On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in a static equilibrium

In this paper the authors study the problem of the title for an elasto-perfectly plastic body subject to Hencky’s law, which states that a material behaves elastically if the deviatoric stress is under a critical value in a von Mises criterion and deforms plastically when this critical value is attained. Earlier workers have solved problems of such materials by first proving the existence of a unique stress tensor for a specific equilibrium configuration by using the Haar-von K´arm´an principle and then determining the displacement field u and the plastic strain field . But here the authors first prove the existence of a u minimizing the energy functional and then construct and . Thus they show that (u, , ) satisfy the conditions of stress symmetry, equilibrium conditions, boundary conditions and Hencky’s constitutive law in a suitable weak sense. Further they show that the Haar-von K´arm´an conditions also hold. As an illustrative example the case of torsion of a cylinder is discussed. In the last section it is shown that the case of a perfectly plastic material can be obtained as the limit of elasto-perfectly plastic problems with yield point converging to zero. This approach may be tried for solutions of some more specific cases and may be compared with solutions obtained by an earlier approach.