Spectral analysis of nonlocal regularization in two-dimensional finite element models

Strain-softening in geomaterials often leads to ill-posed boundary-valued problems (BVP), which cannot be solved with finite element methods without introducing some kind of regularization such as nonlocal plasticity. Hereafter we propose to apply spectral analysis for testing the performance of nonlocal plasticity in regularizing ill-posed BVP and producing mesh-independent solutions when local plasticity usually fails. The spectral analysis consists of examining the eigenvalues and eigenvectors of the global tangential stiffness matrix of the incremental equilibrium equations. Based on spectral analysis, we propose a criterion for passing or failing the test of constitutive regularization in the context of BVP. If the eigenvalues of the tangential operator are all positive then the regularization succeeds, otherwise it fails and may not prevent artificial mesh-dependent solutions from appearing. The approach is illustrated in the particular case of a biaxial compression with strain-softening plasticity. In this particular case, local softening plasticity is found to produce negative eigenvalues in the tangential stiffness matrix, which indicates ill-posed BVP. In contrast, nonlocal softening plasticity always produces positive eigenvalues, which regularizes ill-posed BVP. The dominant eigenvectors, which generate localized deformation patterns, have a bandwidth independent of mesh size, provided that the mesh is fine enough to capture localization. These mesh-independent eigenmodes explain why nonlocal plasticity produces numerical solutions that are mesh-independent. Copyright (c) 2011 John Wiley & Sons, Ltd.