The contraction ratios of perfect elastoplasticity under biaxial controls

We studied contraction ratios, one rate form and one total form, of the Prandtl–Reuss model under combined axial and torsional controls. In the transition point of elasticity and plasticity, the rate form contraction ratio may undergo a discontinuous jump, which, depending on the control paths and initial stresses, may be positive, zero, or negative. For the total form contraction ratio, no similar jump phenomenon is observed in the elasticity–plasticity 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 not be viewed as inputs. This greatly complicates the constitutive model analyses because the resulting differential equations become highly non-linear. To tackle this problem, we devise a new parametrization of the axial and shear stresses, deriving a first order differential equation to solve for the parameter variable, with which the consistency condition and initial conditions are fulfilled automatically. For mixed controls, the responses can be expressed directly in terms of the parameter without solving the first order differential equation. In particular, when control paths are rectilinear exact solutions can be obtained.