Singular solutions in the mechanics of soils

Many rigid plastic models, including models for soils and granular materials, are described by hyperbolic systems of equations. Solutions of boundary value problems for such models can include envelopes of characteristics. It has been already shown for several models that solutions can be singular in the vicinity of surfaces coinciding with such envelopes. In particular, the quadratic invariant of the strain rate tensor approaches infinity according to an inverse square root rule. It is evident that such behaviour of exact solutions can cause difficulties with their numerical approximation. Therefore, it is of importance to obtain the exact asymptotic representation of solutions near singular surfaces. In the present paper, the double slip and rotation model is considered. It is shown that plane strain solutions near envelopes of characteristics are singular and the asymptotic representation of solutions is given. In particular, it is demonstrated that the quadratic invariant of the strain rate tensor approaches infinity according to an inverse square root rule, as in the case of the classical rigid perfectly plastic potential model for pressure-independent materials.