The influence of non-coaxiality on shear banding in viscous-plastic materials

Abstract

Many of the most important and commonly occurring, emergent geological structures such as shear bands, fault zones and folds may be understood as a consequence of changes in the type of the governing model equations. Such changes, or bifurcations, depend strongly on the details of the constitutive relationships including the presence of strain softening, thermal or chemical effects, and whether the flow rule is associated or non-associated, coaxial or non-coaxial. Here we focus on the influence of non-coaxiality of the flow rule on the development and evolution of shear bands. The term non-coaxial refers to the non-coincidence of the principal axes of the stress and of the plastic strain rate tensor. Non coaxial plasticity models were originally proposed by Josselin de Jong and A. J. M. Spencer. The geometric structure of most non-coaxial models is defined by the assumption that the plastic deformation is carried by one (single slip) or two (double slip) slip systems that are inclined at ${\pm (\pi /4+\nu /2), 0\leq \nu \leq \phi }$ to the less compressive principal stress axis; ${\phi }$ is the internal angle of friction of the material. Both cases –single and double slip– are considered here. A particular feature of the double slip model is the appearance of an additional variable requiring an additional constitutive relationship. Geometrically, the additional variable may be interpreted as a spin. Spencer assumed that, in 2D, the additional spin is equal to the rate of rotation of the principal axes of the stress tensor. Here we show that Spencer’s assumption is very similar (up to a factor of proportionality) to the assumption that the internal slip systems are material. We also propose an alternative closure, based on experimental observations, indicating that the ratio between the average particle spin and the spin of a spatial element of the granular assembly is constant, in simple shear. Finally, we propose a closure for the double slip model within the framework of a Cosserat Continuum theory. In a Cosserat continuum, a material point possesses the degrees of freedom of an infinitesimal rigid body: three translations and three rotations. In this case the indeterminacy of the double slip model is removed by the angular momentum balance equation associated with the rotational degrees of freedom. We propose constitutive relationships for the Cosserat model and illustrate the behavior of the model by means of an analytical solution of the simple shear problem.