Simplifi ed mechanics of granular multilayer systems

The careful experiments run by several scientists have shown that the stress transmission in granular matter is a complex problem that embodies mainly the development of strong inhomogeneities and the polarization of the grain contacts. So that some theories based on the statistical mechanics have been proposed to solve some simple problems. In this paper, the principle of the biased mean value is used to determine each separable variable of the stress function. This simplifi es the Burmister's problem, to the point that only second-order transfer matrices are needed to describe the continuity conditions of the vertical displacement and stress at the interface between two consecutive layers. The imposition of the boundary conditions to the overall system not only completes the required number of equations needed to fi nd all the constants of integration but also provides a differential equation for the horizontal plane, much simpler than the Lagrange's equation for plates. Stresses, displacements and strains at any point are determined by means of hyperbolic functions of each layer thickness; for a load uniformly distributed over a circular area and over an arbitrary area as well. The values so calculated fi t well the Boussinesq's solution and the Burmister's rigorous results for small values of the coeffi cients of layer interaction, as befi ts a granular material.