Fracture Mechanics from the Principle of Stationary Action

This study addresses the interplay between fracture and fatigue through a novel application of a simplified stress wave unloading model within the framework of Lagrangian mechanics. The objective is to extend the classical theories of fracture, particularly those established by A. A. Griffith, by proposing that all cracks behave as dynamic entities influenced by stress waves. The methodology involves connecting Griffith’s principles with the concept of Stationary Action (Hamilton’s Principle) and defining a dynamic stress intensity factor applicable to both perfectly elastic and elastic-plastic materials, thus accounting for crack tip plasticity. The results indicate a retrodiction for the Griffith crack that validates the proposed methods, while the exploration of fatigue in elastic-plastic materials reveals that the yield stress ratio (YSR) plays a vital role in fatigue crack growth. Notably, this research provides a new formulation of the fatigue crack growth relationship that clarifies longstanding questions regarding the Paris Law. This findings potentially constitute a significant advancement in our understanding of fatigue behavior in materials subjected to cyclical loading.