Abstract For studying the stress-strain state at singular points and their neighborhoods new concept is proposed. A singular point is identified with an elementary volume that has a characteristic size of the real body representative volume. This makes it possible to set and study the restrictions at that point. It is shown that problems with singular points turn out to be ambiguous, their formulation depends on the combination of the material and geometric parameters of the investigated body. Number of constraints in a singular point is redundant compared to the usual point of the boundary (it makes singular point unique, exclusive). This circumstance determines the non-classical problem formulation for bodies containing singular points. The formulation of a non-classical problem is given, the uniqueness of its solution is proved (under the condition of existence), the algorithm of the iterative-analytical decision method is described. Restrictions on the state parameters at the composite wedge vertex, one generatrix of which is in non-friction contact with a rigid surface are studied under temperature and strength loading. The proposed approach allows to identify critical combinations of material and geometric parameters that define the singularity of stress and strain fields close to singular representative volumes. The constraints on load components needed to solution existence are established. An example of a numerical analysis of the state parameters at the wedge vertex and its neighborhood is considered. Solutions built on the basis of a new concept, directly in a singular point, and its small neighborhood differ significantly from the solutions made with asymptotic methods. Beyond a small neighborhood of a singular point the solutions obtained on the basis of different concepts coincide.