The problem of determining the elastoplastic properties of a cylindrical bar from the experimentally given relation between torsional moments M and torsion angles a is investigated. This problem leads to the inverse coefficient problem for the non-linear differential equation of the second order. The coefficient dependence of the following non-linear direct problem solution is studied. Reformulating the inverse problem as a minimization problem for a certain functional, the existence theorem for the considered inverse problem is presented. A numerical algorithm and examples related to direct and inverse problems are considered.