Next: Original FMM
Up: Application of New Fast
Previous: Introduction
Let
,
or a `crack', be a union of smooth
nonselfintersecting curved surfaces having smooth edges
.
Also let
be the unit normal vector to S. Our
problem is to find a solution
of the equation of
elastostatics
subject to the boundary condition

(1) 
regularity

:= 

(2) 
and an asymptotic condition given by
where
,
C_{ijkl},
,
and
stand for the displacement, elasticity tensor,
traction vector, a solution of the equation of elastostatics in the
whole space and the crack opening displacement, respectively.
Also, the superscript + () indicates the limit on S from the
positive (negative) side of S where the positive side indicates the
one into which the unit normal vector
points. The components of
C_{ijkl} are expressed with Lame's constants
and
Kronecker's delta
as
The solution
to this problem has an integral
representation given by
where
is the fundamental solution of the equation of
elastostatics expressed as




(4) 
Using (1) and (3),
one obtains the following hypersingular integral equation:
where
and p.f. indicate the traction associated
with
and the finite part of a divergent integral.
Eq.(5) can also be written as
where v.p. indicates Cauchy's principal value.
In this paper we use (6) for the direct computation of
(5).
Next: Original FMM
Up: Application of New Fast
Previous: Introduction
Kenichi Yoshida
20010326