The contraction ratios of perfect elastoplasticity under biaxial controls

We studied contraction ratios, one rate form and one total form, of the Pra ndtl-Reuss model under combined axial and torsional controls. In the transi tion point of elasticity and plasticity, the rate form contraction ratio ma y undergo a discontinuous jump, which, depending on the control paths and i nitial stresses, may be positive, zero, or negative. For the total form con traction ratio, no similar jump phenomenon is observed in the elasticity-pl asticity transition point. Depending on initial stresses both ratios may be greater than 1/2. In the simulations of the axial-torsional strain control tests, the hoop and radial strains are not known a priori and hence can no r be viewed as inputs. This greatly complicates the constitutive model anal yses because the resulting differential equations become highly non-linear. To tackle this problem, we devise a new parametrization of the axial and s hear stresses, deriving a first order differential equation to solve for th e parameter variable, with which the consistency condition and initial cond itions are fulfilled automatically. For mixed controls, the responses can b e expressed directly in terms of the parameter without solving the first or der differential equation. In particular, when control paths are rectilinea r exact solutions can be obtained. (C) 2000 Editions scientifiques et medic ales Elsevier SAS.