Application of mathematical method of local variations to solve problems of plastic formification of metal, powder and nanocomposition materials

The paper considers an approach to solving plastic forming problems using a variational approach, which allows to determine the stress-strain state and the technological parameters associated with it, taking into account the combination of rheological properties of the materials being processed. On the basis of the first principle of energy in mechanics, which is built on the theorems on extremal properties, when the absolute minimum of the total power of the form-changing process corresponds to a real velocity field, energy functional has been composed. The energy functional is a power balance of internal and external forces. As the power of internal forces, we understand the cost of power of plastic deformation; power associated with the presence of velocities discontinuity surfaces in the volume of a deformable medium; power of friction forces, on the contact boundary with the tool; inertial component of power expended on the change in kinetic energy. Such a formulation makes it possible to investigate the processes of high-speed strain in the same way. This functional characterizes the state of the material under these processing conditions. To solve this functional, the method of local variations is applied, which refers to the direct numerical methods of the calculus of variations. An algorithm for calculating the power of plastic strains for the process of reverse extrusion of a glass from an isotropic, rigid-plastic material is given as an example.