Granular materials: constitutive equations and strain localization

Abstract

Strain localization into shear bands is commonly observed in natural soil masses, as well as in human-built embankments, footings, retaining walls and other geotechnical structures. Numerical predictions for the process of shear band formation are critically dependent on the constitutive equations employed. In this paper, the plane strain “double-shearing” constitutive model (e.g., Spencer, A.J.M., 1964. A theory of the kinematics of ideal soils under plane strain conditions. Journal of the Mechanics and Physics of Solids 12, 337–351; Spencer, A.J.M., 1982, Deformation of ideal granular materials. In: Hopkins, H.G., Sewell, M.J. (Eds.), Mechanics of Solids. Pergamon Press, Oxford and New York, pp. 607–652; Mehrabadi, M.M., Cowin, S.C., 1978. Initial planar deformation of dilatant granular materials. Journal of the Mechanics and Physics of Solids 26, 269–284; Nemat-Nasser, S., Mehrabadi, M.M., Iwakuma, T. 1981. On certain macroscopic and microscopic aspects of plastic flow of ductile materials. In: Nemat-Nasser, S. (Ed.), Three-dimensional Constitutive Relations and Ductile Fracture. North-Holland, Amsterdam, pp. 157–172; Anand, L., 1983. Plane deformations of ideal granular materials. Journal of the Mechanics and Physics of Solids 31, 105–122) is generalized to three dimensions including the effects of elastic deformation and pre-peak behavior. The constitutive model is implemented in a finite element program and is used to predict the formation of shear bands in plane strain compression, and plane strain cylindrical cavity expansion. The predictions from the model are shown to be in good quantitative agreement with the recent experiments of Han, C., Drescher, A., (1993. Shear bands in biaxial tests on dry coarse sand. Soils and Foundations 33, 118–132) and Alsiny, H., Vardoulakis, I., Drescher, A., (1992. Deformation localization in cavity inflation experiments on dry sand. Geotechnique 42, 395–410) on a dry sand. The constitutive model is also used to predict the stress state in a static sand pile — a topic which has occupied the attention of many investigators in recent years. In our simulations we model an initially loose sand mass as a cohesionless material with a mobilized internal friction coefficient which evolves from an initial value of zero to a saturation value. The formation of a sand pile is numerically modeled as a two-step process: (i) In the first step a conical sand mass is placed between a flat rigid surface and an axi-symmetric conical mold. Interaction between the sand mass and the rigid base plate is modeled using an interface friction coefficient which has the same value as the saturation value of the internal friction coefficient. The sand mass (which is confined between the base plate and the conical mold) is subjected to gravity loading, and the system is allowed to equilibrate. (ii) In the second step the conical mold is quickly lifted and the sand mass allowed to reach a new equilibrate, but slightly slumped configuration. In the process of slight slumping, there is non-homogeneous plastic deformation of the sand pile. This non-homogeneous plastic deformation, coupled with the evolving internal friction coefficient, naturally gives rise to a static stress state which exhibits the interesting feature that the vertical stress distribution at the base of the sand pile does not have a maximum under the apex of the cone, but shows a local dip there.