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 .

Figure 1 . Damage Process for Visco-elastic-plastic Voided Materials
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:

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

Where,

${f}_{N}$
Nucleated void volume fraction
${S}_{N}$
Gaussian standard deviation
${\epsilon}_{N}$
Nucleated effective plastic strain
${\epsilon}_{M}$
Admissible plastic strain
The viscoplastic flow of the porous material is described by:

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

Where,

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

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 .

Figure 2 . Variation of Specific Coalescence Function
The admissible plastic strain rate is computed as:

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

Where,

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

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

$${\Omega}_{evp}={\dot{\Omega}}_{evp}=0$$

From this last expression, it is deduced that:

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

Where,

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