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## Integral equation for crack problems in three-dimensional 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 R3 and assume that D is bounded by two surfaces S1 and S2 (See Fig.3.2). We first consider an elastic state ( ), where ui is a displacement vector and is a stress tensor associated with ui. 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 left-hand side of (3.3) are called the single-layer potential and the double-layer potential, respectively. Also, and are called the single-layer kernel (or the fundamental solution) and the double-layer kernel, respectively.

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 S1 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

Next, letting S2 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 ti and are traction vectors associated with ui 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
Ken-ichi Yoshida
2001-07-28