Generalization of the moment of inertia method to estimate equivalent amplitudes for simplifying the analysis of arbitrary non-proportional multiaxial stress or strain histories

Several models have been proposed in the literature to account for fatigue damage under multiaxial load histories. Most of them require some measure of an equivalent stress or strain amplitude, in the sense of causing the same damage as the original history, which may be difficult to obtain for generic non-proportional multiaxial variable amplitude load histories. To identify individual load cycles, a multiaxial rainflow-like algorithm must be employed. For each rainflow-counted cycle, the equivalent stress or strain amplitude along its path is often computed using the so-called convex enclosure methods, which find minimum spheres, ellipsoids, or rectangular prisms that contain the load path in a deviatoric stress or strain space. However, such procedure involves information loss, in special if the path shape is very different from the shape of the enclosing convex surface, resulting in poor estimates of equivalent stress or strain amplitudes. To overcome this problem, the moment of inertia (MOI) method has been proposed in Meggiolaro and Castro (Int J Fatigue 42:217–226, 2012 ) to calculate equivalent amplitudes and mean components of two-dimensional stress or strain paths, generated, e.g., by tension–torsion or biaxial histories. In this work, the MOI method is extended to deal with generic 6D stress or strain paths, which include all normal and shear components. To accomplish that, the load history path is first represented in a 5D deviatoric stress or strain space and then assumed to be a homogeneous wire with unit mass, whose perimeter centroid is used to estimate the location of the path mean component. Then, the polar moment of inertia (PMOI) of such a hypothetical wire with respect to its mean component is calculated. The PMOI represents the distribution of the path about a single point, the perimeter centroid, giving a measure of how much the path stretches away from its mean component, which is used in the calculation of the equivalent amplitudes. Experimental results for 13 different multiaxial load histories prove the effectiveness of the proposed method to predict equivalent amplitudes and multiaxial fatigue lives.