subject to the boundary condition

regularity condition

and the radiation condition for , where , , , and stand for the displacement, elasticity tensor, traction vector, incident wave and the crack opening displacement, respectively. Also, the superscript () indicates the limit on from the positive (negative) side of where the positive side indicates the one into which the unit normal vector points.

The solution
to this problem has an integral representation
given by

where is the fundamental solution of the equation of elastodynamics in the frequency domain expressed as

and and are the wave numbers of the transverse and longitudinal waves.

Using Eqs. (2) and (3), one obtains a
hypersingular integral equation given by

where p.f. stands for the finite part of a divergent integral and is the traction vector associated with . The hypersingular integral equation in Eq. (5) can be regularised as

where v.p. stands for the Cauchy principal value of a singular integral. We use Eq. (6) for the direct computation in FM-BIEM.