# 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 Figure 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:

$f=\frac{{V}_{a}-{V}_{m}}{{V}_{a}}$

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:

$f={f}_{g}+{f}_{n}$

The growth rate of voids is calculated by:

${f}_{g}=\left(1-f\right)Trace\left[{D}^{p}\right]$

Where, $Trace\left[{D}^{p}\right]$ is the trace of the macroscopic plastic strain rate tensor. The nucleation rate of voids is given by:

${\stackrel{˙}{f}}_{n}=\frac{{f}_{N}}{{S}_{N}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{{\epsilon }_{M}-{\epsilon }_{N}}{{S}_{N}}\right)}^{2}}{\stackrel{˙}{\epsilon }}_{M}$

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:

$\left\{\begin{array}{c}{\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2\text{​}{q}_{1}{f}^{\ast }\mathrm{cosh}\left(\frac{3}{2}{q}_{2}\frac{{\sigma }_{m}}{{\sigma }_{M}}\right)-\left(1+{q}_{3}{f}^{\ast 2}\right)\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{m}>0\\ {\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2{q}_{1}{f}^{\ast }-\left(1+{q}_{3}{f}^{\ast 2}\right)\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}if\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma }_{m}\le 0\end{array}$

Where,
${\sigma }_{eq}$
von Mises is effective stress
${\sigma }_{M}$
${\sigma }_{m}$
Hydrostatic stress
${f}^{*}$
Specific coalescence function which can be written as:
$\left\{\begin{array}{l}{\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2{q}_{1}{f}^{*}\mathrm{cosh}\left(\frac{3}{2}{q}_{2}\frac{{\sigma }_{m}}{{\sigma }_{M}}\right)-\left(1+{q}_{3}{f}^{*2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{\sigma }_{m}>0\\ {\Omega }_{evp}=\frac{{\sigma }_{eq}^{2}}{{\sigma }_{M}^{2}}+2{q}_{1}{f}^{*}-\left(1+{q}_{3}{f}^{*2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}{\sigma }_{m}\le 0\end{array}$
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 Figure 2.

The admissible plastic strain rate is computed as:

${\stackrel{˙}{\epsilon }}_{M}=\frac{\sigma :{D}^{p}}{\left(1-f\right){\sigma }_{M}}$

Where,
$\sigma$
Cauchy stress tensor
${\sigma }_{M}$
${D}^{p}$
Macroscopic plastic strain rate tensor which can be written in the case of the associated plasticity as:
${D}^{p}=\stackrel{˙}{\lambda }\frac{\partial {\Omega }_{evp}}{\partial \sigma }$

with ${\Omega }_{evp}$ the yield surface envelope. The viscoplastic multiplier is deduced from the consistency condition:

${\Omega }_{evp}={\stackrel{˙}{\Omega }}_{evp}=0$

From this last expression, it is deduced that:

$\stackrel{˙}{\lambda }=\frac{{\text{Ω}}_{evp}}{\frac{\partial {\text{Ω}}_{evp}}{2\partial }:{C}^{e}:\frac{\partial {\text{Ω}}_{evp}}{\partial \sigma }-\frac{\partial {\text{Ω}}_{evp}}{\partial {\sigma }_{M}}\frac{\partial {\sigma }_{M}}{\partial {\epsilon }_{M}}{A}_{2}-\frac{\partial {\text{Ω}}_{evp}}{\partial f}\left[\left(1-f\right)\frac{\partial {\text{Ω}}_{evp}}{\partial \sigma }:I+{A}_{1}{A}_{2}\right]}$

Where,

${A}_{2}=\frac{\sigma :\frac{\delta {\Omega }_{evp}}{\delta \sigma }}{\left(1-f\right){\sigma }_{M}};{A}_{1}=\frac{{f}_{N}}{{S}_{N}\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{{\epsilon }_{M}-{\epsilon }_{N}}{{S}_{N}}\right)}^{2}}$

1 Gurson A. L. Continuum theory of ductile rupture by void nucleation and growth: Part I - Yield criteria and flow rules for porous ductile media, Journal of Engineering Materials and Technology, Vol. 99, 2-15, 1977.