Porous Metal Plasticity

The porous metal plasticity material option:

is based on Gurson's porous metal plasticity theory with void nucleation and strain-dependent hardening and strain-rate dependence of the matrix;
can only have isotropic hardening as it deforms plastically;
is well-tuned for tensile applications, such as fracture studies with void coalescence, but it is also useful for compressive cases where the material densifies, and;
defines the inelastic flow of the porous metal on the basis of a potential function that characterizes the porosity in terms of a single state variable, the relative density.

Porous Metal Plasticity

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

You specify the hardening behavior of the fully dense matrix material by defining a metal plasticity model. Only isotropic hardening can be specified (see Plastic Options). 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.

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 $f = 1 - r$ .


Input Data	Description
Relative density value	Initial relative density value, $r_{0}$ .

The yield condition is

Φ = {(\frac{q}{σ_{y}})}^{2} + 2 q_{1} f \cosh (- q_{2} \frac{3 p}{2 σ_{y}}) - (1 + q_{3} f^{2}) = 0

where

S = σ + p I

is the deviatoric stress,

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

is the von Mises stress,

p = - \frac{1}{3} t r a c e (σ)

is the pressure stress,

σ_{y} ({\bar{ε}}_{m}^{p l})

is the yield stress of the fully dense matrix material as a function of the equivalent plastic strain in the matrix,

{\bar{ε}}_{m}^{p l}

, and

q_{1}, q_{2}, q_{3}

are material parameters. For typical metals the ranges of the parameters reported in the literature are

q_{1} = 1.0

1.5

q_{2} = 1.0

q_{3} = q_{1}^{2} = 1.0

2.25

. The original Gurson model is recovered when

q_{1} = q_{2} = q_{3} = 1.0


Input Data	Description
q1	Material parameter $q_{1}$ .
q2	Material parameter $q_{2}$ .
q3	Material parameter $q_{3}$ .
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Void Nucleation

The total change in void volume fraction, $f$ , is given as

\dot{f} = {\dot{f}}_{g r} + {\dot{f}}_{n u c l}

where

{\dot{f}}_{g r}

is change due to growth of existing voids and

{\dot{f}}_{n u c l}

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:

{\dot{f}}_{g r} = (1 - f) {\dot{ε}}^{p l} : I .

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

{\dot{f}}_{n u c l} = A {\dot{\bar{ε}}}_{m}^{p l},

where

A = \frac{f_{N}}{s_{N} \sqrt{2 π}} \exp [- \frac{1}{2} {(\frac{{\bar{ε}}_{m}^{p l} - ε_{N}}{s_{N}})}^{2}] .

The normal distribution of the nucleation strain has a mean value $ε_{N}$ 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 for different values of the standard deviation $s_{N}$ .

The following ranges of values are reported in the literature for typical metals: $ε_{N} = 1.0$ to $0.3$ , $s_{N} = 0.05$ to $0.1$ , and $f_{N} = 0.4$ . You specify these parameters, which can be defined as tabular functions of temperature and predefined field variables.


Input Data	Description
Mean Value of the Nucleation-Strain	$ε_{N}$ , mean value of the nucleation-strain normal distribution.
Standard Deviation of the Nucleation-Strain	$s_{N}$ , standard deviation of the nucleation-strain normal distribution.
Volume Fraction of the Nucleating Voids	$f_{N}$ , volume fraction of nucleating voids.
Use temperature-dependent data	Specifies material parameters that depend on temperature. A Temperature field appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Porous Failure Criteria

The porous metal plasticity model in an explicit analysis allows for failure. In this case the yield condition is written as

Φ = {(\frac{q}{σ_{y}})}^{2} + 2 q_{1} f^{*} \cosh (- q_{2} \frac{3 p}{2 σ_{y}}) - (1 + q_{3} f^{* 2}) = 0

where the function

f^{*} (f)

models the rapid loss of stress carrying capacity that accompanies void coalescence. This function is defined in terms of the void volume fraction:

f^{*} = {\begin{matrix} \begin{matrix} f i f f < f_{c} \\ \begin{matrix} f_{c} + \frac{{\bar{f}}_{F} - f_{c}}{f_{f} - f_{c}} (f - f_{c}) \\ {\bar{f}}_{F} i f f \geq f_{F} \end{matrix} \end{matrix} & i f f_{c} < f < f_{F} \end{matrix}

where

{\bar{f}}_{F} = \frac{q_{1} + \sqrt{q_{1}^{2} - q_{3}}}{q_{3}} .

In the above relationship

f_{c}

is a critical value of the void volume fraction, and

f_{F}

is the value of void volume fraction at which there is a complete loss of stress carrying capacity in the material. The parameters

f_{c}

and

f_{F}

model the material failure when

f_{c} < f < f_{F}

, due to mechanisms such as micro fracture and void coalescence. When

f \geq f_{F}

, total failure at the material point occurs. An element is deleted (or removed) from a mesh upon material failure.


Input Data	Description
Void Volume Fraction at Total Failure	Void volume fraction at total failure, $f_{F} > 0$ .
Critical Void Volume Fraction	Critical void volume fraction (threshold of rapid loss of stress carrying capacity), $f_{c} > 0$ .