Restrictions on stress components and loads at the top of a four-corner pyramid with a rhombic base

Differences between two existing approaches in the study of state parameters (stresses, strains) near singular points are discussed. In the classical (asymptotic) approach, the singular point is excluded from the solution region. Boundary conditions are not specified at this point, and, thus, the state parameters have only asymptotic values. The second, non-classical approach, considers the singular point as a point of the continuum and the elementary volume associated with this point.
The continuum point indicates a singular point location, and the elementary volume defines material characteristics and parameters of the stress-strain state. The number of specified conditions at such a point is greater than that specified at a regular (non-singular) boundary point of a body. Therefore, the problem of solid mechanics with a singular point is non-classical.
In this paper, a non-classical approach is used to study state parameters at the vertex of a quadrangular pyramid with a rhombic base. The case of loading the pyramid near the top by surface forces and the case of the pyramid immersed in an elastic medium are considered. It is shown that the correct formulation of the mechanics problem for such cases is possible only with the implementation of the restrictions established in the work. Particular solutions are presented that are consistent with known analytical results.
For the case of the inclusion of a pyramid in an elastic body, combinations of geometrical and material parameters resulting in an unlimited increase in stresses within elementary volumes containing the top of the pyramid are revealed. The research results will be used in solid mechanics problems in the formulation of specified conditions at the vertices of polyhedra. In particular, they may be applied when analyzing the stress state near the vertices of Knoop indenters used for studying samples, as well as near crystalline inclusions in the mechanics of composites.