Minimum norm ellipsoids as a measure in high-cycle fatigue criteria

1. Abstract This paper focuses on the use of minimum norm ellipsoids in stress-based fatigue criteria. We consider both the theoretical issues involved in constructing mathematical models for high cycle fatigue endurance limits and the computational features concerning the application of the resulting fatigue criteria. In the present approach, the macroscopic level of continuum is enhanced with internal variables corresponding to the accommodated state at the mesoscopic level of a representative volume. That is, elastic shakedown in the representative volume is used in order to explain the mechanisms of fatigue endurance of a body undergoing elastic strain variations at the macroscopic level. We propose a scalar measure for the amplitude of shear stress variations based on the definition of an enclosing hyper-ellipsoid of minimum Frobenius norm. This equivalent shear amplitude is combined with the maximal mean stress, in both linear and quadratic functions, leading to two alternative fatigue criteria. The predictions of these criteria are compared with experimental data. The distinction between minimum norm enclosing ellipsoid and minimum volume enclosing ellipsoid is discussed. We argue in favor of using the Frobenius norm rather than the volume of the hyper-ellipsoid. The computation of the minimum norm ellipsoid is discussed and a practical algorithm is proposed. 2.