NonlinearGurson

Nonlinear General Gurson Porous Model

Yield Function

An extended yield function is used,

$$F=q^2+2q_1f\sigma_y^2\cosh\left(\dfrac{3}{2}\dfrac{q_2p}{\sigma_y}\right)-\sigma_y^2\left(q_1^2f^2+1\right),$$

where

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

Furthermore, $q_1$, $q_2$ and $q_3=q_1^2$ are model constants, $f(\varepsilon_m^p)$ is the volume fraction, $\sigma_y(\varepsilon_m^p)$ is the yield stress, $\varepsilon_m^p$ 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 $\varepsilon_m^p$ is assumed to be governed by the equivalent plastic work expression,

$$\left(1-f\right)\sigma_y\Delta\varepsilon^p_m=\sigma:\Delta\varepsilon^p=2\Delta\gamma\left( \dfrac{q^{tr}}{1+6G\Delta\gamma}\right)^2+3q_1q_2p\Delta\gamma{}f\sigma_y\sinh\left( \dfrac{3}{2}\dfrac{q_2p}{\sigma_y}\right).$$

Evolution of Volume Fraction

The evolution of volume fraction consists of two parts.

$$\Delta{}f=\Delta{}f_g+\Delta{}f_n,$$

where

$$\Delta{}f_g=(1-f)\Delta\varepsilon_v,\qquad\Delta{}f_n=A\Delta\varepsilon_m^p$$

with

$$A=\dfrac{f_N}{s_N\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{\varepsilon_m^p-\varepsilon_N}{s_N}\right)^2\right).$$

Parameters $f_N$, $s_N$ and $\varepsilon_N$ 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.

Recording

This model supports the following additional history variables to be recorded.

variable label	physical meaning
VF	volume fraction