Torsion of prismatic orthotropic elastoplastic rods

Conservation laws were introduced into the theory of differential equations by E. Noether more than 100 years ago and are gradually becoming an important tool for the study of systems of differential equations. They not only allow us to qualitatively investigate the equation, but, as shown by the authors of this article, they allow us to find exact solutions to boundary value problems. For the equations of the isotropic theory of elasticity, the conservation laws were first calculated by P. Olver. For the equations of the theory of plasticity in the two-dimensional case, the conservation laws were found by one of the authors of this article and used to solve the main boundary value problems of the plasticity equations. Later it turned out that the conservation laws can also be used to find the boundaries between elastic and plastic zones in twisted rods, bent beams, and deformable plates. In this paper, we find conservation laws for equations describing the orthotropic elastic-plastic state of a twisted rectilinear rod. It is assumed that the conserved current depends linearly on the components of the voltage tensor. In this paper, we find an infinite series of conservation laws, which allows us to find the elastic-plastic boundary that occurs when an orthotropic rod is twisted.