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Series expansion of the fundamental solution of three-dimensional elastostatics

We introduce a new coordinate system which has following base vectors:

\begin{eqnarray*}\mbox{\boldmath$ v $ } = \mbox{\boldmath$ i $ }_1 + i \ \mbox{\...
... }_2
,\quad \mbox{\boldmath$ i $ }_z = \mbox{\boldmath$ i $ }_3,
\end{eqnarray*}


where $\mbox{\boldmath$\space i $ }_1,\mbox{\boldmath$\space i $ }_2,\mbox{\boldmath$\space i $ }_3$ are base vectors in the Cartesian coordinates. In the new system, we have

\begin{eqnarray*}\mbox{\boldmath$ x $ }= x_n \mbox{\boldmath$ i $ }_n = \frac{\o...
...mbox{\boldmath$ \overline{v} $ } + z_0 \mbox{\boldmath$ i $ }_z,
\end{eqnarray*}


where $\xi = x_1 + i x_2, \ z=x_3, \ \xi_0 = y_1 + i y_2, \ z_0=y_3$. The derivatives with respect to these variables satisfy

\begin{eqnarray*}\frac{\partial}{\partial \xi} = \frac{1}{2}\left(\frac{\partial...
...uad \frac{\partial}{\partial z} = \frac{\partial}{\partial x_3},
\end{eqnarray*}


with which $\mbox{\boldmath$\space \nabla $ }_x$ is written as
 
$\displaystyle \mbox{\boldmath$ \nabla $ }_x=\mbox{\boldmath$ i $ }_n \frac{\par...
...partial \overline{\xi}} + \mbox{\boldmath$ i $ }_z \frac{\partial}{\partial z}.$     (G.1)

The functions $T_{n,m}(\overrightarrow{O\mbox{\boldmath$\space x $ }})$ and $S_{n,m}(\overrightarrow{O\mbox{\boldmath$\space x $ }})$ (See (F.9)) satisfy relations given by

\begin{eqnarray*}T_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}) &=&(2n-1) \xi...
...e{\xi}+ z^2 ) S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}),
\end{eqnarray*}


and the functions $U_{n,m}(\overrightarrow{O\mbox{\boldmath$\space y $ }})$ and $R_{n,m}(\overrightarrow{O\mbox{\boldmath$\space y $ }})$ (See (F.9)) satisfy relations given by

\begin{eqnarray*}U_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}) &=& -(2n+3) \...
...m+2)(n-m+2) R_{n+2,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}).
\end{eqnarray*}


Using above relations, (D.1), (D.2), (D.3), (D.7), (D.8) and (D.11), we differentiate (F.8) using (G.1) to obtain the derivatives of $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ as
 
$\displaystyle \mbox{\boldmath$ \nabla $ }_x \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert$ = $\displaystyle \sum_{n=0}^{\infty} \sum_{m=-n}^{n} \left(\mbox{\boldmath$ v $ } ...
...row{O\mbox{\boldmath$ x $ }}) R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})$  
  = $\displaystyle \sum_{n=0}^{\infty} \sum_{m=-n}^{n} (\mbox{\boldmath$ x $ }-\mbox...
...row{O\mbox{\boldmath$ x $ }})R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}).$ (G.2)

Another way to obtain this result is to use an expansion of $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert^{-1}$ given by[88]
 
$\displaystyle \frac{1}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}...
...ox{\boldmath$ y $ }}\vert < \vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert.$     (G.3)

With (G.3) we can derive (G.2) as follows:

\begin{eqnarray*}\mbox{\boldmath$ \nabla $ }_x \vert\mbox{\boldmath$ x $ }-\mbox...
...dmath$ x $ }})R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}).
\end{eqnarray*}


From (G.2) we obtain the second order derivatives of $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ as
 
$\displaystyle {\mbox{\boldmath$ \nabla $ }_x \otimes \mbox{\boldmath$ \nabla $ }_x \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert} $
  = $\displaystyle \sum_{n=0}^{\infty} \sum_{m=-n}^{n} - \mbox{\boldmath$ \nabla $ }...
..._0 \mbox{\boldmath$ i $ }_z ) R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})$  
    $\displaystyle \qquad \qquad \quad + \mbox{\boldmath$ \nabla $ }_x \overline{S_{...
... z \mbox{\boldmath$ i $ }_z ) R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})$  
    $\displaystyle \qquad \qquad \quad + \overline{S_{n,m}}(\overrightarrow{O\mbox{\...
...oldmath$ v $ }}{2} + \mbox{\boldmath$ i $ }_z \otimes \mbox{\boldmath$ i $ }_z)$  
  = $\displaystyle \sum_{n=0}^{\infty} \sum_{m=-n}^{n} - \mbox{\boldmath$ \nabla $ }...
...{\boldmath$ x $ }})R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}) \ {\bf 1},$ (G.4)

where ${\bf 1}$ is the second order unit tensor. The fundamental solution of the equation of elastostatics is 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$  
  = $\displaystyle \frac{1}{8\pi\mu}\left(
\delta_{ij}\frac{2}{\vert\mbox{\boldmath$...
...}{\partial x_j}
\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert\right).$ (G.5)

Using (G.3), (G.4) and (G.5), we obtain a series expansion of $\Gamma_{ij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ given by
 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })= \frac...
...ox{\boldmath$ y $ }}\vert < \vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert,$     (G.6)

where FSij,n,m and GSi,n,m are functions defined as

\begin{eqnarray*}F^S_{ij,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})= \frac{\...
...\partial x_i}S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}) ,
\end{eqnarray*}



\begin{eqnarray*}G^S_{i,n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }})=
\frac{\...
...partial x_i}
S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}).
\end{eqnarray*}


Also, we can derive (G.6) in another way. We first rewrite then fundamental solution (G.5) as

 
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = \fra...
...th$ y $ }})_j}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right).$     (G.7)

Substituting (G.3) into (G.7), we obtain
$\displaystyle \Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })$ = $\displaystyle \frac{1}{8\pi\mu}\left(
\delta_{ij}\frac{2}{\vert\mbox{\boldmath$...
...ath$ y $ }})_j}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right)$  
  = $\displaystyle \frac{1}{8\pi\mu}\left(
\delta_{ij}\frac{2}{\vert\mbox{\boldmath$...
...ath$ y $ }})_j}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right)$  
  = $\displaystyle \frac{1}{8\pi\mu}\left(
\frac{\lambda+3\mu}{\lambda+2\mu}
\frac{\...
...ath$ y $ }})_j}{\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}\right)$  
  = $\displaystyle \frac{1}{8\pi\mu}\sum_{n=0}^{\infty} \sum_{m=-n}^{n}\left(
\frac{...
...box{\boldmath$ x $ }})
R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})\right.$  
    $\displaystyle \left. - \frac{\lambda+\mu}{\lambda+2\mu}(\overrightarrow{O\mbox{...
...x{\boldmath$ y $ }})_j R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})\right)$  
  = $\displaystyle \frac{1}{8\pi\mu}\sum_{n=0}^{\infty}
\sum_{m=-n}^{n}\left(\overli...
...boldmath$ y $ }})_{j}R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})
\right).$  


next up previous contents
Next: M2M, M2L, L2L in Up: Applications of Fast Multipole Previous: Series expansion of |x-y|
Ken-ichi Yoshida
2001-07-28