Analysis of Fracture and Fatigue using Lagrangian Mechanics

TL;DR

This paper extends Griffith's fracture theory using Lagrangian mechanics, introducing a dynamic stress intensity factor and applying it to fatigue, revealing new insights into crack growth in elastic and elastic-plastic materials.

Contribution

It develops a Lagrangian-based approach to fracture and fatigue, incorporating stress wave effects and crack tip plasticity, providing new theoretical insights and models.

Findings

Introduction of a dynamic stress intensity factor.

Retrodiction of Griffith's crack and dynamic cases.

New fatigue crack growth relationship explaining the Paris Law.

Abstract

Many people are aware of the theory of elastic fracture originated by AA Griffith, and although Griffith used the theorem of minimum potential energy most people seem unaware of the broader implications of this theorem. If it is set within its classical mechanics roots, it is clear that it is a restricted form of a Lagrangian. In advanced texts on fracture cracks are treated as dynamic entities, and the role of stress waves is clearly articulated. However, in most non-advanced texts on fracture and fatigue the role of stress waves are either not included or not emphasised, often leading to a possible misunderstanding of the fundamentals of fracture. What is done here is to extend Griffiths approach by setting it within the concept of Stationary Action, and introducing a quasi-static stress wave unloading model, which connects the energy release mechanism with the stress field. This leads to a definition of a dynamic stress intensity factor, and this model is then applied to fatigue of perfectly elastic and elastic-plastic materials to include crack tip plasticity. The results for the Griffiths crack and the dynamic case are retrodiction, to establish the validity of the methods used. The extension to fatigue gives significant new results, which show that for elastic-plastic materials the influence of the maximum stress in the cycle as a fraction of the yield stress, called the yield stress ratio, has not been recognised. The new form of the fatigue crack growth relationship derived answers many of the long standing questions about the Paris Law.