We present a brief survey of the history of development of the mechanical theories of strength of materials, their advantages, and drawbacks. A physically reliable and experimentally justified criterion of the equivalence of quasihomogeneous materials is established. The structure of the criterion is determined according to the results of analysis of the equation of the limiting surface whose shape satisfies the necessary and sufficient requirements imposed by the principal regularities and postulates of the mechanics of deformed body. The corresponding equation contains invariants used to control shear and tensile fracture. The degrees of responsibility of each of these mechanisms are put in correspondence with the parameter of plasticity of the material appearing in the equation in the form of a weight coefficient determined according to the results of testing by tension and compression. The condition of limiting state of materials with inhomogeneous structure is established by introducing in the criterion the influence function taking into account the statistical aspects of strength of materials in connection with the presence of "weak places" (defects). The analysis of the structure of the influence function is reduced to the solution of the problem of probability theory of repeated sampling of a given volume.