The stress intensity factors at a crack for a linear elastic fracture mechanics analysis can be calculated (via 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):
 | (6–1) |
 | (6–2) |
 | (6–3) |
where:
| u, v, w = displacements in a local Cartesian coordinate system as shown in Figure 6.1: Local Coordinates Measured From a 3D Crack Front. |
| r, θ = coordinates in a local cylindrical coordinate system also shown in Figure 6.1: Local Coordinates Measured From a 3D Crack Front. |
| G = shear modulus |
| KI, KII, KIII = stress intensity factors relating to deformation shapes shown in Figure 6.2: The Three Basic Modes of Fracture |
|
| ν = Poisson's ratio |
| 0(r) = terms of order r or higher |
Evaluating Equation 6–1 through Equation 6–3 at θ = ± 180.0° and dropping the higher order terms yields:
 | (6–4) |
 | (6–5) |
 | (6–6) |
The crack width is shown greatly enlarged, for clarity.
For models symmetric about the crack plane (half-crack model, Figure 6.3: Nodes Used for the Approximate Crack-Tip Displacements(a)), Equation 6–4 to Equation 6–6 can be reorganized to give:
 | (6–7) |
 | (6–8) |
 | (6–9) |
and for the case of no symmetry (full-crack model, Figure 6.3: Nodes Used for the Approximate Crack-Tip Displacements(b)),
 | (6–10) |
 | (6–11) |
 | (6–12) |
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 must be evaluated based on the nodal displacements and
locations. As shown in Figure 6.3: 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
 | (6–13) |
at points J and K. Next, let r approach 0.0:
 | (6–14) |
Thus, Equation 6–7 becomes:
 | (6–15) |
Equation 6–8 through Equation 6–12 are also fit in the same manner.