Theoretical Fundamentals for Formulation of Discrete Equations of Motion in Thermodynamic Approach for Description of Metal Forming Processes

This paper concerns the application of variational methods for the formulation of object motion equations as well as generation heat and its transfer formulated in the updated Lagrange description. These physical phenomena occur in the processing of metals in which there are thermal, plastic viscous and phase deformations. The obtained incremental variation equation of the object's motion and the incremental variation of the heat conduction equation were discretized by the Finite Element Method to obtain discrete systems of equations. These equations are solved by known methods for given boundary and initial conditions. As a result, a column vector of the displacement increment and a column vector of the temperature increment in the object nodes are obtained at each time step. Examples of the use of these methods for the burnishing rolling process with electric current are shown.