Simplification of the Burmister's problem by means of skew functions of discontinuum mechanics

As a standard procedure, the dissipation of stresses in the Burmister's space is obtained from the Love's function, by using the method of the separation of variables, which has the advantage to allow for the excluding imposition of the boundary conditions in the horizontal plane and in the vertical direction. Unfortunately, the solution is so complicated that it is not directly applicable in engineering practice. In the present paper, the principle of the mean value to find a function of state in a statistically isotropic discontinuous substance is used to separate the variables of a strain potential by means of skew functions. By skewing the stresses, strains and Hooke's law, the Germain-Lagrange's equation is established for the horizontal plane, and two kinds of ordinary differential equation are gotten for the vertical direction: one biased and other systematic skewed. The solution to the first one coincides with the well-known polynomials for a thin plate that holds the Kirchhoff's simplifications. The solution to the second one is suitable for a slab of any thickness, comprises the well-known solution of the mechanics of materials when the layer thickness is comparatively small, and adjusts very well to the Boussinesq's solution when the layer thickness is very large.