Similarity 'hot-spot' solutions for a hypoplastic granular material

Hyoplasticity is a theory which has been developed by the Institute for Soil and Rock Mechanics at the University of Karlsruhe to accurately model the behaviour of granular materials such as sand, soil and certain powders. The earlier hypoplasticity theories were formulated on the basis of the two assumptions of rate independence and homogeneous dependence of the stress rate on the stress. These two assumptions mean that the governing partial differential equations remain invariant under a general family of stretching similarity transformations, which in turn imply the existence of similarity solutions for which the partial differential equations can be reduced to ordinary differential equations. In two recent papers involving uniaxial compaction and cylindrical and spherical cavity expansion problems, these similarity solutions have been identified and the appearance of uniaxial 'hot-spot' solutions has been noted. In this context,'hot-spot' refers to the material undergoing an infinite stress in a finite time. Here, for planar, cylindrical and spherical geometries, we summarize the general picture relating to these similarity 'hot-spot' solutions and we give a number of simple analytical expressions for them not included in the recent papers.