Post-critical plastic deformation in incrementally nonlinear materials

The formation of multiple macroscopic shear bands is investigated as a mechanism of advanced plastic flow of polycrystalline metals. The overall deformation pattern and material characteristics are determined beyond the critical instant of ellipticity loss, without the need of introducing an internal length scale. This novel approach to the modelling of post-critical plastic deformation is based on the concept of a representative nonuniform solution in a homogeneous material. The indeterminacy of a post-critical representative solution is removed by eliminating unstable solution paths with the help of the energy criterion of path instability. It is shown that the use of micromechanically based, incrementally nonlinear corner theories of time-independent plasticity leads then to gradual concentration of post-critical plastic deformation. The volume fraction occupied by shear bands is found to have initially a well-defined, finite value insensitive to the mesh size in finite element calculations. Further deformation depends qualitatively on details of the constitutive law. In certain cases, the volume fraction of active bands decreases rapidly to zero, leading to material instability of dynamic type. However, for physically hardening materials with the yield-vertex effect, the localization volume typically remains finite over a considerable deformation range. At later stages of the plane strain simulation, differently aligned secondary bands are formed in a series of bifurcations.