In the paper the existence of a solution to the three-dimensional elas
toplastic problem with the Prandtl-Reuss constitutive law and the Neum
ann boundary conditions is established. The proof is based on a suitab
le combination of the parabolic regularization of equations and the pe
nalty method for the elastoplastic yield condition. The method is appl
ied in the case of the domain with smooth boundary as well as in the c
ase of an interior crack. It is shown that the weak solutions to the e
lastoplastic problem satisfying the variational inequality meet all bo
undary conditions.