The governing equation of elastostatic boundary value problems in
the domain
is given by

(7) |

where is the Navier's operator defined by with elastic constant tensor , are Lame's constants, is Kronecker's delta and stands for the partial differentiation with respect to

where

where denotes the partial differentiation by

According to the formulation of FMBEM for 3D elastostatic crack problems
developed by Nishimura et al.[2],
the first fundamental solution given by Eq.(9) is expanded as

where denotes the complex conjugate. And the functions

where, in Eq.(14) and Eq.(15), denote the polar coordinates of point

Now, by using Eq.(11), the integration
on
involved in Eq.(8) is evaluated as

assuming that

M_{p,N,M}(O) | |||

(17) | |||

(18) |

Eq.(16) is the multipole expansion formula for 3D elastostatic problem, which is entirely similar to one for the 3D elastostatic crack problem formulated by Nishimura et al.[2]. Therefore, some translation formulas are deduced similarly. In this paper, only the results are shown bellow without the proof.

When the center of the multipole moments are translated from
*O* to *O*', the current multipole moments are translated as

(20) |

Evaluating the multipole expansion formula given by Eq.(16),
for a point *x* with respect to its neighboring point *x*_{0},
the following local expansion is used.

where

(23) |

Finally, with an assumption
,
when the center of the local expansion is translated from *x*_{0} to *x*_{1},
the coefficients of the local expansion are translated as

By using some formulas mentioned above, the FMBEM for 3D elastostatic problem can be carried out similarly described in Nishimura et al.[2]. Following remarks should be mentioned about numerical analysis applied in the present paper. 1) the constant boundary elements are employed for both displacements and tractions, 2) non-restart GMRES[3] is used for solving the discretized linear equations, 3) the block diagonal scaling technique proposed by Nishida and Hayami[4] is applied for preconditioning the matrix.

To verify the program based on the above theory,
a sample test problem is solved, i.e. an elastic rectangular block
subjected to the uniaxial tension. In this example,
the truncation of infinite series involved in
Eqs.(21)-(25) is made at 10 terms and
the maximum number of boundary elements included in a cell is 50.
Fig.1 shows the computation time (CPU time[*sec*]) per one iteration process are plotted
to the degrees of freedom(DOF). It reveals that the CPU time is proportional to the degrees of freedom.
This fact implies that the present FMBEM is well done and that
it is confirmed that the solutions obtained in this analysis
are in good agreement with the theoretical ones.