Plastic instability under plane stress

TLDR: In this paper, conditions for instability of plastic strain under plane stress for a material conforming to the Mises-Hencky yield condition and strain-hardening according to a unique relationship between root-mean-square values of shear stress (q) and incremental strain (δψ).

Abstract: This paper examines the conditions for instability of plastic strain under plane stress for a material conforming to the Mises-Hencky yield condition and strain-hardening according to a unique relationship between root-mean-square values of shear stress (q) and incremental strain (δψ). If, under fixed loading conditions, the material undergoes a strain increment which is consistent with the applied stress system, the conditions are stable or unstable according as the increment in representative yield stress is greater or less than the increment in representative induced stress. The strain at which instability arises is found in terms of the biaxial stress ratio p2/p1 under different conditions of applied loading, and the effect is demonstrated of strain-hardening according to an empirical relation of the type q = c (a + ψ)n. The analysis is also applied to certain cases of non-uniform stress distribution. In the case of the hydrostatic bulge results are obtained showing a critical thinning ranging from 26 per cent for a non-hardening material to about 45 per cent for typical strain-hardening materials, values in general agreement with experimental data. Conditions over the punch head in the pressing of a cylindrical shell are discussed but computations are not attempted.