Undeservedly little attention is paid in the vast literature on the theories of vibration and plasticity to the problem of steady-state vibrations in elastoplastic bodies. This problem, however, is of considerable interest and has many important applications. The problem of low-cyclic fatigue of metals, which is now in a well developed state is one such application. The investigations within this area are actually directed to collecting experimental facts about repeated cyclic loadings, cf. [47J. Theoretical investigations within this area usually con­ sider the hysteretic loops and the construction of models of plasticity theory which are applicable to the analysis of repeated loadings and the study of the simplest dynamic problems. Another area of application of the theory of the vibration of elastoplas­ tic bodies is the applied theory of amplitude-dependent internal damping. Another name for this theory is the theory of energy dissipation in vibrat­ ing bodies. In accordance with the point of view of Davidenkov "internal damping" in many metals, alloys and structural materials under consider­ able stress presents exactly the effect of micro plastic deformations. There­ fore, it may be described by the methods of plasticity theory. This point of view is no doubt fruitful for the theory of energy dissipation in vibrating bodies, as it allows one to write down the constitutive equations appropri­ ate both for vibrational analysis of three-dimensional stress states and an investigation of nonharmonic deformation. These problems are known to be important for the theory of internal damping.