NonlinearGurson.md

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