The punch problem for shear-index granular materials

Experimental evidence indicates that a wide range of powders and granular materials have shear-index yield conditions of the form (tau/c)(n) = 1 -(sigma/t), where tau and sigma denote the shear and normal components of stress and c, t and n are experimentally-determined positive constants that are referred to as the cohesion, tensile strength and shear index respectively. This yield function is also known as the Warren Spring equation and experimentally-determined values of the shear index n indicate that n lies between the values 1 and 2. The value n = 1 corresponds to the well-known Coulomb-Mohr yield function and previous work shows that the special value n = 2 gives the simplest theory amongst those values of n which lie in the range of physical interest. We extend the known Coulomb-Mohr solution for the indentation of a granular material by a flat rigid punch to the case of shear-index materials and we determine an associated velocity field assuming a previously proposed dilatant double-shearing theory. The results obtained are illustrated numerically and it is shown how for general values of n in the range 1 < n < 2 the theories corresponding to n = 1 and n = 2 bound the various quantities arising in the stress and velocity fields. The proposed solution is by no means unique and an alternative solution is noted in an Appendix which although a bonafide solution of the governing equations may not arise physically.