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Boundary Integral Equation

Let , or a `crack', be a smooth non-self-intersecting curved surface having a smooth edge . Also let be the unit normal to . Our problem is to find a solution of the equation of three dimensional elastodynamics

 in

subject to the boundary condition
 on (1)

regularity condition
 on

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

 (2)

where is the fundamental solution of the equation of elastodynamics in the frequency domain expressed as
 (3)

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

 p.f. (4)

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
 v.p. (5)

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

Next: FM-BIEM Up: Application of a diagonal Previous: Introduction
2001-12-14