Westergaard Approach

The Westergaard approach is derived for a crack that propagates in Mode I, therefore _{xy}=0
along

Using this condition and employing (A.29) we can write

| (B.1) |

which can be satisfied only if

| (B.2) |

where

As it is possible to express the stresses and displacements by a single analytic function we write

| (B.3) |

which can be plugged into (A.27) using (A.37). Solving the resulting equation leads to the stresses

| (B.4) |

and the displacements

| (B.5) |

If we define

| (B.6) |

where the hat in _{I}_{I}

| (B.7) |

where

Therefore, the stresses and displacements (B.7) can be derived by choosing the following Airy stress function

| (B.8) |

where the derivatives of the function are defined as

| (B.9) |