# How Soft Polymers Cope with Cracks and Notches
## 3 Discussion  
### Model for Notch Blunting  
Cracks and notches in soft materials become distorted under loading due to large strain effects, with an evident tip blunting, as is clearly shown in the figures obtained from the DIC on the experimental tests. In order to quantify the flaw severity and account for the geometrical effect of blunting, we resort to the concept of the stress concentration factor. Considering a notched sample under remote uniform stress acting parallel to the \(y\)-axis, the stress concentration factor can be defined as  
\[K_{t}=\sigma_{\text{max}}/\sigma_{0} \tag{2}\]  
where \(\sigma_{\text{max}}=\sigma_{yy,\text{max}}\) is the maximum notch root stress and \(\sigma_{0}\) is the corresponding remote stress, measured with respect to a uniformly stressed (gross) section of the sample. In the following, we propose a simplified analytical model, capable of relating the stress concentration factor with the tip radius \(\rho\) of the blunted notch and explore the variation of \(K_{t}(\rho)\) under increasing applied loading. Such an approach, conceived for the analysis of blunting of elliptical notches [29], is extended to the tip blunting of cracks (Section 3.2). Figure 9: Ultimate stretch vs the initial relative crack size for the edge-cracked samples. (**a**) Effect of the applied strain rate in the specimens EC1–EC4 and (**b**) influence of the material defects, for a strain rate \(\dot{\epsilon}_{0}=1.9\cdot 10^{-3}\) s\({}^{-1}\) (results from samples EC1–EC4 are shown as solid circles, and D1–D4 as hollow circles). The starting point is the solution for an elliptical notch in an infinite elastic plate, having semi-axes \(a\) and \(b\). Its equation can be written as follows  
\[y(x)=\frac{b}{a}\sqrt{a^{2}-x^{2}} \tag{3}\]  
and the radius of curvature at the tips (\(x=\pm a\)) is equal to  
\[\rho=b^{2}/a \tag{4}\]  
The stress concentration factor \(K_{t}\) for an elliptical notch is obtained from the renowned Inglis' solution [37]:  
\[K_{t}(\rho)=\left(1+2\sqrt{\frac{a}{\rho}}\right)Y\left(\frac{a}{W},\frac{a}{ b}\right) \tag{5}\]  
where \(Y\) is a corrective factor introduced to account for the finite width of the specimens [38] (considering the initial geometry of the tested notched specimens, the values of \(Y\) range from 1.09 to 1.10). For the sake of simplicity, this factor is taken equal to unity in the following equations. In order to describe the change of the stress concentration factor with the increasing deformation of the blunted tip under loading, we need to consider the variation of the radius of curvature. Such a variation depends on the deformation state in proximity of the notch root, which has to be evaluated in the deformed configuration (see Figure 10a). Let us consider a small square element of edge length equal to \(h\), located in the proximity of the notch root, with edges inclined by an angle \(\alpha\) with respect to the horizontal axis (Figure 10b). The small strain tensor \(\varepsilon^{\prime}\) in the local reference system 1-2 (with components \(\varepsilon_{11},\varepsilon_{22},\varepsilon_{12}\)) is related to the corresponding tensor \(\varepsilon\) (with components \(\varepsilon_{tt},\varepsilon_{nn},\varepsilon_{tn}\)) in the reference system \(t\)-\(n\), with its origin located at the notch root, through the well-known relationships [33]:  
\[\varepsilon_{11}=c^{2}\varepsilon_{tt}+s^{2}\varepsilon_{nn},\varepsilon_{22}= s^{2}\varepsilon_{tt}+c^{2}\varepsilon_{nn},\varepsilon_{12}=cse_{tt}-cse_{nn} \tag{6}\]  
with \(c=\cos\alpha,s=\sin\alpha\).