A review of stress-strain paths and constitutive relations in the plastic range

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Abstract
Plastic strain paths produced by both radial and non-radial stress paths are reviewed for metals and alloys under uniaxial and biaxial stress. The present interpretation illustrates collective trends and offers theoretical solutions for each of several stress paths. In general the strain path is linear for a radial stress path for most materials at low to medium temperatures. The isotropic quadratic hardening rule of von Mises rarely describes the strain behaviour accurately. but a simple anisotropic quadratic form provides good agreement by the uniform hardening rule. Exceptions are found for magnesium and an associated alloy at room temperature and for the high temperature deformation in an aluminium alloy. Descriptive yield functions are presented and the anistropic constants determined for one particular function of the Bailey form. Strain paths produced by outward non radial zig-zag and stepped stress paths are approximately linear provided that they are confined to a narrow band in stress space. It is shown that an equivalent radial stress path analysis provides good agreement with experiment. The strain paths resulting from stress paths containing corners and from paths composed of partial or complete unloading followed by reloading in a different direction are non-linear. The behaviour is attributed to strain history; namely, a Bauschinger effect where unloading occurs and an interaction between direction dependent plastic strains for stress paths containing corners. For unloading-reloading paths a rotation in the strain path towards a linear direction is indicative of the transient nature of strain history. Good agreement is found with the prediction from each of three rules of anisotropic hardening. They are thus generalized to provide the plastic strain response to any stress path that is composed of linear segments. Modifications that account for the effects of non-linear work hardening and creep are discussed.

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