Superposition of movements with irreversible deformation

Abstract

Аlyushin Y. А. Superposition of movements with irreversible deformation. Material working by pressure. 2020. № 1 (50). Р. 3-17.

The principle of superposition in the description of motion in the form of Lagrange, which consists in replacing the Lagrange variables of external motion with equations for the corresponding Euler variables of internal motion, is justified. The rule of geometric addition of velocities and accelerations is fulfilled for each of the moving particles at any time. Examples of superposition in motion of absolutely solid bodies, linear stretching with torsion and bending under elastic and plastic deformations, for screw rolling are given. It is noted that for absolutely solid bodies there are no restrictions on combined movements, the implementation of combined movements is provided by external forces determined by the equations of dynamics. In the case of superposition of movements for processes with elastic deformation, the correct solutions must satisfy the differential equation that ensures the independence of energy from the choice of the velocity reference system. The possibility of analytical determination of equations of motion for irreversible deformations based on Laplace differential equations is proved. It is noted that checking the correctness of the combined motion under equilibrium conditions allows the loss of possible correct solutions. In the field of plastic deformations, the choice of external and internal movements due to small movements practically does not affect the accuracy of the calculation results. It is recommended to choose external and internal movements so that the final equations of combined motion have a simpler form. The superposition for irreversible inhomogeneous deformations is recommended to be considered as kinematically possible variants of movements that can be used to determine the upper estimate of the power of external forces required for the implementation of combined movements.