Endowing Griffith's fracture theory with the ability to describe fatigue cracks

In this paper, a possible extension of Griffith's fracture theory to describe fatigue-induced crack propagation is proposed. To this end, an energy-based model is presented, taking Griffith's model as the point of departure and employing the concept of state-dependent fracture toughness. Therein, fatigue degradation is achieved through a suitable history variable endowed with a functional form that is able to consider crucial aspects of fatigue, including the crack-tip singularity, the identification of fatigue-inducing loading conditions, and mean load effects. Simple paradigmatic examples indicate that the model provides a unified description of different fatigue responses and unveils peculiar regimes in the crack propagation process, always preserving the link to classical fracture mechanics. In this context, analytical results establish a clear relation between Griffith's fracture theory and Paris' law. The proposed modeling framework paves the way for future developments in modern fracture mechanics, e.g., to derive a novel generation of variational fatigue phase-field fracture models suitably rooted in a Griffith-based theory.