## 3 Cohesive Model Applied to the SENT Specim in Tension
\[K_{\infty}=\int_{0}^{a}\sigma_{\infty}G(a,t)dt=\sigma_{\infty}\sqrt{W}\,k \Big{(}\frac{a}{W}\Big{)}[h(a,t)]_{0}^{a}=\sigma_{\infty}\sqrt{W}\,k\Big{(} \frac{a}{W}\Big{)} \tag{6}\]  
The substitution in Equation (6) of the stress intensity factor \(K\infty\) given in [28] leads to the following expression for the function \(k(a/W)\):  
\[k\Big{(}\frac{a}{W}\Big{)}=\frac{K_{\infty}}{\sigma_{\infty}\sqrt{W}} = \sqrt{2\text{tan}\frac{\pi\,a}{2W}}\frac{0.752+2.02\frac{a}{W}+0. 37\big{(}1-\text{sen}\frac{\pi a}{W}\big{)}^{3}}{\cos\frac{\pi a}{2W}} \tag{7}\]  
The condition of zero stress intensity factor \(K\infty-K_{Y}\) due to the combined action of the remote tensile stresses \(s\infty\) and the stresses \(Y\) developed in the cohesive zone of the crack determines the total depth at that the cohesive crack acquires when the SENT specimen is tensile-loaded (Figure 4a). \[ \begin{array}{l}K_{Y}=\int_{0}^{A}YG(A,t)dt=Y\sqrt{W}k\Big{(}\frac{A}{W} \Big{)}[1-h(A,a)]\\ K_{\infty}=K_{Y}\quad\Rightarrow\quad\sigma_{\infty}=Y[1-h(A,a)]\end{array}  \tag{8}\]  
Thus, the depth a results from the function \(h(A,a)\) given by Equation (5) in terms of the relative crack size \(\tilde{a}=a/W\):  
\[A=\frac{a}{\widetilde{a}+\Big{(}1-\widetilde{a}\Big{)}\text{cos}\bigg{(}\frac {\pi\sigma_{\infty}}{2Y}\ \frac{1}{\big{(}1-\widetilde{a}\big{)}^{2}}\bigg{)}} \tag{9}\]  
The field \(\omega(x)\) of crack-opening displacements along the cohesive crack (Figure 4a) is the difference between those given by Equation (2) when \(K_{S}(t)\) is respectively particularized for the stress intensity factors \(K\infty\) and \(K_{Y}\), and the crack size is particularized for \(A\):  
\[\omega=\omega(x)=\frac{2}{E}\int_{x}^{A}G(t,x)(K_{\infty}(t)-K_{Y}(t))dt= \frac{2 WY}{E}\int_{x}^{A}k^{2}\bigg{(}\frac{t}{W}\bigg{)}\ h_{2}(t,x)[h(t,a)-h( A,a)]dt \tag{10}\]  
Figure 4: Stress intensity factors of the SENT specimen due to: (a) arbitrary compressive symmetric load on the cohesive crack faces; (b) applied load and cohesive stresses. The respective values \(\omega_{\rm M}\) and \(\omega_{\rm t}\) of this field the positions x = a and x = 0 are the crack-opening displacements at the physical crack tip (CTOD) and at the mouth of the notch (CMOD):  
\[\rm CTOD=\omega_{\rm t}=\frac{2 WY}{E}\int_{a}^{A}k^{2}\left(\frac{t}{W}\right) h_{2}(t,a)[h(t,a)-h(A,a)]dt \tag{11}\]  
\[\rm CMOD=\omega_{\rm M}=\frac{2 WY}{E}\int_{0}^{A}k^{2}\left(\frac{t}{W}\right) h_{2}(t,0)[h(t,a)-h(A,a)]dt \tag{12}\]  
Given that the cohesive crack does not entail any stress singularity, the integration contour to determine the J-integral can be infinitely close to the crack faces. In this case, its value is the product \(Y\omega_{\rm t}\) and, consequently, Equation (11) provides the quotient J/Y. The integrals of Equations (11) and (12) were numerically solved and combined with Equation (10) in order to determine the applied remote stress and the value of the J-integral as a function of CMOD and the relative crack size \(\tilde{a}=a/W\), with generic values of the elastic modulus E and the cohesive resistance Y. Figure 5 shows these results. Note that the values of the dimensionless J-integral are also those of the dimensionless CTOD (\(\omega_{\rm t}E\))/(WY). The asymptotic maximum of the load-CMOD curves corresponds to the condition of A = W and means that the physical crack and the cohesive zone completely occupies the cracked cross-section. The most notable feature of the J-integral is the tendency to increase linearly once the parabolic growth typical to the SSY regime is exceeded. According to Equation (9), the remote stress limit soy is:  
\[A=W\Rightarrow\sigma_{\omega Y}=Y\left(1-\frac{a}{W}\right)^{2} \tag{13}\]  
The dependence between the relative crack size and the slope of the straight part of the curve giving the J-integral can be fully fitted by the linear equation:  
\[\frac{dJ}{d\omega_{\rm M}}=\left(1.00-1.03\frac{a}{W}\right)Y \tag{14}\]  
Figure 5: Load–CMOD and J-integral–CMOD curves of the SENT specimen as predicted by the cohesive model.