Forman–Kearney–Engle fractal propagation law of fatigue crack

Purpose The objective of the present paper is to propose a new approach to modeling the dependence of the number of cycles to failure on the initial crack length and stress amplitude in the context of linear fracture mechanics. Design/methodology/approach The anticipated Forman–Kearney–Engle propagation function describes the infinitesimal crack length growth per increasing number of load cycles, assuming that the load ratio remains constant over the load history. Two types of Forman–Kearney–Engle functions, each with a different number of fitting parameters, are put forth for consideration. Findings The paper presents a closed-form analytical expression for the Forman–Kearney–Engle fractional-differential relationship between crack length and the number of cycles. Two novel functions are introduced, which express the damage growth per cycle. These functions facilitate the unification of disparate fatigue laws within a single expression, thereby providing a unified framework for analysis. The unified fatigue law provides closed-form analytical solutions for the relationship between crack length and the mean value and range of cyclic variation of the stress intensity factor. The solution presents the number of cycles to failure as a function of the initial size of the crack, thereby obviating the necessity to solve a nonlinear fractional differential equation of the first order. The explicit formulas for stress versus the number of cycles to failure are provided for both proposed unified fatigue laws. Research limitations/implications The original formulation of Paris Law (Paris and Erdogan, 1963), (Paris, 1964) was subsequently revised by Forman et al . (1967). As demonstrated in this study, the crack growth rate increases exponentially as the stress intensity factor approaches the fracture toughness. Practical implications The principal benefit of the proposed functions is the potential for obtaining closed-form analytical solutions for the unified fractional crack growth law. In order to apply the aforementioned concepts, it is imperative to examine the relationship be-tween the propagation of cracks in ordinary processes and in fractal processes. The formulation of this relationship can be expressed for traditional Forman–Kearney–Engle-propagation and thermodynamically motivated function. Originality/value The objective of the present study is to provide a mathematical description of the crack propagation process in a fractal medium, which obeys the Forman–Kearney–Engle law. The length of a crack that is propagated in a cyclical manner is contingent upon the fractal dimension of the medium, the number of load cycles and other conventional mechanical parameters. The fractal dimension of the microstructure, derived from the fractional derivative of the initially unknown order, is a critical component in this regard. The equations developed in this article are instrumental in facilitating the calculation of the fractional derivative of the Forman–Kearney–Engle cyclic crack elongation process from standard fatigue tests.