A stress-strain state of the thin square workpiece during upsetting by the rough plates

In this paper, a stress-strain state of the thin square rigid-plastic plate upsetting by the rough plates has been investigated using the energy minimization method. It is known that the total strain energy $\Pi$ is expressed in terms of the strain rate, which according to the Ritz method must be chosen so that the boundary conditions and incompressibility conditions are both satisfied. The obtained dependences for the strain rates contain the c coefficient depending on the plate dimensions and external friction coefficient. The value of the c is obtained from the condition of zero partial derivative of $\Pi$ with respect to $c$. Having determined the $c$, it is possible to obtain the strain rate field, i.e. the solution of the problem of strain state of the plate. The strain state as detailed in the paper is determined using the Levy–Mises equations, which integrate the stress with the strain rates. The authors of the paper have previously considered a similar problem and it has been solved using the equilibrium equations and physical equations of the plastic flow. The results in the form of strain state parameters of the workpieces made of different materials have been experimentally obtained. In both cases, there is a good agreement with the theoretical dependences for the strain rates. The obtained dependences for velocities and strain rates, which are in a good agreement with the experimental data, allow determining both the workpiece configuration depending on the upsetting value, the voltage and the external friction coefficient.