Work inequalities for plastic fracturing materials

Studied are second-order work inequalities for stability of plastic strain increments and fracturing stress decrements. It is found that the positiveness of the second-order work during small loading cycles does not necessarily require normality; it also allows non-normal plastic strains or fracturing relaxations which do no work, as well as non-normal ones which always do non-negative work. The latter ones include plastic strains and fracturing relaxations that are tangential to the loading surface. It is shown that the endochronic theory follows from Drucker's postulate by the same arguments as classical plasticity. The endochronic loading surface has the significance of separating the directions for which Drucker's postulate is satisfied from those for which it is not, whereas in classical plasticity it separates the stress increment directions for which the plastic strain increment vector points outside the loading surface from those for which it would point inward. The incremental linearity of classical plasticity is shown to be a tacitly implied hypothesis which does not follow from Drucker's postulate and the existence of the loading surface. Various incrementally nonlinear stress-strain relations satisfying Drucker's postulate, both such that do and do not obey normality, are demonstrated. Furthermore, it is found that for frictional materials there exists, in addition to Drucker's (or Il'yushin's) postulate, another inequality that also suffices for stability and reflects the fact that a release of elastic energy blocked by friction or by resistance to fracturing due to compression cannot cause instability. This enlarges the domain of all stable stress increment vectors from a halfspace to a reentrant wedge. The corresponding plastic strain increment vectors have no unique direction and occupy a fan, one boundary of which is the normal vector. Dependence of the second-order work in loading cycle upon the angle between the strain increment vector and the normal is useful for comparing various theories. For the incrementally linear vertex model one needs to introduce at the tangential direction a discontinuity in this dependence. Finally, some related questions of uniqueness or continuity of response are discussed, particularly for the case of a staircase path in strain space approaching a straight path as the number of stairs tends to infinity. For endochronic theory as well as some vertex models and for plasticity with a corner on the loading surface, the response to the staircase path in the limit does not approach the response to the straight path. Although this is not physically unreasonable, it is nevertheless possible to slightly adjust the definition of intrinsic time so that continuity (uniqueness) is achieved.