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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.