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Integral equation for crack problems in threedimensional
elastostatics
In this section we describe the derivation of the
integral equation for elastostatic crack problems in the infinite
domain.
We consider a domain D in the infinite space R^{3} and assume that
D is bounded by two surfaces S_{1} and S_{2} (See
Fig.3.2). We first consider an elastic state (
), where u_{i} is a displacement vector and
is a stress tensor associated with u_{i}.
satisfies the
equilibrium equation given by



(3.1) 
We also consider another elastic state
(
), where
is
a displacement vector at
in the direction i when a unit
concentrated load is applied at
in the direction k and is
called the fundamental solution of elastostatics.
is the stress tensor associated with
,
which
is given by
and satisfies the equilibrium equation given by



(3.2) 
in D.
Now we introduce Betti's reciprocal identity




(3.3) 
The first term and the second term in the lefthand side of
(3.3) are called the singlelayer potential and the doublelayer
potential, respectively. Also,
and
are
called the singlelayer kernel (or the fundamental solution) and the
doublelayer kernel, respectively.
Figure 3.2:
Domain D and surfaces S_{1} and S_{2}

Substituting (3.1) and (3.2) into (3.3),
we obtain the Somigliana formula



(3.4) 
Now we consider the case where
is in D and
assume that the boundary S_{1} is the surface of a crack with
and
(See
Fig.3.3). In numerical examples in this thesis we assume that a
surface of a crack is free from traction and, hence, tractions on
S^{+} and S^{} are set to be zero. Taking the above conditions into
consideration, we rewrite (3.4) as




(3.5) 
where
and
are the
displacement vector and the unit normal vector on S^{+} (S^{}).
We set S^{+}=S^{}=S and
in (3.5),
because the crack is considered to be a slit whose thickness is zero,
to obtain the following integral representation for
in the
finite domain which contains a crack:







(3.6) 
where
is the crack opening displacement vector defined by
Figure 3.3:
Domain D and surfaces S_{2}, S^{+} and S^{}

Next, letting S_{2} tend to infinity in (3.6), we have
the following integral representation



(3.7) 
where
is a solution of the equation of
elastostatics in the whole space. Physically,
we can interpret
as the ``no crack solution''.
Applying the traction operator
defined as
on both sides of (3.7), we obtain



(3.8) 
where t_{i} and
are traction vectors associated with
u_{i} and
,
respectively. Finally, letting
approach
S in (3.8) and noting that S is free from traction,
we have the following hypersingular integral equation



(3.9) 
where stands for the finite part of a divergent integral.
Hypersingular integral equations of this type are the basic equations
for crack problems considered in this thesis.
Next: A note on integral
Up: Crack
Previous: Notation
Kenichi Yoshida
20010728