Dugdale strip-yield model

Remarks

Clause 3 gives the plastic zone length through $$\frac{a}{a + \rho} = \cos\left( \frac{\pi \sigma}{2 \sigma_Y} \right).$$ Check for small loads: $\cos x \approx 1 - x^2/2$ gives $\rho \approx \frac{\pi^2 \sigma^2 a}{8 \sigma_Y^2} = \frac{\pi}{8} \left( \frac{K_I}{\sigma_Y} \right)^2$ with $K_I = \sigma \sqrt{\pi a}$, so the plastic zone is small compared with $a$. As $\sigma \to \sigma_Y$, $\rho \to \infty$, which describes general yielding of the plate. The opening of the fictitious crack at the physical tip $x_1 = a$ is $\delta = \frac{8 \sigma_Y a}{\pi E} \ln \sec\left( \frac{\pi \sigma}{2 \sigma_Y} \right)$, with $E$ the Young's modulus. Non-example: the Irwin plastic zone correction, which only shifts the elastic crack tip by a length proportional to $(K_I / \sigma_Y)^2$, does not impose clause 2 and is not the Dugdale model.