Localization Analysis in Finite Deformation Elasto-Plasticity
A failure phenomenon which is frequently observed in laboratory experiments as well as in nature is the localization of deformations within narrow bands. Thereby, the underlying physical mechanisms vary with the particularly material under consideration. Large inelastic strains accumulate inside the localization zone and form a precursor to fracture. On the theoretical side, the micro-structure of grain particle interfaces is traditionally homogeneized into a continuum, allowing point-measures of strains and stresses to describe the response of a given configuration under prescribed load histories. Classical continuum theory signals the onset of localization via bifurcation at the constitutive level, i.e. the velocity gradient may develop a discontinuity across a singularity surface with orientation N. Discontinuous bifurcation, which is synonymous with the loss of ellipticity of the governing equations, is reflected by a singularity of the acoustic tensor Q. Localization analysis of constitutive models furnishes the critical directions N which are often in close agreement with the inclination of the observed failure bands. Nevertheless, continuum theory fails to describe the post-peak behaviour since discontinuities rather than steep gradients are embedded in the solution of the localization problem. This leads to severe mesh-dependence of numerical computations. Thus, the validity of numerical solutions of localization problems is very difficult to assess once localization has fully developed. Hence it is of utmost importance that the finite element formulation is capable of reflecting the singularity of the acoustic tensor not only at the element level but also throughout the spatial discretization. In this study we will extend the weak localization test   to finite deformations and we will verify the localization capturing abilities of single elements. Several variations of low order four noded elements will be studied within the finite deformation theory of hyperelastic-plastic solids. These findings will lead to the formulation of a singularized finite element which is designed to capture localization in the weak sense.
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