As is known, ductile strength is characterized by the shear resistance of a material Sshf, and brittle strength is characterized by the breakaway resistance of a material Sref. Under a complex stress state, the ductile and brittle strength criteria of a material are functions of all three true stresses S1, S2 and S3. If the functions from S1, S2, S3, expressing shear and tear resistance, are chosen correctly, then the mentioned strength criteria for a given material and its physical state before loading will be constant, independent of the type of stress state ...

The mathematical theory of plasticity is based on the law of the existence of a generalized function of the mechanical state of a material at constant temperature and strain rate. The essence of this law is reduced to the recognition for each material of a generalized, independent of the type of stress state, curve of plastic flow in certain coordinates. The conditions of plasticity and strength theory express this law by various equations characterizing the hardening of the material during its plastic deformation.The influence of normal stresses on the resistance of a material to deformation is quite obvious, since normal stresses in shear planes cause frictional forces that facilitate or impede the plastic flow of the material. In order to at least approximate the strength of a material under a complex stress state, it is necessary to perform at least two comparative tests for two types of stress states: tension-compression, tension-torsion or, in extreme cases, tension-cutting. The initial tests should be as simple and reliable as possible.

According to modern views, all metals are a crystalline spatial lattice of ions in its nodes and free (socialized) electrons. The process of metal deformation is usually divided into three stages: 1) elastic (reversible) deformation, in which the distances between the elements of the atomic lattice change without changing the structure of the substance; 2) plastic (irreversible) deformation, in which local changes occur in the structure of the metal, associated mainly with a change in the size and shape of crystals and deformation of the intermediate layers; 3) destruction. In order to form a correct idea of ​​the overall strength of the material, it is necessary to evaluate its behavior in all three stages. The division of the deformation process into three stages is purely conditional. In reality, in the elastic section, there is also a small plastic deformation, in the section of plastic deformation, both elastic deformation occurs and processes directly associated with the destruction of the material are generated (see article).

Experimental studies have established that during plastic bending of a rod in the region of small plastic deformations, there is no curvature of the planes of cross sections that have two axes of symmetry. This is true for materials that equally resist tension and compression and for materials that do not equally work under tension and compression. Bending is a very common, if not the most common, type of deformation in structures and machines. Therefore, for the full use of the material and the lightening of structures, the correct approach to calculating the bearing capacity of a rod during bending is one of the pressing issues. The need to determine bending moments for any values ​​of plastic deformation also arises during cold stamping. The approximate formulas obtained in this work for calculating bending moments throughout the plastic deformation of a rod can be used in practice.