Some extremum properties of finite-step solutions inelastoplasticity

For the nonholonomic elastic–plastic problem under a given external action history overa time interval, an extremal formulation is given in terms of the complete solution over the wholeinterval. The assumed elastic–plastic behaviour is of the associated type with piecewiselinearized yield surface and linear hardening.When the loading history is reduced to an infinitesimal increment of the external actions(incremental problem) or when the material behaviour is assumed to be of the holonomic type (finite holonomic step) problem, the functional of the extremal formulation may be split into thesum of two other simpler functionals (previously introduced) whose minimum, for both of them,gives the problem solution under less constraints than in the original problem.For general non-holonomic loading histories the above splitting is shown to be still possiblewhen a particular change of the complementarity condition of the constitutive law is considered,which leads to a new class of holonomic problems.It is shown that some problems of this new class, together with a suitable time discretization,represent the schematization of the original problem corresponding to well known numericalintegration schemes.