Numerical methods for porous metals with deformation-induced anisotropy

Abstract

A constitutive model for porous metals subjected to general three-dimensional finite deformations is presented. The model takes into account the evolution of porosity and the development of anisotropy due to changes in the shape and the orientation of the voids during deformation. Initially, the pores are assumed to be ellipsoids distributed randomly in an elastic–plastic matrix (metal). This includes also the special case in which the initial shape of the voids is spherical and the material is initially isotropic. Under finite plastic deformation, the voids are assumed to remain ellipsoids but to change their volume, shape and orientation. At every material point, a “representative” ellipsoid is considered and the homogenized continuum is assumed to be locally orthotropic, with the local axes of orthotropy coinciding with the principal axes of the representative local ellipsoid. The orientation of the principal axes is defined by the unit vectors n (1) , n (2) , n (3)= n (1)× n (2) and the corresponding lengths are 2 a 1, 2 a 2 and 2 a 3. The basic “internal variables” characterizing the state of the microstructure at every point in the homogenized continuum are given by the local equivalent plastic strain ϵ ̄ p in the metal matrix, the local void volume fraction (or porosity) f, the two aspect ratios of the local representative ellipsoid ( w 1= a 3/ a 1, w 2= a 3/ a 2) and the orientation of the principal axes of the ellipsoid ( n (1), n (2), n (3)) . A methodology for the numerical integration of the elastoplastic constitutive model is developed. The problems of uniaxial tension, simple shear, plastic flow localization and necking in plane strain tension, and ductile fracture initiation at the tip of a blunt crack are analyzed in detail; comparisons with the isotropic Gurson model are made.