Example 3: Plasticity in plate with circular hole

This example is taken from [13] (page 245). A plate ( $20 \; {\rm mm}$ wide, $36 \; {\rm mm}$ long) is perforated at the center with a hole with radius $5 \; {\rm mm}$. The plate is stretched in length direction by increasing the overall strain $\epsilon$ in time. The other two boundaries are free to deform. Because of symmetry, only a quarter needs to be modeled. A mesh with 48 linear quadrilateral elements is used

The Young's modulus $E=7000 \; \frac{kg}{mm^2}$, Poisson's ratio $\nu = 0.2$, the Von Mises yield stress $\sigma_y = 24.3 \; \frac{kg}{mm^2}$ and the strain hardening $h=0.032E=224 \; \frac{kg}{mm^2}$. Plane stress conditions hold. The plasticity influence is characterized by $\sigma^{\rm mean}$ at the upper edge for $v=0.0625$. The elasto-plastic analysis gives $\sigma^{\rm mean}=12.4 \; \frac{kg}{mm^2}$ for $v=0.0625$. A purely elastic calculation would have given $\sigma^{\rm mean}=16.13 \; \frac{kg}{mm^2}$ .