# How Soft Polymers Cope with Cracks and Notches
## 3 Discussion
### Crack Tip Blunting  
Linear Elastic Fracture Mechanics (LEFM) is grounded on the hypothesis of infinitesimal displacements, with the near-tip region being described by the K-dominated fields, which imply, for instance, a square root singularity in the strain and stress components at the crack tip. In soft materials, the large strain region near the tip can be relevant, and the local fields need to be defined within the framework of finite strain elastostatic, showing for instance stress singularities different from the well-known square-root [39]. However, tip blunting can still be described according to the simplified model introduced in Section 3.1, provided that the local strain is not too large.  
According to LEFM, the deformed shape of a generic crack of semilength \(a\), subjected to a uniform stress \(\sigma_{0}\), is described by the following expression of the crack flank displacement [40].  
\[y(x)=2\frac{\sigma_{0}}{E}\sqrt{a^{2}-x^{2}} \tag{11}\]  
which is the equation of an ellipse, having the major semi-axis equal to \(a\) and the minor semi-axis equal to \(b_{0}=2a\frac{\sigma_{0}}{E}\) (Figure 10c). Retrieving the expression of the tip radius of an ellipse from Equation (4), the equivalent radius of curvature is  
\[\rho_{0}=4a\Big{(}\frac{\sigma_{0}}{E}\Big{)}^{2} \tag{12}\]  
A direct relationship between the applied remote stretch and the stress concentration factor predicted by the Inglis solution, Equation (5), is derived as  
\[K_{t}(\lambda_{0})=1+2\sqrt{\frac{a}{\rho_{0}}}=\frac{\lambda_{0}}{\lambda_{0 }-1} \tag{13}\]  
where we have used the standard relation for linear elastic materials: \(\lambda_{0}=1+\frac{\sigma_{0}}{E}\). The noticeable feature of such an expression is that it does not depend on the initial length of the crack. According to LEFM, the ultimate condition at failure occurs when the remote stress equals  
\[\sigma_{\text{C}}=\frac{1}{Z\big{(}\frac{a}{W}\big{)}}\sqrt{\frac{G_{\text{C }}E}{\pi a}} \tag{14}\]  
where \(Z\big{(}\frac{a}{W}\big{)}\) is a corrective factor for the finite width of the cracked specimens [41] (considering the initial geometry of the tested notched specimens, the values of \(Z\) range from 1.02 to 1.69) and \(G_{\text{C}}\) is the fracture resistance of the material. The critical stretch, omitting the corrective factor \(Z\), is then obtained as  
\[\lambda_{C}=1+\sqrt{\frac{G_{\text{C}}}{\pi E}}a^{-\frac{1}{2}} \tag{15}\]  
Up to this point, we have not considered the deformation of the blunted crack under loading. The predicted stress concentration factor, as is described from this 'pure' LEFM model in Equation (13), is illustrated by the continuous line in Figure 11.  
The effects of the blunted tip deformation under increasing loading and the resulting stress concentration factor are described adopting the'modified' model introduced in Section 3.1, where the current tip radius is computed on the deformed configuration from Equation (10). We assume that the initial configuration of the crack is the critical condition of LEFM, Equation (14), when \(\sigma_{0}=\sigma_{\text{C}}\) and the corresponding value of the radius of curvature, from Equation (12), is  
\[\rho_{\text{C}}=\frac{4}{\pi}\frac{G_{\text{C}}}{E} \tag{16}\]  
The results of the'modified' model are also shown in Figure 11. We have considered the center-cracked specimens CC1 and CC4, which present two different lengths of the crack: we notice that, contrary to the LEFM model results, there is a dependence of \(K_{t}\) on \(a\).