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Abstract:
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.

Keywords:
Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation

Abstract:
This work is concerned with the formation of multiple macroscopic shear bands viewed as a mechanism of large plastic deformation of polycrystalline metals. The plastic deformation pattern in a time-independent material with a yield-surface vertex effect is investigated numerically in plane strain beyond the critical instant of ellipticity loss under quasi-static loading. The energy criterion of path instability applied to a family of post-critical solutions eliminates unstable paths and enables the overall deformation pattern to be determined, although the solutions remain locally indeterminate due to the absence of an internal length scale. In particular, the volume fraction of incipient shear bands is found to have a well-defined value irrespective of the mesh size in finite element calculations. As an apparently novel qualitative result, the formation of coarse, differently aligned secondary bands is observed at later stages of simulation.

Keywords:
Plasticity, Shear bands, Material instability, Energy criterion, Bifurcation

Abstract:
A theoretical and numerical analysis of the formation of a localized neck in a biaxially stretched sheet is presented. A time-independent constitutive law is assumed to be incrementally non-linear as suggested by micromechanical studies of the elastoplastic deformation of polycrystalline metals. The incipient width of a necking band in an infinitely thin perfect sheet of a time-independent material is found here to have a well-defined initial value, proportional to the in-plane sheet dimension. During subsequent post-critical deformation the boundary of the necking band moves with respect to the material until the transition to localized necking is completed. These conclusions are derived on a theoretical route from the condition of stability of the post-bifurcation deformation process and are confirmed by the numerical analysis performed for a sheet of finite thickness.

Abstract:
A spatially discretized non-linear rate problem for a time-independent plastic solid is examined with particular reference to bifurcation. Constitutive non-linearity in a general form encompassing the yield-surface vertex effect is considered under the restriction that the tangent stiffness matrix for the whole system is symmetric. Theorems concerning existence, uniqueness and stability of solutions are presented. As an outcome of the theoretical analysis, a computational method is proposed for crossing bifurcation points with automatic rejection of an unstable postbifurcation branch. An illustrative example of plane strain tension is calculated by using the finite element method.

Abstract:
Second-order rate constitutive equations are formulated for a time-independent elastic-plastic material, obeying the normality flow rule with a smooth yield surface. Under specified regularity restrictions imposed on the involved fields, the regular second-order rate boundary value problem with quasistatic accelerations as unknowns is posed. It is shown that every solution of this generally non-linear rate problem is governed by a variational principle and that the corresponding functional reaches a strict absolute minimum, provided the solution satisfies a sufficient uniqueness condition. With the same incrementally linear comparison solid, Hill's exclusion condition rules out not only a first- but also a second-order bifurcation. The criticality of the exclusion condition is discussed and conditions are indicated under which a second-order bifurcation becomes possible, while the first-order rate problem is still uniquely solvable.