Linear Elastic Fracture Mechanics (LEFM): Part One

INTRODUCTION

For decades, fracture mechanics has been used to study the fatigue crack propagation of mechanical components. Great efforts have been made to investigate the growth rate and mechanism of the fatigue crack in different conditions. Many of the studies focused on establishing a method using a few clear parameters to describe the fatigue crack growth rate.

Linear Elastic Fracture Mechanics (LEFM) is the basic theory of fracture, originally developed by Griffith (1921 to1924) and completed in its essential form by Irwin (1957, 1958) and Rice (1968 a,b). LEFM is a highly simplified, yet sophisticated, theory that deals with sharp cracks in elastic bodies.

LEFM is applicable to any material as long as certain conditions are met. These conditions rely on the presence of all basic ideal conditions analyzed in LEFM in which all materials are elastic except in a vanishingly small region (a point) at the crack tip. In fact, the stress near the crack tip is so high that some kind of inelasticity must take place in the immediate vicinity of the crack tip.

However, if the size of the inelastic zone is small relative to the linear dimensions of the body (including the size of the crack itself), then the disturbance introduced by this small inelastic region is also small. Therefore LEFM and point of failure can be verified exactly. Using this information LEFM theory can therefore be used as the basic foundation for the behavior of any material prone to cracking (such as concrete) even though precise LEFM calculations cannot be made.

The theory of Linear Elastic Fracture Mechanics (LEFM) has been developed using a stress intensity factor (K) determined by the stress analysis, and expressed as a function of stress and crack size i.e. (stress) x (length)^1/2. The strain energy release rate, or the stress intensity at the crack tip (K_C), eventually, will inevitably reach a critical value.

There are three modes of fracture, Mode I being identified as the opening mode, in which the crack surfaces move opposite and perpendicular to each other (as when opening by driving in a wedge).This mode is the most important from the low stress fracture point of view and has been studied more extensively than modes II and III, which involve sliding and lateral tearing respectively. The basic assumption is that crack propagation will occur when the strain energy release rate, or the stress intensity at the crack tip K_C reaches a critical value. As previously mentioned, mode I is the most important when considering the low stress fracture.

Plane strain is defined as a state of two dimensional strain, i.e. there is no strain in the through-thickness direction, and is therefore classed as a state of triaxial stress. The ideal conditions of stress are not usually realized in practice and a mixed mode state of stress exists. Even in very brittle fractures, some plastic flow may occur at the tip of a sharp defect.

In order to establish the critical stress intensity by Linear Elastic Fracture Mechanics, the plastic zone must be kept small in comparison to the other dimensions of the specimen. For what is essentially plane strain conditions, the inherent fracture toughness of a material can be expressed in terms of the critical value of the stress intensity factor K_IC at which crack instability occurs. The value of K_IC has to be determined experimentally but, once properly determined under one set of conditions, it is equally applicable to other conditions.

The value of K_IC does, of course, vary markedly with metallurgical variables, such as steelmaking practice and inclusions, and heat treatment and microstructure, but it can be used to compare steels of different strength levels by use of the same parameter, which is not possible when using alternative approaches, such as those based on transition temperature.

The elements of fracture mechanics may be summarized in the form of a triangle having the 3 critical parameters situated at each apex: working stress (σ_w), fracture toughness (K_IC) and critical flaw size (a_crit). If two of the three parameters are known, the third can be calculated; this simple concept emphasizes the importance of being able to accurately measure the size of the discontinuity.

Furthermore, the exact magnitude and distribution of the stresses in a component may not be known and therefore, it is obvious why a safety factor has to be applied. But residual stresses in a casting can be of a surprisingly high order before the working stress is applied. Consequently, some knowledge of the magnitude of possible residual stress is essential.