Products: ABAQUS/Standard ABAQUS/Explicit
The porous metal plasticity model is intended for use with mildly voided metals. Even though the material that contains the voids (also known as the matrix material) is assumed to be plastically incompressible, the plastic behavior of the bulk material is pressure-dependent due to the presence of voids. The model is described in the following paragraphs, followed by a brief description of the material point calculations.
For a metal containing a dilute concentration of voids, based on a rigid-plastic upper bound solution for spherically symmetric deformations of a single spherical void, Gurson (1977) proposed a yield condition of the form
It should be noted that implies that the material is fully dense, and the Gurson yield condition reduces to that of von Mises; implies that the material is fully voided and has no stress carrying capacity. This is illustrated in Figure 4.3.61, where the yield surfaces for different levels of porosity are shown in the p–q plane.
Figure 4.3.62 compares the behavior of a porous metal (which has an initial void volume fraction of ) in tension and compression against the perfectly plastic matrix material; the initial yield stress of the porous metal is . 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.The plastic strains are derived from the yield potential; the presence of the first invariant of the stress tensor in the yield condition results in nondeviatoric plastic strains:
The hardening of the (fully dense) matrix material is described through . The evolution of is assumed to be governed by the equivalent plastic work expression; i.e.,
The change in volume fraction of the voids is due partly to the growth of existing voids and partly to the nucleation of voids. Growth of existing voids is based on the law of conservation of mass and is expressed as
Nucleation of voids can occur due to micro-cracking and/or decohesion of the particle-matrix interface. ABAQUS assumes that the nucleation of new voids is plastic strain controlled (see Chu and Needleman, 1980), so that
The nucleation function , which is assumed to have a normal distribution, is shown in Figure 4.3.63 for different values of the parameter . Figure 4.3.64 shows the extent of softening in a uniaxial tension test of a porous material for different values of .
The integration of the elastoplastic equations for the porous plasticity model is carried out using the backward Euler scheme proposed by Aravas (1987). This method is briefly discussed in the following paragraphs; the user can refer to the paper for further details.
During the constitutive calculations in an increment, the stress and state variables are known at time t (beginning of the increment). Given a total incremental strain , the stress and state variables need to be updated at (end of the increment) so that they satisfy the yield condition, flow rules, and evolution equation of the state variables. To do this, consider the elasticity equations
whereThe yield condition, the flow rule, and the evolution of the state variables are rewritten as
Equation 4.3.6–6 and Equation 4.3.6–7 are solved for and using Newton's method; p and q are updated using Equation 4.3.6–8 and Equation 4.3.6–9; the state variables are updated using Equation 4.3.6–10.
In the implicit finite element method of solving large-deformation problems, the discretized equilibrium equations result in a set of nonlinear equations for the nodal unknowns at the end of the increment. ABAQUS/Standard uses Newton's method to solve these equations, which requires the computation of linearization moduli