SPATIAL PATTERNS AND SIZE EFFECTS IN SHEAR ZONES - A HYPERELASTIC MODEL WITH HIGHER-ORDER GRADIENTS

THE ONSET of localization of the deformation is analysed from the results of a two-dimensional bifurcation analysis for a shear zone of infinite extent and of finite thickness. The material is assumed to be incompressible and hyperelastic and, to account for micro-structural effects, to have an energy density that is a function of second-order deformation gradients. With such a model, the onset of strain localization is associated with a bifurcation but not with a change of type of the governing partial differential equations. It is demonstrated that both layer thickness and the exact nature of the boundary conditions affect that bifurcation. As expected, the incremental displacement field that characterizes the first loss of uniqueness has a spatial variation across but not along the shear zone. However, and contrary to a common belief, it is shown that the displacement field could vary in these two directions, for a range of values of the material properties. Consequences of this wavy pattern for the onset of failure during localization are discussed.