Fundamental Issues in Finite Element Analyses of Localization of Deformation

Abstract

Classical continuum models, i.e. continuum models that do not incorporate an internal length scale, suffer from excessive mesh dependence when strain-softening models are used in numerical analyses and cannot reproduce the size effect commonly observed in quasi-brittle failure. In this contribution three different approaches will be scrutinized which may be used to remedy these two intimately related deficiencies of the classical theory, namely (i) the addition of higher-order deformation gradients, (ii) the use of micropolar continuum models, and (iii) the addition of rate dependence. By means of a number of numerical simulations it will be investigated under which conditions these enriched continuum theories permit localization of deformation without losing ellipticity for static problems and hyperbolicity for dynamic problems. For the latter class of problems the crucial role of dispersion in wave propagation in strain-softening media will also be highlighted.

Localisation issues in local and nonlocal continuum approaches to fracture

“…A number of methods have been developed which aim at avoiding the pathological localisation of deformation and damage growth (or plastic flow) by improving the continuum modelling, see for instance (Pijaudier-Cabot et al, 1988;Sluys, 1992) or (de Borst et al, 1993) for detailed discussions and comparisons. Here the class of nonlocal and gradient models is considered.…”

“…It has since been found, however, that finite element solutions of standard damage problems often do not seem to converge upon mesh refinement. As a matter of fact, they do converge to a solution, but this solution is physically meaningless as a consequence of the inability of the modelling to properly describe the physical phenomena that take place (Bažant et al, 1984;de Borst et al, 1993). This can be understood if one realises that the concept of a continuous damage variable presumes a certain local homogeneity -or at least smoothness -of the microstructural damage distribution.…”

Computing strain fields from discrete displacement fields in 2D-solids

“…In recent years, it has been recognized that the mathematically consistent description of failure and the accompanying localization phenomenon requires higher-order continuum models or the addition of viscosity in the constitutive description (for an overview see, for instance, de Borst et al, 1993). If such an enhancement is not made, ill-posed boundary value problems result, which cannot properly represent the physical failure process.…”

Nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity, viscoplasticity and damage

“…This corresponds to the localization of plastic strain into a band of finite width in the form of a sinus arch. This regularization property has been widely used, for instance, in the simulation of strain localization phenomena like shear banding [39,64]. character of the original model: smoothing of discontinuities at interfaces and boundary layers in hardening materials, and description of finite width localization bands in softening materials.…”