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Integral equation

Our problem is to find a solution $\mbox{\boldmath$\space u $ }$ of the equation of three-dimensional elastodynamics

  \begin{eqnarray*}C_{ijkl}u_{k,lj}(\mbox{\boldmath$ x $ })+\rho \omega^2 u_i(\mbo...
...oldmath$ 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} (3.135)

regularity condition
 
$\displaystyle \mbox{\boldmath$ \phi $ }(\mbox{\boldmath$ x $ }) := \mbox{\boldm...
...{\boldmath$ u $ }^{-}(\mbox{\boldmath$ x $ }) = 0 \quad
\mbox{on} \ \partial S,$     (3.136)

and the radiation condition for $\mbox{\boldmath$\space u $ }(\mbox{\boldmath$\space x $ }) - \mbox{\boldmath$\space u $ }^I(\mbox{\boldmath$\space x $ })$ (See Kupradze[51]), where $\mbox{\boldmath$\space u $ }$, Cijkl, $\mbox{\boldmath$\space t $ }$, $\mbox{\boldmath$\space u $ }^I$ and $\mbox{\boldmath$\space \phi $ }$ stand for the displacement, elasticity tensor, traction vector, an incident wave and the crack opening displacement, respectively.

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

 
$\displaystyle u_{i}(\mbox{\boldmath$ x $ })= u_{i}^{I}(\mbox{\boldmath$ x $ })+...
...dmath$ y $ }) dS_y, \quad \mbox{\boldmath$ x $ }\in R^3 \setminus \overline{S},$     (3.137)

where $\Gamma_{ij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is the fundamental solution of the equation of elastodynamics in the frequency domain expressed as
 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) =
\fra...
...\right)\right),
\quad R=\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert$     (3.138)

and kT and kL are the transverse wave number and the longitudinal wave number given by

\begin{eqnarray*}k_T=\sqrt{\frac{\rho}{\mu}}\omega,\quad
k_L=\sqrt{\frac{\rho}{\lambda+2\mu}}\omega.
\end{eqnarray*}


Using (3.136) and (3.138), one obtains the following hypersingular integral equation given by
 
$\displaystyle t^{I}_{a}(\mbox{\boldmath$ x $ })=- n_{b}(\mbox{\boldmath$ x $ })...
... $ })\phi_{j}(\mbox{\boldmath$ y $ }) dS_{y},\quad \mbox{\boldmath$ x $ }\in S,$     (3.139)

where $\pfint_{}$  stands for the finite part of a divergent integral and $\mbox{\boldmath$\space t $ }^{I}$ is a traction vector associated with $\mbox{\boldmath$\space u $ }^I$. The hypersingular integral equation (3.140) can be regularised as (See Appendix L.5)
 
$\displaystyle t_{a}^{I}(\mbox{\boldmath$ x $ })$ = $\displaystyle n_{b}(\mbox{\boldmath$ x $ }) C_{ablm}
\vpint_{S} e_{rkl}C_{jknp}...
... })
e_{riq} \phi_{j,i}(\mbox{\boldmath$ y $ }) n_q(\mbox{\boldmath$ y $ }) dS_y$  
    $\displaystyle - n_{b}(\mbox{\boldmath$ x $ }) C_{ablm} \rho \omega^2 \int_{S} \...
...ldmath$ y $ }) n_l(\mbox{\boldmath$ y $ }) \phi_j(\mbox{\boldmath$ y $ }) dS_y.$ (3.140)

where $\vpint_{} $  stands for the Cauchy principal value of a singular integral. We use (3.141) for a direct computation in FM-BIEM.
next up previous contents
Next: FM-BIEM Up: Three-dimensional scattering of elastic Previous: Three-dimensional scattering of elastic
Ken-ichi Yoshida
2001-07-28