The perfectly plastic problem

Let us consider the case of contact between a perfectly plastic rod and a rigid punch. This corresponds to the case when ci = C2 = 0 in (5.247)-(5.252) or, equivalently, when Ai = A2 = 0. 

It is obvious that in the case under consideration the elements 7r(n,m) are bounded in L T). Therefore, from the obtained inequality it follows that 

After all, we can get the answer to the question on the convergence for solutions of the elastoplastic problem to a solution of the perfectly plastic problem provided that the assumptions of Theorem 5.9 are fulfilled. To this end, consider the problem (5.255)-(5.259), where /xAi and /xA2 are used instead of Ai and A2, /x 0. A priori estimates are of the same form as in the case of the perfectly plastic problem. It is important to note that the estimates are uniform not only in q, (5 (5o, A, but also in /x /xq. For each fixed /x it is possible to pass to the limit as — 0, 5 — 0, A — 0 in (5.255)-(5.259) and to derive, therefore, that the solution v, w, n, satisfies the inequality 

Theorem 5.10. Let all the assumption of Theorem 5.9 he fulfilled. Then, from the solutions v, w, n, of the elastoplastic contact problem 

Note that different perfectly plastic models for three dimensional case are considered in (Mosolov, Myasnikov, 1971). 

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