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Boundary Integral Equation

Let $ S \subset \mathbb{R}^3 $, or a `crack', be a smooth non-self-intersecting curved surface having a smooth edge $ \partial
S$. Also let $ \mbox{$\mathbf n $}$ be the unit normal to $ S$. Our problem is to find a solution $ \mbox{$\mathbf u $}$ of the equation of three dimensional elastodynamics

$\displaystyle C_{ijkl}u_{k,lj}(x)+\rho \omega^2 u_i(x) = 0$   in $\displaystyle \mathbb{R}^3 \setminus \overline{S},$    

subject to the boundary condition
$\displaystyle t_i^{\pm}(x) := C_{ijkl} u_{k,l}^{\pm}(x)n_j(x) = 0$   on$\displaystyle  S,$     (1)

regularity condition
$\displaystyle \phi_i(x) := u_i^{+}(x) - u_i^{-}(x) = 0$   on$\displaystyle  \partial S,$      

and the radiation condition for $ \mbox{$\mathbf u $}$$ (x) -$   $ \mbox{$\mathbf u $}$$ ^I(x)$, where $ u_i$, $ C_{ijkl}$, $ t_i$, $ u^I_i$ and $ \phi_i$ stand for the displacement, elasticity tensor, traction vector, incident wave 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{$\mathbf n $}$ points.

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

$\displaystyle u_{i}(x)= u_{i}^{I}(x)+
\int_{S} C_{jknp} \frac{\partial}{\partia...
n_{k}(y) \phi_{j}(y) dS_y, \quad x \in \mathbb{R}^3 \setminus \overline{S},$     (2)

where $ \Gamma_{ij}(x,y)$ is the fundamental solution of the equation of elastodynamics in the frequency domain expressed as
$\displaystyle \Gamma_{ij}(x,y) =
\frac{1}{4\pi\mu}\left( \frac{e^{i k_T r}}{r}\...
... \partial y_j}
\left(\frac{e^{i k_T r}}{r}-\frac{e^{i k_L r}}{r}\right)\right),$     (3)

$ r=\vert$$ \mbox{$\mathbf x $}$$ -$$ \mbox{$\mathbf y $}$$ \vert$ and $ k_T$ and $ k_L$ are the wave numbers of the transverse and longitudinal waves.

Using Eqs. (2) and (3), one obtains a hypersingular integral equation given by

$\displaystyle t^{I}_{a}(x)=- n_{b}(x) C_{ablm}C_{jknp}$   p.f.$\displaystyle \int_{S} \frac{\partial^2}{\partial x_{l}\partial
y_{n}} \Gamma_{mp}(x,y)
n_{k}(y)\phi_{j}(y) dS_{y},\quad x \in S,$     (4)

where p.f. stands for the finite part of a divergent integral and $ \mbox{$\mathbf t $}$$ ^{I}$ is the traction vector associated with $ \mbox{$\mathbf u $}$$ ^{I}$. The hypersingular integral equation in Eq. (5) can be regularised as
$\displaystyle t_{a}^{I}($$\displaystyle \mbox{$\mathbf x $}$$\displaystyle )$ $\displaystyle =$ $\displaystyle n_{b}($$\displaystyle \mbox{$\mathbf x $}$$\displaystyle ) C_{ablm}$   v.p.$\displaystyle \int_{S} e_{rkl}C_{jknp}
\frac{\partial}{\partial y_n}\Gamma_{mp}(x,y)
e_{riq} \phi_{j,i}($$\displaystyle \mbox{$\mathbf y $}$$\displaystyle ) n_q(y) dS_y$  
    $\displaystyle - n_{b}(x) C_{ablm} \rho \omega^2 \int_{S} \Gamma_{mj}
(x,y) n_l(y) \phi_j(y) dS_y.$ (5)

where v.p. stands for the Cauchy principal value of a singular integral. We use Eq. (6) for the direct computation in FM-BIEM.

next up previous
Next: FM-BIEM Up: Application of a diagonal Previous: Introduction