The model referred to above is called the linear elastic fracture mechanics model and has found wide acceptance as a method for determining the resistance of a material to below-yield strength fractures.  The model is based on the use of linear elastic stress analysis; therefore, in using the model one implicitly assumes that at the initiation of fracture any localized plastic deformation is small and considered within the surrounding elastic stress field.  Application of linear elastic stress analysis tools to cracks of the type shown in Figure 2.2.2 shows that the local stress field (within r < a/10) is given by [Irwin, 1957; Williams, 1957; Sneddon & Lowengrub, 1969; Rice, 1968a]:

The stress in the third direction are given by s_z = s_xz = s_yz = 0 for the plane stress problem, and when the third directional strains are zero (plane strain problem), the out of plane stresses become s_xz = s_yz = 0 and s_z = n (s_x + s_y).  While the geometry and loading of a component may change, as long as the crack opens in a direction normal to the crack path, the crack tip stresses are found to be as given by Equations 2.2.1.  Thus, the Equations 2.2.1 only represent the crack tip stress field for the Mode 1 crack extension described by Figure 2.2.2.

Figure 2.2.2.  Infinite Plate with a Flaw that Extends Through Thickness

Three variables appear in the stress field equation:  the crack tip polar coordinates r and q and the parameter K.  The functions of the coordinates determine how the stresses vary with distance from the right hand crack tip (point B) and with angular displacement from the x-axis.  As the stress element is moved closer to the crack tip, the stresses are seen to become infinite.  Mathematically speaking, the stresses are said to have a square root singularity in r.  Because most cracks have the same geometrical shape at their tip, the square root singularity in r is a general feature of most crack problem solutions.

The parameter K, which occurs in all three stresses, is called the stress intensity factor because its magnitude determines the intensity or magnitude of the stresses in the crack tip region.  The influence of external variables, i.e. magnitude and method of loading and the geometry of the cracked body, is sensed in the crack tip region only through the stress intensity factor.  Because the dependence of the stresses (Equation 2.2.1) on the coordinate variables remain the same for different types of cracks and shaped bodies, the stress intensity factor is a single parameter characterization of the crack tip stress field.

The stress intensity factors for each geometry can be described using the general form:

where the factor b is used to relate gross geometrical features to the stress intensity factors.  Note that b can be a function of crack length (a) as well as other geometrical features

It is seen from Equation 2.2.2 that the intensity of the stress field and hence the stresses in the crack tip region are linearly proportional to the remotely applied stress and proportional to the square root of the half crack length.

A structural analyst should be able to determine analytically, numerically, or experimentally the stress-intensity factor relationship for almost any conceivable cracked body geometry and loading.  The analysis for stress-intensity factors, however, is not always straightforward and information for determining this important structural property will be presented subsequently in Section 11.  A mini-handbook of stress-intensity factors and some methods for approximating stress-intensity factors are also presented in Section 11.