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Integral Equation

Our problem is to find a solution u of Laplace's equation

 
$\displaystyle \Delta u(\mbox{\boldmath$ x $ }) = 0 \quad \mbox{in } R^3 \setminus \overline{S},$     (3.11)

subject to the boundary condition

 \begin{displaymath}\frac{\partial u^{\pm}(\mbox{\boldmath$ x $ })}{\partial n}= 0 \quad \mbox{on} \ S,
\end{displaymath} (3.12)

regularity condition
 
$\displaystyle \phi(\mbox{\boldmath$ x $ }) := u^{+}(\mbox{\boldmath$ x $ }) - u^{-} (\mbox{\boldmath$ x $ })= 0 \quad
\mbox{on} \ \partial S,$     (3.13)

and an asymptotic condition

\begin{eqnarray*}u(\mbox{\boldmath$ x $ }) \to u^{\infty}(\mbox{\boldmath$ x $ }...
...\mbox{as} \quad
\vert\mbox{\boldmath$ x $ }\vert \to \infty ,
\end{eqnarray*}


where $u^{\infty}$ and $\phi$ stand for a solution of Laplace's equation in the whole space and the crack opening displacement, respectively. Physically, the function $u^{\infty}$ can be viewed as the ``no crack solution'' of the problem.

The solution u to this problem has an integral representation given by

 
$\displaystyle u(\mbox{\boldmath$ x $ })= u^{\infty}(\mbox{\boldmath$ x $ }) + \...
...dmath$ y $ }) dS_{y},\quad \mbox{\boldmath$ x $ }\in R^3\setminus \overline{S},$     (3.14)

where $G(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is the fundamental solution of Laplace's equation expressed as

\begin{eqnarray*}G(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })= \frac{1}{4 \pi \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}.
\end{eqnarray*}


From the boundary condition (3.12) and (3.14), one obtains the following hypersingular integral equation given by
 
$\displaystyle -\frac{\partial u^{\infty}(\mbox{\boldmath$ x $ })}{\partial n_x}...
...oldmath$ y $ }) dS_y,\quad \mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }
\in S,$     (3.15)

where $\pfint_{}$  stands for the finite part of a divergent integral. Also, using Stokes' theorem, one can regularise (3.15) as (See Appendix L.1)

 \begin{displaymath}-\frac{\partial u^{\infty}(\mbox{\boldmath$ x $ })}{\partial ...
...dS_y,\quad \mbox{\boldmath$ x $ },\mbox{\boldmath$ y $ }\in S.
\end{displaymath} (3.16)

where $\vpint_{} $  stands for the Cauchy principal value of a singular integral. We use (3.16) for a direct computation of (3.15) in the conventional BIEM as well as in FM-BIEM. In the following sections we describe two types of FM-BIEM; one is FM-BIEM with the hypersingular integral equation and the other is FM-BIEM with the regularised integral equation.


next up previous contents
Next: FM-BIEM with hypersingular integral Up: Crack problems for three-dimensional Previous: Crack problems for three-dimensional
Ken-ichi Yoshida
2001-07-28