# Ductile Damage Model for Porous Materials (LAW52)

The Gurson constitutive law 1 models progressive microrupture through void nucleation and growth. It is dedicated to high strain rate elasto-viscoplastic porous metals. A coupled damage mechanical model for strain rate dependent voided material is used. The material undergoes several phases in the damage process as described in 図 1.
The constitutive law takes into account the void growth, nucleation and coalescence under dynamic loading. The evolution of the damage is represented by the void volume fraction, defined by:(1)
$f=\frac{{V}_{a}-{V}_{m}}{{V}_{a}}$
Where,
${V}_{a}$ , ${V}_{m}$
Respectively, are the elementary apparent volume of the material and the corresponding elementary volume of the matrix.
The rate of increase of the void volume fraction is given by:(2)
$f={f}_{g}+{f}_{n}$
The growth rate of voids is calculated by:(3)
${f}_{g}=\left(1-f\right)Trace\left[{D}^{p}\right]$
Where, $Trace\left[{D}^{p}\right]$ is the trace of the macroscopic plastic strain rate tensor. The nucleation rate of voids is given by:(4)
Where,
${f}_{N}$
Nucleated void volume fraction
${S}_{N}$
Gaussian standard deviation
${\epsilon }_{N}$
Nucleated effective plastic strain
${\epsilon }_{M}$
The viscoplastic flow of the porous material is described by:(5)
$\left\{\begin{array}{c}{\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2\text{​}{q}_{1}{f}^{\ast }\mathrm{cosh}\left(\frac{3}{2}{q}_{2}\frac{{\sigma }_{m}}{{\sigma }_{M}}\right)-\left(1+{q}_{3}{f}^{\ast 2}\right)\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{m}>0\\ {\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2{q}_{1}{f}^{\ast }-\left(1+{q}_{3}{f}^{\ast 2}\right)\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{m}\le 0\end{array}$
Where,
${\sigma }_{eq}$
von Mises is effective stress
${\sigma }_{M}$
${\sigma }_{m}$
Hydrostatic stress
${f}^{*}$
Specific coalescence function which can be written as:
(6)
$\left\{\begin{array}{l}{\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2{q}_{1}{f}^{*}\mathrm{cosh}\left(\frac{3}{2}{q}_{2}\frac{{\sigma }_{m}}{{\sigma }_{M}}\right)-\left(1+{q}_{3}{f}^{*2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{\sigma }_{m}>0\\ {\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2{q}_{1}{f}^{*}-\left(1+{q}_{3}{f}^{*2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{\sigma }_{m}\le 0\end{array}$
Where,
${f}_{c}$
Critical void volume fraction at coalescence
${f}_{F}$
Critical void volume fraction at ductile fracture
${f}_{u}$
Corresponding value of the coalescence function ${f}_{u}=\frac{1}{{q}_{1}}$ , ${f}^{*}\left({f}_{F}\right)={f}_{u}$
The variation of the specific coalescence function is shown in 図 2.
The admissible plastic strain rate is computed as:(7)
${\stackrel{˙}{\epsilon }}_{M}=\frac{\sigma :{D}^{p}}{\left(1-f\right){\sigma }_{M}}$
Where,
$\sigma$
Cauchy stress tensor
${\sigma }_{M}$
${D}^{p}$
${D}^{p}=\stackrel{˙}{\lambda }\frac{\partial {\Omega }_{evp}}{\partial \sigma }$
with ${\Omega }_{evp}$ the yield surface envelope. The viscoplastic multiplier is deduced from the consistency condition:(9)
${\Omega }_{evp}={\stackrel{˙}{\Omega }}_{evp}=0$
$\stackrel{˙}{\lambda }=\frac{{\text{Ω}}_{evp}}{\frac{\partial {\text{Ω}}_{evp}}{2\partial }:{C}^{e}:\frac{\partial {\text{Ω}}_{evp}}{\partial \sigma }-\frac{\partial {\text{Ω}}_{evp}}{\partial {\sigma }_{M}}\frac{\partial {\sigma }_{M}}{\partial {\epsilon }_{M}}{A}_{2}-\frac{\partial {\text{Ω}}_{evp}}{\partial f}\left[\left(1-f\right)\frac{\partial {\text{Ω}}_{evp}}{\partial \sigma }:I+{A}_{1}{A}_{2}\right]}$