A non-local plasticity model for porous metals with deformation-induced anisotropy: Mathematical and computational issues

A non-local (gradient) plasticity model for porous metals that accounts for deformation-induced anisotropy is presented. The model is based on the work of Ponte Castaneda and co-workers on porous materials containing randomly distributed ellipsoidal voids. It takes into account the evolution of porosity and the evolution/development of anisotropy due to changes in the shape and the orientation of the voids during plastic deformation. A "material length" l' is introduced and a "non-local" porosity is defined from the solution of a modified Helmholtz equation with appropriate boundary conditions, as proposed by Geers et al. (2001); Peerlings et al. (2001). At a material point located at x, the non-local porosity f (x) can be identified with the average value of the "local" porosity f(loc)(x) over a sphere of radius R similar or equal to 3 l' centered at x. The same approach is used to formulate a non-local version of the Gurson isotropic model. The mathematical character of the resulting incremental elastoplastic partial differential equations of the non-local model is analyzed. It is shown that the hardening modulus of the non-local model is always larger than the corresponding hardening modulus of the local model; as a consequence, the non-local incremental problem retains its elliptic character and the possibility of discontinuous solutions is eliminated. A rate-dependent version of the non-local model is also developed. An algorithm for the numerical integration of the non-local constitutive equations is developed, and the numerical implementation of the boundary value problem in a finite element environment is discussed. An analytical method for the required calculation of the eigenvectors of symmetric second order tensors is presented. The non-local model is implemented in ABAQUS via a material "user subroutine" (UMAT or VUMAT) and the coupled thermo-mechanical solution procedure, in which temperature is identified with the non-local porosity. Several example problems are solved numerically and the effects of the non-local formulation on the solution are discussed. In particular, the problems of plastic flow localization in plane strain tension, the plane strain mode-I blunt crack tip under small-scale-yielding conditions, the cup-and-cone fracture of a round bar, and the Charpy V-notch test specimen are analyzed.