Stress-strain relations and necking criteria for triaxial loading, two principal stresses being equal

In a preceding paper a theoretical derivation of the axial stress-strain relation during monotonic uniform straining under uniaxial loading, the fundamental stress-strain relation, has been given. Since a hydrostatic tension or compression hardly influences the relation between the maximum shear stress and the principle strains, the axial and the radial stress-strain relations for uniform straining under bi- and triaxial loading can be derived from this formula. It is shown how the uniform strains at necking in axial and radial direction respectively depend on the principle stresses when the loading system is of such a kind that radial loading does not affect the axial load and vice versa. The necking criterion than leads to two kinds of necking, positive necking (local contraction) and negative necking (local bulging). However, the condition for negative necking in one principal direction is always preceded by the condition for positive necking in an other principal direction and thus belongs to states of stress which have already become unstable. Only a qualtitative picture of the stress-strain relations during necking can be given.