs at both ends of the adhesive area. The differences between the outcomes of the analytical solution (Equation (29)), FEM, and the result obtained by [PERSON] [9] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material Name** & \(\mathbf{\
u}\) & \(\mathbf{G}\) (GPa) & \(\mathbf{E_{z}}\) (GPa) & \(\mathbf{E_{x}}\) (GPa) \\ \hline Graphite-Epoxy & 0.280 & 7.170 & 10.300 & 181 \\ Method & 0.343 & 0.358 & - & 0.96 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mechanical properties of selected materials. Figure 5: Length of the top adhesive and adherend layers (adherent layer No 1). Figure 6: Variation of peel stress along bond length based on analytical and finite element solutions. for a classic double lap joint are illustrated in Figure 7. According to the graphs, the dimensionless maximum peel stress obtained from the analytical solution is 0.36, whereas the FEM and Mortensen methods reported it as 0.35 and 0.33, respectively. Therefore, the analytical model which is used in this paper is more accordant with the FEM.