next up previous contents
Next: FM-BIEM with hypersingluar integral Up: Crack problems for three-dimensional Previous: Crack problems for three-dimensional

Integral equation

Our problem is to find a solution $\mbox{\boldmath$\space u $ }$ of the equation of elastostatics

\begin{eqnarray*}C_{ijkl} u_{k,lj}(\mbox{\boldmath$ x $ }) &=& 0 \quad \mbox{in } R^3 \setminus
\overline{S},
\end{eqnarray*}


subject to the boundary condition

 \begin{displaymath}t_i^{\pm}(\mbox{\boldmath$ x $ }) := C_{ijkl} u_{k,l}^{\pm}(\...
... x $ })n_j(\mbox{\boldmath$ x $ }) = 0
\quad \mbox{on} \ S,
\end{displaymath} (3.42)

regularity condition
 
$\displaystyle \mbox{\boldmath$ \phi $ }(\mbox{\boldmath$ x $ })$ := $\displaystyle \mbox{\boldmath$ u $ }^{+}(\mbox{\boldmath$ x $ }) - \mbox{\boldmath$ u $ }^{-}(\mbox{\boldmath$ x $ }) = 0 \quad
\mbox{on} \ \partial S,$ (3.43)

and an asymptotic condition

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


where $\mbox{\boldmath$\space u $ }$, Cijkl, $\mbox{\boldmath$\space t $ }$, $\mbox{\boldmath$\space u $ }^{\infty}$ and $\mbox{\boldmath$\space \phi $ }$ 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. The components of Cijkl are expressed with Lame's constants $\lambda,\mu$ and Kronecker's delta $\delta_{ij}$ as

\begin{eqnarray*}C_{ijkl}=\lambda \delta_{ij}\delta_{kl} + \mu (\delta_{ik}\delta_{jl}
+ \delta_{il}\delta_{jk}).
\end{eqnarray*}


The solution $\mbox{\boldmath$\space u $ }$ to this problem has an integral representation given by (See (3.7))

 
$\displaystyle u_{i}(\mbox{\boldmath$ x $ })$ = $\displaystyle u_{i}^{\infty}(\mbox{\boldmath$ x $ })
+ \int_{S} C_{cd\/jl} \fra...
...ath$ y $ }) dS_{y}, \qquad \mbox{\boldmath$ x $ }\in R^3\setminus \overline{S},$ (3.44)

where $\Gamma_{ij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is the fundamental solution of the equation of elastostatics expressed as
 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })=
\frac...
...}{\partial x_j}\right) \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert.$     (3.45)

Using (3.42) and (3.44), one obtains the following hypersingular integral equation given by (See (3.9))

 
$\displaystyle t^{\infty}_{a}(\mbox{\boldmath$ x $ })= - \pfint_{S}
n_{b}(\mbox{...
... $ }) \phi_d(\mbox{\boldmath$ y $ }) dS_{y}, \quad \mbox{\boldmath$ x $ }\in S,$     (3.46)

where $\pfint_{}$  stands for the finite part of a divergent integral and $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })$ indicates the traction associated with $\mbox{\boldmath$\space u $ }^{\infty}(\mbox{\boldmath$\space x $ })$. The hypersingular integral equation (3.46) can be regularised as (See Appendix L.2)
 
$\displaystyle t_{a}^{\infty}(\mbox{\boldmath$ x $ })=
\vpint_{S} n_{b}(\mbox{\b...
...d}(\mbox{\boldmath$ y $ })}{\partial y_q}
n_{s}(\mbox{\boldmath$ y $ }) dS_{y},$     (3.47)

where $\vpint_{} $  stands for the Cauchy principal value of a singular integral. We use (3.47) for a direct computation of (3.46). 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 hypersingluar integral Up: Crack problems for three-dimensional Previous: Crack problems for three-dimensional
Ken-ichi Yoshida
2001-07-28