LEFM and Stress Intensity Factor
It was a great achievement of Irwin to reformulate LEFM problem in terms of the stress states in the material close to the crack tip rather than energetically and prove that this, so-called local, approach was essentially equivalent to the Griffith energetic (or global) approach.
The essential fact is that when a body contains a crack, a strong stress concentration develops around the crack tip. If the behavior of the material is isotropic and linear elastic except in a vanishingly small fracture process zone, it happens that this stress concentration has the same distribution close to the crack tip whatever the size, shape, and specific boundary conditions of the body. Only the intensity of the stress concentration varies. For the same intensity, the stresses around and close to the crack tip are identical.
To prove this assertion and to be able to solve problems for cracked structures of interest in engineering, mathematical tools specially suited to handle problems of elastic bodies with cracks were developed in the theory of elasticity. However, a user of LEFM (even a proficient one) does not need to make use of the sophisticated mathematical formalisms required to prove the most general properties of the elastic fields in cracked bodies. Therefore, in this section we present the most important results regarding linear elastic bodies with cracks. Chapter 4 gives the mathematical treatment and derivation of these results, intended only for those readers who wish to understand the sources of LEFM in greater depth.
We also restrict the analysis to the so-called mode /, by far the most often encountered mode in engineering practice. This is the mode where the crack lies in a plane of geometrical and loading symmetry of the structure and, therefore, no shear stresses act on the crack plane. The shear loading modes (II and III) and the fracture criteria associated with them will be analyzed in Chapter 4.