Asymptotic axes of stress tensors and strain increment tensors in mechanics of compressible continua

Abstract

New tensor representations of the stress state and the kinematics of compressible flows are obtained in the paper with the use of the notion of asymptotic directions of the symmetric stress tensor and the strain increment tensor. The exposition is based on terminology and notation typical of the mathematical theory of plasticity, but all main results remain valid for stresses and strains in compressible continua. The simplest and most efficient forms of the stress tensor for “completely plastic,” “semiplastic,” and “nonplastic” spatial stress states are found, where the asymptotic stress axes serve as the most natural reference frame ensuring new symmetric tensor representations of stresses different from the spectral ones. Similar representations can be extended to the stress increment tensor. Two-dimensional curvilinear grids such that the strain rates of their elements are always zero are chosen on the surfaces orthogonal to the directions of the “intermediate” principal strain increment. Incremental relations for the sliding rates along the grid lines are obtained, and these relations generalize the Geiringer equations along the characteristic lines, which are well known in the theory of plane deformation of perfectly plastic bodies. The generalization readily applies to spatial flows, and the possible flow compressibility is taken into account as well.