## 5 Results
### Energy Release Rate  
Figure 9 presents the ERR versus the external load for the cases without thermal loads ((a) and (d)), and with +135-degree ((b) and (e)), and \(-\)135-degree ((c) and (f)) temperature differences. Figure 9a-c present the solutions of the ERR considering the contact force evaluated by each respective joint model; for example, the formulation of the flexible joint model for the ERR was estimated by using the formulation of the flexible joint model for the contact force. On the other hand, Figure 9a-c present the solutions of the ERR considering the contact force evaluated by the rigid joint model, as it is commonly done in the literature. The value of the coefficient of friction was considered to be 0.01 to limit the influence of the friction effects on the ERR and we only considered the effects of the contact force that is presented in Section 5.1.  
Figure 8: Contact force versus the external load for the cases without thermal loads (**a**), and with +135-degree (**b**), and \(-\)135-degree (**c**) temperature differences.  
In Figure 9a-c, the comparison of the joint models, including the rigid, semi-rigid, and flexible models, with the VCCT and CZM models shows that the rigid and semi-rigid models were in good agreement with the VCCT, as they assumed no relative translation at the interface between the upper and lower sub-beams. Similarly, the flexible joint model was found to be comparable to the CZM, as it used cohesive elements between the two sub-beams that behaved like a series of springs. All three models were found to be consistent with the analytical theories and the numerical simulations. The addition of \(-135\) degrees of temperature difference resulted in a decrease in the ERR whereas the addition of a \(-135\)-degree temperature difference resulted in an increase in the ERR. The decrease and increase in the ERR due to the thermal effects were of equal magnitude.  
Figure 9d-f demonstrates that the consideration of the contact force evaluated by the rigid model in the evaluation of the ERR for all joint models did not influence the predictions of the semi-rigid model since the contact forces estimated by both the rigid and the semi-rigid model were almost equal. However, considering the contact force in the formulation of the flexible joint model considerably changed the estimation of the ERR. In fact, the ERR estimated by the flexible joint model was then almost identical to the one of the rigid and the semi-rigid joint model. Therefore, the application of the appropriate contact force is indeed important to obtain accurate results when using analytical models.  
The impact of the coefficient of friction on the ERR is illustrated in Figure 10. The value of the ERR when a coefficient of friction of 0.01 was used is represented by \(G_{0}\), whereas \(G\) represents the ERR when a different value of the coefficient of friction was applied. The study examined coefficient of friction values of up to 0.2, which is a typical value for an epoxy composite in contact with aluminum [49]. The results were only obtained for the case of +135 degrees, which considered both residual thermal stresses and a point force, consistent with the assumptions of the analytical models. The results showed that an increase in friction led to a linear decrease in the ERR. However, the decrease was minimal and for a coefficient of friction of 0.2, the expected reduction in the ERR was 0.8 %. It is worth noting that the external load was taken as 10 kN/m as the percentage-wise difference in the ERR was constant for all mechanical loads.  
Figure 9: ERR versus the external load for the cases without thermal loads (**a**,**d**), and with +135-degree (**b**,**e**), and \(-135\)-degree (**c**,**f**) temperature differences. ERR estimated by the respective contact force formulation for each joint model (**a–c**). ERR estimated by the rigid joint model (**d–f**).