Variational Modelling of Strain Localization in Solids: A Computational Mechanics Point of View

Strain localization is one of the most challenging phenomena in solid mechanics. It occurs when strains concentrate in very narrow bands within a solid, typically referred to as localization bands. This behaviour is related to different types of failure mechanisms: fracture, shear bands and slip lines. Being a dissipative process, the modelling of strain localization could seem to preclude the use of variational methods. However, a sound framework to deal with this issue has been developed in the last few decades. In this contribution, we review the modelling of strain localization by means of variational methods systematically, presenting the main underlying theoretical concepts. The issue of irreversibility is approached by means of the theory of generalized standard materials, which constitutes the basis for the variational approach. Then, typical problems occurring in the modelling of strain localization are analyzed: the tendency to localize in a band of zero thickness with no dissipation, the determination of the geometry of the localization band, and the orientation bias of such band with respect to mesh alignment in finite element discretizations. We discuss solutions for these problems, focusing on the approach that tackles the description of the localization band using phase fields.