Investigation of the formulation of the boundary-value problem of the local theory of elastoplastic processes

Abstract

For a complex stressed state, a three-term defining relation [1, 2] is used which implies that the five-dimensional stress, stress rate and strain rate vectors are coplunar. On the hypothesis of local definiteness [3], the two coefficients occurring in the three-term relation are taken as functions of three functionals of the process—the stress intensity, the length of the arc of the deformation path and the angle of approach. For bounded derivatives of these two functions with respect to each argument, conditions securing a correct formulation of the static boundary-value problem in terms of rates of each instant of the elastoplastic process are determined. A formulation is given of the quasistatie global boundary-value problem for the whole process. It is proved that the operator of the global problem, an operator of the variational calculus [4], is positive definite, strictly monotonic in the main and possesses the ( S) i -property [5]. Using the theorem of Leray and Lions [4], it is shown that a generalized solution exists. It is proved that the global solution is unique and continuously dependent on the external loads. For the step method, using discretization of the process with respect to the load parameter, and iterational methods (of the type of SN-EVM method [2]), convergence of the approximate solutions to the exact solution of the global problem is proved.