Numerical solutions of strain localization with nonlocal softening plasticity

Softening plasticity, when it is deprived of a characteristic length, often leads to boundary value problems that are ill-posed and produce unreliable numerical solutions. As meshes are refined, plastic strains localize into bands that become narrower and narrower, and generate load–displacement responses with excessive softening, which may even degenerate into unrealistic snapbacks. In the case of discrete systems, the underlying reasons for this poor numerical performance can be examined using a spectral analysis of the tangential stiffness matrix derived from incremental equilibrium equations. In one-dimensional boundary-valued problems, the introduction of a weak softening element into a mesh of elasto-plastic elements produces a dominant eigenvector that clearly exhibits a strain localization width directly related to the weak element size. The dominant eigenvector that controls the incremental solution varies spuriously when the mesh is refined, which results in mesh dependency. Over-nonlocal softening plasticity introduces a length scale, forces more elements to become plastic, and preserves the width of the plastic zone when the mesh is refined. When one-dimensional meshes are refined, spectral analysis shows that the dominant eigenvector does not vary erratically, and that its deformation patterns are smooth and display a constant localization width. This explains why the spatial distribution of plastic strains and the global load–displacement responses converge to analytical solutions in one dimension. The generalization to higher dimensions will be the object of future studies.