Two Dimensional Plastic-Elastic Problems with Special Reference to the Difference of Plastic Deformation Theory and Flow Theory

Abstract

In the “plastic-elastic problem” the author showed a solution of general two-dimensional boundary value problem by small perturbation method. Using conjugate complex variables z=x +iy and z=x-iy the stress function φ is expressed as (21) in the case of plastic deformation theory and as (44) in flow theory, where A is a constant of strain hardening, τ2 is invarient of stress deviator tensor szj, and σzz is stress perpendicular to the stress field.The problems are divided into next four cases;(I-A) plane stress problem (σzz=0) in plastic deformation theory(I-B) plane strain problem (εzz=0) in plastic deformation theory(II-A) plane stress problem in plastic flow theory(II-B) plane strain problem in plastic flow theoryIntroducing parameter p the fundamental equation are expanded and successive approximation are obtained as (28) (29) (30) etc. The Oth approximation is elastic solution, and the first approximation coincides in two theories. The second approximation has difference in two theories. A solution having no difference in two theories was shown in Fig.1. or (66). Following to the example of stress field of infinite plate with a circular hole under uniform tension (author's last report), an example of stress and residual stress distribution of the same plate under inner pressure P at the hole was calculated as Fig2-Fig5.