Relaxation methods applied to engineering problems XIV. Plastic straining in two-dimensional stress-systems

Abstract
Only experiment can decide the criterion of elastic failure, and the relation of stress to strain during plastic distortion, in real materials such as steel; and (since stress is not directly measurable) it can do this only by verifying relations deduced from theory in regard to total displacements and resultant actions. Consequently practical value attaches to computational methods whereby, on the basis of some assumed criterion, relations of that kind can be formulated. This paper draws conclusions regarding two-dimensional systems (plane stress and plane strain) from the ‘Mises-Hencky hypothesis’, according to which failure occurs when (р 2 - р 3 ) 2 + (р 3 - р 1 ) 2 + (р 1 - р 2 ) 2 = const. (р 1 , р 2 , р 3 denoting the principal stresses), and from the relation ∆ ϒ1 : ∆ ϒ2 : ∆ ϒ3 = q 1 : q 2 : q 3 assumed to hold during the subsequent plastic distortion (∆ ϒ1 , ∆ ϒ2 , ∆ ϒ3 denoting the incremental plastic shear-strains and q 1 , q 2 , q 3 the principal shear stresses). Its methods could be applied to other hypotheses. In its worked examples some regions remain elastic while in other regions (here termed enclaves ) the strain is partly plastic. Such cases present special difficulty in an orthodox treatment.

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