Restrictions on stress components in the top of round cone

The stress state at a vertex of homogeneous or composite cone is studied using the concept of the point of deformed body considered as an elementary volume contractible to this point. Elementary volume represents an assembly of continuum points with a linear characteristic dimension equal to that of representative volume element of the material of modeled body. Elementary volume is a carrier of physical properties and state characteristics of continuous medium. The state characteristics are uniform in such a volume and, therefore, the values retain constant while the volume is being contracted to the vertex of the cone. The accepted concepts about singular point — the vertex of a cone — allow one to formulate restrictions on the state parameters in it. These restrictions are imposed in the cases when the cone is under an axisymmetric load, its generator slides without friction along a rigid surface or it is clamped, and also when the vertex of the cone is an internal point. The number of restrictions at a singular point is more than that imposed at typical surface points in the classical problem. This circumstance makes it necessary to consider the problem for a body containing singular point as non-classical problem. The proposed concept was used to define the conditions for material and geometric parameters under which the stress state at the vertex of the cone is completely determined. The cases of singular behavior of stresses in the vicinity of the cone vertex are found. The limits for load components providing a correctness of considered problem formulation are obtained. It is shown that the stresses and strains are continuous at the internal singular point.
The presented results will find an application in the formulation of non-classical problems of studying state parameters near the cone vertex. In particular, they will be useful in researching interaction of conical indenters with a test sample.