In 1920, Griffith formulated the concept that a crack in an ideally brittle component will propagate if the total energy of the system is lowered with crack propagation. In other words, if the change in elastic strain energy due to crack extension is larger than the energy required to create new crack surfaces, the crack will propagate. The Griffith theory in equation form is:

-dP/da = dWs/da

where a is the crack length, Ws is the work done by external forces per unit thickness, and P is the potential energy supplied by the internal strain energy and external forces. The crack will propagate if -dP/da is greater than or equal to dWs/da.

In 1949, Irwin extended Griffith's theory to ductile materials by including the energy dissipated by local plastic flow. He postulated that the energy due to plastic deformation must be added to the surface energy associated with the creation of new crack surfaces. For ductile materials, the surface energy term is often negligible compared to the energy associated with plastic deformation. In 1956, Irwin defined a new quantity, G, the strain energy release rate (also called the crack extension force or the crack driving force). The strain energy release rate for a linear elastic material is defined as the total energy absorbed during cracking per unit increase in the crack length and per unit thickness. The crack will propagate when a critical strain energy release rate, Gc, is achieved (when G is greater than or equal to the crack resistance energy, Gc). The equation below shows Irwin's modification to the Griffith theory, where U is the strain energy stored in the body per unit thickness.

G = -dP/da = dWs/da - dU/da

In 1957, Irwin used the Westergaard approach to show that the local stresses and displacements near the crack tip had a general closed form solution that could be described by a single constant that was related to the strain energy release rate (see crack singular stress and displacement fields module). This crack tip characterizing parameter later became known as the stress intensity factor, K. The stress intensity factor completely characterizes the crack tip conditions for a linear elastic material. K relates the global applied stress and the local stress near the crack tip. The stress intensity approach is equivalent to the strain energy approach, and crack propagation occurs when a critical stress intensity is achieved. The critical stress intensity is called the fracture toughness, Kc. Kc is a material parameter that depends on the temperature and the thickness of the specimen. Kc is normally determined from an Izod Impact Test. This module can calculate the strain energy release rate for a given material if the stress intensity factor and mode of loading are known. Note that for a thick specimen the plane strain G value should be used and for a thin specimen the plane stress G value should be used (plane strain and plane stress G values are identical for Mode III failure).

The three basic types of loading that a crack can experience are shown in Figure 2.

• Mode I loading is normal to the crack plane, and tends to open the crack. The crack surfaces tend to separate symmetrically with respect to the crack plane.

• Mode II corresponds to in-plane shear loading and tends to slide one crack face with respect to the other (shearing mode). The stress is parallel to the crack growth direction.

• Mode III corresponds to out-of-plane shear, or tearing.

A cracked body can be loaded in any of these modes, or a combination of two or three modes. This module provides solutions for the stress intensity factors for through cracks in flat plates under Mode I loading. For solutions for other modes or other geometries (flawed cylinders or part-through cracks) see the references.

Figure 1: The three basic modes of fracture.

Stress Intensity Factors (SIFs) are used to define the magnitude of the singular stress and displacement fields (local stresses and displacements near the crack tip). The SIF depends on the loading, the crack size, the crack shape, and the geometric boundaries of the specimen. Closed form SIF solutions have been obtained for a wide variety of problems and are published in handbook form (see references). Many of these solutions incorporate numerical methods (including finite element methods) to determine the stress intensity factor, K. Many commercially available finite element computer program include subroutines to calculate K. The SIFs calculated from these above solutions are approximations. However, the error associated with the calculation of K is small compared to the other uncertainties in a fatigue analysis, such as material properties, load levels, load history, and environment.

Using the principle of superposition, SIFs for a single loading mode can be added algebraically. Therefore, SIFs for complex loading conditions of the same mode can be determined from the superposition of simpler closed form solutions, such as those used in this module and readily obtainable from handbooks. For example, the stress intensity factor for a plate under both axial tension and a bending moment can be calculated by summing the SIF for the same plate under pure axial tension and the SIF for the same plate under pure bending.  For example,

KI_total = KI_axial + KI_bending

Many closed form solution for K consist of a crack with a simple shape (ellipse or rectangle) in an infinitely wide plate (the crack dimensions are small compared to the size of the plate). In an infinitely wide plate, the crack tip conditions are not influenced by external boundaries. However, as the crack size increases, or as the plate's width decreases, the outer boundaries begin to exert influence on the crack tip. The closed-form solutions utilized in this module account for the effects of finite width; to calculate K for an infinitely wide plate, simply enter a very large value for the plate width, W. The K values calculated in this module follow one of the general forms:

1) K = b * s * (p * a)^0.5

2) K = (b2 * F) / (t * W^0.5)

Where b and b2 are the stress intensity factor constants, s is the applied stress, F is the applied force, t is the plate thickness, W is the plate width, and p is the mathematical constant ~ 3.14. To convert b2 to a b value (used in fatigue crack growth module), multiply b2 by the following constant:

A / (t *(W * p * a)^0.5)

where A is the cross sectional area of the surface on which the force F acts:

Stress Intensity Factor Near a Notch:

Fracture mechanics approaches may also be used in the fatigue analysis of notched components. For notched components, the near notch stress-strain field dominates the stress intensity solution and must be considered. Important aspects of the analysis include the determination of the size of the region over which this field is effective and the stress intensity factor in this region. The notch produces a stress gradient, and as the distance from the notch increases, the evaluation of the stresses due to the notch decreases. Consequently, at some distance from the notch, the local stress approaches the value of the bulk stress. The crack length corresponding to this distance is defined as the transition crack length, l_t.

Once the crack length is larger than l_t, the effective crack length, a, is assumed to be the sum of the notch width, R, and the actual crack length, l, growing from the notch. This module can be used to calculate K for long cracks (l > l_t) if the effective crack length is used for a and the proper geometry is selected.

a = R + l

When the crack is short (l < l_t), it behaves like a crack growing from the edge of a plate subjected to the nominal stress of Kt*S, where Kt is the theoretical elastic stress concentration factor. For the general case, all short cracks behave as cracks growing from an edge and therefore the short crack SIF calculation is independent of geometry (geometry does effect Kt value). For short cracks set the effective crack length equal to l.

Dowling derived an expression for the transition crack length by analyzing a circular hole in an infinite plate and equating the long crack and short crack SIF solutions. Smith and Miller developed another expression for the transition crack length based on an empirical curve fit to numerical solutions. Dowlimg showed that the SIF solution discussed above for short cracks has limited usefulness for fatigue problems because the linear elastic fracture mechanics assumption of small-scale plasticity (the plastic zone is small compared to the crack length) is usually violated. A plastic zone often develops in the notch root due to stress concentrations associated with the notch. Dowling proposed a method to estimate the total fatigue life of a notched component (good agreement with experimental test results for both sharp and blunt notches). The method combines the strain-life approach and fatigue crack growth. When the crack is smaller than l_t, crack initiation or early crack growth from the notch can be estimated using the strain-life method with a form of Neuber's rule for nominally elastic behavior (for sharp notches this initiation life is negligible, act as cracks). Once the crack is larger than l_t, crack growth can be modeled with a standard fracture mechanics approach with the initial crack size taken to be l_t.

The module will also calculate the residual stress or maximum permissible crack length if K and Beta are known. The fracture toughness can be determined if Beta is known and either the critical stress at a certain crack length or the critical crack length at a given stress are known.

Figure 1: The module input form.

The stress intensity factor analysis problem is defined by first selecting the proper type of solution from the Solution Type pull-down menu. The required inputs fields for the analysis will become white in color so they can be edited. All the text fields that are gray in color are not required for that particular solution type.

The user must be consistent with the units to get correct answers. For example, if the applied stress is entered in kPa, and the crack length and plate width are entered in m, then the calculated K value will be displayed in kPa(m)^0.5.

The module will verify that proper loading and dimensions were entered. If the second biaxial stress is larger than the first, if any of the dimensions are less than or equal to zero, if the modulus of elasticity or Poisson's ratio are less than or equal to zero, or if the crack minor axis is larger than the major axis an error message will be displayed in a separate window.

• All the inputs for the problem (loading and geometry)
• The stress intensity factor constant, Beta or Beta2
• The stress intensity factor

Figure 2. Module tabulated results.