# Classical Linear Theory

The stimulus for fracture mechanics was provided by a classical paper of Inglis (1913), who obtained the elastic solution for stresses at the vertex of an ellipsoidal cavity in an infinite solid and observed that, as the ellipse approaches a line crack (i. e., as the shorter axis tends to zero), the stress at the vertex of the ellipse tends to infinity (Fig. 1.1.1). Noting this fact, Griffith (1921, 1924) concluded that, in presence of a crack, the stress value cannot be used as a criterion of failure since the stress at the tip of a sharp crack in an elastic continuum is infinite no matter how small the applied load (Fig. 1.1.1b).

Figure 1.1.1 The stress at the ellipse vertices is finite in an elastic plate with an elliptical hole (a); but the stress concentration tends to infinity as the ellipse shrinks to a crack (b).

This led him to propose an energy criterion of failure, which serves as the basis of the classical linear elastic fracture mechanics (LEFiM) or of the more general elastic fracture mechanics (EFM, in which linearity is not required). According to this criterion, which may be viewed as a statement of the principle of balance of energy, the crack will propagate if the energy available to extend the crack by a unit surface area equals the energy required to do so. Griffith took this energy to be equal to where ys is the specific surface energy of the elastic solid, representing the energy that must be supplied to break the bonds in the material microstructure and, thus, create a unit area of new surface.

Soon, however, it was realized that the energy actually required for unit crack propagation is much larger than (his value, due to the fact that cracks in most materials are not smooth and straight but rough and tortuous, and are accompanied by microcracking, frictional slip, and plasticity in a sizable zone around the fracture tip. For this reason, the solid state specific surface energy 2ys was replaced by a more general crack growth resistance, 7Z, which, in the simplest approximation, is a constant. The determination of 7Z has been, and still is, a basic problem in experimental fracture mechanics. The other essential problem of LEFM is the determination, for a given structure, of the energy available to advance the crack by a unit area. Today, this magnitude is called the energy release rate, and is usually called Q (note that the rate is with respect to crack length, not time).

The early Griffith work was considered of a rather academic nature because it could only explain the failure of very brittle materials such as glass. Research in this field was not intensely pursued until the 1940s. The development of elastic fracture mechanics essentially occurred during 1940-1970, stimulated by some perplexing failures of metal structures (e. g., the fracture splitting of the hulls of the “Liberty” ships in the U. S. Navy during World War II). During this period, a good deal of theoretical, numerical, and experimental work was accomplished to bring LEFM to its present state of mature scientific discipline.

In a highly schematic vision, the essence of the theoretical work consisted in generalizing Griffith’s ideas, which he had worked out only for a particular case, to any situation of geometry and loading, and to link the energy release rate Q (a structural, or global, quantity) to the elastic stress and strain fields. The essence of the experimental work consisted in setting up test methods to measure the crack growth resistance 7Z. In the energetic approach, the last theoretical step was the discovery of the J-integral by Rice (1968a, b). It gave a key that dosed, on very general grounds, the circle relating the energy release rate to the stress and strain fields close to the crack tip for any elastic material, linear or not, and supplied a logical tool to analyze fracture for more general nonlinear behaviors. Today, it is one of the cornerstones of elastoplaslic fracture mechanics, the branch of fracture mechanics dealing with fracture of ductile materials.

The second major achievement in the theoretical foundation of LEFM was due to Irwin (1957), who introduced the concept of the stress intensity factor К as a parameter for the intensity of stresses close to the crack tip and related it to the energy release rate. Irwin’s approach had the enormous advantage that the stress intensity factors arc additive, while Griffith’s energy release rates were not. However, his approach was limited to linear elasticity, while the concept of energy release rate was not.