Transition conditions: the yield condition

Abstract

The transformation of a continuum from a state A through a state T into a state B may be represented as a mapping. T is asymptotic in character, and hence is characterised by a contraint on the invariants of the strain or stress tensor of the field, giving rise to a functional relation between the otherwise independent invariants, I₁,I₂,I₃. At present ad-hoc forms are used for this relation. It should be obtained from the condition that the modulus of transformation from A to B tends to become zero or infinite. As a consequence the corresponding reciprocal deformation ellipsoid degenerates into a cylinder, a plane or a point. For an isotropic medium the normalised yield condition is found to be; 3I'₂ + 2(27I₃ + 3/2I₁I'₂ - I³)=2, I'₂=2I²₁-6I₂. This includes the classical yield conditions and also the Bauchinger effect. A similar condition is obtained for orthotropic bodies.