| Strain localization is a commonly observed feature in many models of solid behavior. Examples include adiabatic sliplines in hot worked metals, strain softening plasticity leading to shear banding, localized bands of deformation accompanying damage degradation, and shear banding in soils to name just a few. This work approaches the problem by mathematically idealizing the localized region as a set of zero measure (a surface in 3 dimensions). The displacements are then discontinuous across this surface. Discontinuities in the deformation field are referred to as Strong Discontinuities. In a distributional framework, the strains are Dirac-delta functions. In the light of inelastic constitutive laws (e.g., plasticity, damage degradation), necessary conditions for the appearance of strong discontinuities are deduced for the inelastic consistency parameter and the softening relation. Staying entirely within the continuum framework, a stress-displacement relation is derived for the evolution of the discontinuous component. The analysis is performed for the infinitesimal and finite deformation regimes. The approach provides insight into several others that preceded it, unifies them and places them on a strong theoretical footing.;Standard finite element methods for this problem yield strongly mesh dependent results. The dissipation is inconsistently resolved, leading to solutions that display spurious variation with the element size. Displacement fields obtained using traditional finite element techniques are also highly dependent upon orientation of element edges. A numerical implementation of the aforementioned analysis is carried out in the framework of the Assumed Enhanced Strain Method and a new class of finite elements is proposed which render the results entirely independent of mesh orientation. Solutions insensitive to element size are obtained without introducing ad-hoc parameters like a characteristic length. Numerical simulations of standard bench mark examples are presented. In all cases, strain localization is resolved sharply and in a consistent manner. |
