# Essentials of LEFM

Linear elastic fracture mechanics (LEFM) is the basic theory of fracture, originated by Griffith (1921, 1924) and completed in its essential aspects by Irwin (1957, 1958) and Rice (1968a, b).

LEFM is a highly simplified, yet sophisticated, theory, that deals with sharp cracks in elastic bodies. As we shall see, LEFM is applicable to any material as long as certain conditions are met. These conditions are related to the basic ideal situation analyzed in LEFM in which all the material is elastic except in a vanishingly small region (a point) at the crack tip. In fact, the stresses near the crack lip arc so high that some kind of inelasticity must take place in the immediate neighborhood of the crack tip; however, if the size of the inelastic zone is small relative to linear the dimensions of the body (including the size of the crack itself), the disturbance introduced by this small inelastic region is also small and, in the limit, LEFM is verified exactly.

Thus, LEFM is the basic theoretical reference to describe the behavior of any material with cracks, even if, as it happens for concrete, the geometry and dimensions of structures built in practice do not allow direct use of LEFM.

This and the next two chapters give an overall view of the mathematical theory of LEFM with some straightforward applications to idealized eases. They are not intended as a substitute for handbooks or treatises on LEFM. Their objective is to simplify the access of the reader to the concepts required in the remaining chapters and, at the same time, to provide in this book a self-contained presentation, so that recourse to external references be minimized.

This introductory chapter gives a short account of the most essential concepts in LEFM. Section 2.1 develops the energetic approach to fracture —the Griffith approach. It introduces the concept of energy release rate Q, representing the energy available for fracture, and the fracture energy or crack resisting force R-, representing the energy required for fracture. Also included in this section arc the basic expressions for G, and some techniques to describe fracture processes (Section 2.1). The systematic analysis of the techniques available to compute G and to measure IZ is deferred until the next chapter.

Section 2.2 introduces the concept of stress intensity factor Кi based on a simple example and describes the general properties of the stresses and displacements near the crack tip (the formal derivation of such properties is skipped in this introductory chapter and postponed until Chapter 4). It also shows that the energetic approach and the approach based on Кj —Irwin’s approach— are equivalent, and rewrites the crack growth criterion in terms of the stress intensity factor. The presentation of the methods to compute stress intensity factors and other jelated quantities is postponed until the next chapter.

The final Section 2.3 deduces the size effect laws for classical plasticity and for LEFM. As explained in the previous Chapter, these are the reference laws for any nonlinear fracture model, and are extensively used for comparison with experimental as well as theoretical nonlinear size effects in the remainder of the book.