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:

$$f=\frac{V_a-V_m}{V_a}$$

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:

$$\{\begin{array}{c}{\Omega }_{evp}=\frac{{\sigma }_{eq}^2}{{\sigma }_M^2}+2\text{}{q}_1{f}^{\ast }\cosh \left(\frac{3}{2}{q}_2\frac{{\sigma }_m}{{\sigma }_M}\right)-\left(1+q_3{f}^{\ast 2}\right)\text{}\text{}if{\sigma }_m>0\\ {\Omega }_{evp}=\frac{{\sigma }_{eq}^2}{{\sigma }_M^2}+2q_1{f}^{\ast }-\left(1+q_3{f}^{\ast 2}\right)\text{}\text{}if{\sigma }_m\le 0\end{array}$$

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:

$$\{\begin{array}{l}{\Omega }_{evp}=\frac{{\sigma }_{eq}^2}{{\sigma }_M^2}+2q_1{f}^{*}\cosh \left(\frac{3}{2}{q}_2\frac{{\sigma }_m}{{\sigma }_M}\right)-\left(1+q_3{f}^{*2}\right)\text{if}{\sigma }_m>0\\ {\Omega }_{evp}=\frac{{\sigma }_{eq}^2}{{\sigma }_M^2}+2q_1{f}^{*}-\left(1+q_3{f}^{*2}\right)\text{if}{\sigma }_m\le 0\end{array}$$

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:

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

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:

$$\dot{\lambda }=\frac{{\text{Ω}}_{evp}}{\frac{\partial {\text{Ω}}_{evp}}{2\partial }:C^e:\frac{\partial {\text{Ω}}_{evp}}{\partial \sigma }-\frac{\partial {\text{Ω}}_{evp}}{\partial {\sigma }_M}\frac{\partial {\sigma }_M}{\partial {\epsilon }_M}{A}_2-\frac{\partial {\text{Ω}}_{evp}}{\partial f}\left[\left(1-f\right)\frac{\partial {\text{Ω}}_{evp}}{\partial \sigma }:I+A_1{A}_2\right]}$$

Where,

$$A_2=\frac{\sigma :\frac{\delta {\Omega }_{evp}}{\delta \sigma }}{(1-f){\sigma }_M};A_1=\frac{f_N}{S_N\sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{{\epsilon }_M-{\epsilon }_N}{S_N}\right)}^2}$$