The stress intensity factors at a crack for a linear elastic fracture mechanics analysis may be computed (using the KCALC command). The analysis uses a fit of the nodal displacements in the vicinity of the crack. The actual displacements at and near a crack for linear elastic materials are (Paris and Sih([106])):

(17–98)

(17–99)

(17–100)

where:

u, v, w = displacements in a local Cartesian coordinate system as shown in Figure 17.11: Local Coordinates Measured From a 3-D Crack Front.

r, θ = coordinates in a local cylindrical coordinate system also shown in Figure 17.11: Local Coordinates Measured From a 3-D Crack Front.

G = shear modulus

KI, KII, KIII = stress intensity factors relating to deformation shapes shown in Figure 17.12: The Three Basic Modes of Fracture

ν = Poisson's ratio

0(r) = terms of order r or higher

Evaluating Equation 17–98 through Equation 17–100 at θ = ± 180.0° and dropping the higher order terms yields:

(17–101)

(17–102)

(17–103)

Figure 17.11:  Local Coordinates Measured From a 3-D Crack Front

The crack width is shown greatly enlarged, for clarity.

Figure 17.12:  The Three Basic Modes of Fracture

For models symmetric about the crack plane (half-crack model, Figure 17.13: Nodes Used for the Approximate Crack-Tip Displacements(a)), Equation 17–101 to Equation 17–103 can be reorganized to give:

(17–104)

(17–105)

(17–106)

and for the case of no symmetry (full-crack model, Figure 17.13: Nodes Used for the Approximate Crack-Tip Displacements(b)),

(17–107)

(17–108)

(17–109)

where Δv, Δu, and Δw are the motions of one crack face with respect to the other.

As the above six equations are similar, consider only the first one further. The final factor is , which needs to be evaluated based on the nodal displacements and locations. As shown in Figure 17.13: Nodes Used for the Approximate Crack-Tip Displacements(a), three points are available. v is normalized so that v at node I is zero. Then A and B are determined so that

(17–110)

at points J and K. Next, let r approach 0.0:

(17–111)

Figure 17.13:  Nodes Used for the Approximate Crack-Tip Displacements

(a) Half Model, (b) Full Model

Thus, Equation 17–104 becomes:

(17–112)

Equation 17–105 through Equation 17–109 are also fit in the same manner.