NonlinearGurson

Nonlinear General Gurson Porous Model

Yield Function

An extended yield function is used,

$$F=q^2+2q_{1}f{\sigma }_y^2\cosh \left({\displaystyle \frac{3}{2}}{\displaystyle \frac{q_{2}p}{{\sigma }_y}}\right)-{\sigma }_y^2(q_1^2{f}^2+1),$$

where

$$s=\mathrm{dev}\sigma ,p={\displaystyle \frac{\mathrm{tr}\sigma }{3}}={\displaystyle \frac{I_1}{3}},q=\sqrt{3J_2}=\sqrt{{\displaystyle \frac{3}{2}}s:s}=\sqrt{{\displaystyle \frac{3}{2}}}|s|.$$

Furthermore, $q_1$, $q_2$ and $q_3=q_1^2$ are model constants, $f({\epsilon }_m^p)$ is the volume fraction, ${\sigma }_y({\epsilon }_m^p)$ is the yield stress, ${\epsilon }_m^p$ is the equivalent plastic strain.

• $q_1=q_2=1$ The original Gurson model is recovered.
• $q_1=0$ The von Mises model is recovered.

Evolution of Equivalent Plastic Strain

The evolution of ${\epsilon }_m^p$ is assumed to be governed by the equivalent plastic work expression,

$$(1-f){\sigma }_y\mathrm{\Delta }{\epsilon }_m^p=\sigma :\mathrm{\Delta }{\epsilon }^p=2\mathrm{\Delta }\gamma {\left({\displaystyle \frac{q^{tr}}{1+6G\mathrm{\Delta }\gamma }}\right)}^2+3q_1{q}_{2}p\mathrm{\Delta }\gamma f{\sigma }_y\sinh \left({\displaystyle \frac{3}{2}}{\displaystyle \frac{q_{2}p}{{\sigma }_y}}\right).$$

Evolution of Volume Fraction

The evolution of volume fraction consists of two parts.

$$\mathrm{\Delta }f=\mathrm{\Delta }{f}_g+\mathrm{\Delta }{f}_n,$$

where

$$\mathrm{\Delta }{f}_g=(1-f)\mathrm{\Delta }{\epsilon }_v,\mathrm{\Delta }{f}_n=A\mathrm{\Delta }{\epsilon }_m^p$$

with

$$A={\displaystyle \frac{f_N}{s_N\sqrt{2\pi }}}\exp (-{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\epsilon }_m^p-{\epsilon }_N}{s_N}}\right)}^2).$$

Parameters $f_N$, $s_N$ and ${\epsilon }_N$ controls the normal distribution of volume fraction. If ($f_N=0$, the nucleation is disabled. In this case, when ($f_0=0$, the volume fraction will stay at zero regardless of strain history.