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Formulation for integral equation

Let $S \subset R^3$, or a `crack', be a union of smooth non-self-intersecting curved surfaces having smooth edges $\partial
S$. Also let $\mbox{\boldmath$\space n $ }$ be the unit normal vector to S. Our problem is to find a solution $\mbox{\boldmath$\space u $ }$ of the equation of elastostatics

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


subject to the boundary condition

 \begin{displaymath}t_i^{\pm} := C_{ijkl} u_{k,l}^{\pm}n_j = 0 \quad \mbox{on} \ S
\end{displaymath} (1)

regularity
 
$\displaystyle \mbox{\boldmath$ \phi $ }(\mbox{\boldmath$ x $ })$ := $\displaystyle \mbox{\boldmath$ u $ }^{+}(\mbox{\boldmath$ x $ }) - \mbox{\boldmath$ u $ }^{-}(\mbox{\boldmath$ x $ }) = 0 \quad
\mbox{on} \ \partial S$ (2)

and an asymptotic condition given by

\begin{eqnarray*}\mbox{\boldmath$ u $ }(\mbox{\boldmath$ x $ }) &\to& \mbox{\bol...
...ad \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. Also, the superscript + (-) indicates the limit on S from the positive (negative) side of S where the positive side indicates the one into which the unit normal vector $\mbox{\boldmath$\space n $ }$ points. 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

 
$\displaystyle u_{i}(\mbox{\boldmath$ x $ })$ = $\displaystyle u_{i}^{\infty}(\mbox{\boldmath$ x $ })$  
  + $\displaystyle \int_{S}
C_{cd\/jl} \frac{\partial}{\partial y_{l}} \Gamma_{ij}(\...
...ath$ y $ }) n_{c}(\mbox{\boldmath$ y $ })
\phi_d(\mbox{\boldmath$ y $ }) dS_{y}$  
    $\displaystyle \qquad \qquad \mbox{\boldmath$ x $ }\in R^3\setminus \overline{S}$ (3)

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 $ })=}$
    $\displaystyle \frac{1}{8\pi\mu}\left(\delta_{ij}\frac{\partial}{\partial x_l}
\...
...l}{\partial x_j}\right)
\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert$ (4)

Using (1) and (3), one obtains the following hyper-singular integral equation:
 
$\displaystyle {t^{\infty}_{a}(\mbox{\boldmath$ x $ })=}$
    $\displaystyle - \mbox{p.f.}\int_{S}
n_{b}(\mbox{\boldmath$ x $ }) C_{abik} \fra...
...rtial}{\partial y_l} \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })$  
    $\displaystyle \qquad \qquad
C_{cd\/jl} \/ n_c(\mbox{\boldmath$ y $ }) \phi_d(\mbox{\boldmath$ y $ }) dS_{y},\quad \mbox{\boldmath$ x $ }\in S$ (5)

where $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })$ and p.f. indicate the traction associated with $\mbox{\boldmath$\space u $ }^{\infty}(\mbox{\boldmath$\space x $ })$ and the finite part of a divergent integral. Eq.(5) can also be written as
 
$\displaystyle { t_{a}^{\infty}(\mbox{\boldmath$ x $ })=}$
    $\displaystyle \mbox{v.p.}\int_{S} n_{b}(\mbox{\boldmath$ x $ }) C_{abik} e_{rck...
...rtial}{\partial y_l}
\Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })$  
    $\displaystyle \qquad \qquad \qquad e_{rqs}\frac{\partial\phi_{d}(\mbox{\boldmath$ y $ })}{\partial y_q}
n_{s}(\mbox{\boldmath$ y $ }) dS_{y}$ (6)

where v.p. indicates Cauchy's principal value. In this paper we use (6) for the direct computation of (5).


next up previous
Next: Original FMM Up: Application of New Fast Previous: Introduction
Ken-ichi Yoshida
2001-03-26