Crack tip opening displacement

Remarks

Example: in the Dugdale model of a crack of length $2a$ in a thin plate under remote tension $\sigma$, $\delta = \frac{8 \sigma_Y a}{\pi E} \ln \sec\left( \frac{\pi \sigma}{2 \sigma_Y} \right)$, where $\sigma_Y$ is the yield stress and $E$ Young's modulus. For small $\sigma / \sigma_Y$, $\ln \sec x \approx x^2/2$ gives $\delta \approx \pi \sigma^2 a / (E \sigma_Y) = K_I^2 / (E \sigma_Y)$ with $K_I = \sigma \sqrt{\pi a}$. Non-example: the opening of a crack in a purely linear elastic body at its tip is zero, since the faces close like $\sqrt{r}$. A positive CTOD requires blunting or a cohesive zone. The second definition in the list is called the $90^\circ$ intercept definition. In general $\delta = d_n J / \sigma_Y$, where $J$ is the J-integral and $d_n$ a dimensionless factor of order one that depends on the strain hardening and on plane stress or plane strain.