NonlinearGurson

Nonlinear General Gurson Porous Model

See also the corresponding section in Constitutive Modelling Cookbook.

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.

The following is a visualization of the yield surface in the $p-q$ plane.

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.

There is no consideration of coalescence in the current implementation.

Recording

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

variable label	physical meaning
VF	volume fraction