Fracture Mechanics

The fatigue life of a component is made up of initiation and propagation stages. This is illustrated schematically in Fig. 1

The size of the crack at the transition from initiation to propagation is usually unknown and often depends on the point of view of the analyst and the size of the component being analyzed. For example, for a researcher equipped with microscopic equipment it may be on the order of a crystal imperfection, dislocation, or a 0,1 mm-crack, while to the inspector in the field it may be the smallest crack that is readily detectable with nondestructive inspection equipment.

Nevertheless, the distinction between the initiation life and propagation life is important. At low strain amplitudes up to 90% of the life may be taken up with initiation, while at high amplitudes the majority of the fatigue life may be spent propagating a crack. Fracture mechanics approaches are used to estimate the propagation life.

Fracture mechanics approaches require that an initial crack size be known or assumed. For components with imperfections or defects (such as welding porosities, inclusions and casting defects, etc.) an initial crack size may be known. Alternatively, for an estimate of the total fatigue life of a defect-free material, fracture mechanics approaches can be used to determine propagation. Strain-life approaches may then be used to determine initiation life, with the total life being the sum of these two estimates.

Linear Elastic Fracture Mechanics Background

Linear elastic fracture mechanics (LEFM) principles are used to relate the stress magnitude and distribution near the crack tip to:

• Remote stresses applied to the cracked component
• The crack size and shape
• The material properties of the cracked component

Historical Overview

In the 1920s, Griffith formulated the concept that a crack in a component will propagate if the total energy of the system is lowered with crack propagation. That is, if the change in elastic strain energy due to crack extension is larger than the energy required to create new crack surfaces, crack propagation will occur.

Griffith`s theory was developed for brittle materials. In the 1940s, Irwin extended the theory for ductile materials.
He postulated that the energy due to plastic deformation must be added to the surface energy associated with the creation of new crack surfaces. He recognized that for ductile materials, the surface energy term is often negligible compared to the energy associated with plastic deformation. Further, he defined a quantity, G, the strain energy release rate or "crack driving force," which is the total energy absorbed during cracking per unit increase in crack length and per unit thickness.

In the mid-1950s, Irwin made another significant contribution. He showed that the local stresses near the crack tip are of the general form

(1)

where r and q are cylindrical coordinates of a point with respect to the crack tip (see Fig. 2) and K is the stress intensity factor. He further showed that the energy approach (the "G" approach above) is equivalent to the stress intensity approach and that crack propagation occurs when a critical strain energy release rate, G, (or in terms of a critical stress intensity, K_c) is achieved.

LEFM Assumptions

The general form of the LEFM equations is given in Eq. 1. As seen, a singularity exists such that as r, the distance from the crack tip, tends toward zero, the stresses go to infinity. Since materials plastically deform as the yield stress is exceeded, a plastic zone will form near the crack tip. The basis of LEFM remains valid, though, if this region of plasticity remains small in relation to the overall dimensions of the crack and cracked body.

Loading Modes

There are generally three modes of loading, which involve different crack surface displacements (see Fig. 3). The three modes are:

Mode 1:	opening or tensile mode (the crack faces are pulled apart)
Mode 2:	sliding or in-plane shear (the crack surfaces slide over each other)
Mode 3:	tearing or anti-plane shear (the crack surfaces move parallel to the leading edge of the crack and relative to each other)

The following discussion deals with Mode 1 since this is the predominant loading mode in most engineering applications. Similar treatments can readily be extended to Modes 2 and 3.

Stress Intensity Factor

The stress intensity factor, K, which was introduced in Eq. 1, defines the magnitude of the local stresses around the crack tip. This factor depends on loading, crack size, crack shape, and geometric boundaries, with the general form given by

(2)

where:

s = remote stress applied to component (not to be confused with the local stresses, s_ij, in Eq. 1)
a = crack length
f (a/w) = correction factor that depends on specimen and crack geometry

Figure 4 gives the stress intensity relationships for a few of the more common loading conditions. Stress intensity factors for a single loading mode can be added algebraically. Consequently, stress intensity factors for complex loading conditions of the same mode can be determined from the superposition of simpler results, such as those readily obtainable from handbooks.