Mixed stabilized finite element methods in nonlinear solid mechanics: Part II: Strain localization

This paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining. A standard local isotropic Rankine damage model with strain-softening is used as exemplary constitutive model. Both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed. Lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated. From this analysis, the issue of local discretization error in the pre-localization regime is deemed as the main difficulty to be overcome in the discrete problem. Focus is placed on low order finite elements with continuous strain and displacement fields (triangular P1P1 and quadrilateral Q1Q1), although the presented approach is very general. Numerical examples show that the resulting procedure is remarkably robust: it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh-bias dependence.