Integral equation

Our problem is to find a solution $\mbox{\boldmath$\space u $ }$ of the equation of elastostatics

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

subject to the boundary condition

$$t_i^{\pm}(\mbox{\boldmath$ x $ }) := C_{ijkl} u_{k,l}^{\pm}(\... ... x $ })n_j(\mbox{\boldmath$ x $ }) = 0 \quad \mbox{on} \ S,$$ (3.42)

regularity condition

 

$$\mbox{\boldmath$ \phi $ }(\mbox{\boldmath$ x $ }) \mbox{\boldmath$ u $ }^{+}(\mbox{\boldmath$ x $ }) - \mbox{\boldmath$ u $ }^{-}(\mbox{\boldmath$ x $ }) = 0 \quad \mbox{on} \ \partial S,$$ := (3.43)

and an asymptotic condition

$$\begin{eqnarray*}\mbox{\boldmath$ u $ }(\mbox{\boldmath$ x $ }) &\to& \mbox{\bol... ... \mbox{as} \quad \vert\mbox{\boldmath$ x $ }\vert\to \infty, \end{eqnarray*}$$

where $\mbox{\boldmath$\space u $ }$, C_ijkl, $\mbox{\boldmath$\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. The components of C_ijkl 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 (See (3.7))

 

$$u_{i}(\mbox{\boldmath$ x $ }) u_{i}^{\infty}(\mbox{\boldmath$ x $ }) + \int_{S} C_{cd\/jl} \fra... ...ath$ y $ }) dS_{y}, \qquad \mbox{\boldmath$ x $ }\in R^3\setminus \overline{S},$$ = (3.44)

where $\Gamma_{ij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is the fundamental solution of the equation of elastostatics expressed as

 

$$\Gamma_{ij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })= \frac... ...}{\partial x_j}\right) \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert.$$ (3.45)

Using (3.42) and (3.44), one obtains the following hypersingular integral equation given by (See (3.9))

 

$$t^{\infty}_{a}(\mbox{\boldmath$ x $ })= - \pfint_{S} n_{b}(\mbox{... ... $ }) \phi_d(\mbox{\boldmath$ y $ }) dS_{y}, \quad \mbox{\boldmath$ x $ }\in S,$$ (3.46)

where $\pfint_{}$  stands for the finite part of a divergent integral and $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })$ indicates the traction associated with $\mbox{\boldmath$\space u $ }^{\infty}(\mbox{\boldmath$\space x $ })$. The hypersingular integral equation (3.46) can be regularised as (See Appendix L.2)

 

$$t_{a}^{\infty}(\mbox{\boldmath$ x $ })= \vpint_{S} n_{b}(\mbox{\b... ...d}(\mbox{\boldmath$ y $ })}{\partial y_q} n_{s}(\mbox{\boldmath$ y $ }) dS_{y},$$ (3.47)

where $\vpint_{}$  stands for the Cauchy principal value of a singular integral. We use (3.47) for a direct computation of (3.46). In the following sections we describe two types of FM-BIEM; one is FM-BIEM with the hypersingular integral equation and the other is FM-BIEM with the regularised integral equation.