On the use of tensor paths to estimate the nonproportionality factor of multiaxial stress or strain histories under free-surface conditions

Abstract

Nonproportional (NP) strain hardening is caused by multiaxial load histories that induce variable principal stress/strain directions, activating cross-slip bands in several directions, due to the associated rotation of the maximum shear planes. This effect increases the strain-hardening behavior observed under proportional loads, those with fixed principal directions, and must be considered in multiaxial fatigue calculations, especially for materials with low stacking fault energy, such as austenitic stainless steels. NP hardening depends on the material and on the shape of the multiaxial load history path in a stress or strain diagram as well. It can be evaluated by a nonproportionality factor ${F_{\rm NP}}$ that varies from zero, for a proportional load history, to one, for a ${90^{\circ}}$ out-of-phase tension–torsion loading with the same normal and effective shear amplitudes. Originally, ${F_{\rm NP}}$ was estimated from the aspect ratio of a convex enclosure that contains the load history path, such as an ellipse or a prismatic enclosure, but such convex enclosure estimates can lead to poor predictions of ${F_{\rm NP}}$. Another approach consists on evaluating the shape of the six-dimensional (6D) path described by the six normal and shear components of the stress tensor, where the stress path contour is interpreted as a homogeneous wire with unit mass. The moment of inertia (MOI) tensor of this hypothetical wire is then calculated and used to estimate ${F_{\rm NP}}$. The use of 6D stress paths to estimate ${F_{\rm NP}}$ is questionable, since 6D formulations implicitly include the effect of the hydrostatic stress, while NP hardening is caused by the deviatoric plastic straining, not by stresses alone or by their hydrostatic component. In this work, the NP factor ${F_{\rm NP}}$ of a multiaxial load history is estimated from the eigenvalues of the MOI tensor of the plastic strain path, which are associated with the accumulated plastic straining in the principal directions defined by the associated eigenvectors. The presented formulation assumes free-surface conditions, but allows a surface pressure, covering the conditions of most critical points, which indeed are located on free surfaces. Experimental results for 14 different tension–torsion multiaxial histories prove the effectiveness of the proposed method.