Remarks on Il'iushin's postulate

Abstract

In its original version (strong form), the postulate of Il'iushin states that the integral of the stress power of an elastic-plastic material must be non-negative for any closed strain path. As a consequence, it has been shown that the so-called simple endochronic theory of plasticity violates this postulate of "material stability". The characteristic feature of this theory is that yield surfaces and related loading conditions are not involved in the governing constitutive equations. In the present paper it is shown, with reference to a well-established class of plasticity laws, that the strong form of Il'iushin's postulate may be violated as well, if the constitutive theory is constructed on the basis of a yield surface and related loading conditions. The question arises if such strain-stress relations preserve some weaker stability conditions in the sense of Il'iushin. It turns out that they satisfy a weaker form of Il'iushin's postulate, in which the integral of the stress power is required to be non-negative only for special, so-called small cycles of deformation, as defined by Lucchesi and Silhavy. From a physical point of view on a phenomenological level, it seems that there is no experimental evidence to exclude from a general theory of plasticity such material behaviour which complies with the weak form of Il'iushin's postulate. Moreover, if the validity of the weak form of Il'iushin 's postulate is assumed, then it is shown that the simple endochronic theory of plasticity is no longer in conflict with this version of the postulate.