On behavior of perfect elastoplasticity under rectilinear paths

Abstract

A perfect elastoplastic model for the relation of generalized stresses and their conjugate generalized strain rates is investigated in the paper. The exact solutions are solved for the model subjected to rectilinear generalized strain paths1 and the concepts of response subspace and visualization are introduced such that the n-dimensional problem is reduced to a two-dimensional problem in the response subspace. The response path is found to be, geometrically speaking, a geodesic in the n-dimensional closed ball Bn of generalized stresses. The phase portrait of dissipation is studied and it is a remarkable fact that in the response subspace the coordinate x is exactly the dissipation power per unit generalized strain rate when x ⩾ xon [see eqn (63)]. The generalized stress-strain curves are classified by geometrical shapes into ten types, in contrast to the superficial impression of only one type of shape, an inclined straight line followed by a horizontal line. The existence of a limit strength vector and a limit power of dissipation is demonstrated by the long-term behavior based on the exact solutions, and the existence together with the response's stability is further verified by Lyapunov's direct method. The closeness to the limit strength point is also estimated and the exact formula accounted for prestress is derived. The dependence of the responses on the product space of initial values, input paths, property constants and closeness to the limit is thoroughly investigated with control parameters identified.