Porous Metal Plasticity

You specify the elastic part of the response separately; only linear isotropic elasticity can be specified (see Linear Elastic Behavior). The porous metal plasticity model cannot be used in conjunction with porous elasticity (Elastic Behavior of Porous Materials).

You specify the hardening behavior of the fully dense matrix material by defining a metal plasticity model (see Classical Metal Plasticity). Only isotropic hardening can be specified. The hardening curve must describe the yield stress of the matrix material as a function of plastic strain in the matrix material. In defining this dependence at finite strains, “true” (Cauchy) stress and log strain values should be given. Rate dependency effects for the matrix material can be modeled (see Rate-Dependent Yield).

The relative density of a material, r, is defined as the ratio of the volume of solid material to the total volume of the material. The relationships defining the model are expressed in terms of the void volume fraction, f, which is defined as the ratio of the volume of voids to the total volume of the material. It follows that $f=1-r.$ For a metal containing a dilute concentration of voids, Gurson (1977) proposed a yield condition as a function of the void volume fraction. This yield condition was later modified by Tvergaard (1981) to the form

where

$\mathbf{S}=p\mathbf{I}+\sigma$

is the deviatoric part of the Cauchy stress tensor $\mathit{\sigma }$;

$q=\sqrt{\frac{3}{2}\mathbf{S}:\mathbf{S}}$

is the effective Mises stress;

$p=-\frac{1}{3}\mathit{\sigma }:\mathbf{I}$

is the hydrostatic pressure;

${\sigma }_y\left({\overline{\epsilon }}_m^{pl}\right)$

is the yield stress of the fully dense matrix material as a function of ${\overline{\epsilon }}_m^{pl}$, the equivalent plastic strain in the matrix; and

$q_1$, $q_2$, $q_3$

are material parameters.

The Cauchy stress is defined as the force per “current unit area,” comprised of voids and the solid (matrix) material.

f = 0 (r = 1) implies that the material is fully dense, and the Gurson yield condition reduces to the Mises yield condition. f = 1 (r = 0) implies that the material is completely voided and has no stress carrying capacity. The model generally gives physically reasonable results only for $f<$0.1 ($r>$0.9).

The model is described in detail in Porous metal plasticity, along with a discussion of its numerical implementation.

If the porous metal plasticity model is used during a pore pressure analysis (see Coupled Pore Fluid Diffusion and Stress Analysis), the relative density, r, is tracked independently of the void ratio.

The presence of pressure in the yield condition results in nondeviatoric plastic strains. Plastic flow is assumed to be normal to the yield surface:

The hardening of the fully dense matrix material is described through ${\sigma }_y={\sigma }_y\left({\overline{\epsilon }}_m^{pl}\right)$. The evolution of the equivalent plastic strain in the matrix material is obtained from the following equivalent plastic work expression:

The model is illustrated in Figure 1, where the yield surfaces for different levels of void volume fraction are shown in the p–q plane.

Figure 1. Schematic of the yield surface in the p–q plane.

Figure 2 compares the behavior of a porous material (whose initial yield stress is ${\sigma }_{y_0}$) in tension and compression against the behavior of the perfectly plastic matrix material. In compression the porous material “hardens” due to closing of the voids, and in tension it “softens” due to growth and nucleation of the voids.

Figure 2. Schematic of uniaxial behavior of a porous metal (perfectly plastic matrix material with initial volume fraction of voids = $f_0$).

The total change in void volume fraction is given as

where ${\dot{f}}_{\mathrm{gr}}$ is change due to growth of existing voids and ${\dot{f}}_{\mathrm{nucl}}$ is change due to nucleation of new voids. Growth of the existing voids is based on the law of conservation of mass and is expressed in terms of the void volume fraction:

The nucleation of voids is given by a strain-controlled relationship:

where

The normal distribution of the nucleation strain has a mean value ${\epsilon }_N$ and standard deviation $s_N$. $f_N$ is the volume fraction of the nucleated voids, and voids are nucleated only in tension.

The nucleation function $A/f_N$ is assumed to have a normal distribution, as shown in Figure 3 for different values of the standard deviation $s_N$.

Figure 3. Nucleation function $A/f_N$.

Figure 4 shows the extent of softening in a uniaxial tension test of a porous material for different values of $f_N$.

Figure 4. Softening (in uniaxial tension) as a function of $f_N$.

The following ranges of values are reported in the literature for typical metals: ${\epsilon }_N$ = 0.1 to 0.3, $s_N=$ 0.05 to 0.1, and $f_N$ = 0.04 (see Necking of a round tensile bar). You specify these parameters, which can be defined as tabular functions of temperature and predefined field variables. Abaqus will include void nucleation in a tensile field only when you include it in the material definition.

In Abaqus/Standard the accuracy of the implicit integration of the void nucleation and growth equation is controlled by prescribing the maximum allowable time increment in the automatic time incrementation scheme.

VOID NUCLEATION

Property module: material editor: MechanicalPlasticityPorous Metal Plasticity: SuboptionsVoid Nucleation