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Integral equation for crack problems in three-dimensional elastostatics

In this section we describe the derivation of the integral equation for elastostatic crack problems in the infinite domain.

We consider a domain D in the infinite space R3 and assume that D is bounded by two surfaces S1 and S2 (See Fig.3.2). We first consider an elastic state ( $u_i,
\sigma_{ij}$), where ui is a displacement vector and $\sigma_{ij}$ is a stress tensor associated with ui. $\sigma_{ij}$ satisfies the equilibrium equation given by

 
$\displaystyle \sigma_{ij,i}(\mbox{\boldmath$ y $ }) = 0.$     (3.1)

We also consider another elastic state ( $\Gamma_{ki}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }),\Sigma_{kij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$), where $\Gamma_{ki}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is a displacement vector at $\mbox{\boldmath$\space y $ }$ in the direction i when a unit concentrated load is applied at $\mbox{\boldmath$\space x $ }$ in the direction k and is called the fundamental solution of elastostatics. $\Sigma_{kij}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$ is the stress tensor associated with $\Gamma_{ki}(\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ })$, which is given by

\begin{eqnarray*}\Sigma_{kij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = C...
...y_m} \Gamma_{kn}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ })
\end{eqnarray*}


and satisfies the equilibrium equation given by
 
$\displaystyle \Sigma_{kij,i}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }) = -\delta_{kj} \delta(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }),$     (3.2)

in D.

Now we introduce Betti's reciprocal identity

 
$\displaystyle { \int_{S_1 + S_2} \sigma_{ij}(\mbox{\boldmath$ y $ }) n_i(\mbox{...
... }) u_j(\mbox{\boldmath$ y $ }) dS \quad
(\mbox{\boldmath$ y $ }\in S_1 + S_2)}$
    $\displaystyle = \int_D \bigl\{\sigma_{ij,i}(\mbox{\boldmath$ y $ }) \Gamma_{kj}...
...}) u_j(\mbox{\boldmath$ y $ })\bigr\} dV,
\quad (\mbox{\boldmath$ y $ }\in D ).$ (3.3)

The first term and the second term in the left-hand side of (3.3) are called the single-layer potential and the double-layer potential, respectively. Also, $\Gamma_{kj}$ and $\Sigma_{kij}n_i$ are called the single-layer kernel (or the fundamental solution) and the double-layer kernel, respectively.
  
Figure 3.2: Domain D and surfaces S1 and S2
\begin{figure}
\begin{center}
\epsfile{file=FIG/domain2.eps,scale=0.55} \end{center} \end{figure}

Substituting (3.1) and (3.2) into (3.3), we obtain the Somigliana formula

 
$\displaystyle \int_{S_1 + S_2} \bigl\{\sigma_{ij}(\mbox{\boldmath$ y $ }) n_i (...
...in D) \\
0, \ \ \ \qquad (\mbox{\boldmath$ x $ }\notin D)
\end{array}\right. .$     (3.4)

Now we consider the case where $\mbox{\boldmath$\space x $ }$ is in D and assume that the boundary S1 is the surface of a crack with $S_1
= S^+ \cup S^- $ and $S^+ \cap S^- = \emptyset$ (See Fig.3.3). In numerical examples in this thesis we assume that a surface of a crack is free from traction and, hence, tractions on S+ and S- are set to be zero. Taking the above conditions into consideration, we rewrite (3.4) as

 
$\displaystyle {u_k(\mbox{\boldmath$ x $ }) =
\int_{S_2} \bigl\{ \sigma_{ij}(\mb...
...ath$ y $ }) n_i(\mbox{\boldmath$ y $ }) u_j(\mbox{\boldmath$ y $ }) \bigr\}dS }$
    $\displaystyle - \int_{S^+} \Sigma_{kij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$...
...h$ y $ }) u^-_j(\mbox{\boldmath$ y $ }) dS
, \quad \mbox{\boldmath$ x $ }\in D,$ (3.5)

where $\mbox{\boldmath$\space u $ }^+ (\mbox{\boldmath$\space u $ }^-)$ and $\mbox{\boldmath$\space n $ }^+ (\mbox{\boldmath$\space n $ }^-)$ are the displacement vector and the unit normal vector on S+ (S-). We set S+=S-=S and $\mbox{\boldmath$\space n $ }^-=-\mbox{\boldmath$\space n $ }^+=\mbox{\boldmath$\space n $ }$ in (3.5), because the crack is considered to be a slit whose thickness is zero, to obtain the following integral representation for $\mbox{\boldmath$\space u $ }$ in the finite domain which contains a crack:
 
$\displaystyle {u_k(\mbox{\boldmath$ x $ }) =
\int_{S_2} \bigl\{ \sigma_{ij}(\mb...
...ath$ y $ }) n_i(\mbox{\boldmath$ y $ }) u_j(\mbox{\boldmath$ y $ }) \bigr\}dS }$      
$\displaystyle + \int_{S} \Sigma_{kij}(\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y...
...$ y $ })
\phi_j(\mbox{\boldmath$ y $ }) dS , \quad \mbox{\boldmath$ x $ }\in D,$     (3.6)

where $\mbox{\boldmath$\space \phi $ }$ is the crack opening displacement vector defined by

\begin{eqnarray*}\mbox{\boldmath$ \phi $ }(\mbox{\boldmath$ y $ }) := \mbox{\bol...
...dmath$ y $ })-\mbox{\boldmath$ u $ }^-(\mbox{\boldmath$ y $ }).
\end{eqnarray*}



  
Figure 3.3: Domain D and surfaces S2, S+ and S-
\begin{figure}
\begin{center}
\epsfile{file=FIG/domain.eps,scale=0.55} \end{center} \end{figure}

Next, letting S2 tend to infinity in (3.6), we have the following integral representation
 
$\displaystyle u_k(\mbox{\boldmath$ x $ }) = u^{\infty}_k(\mbox{\boldmath$ x $ }...
...$ y $ })
\phi_j(\mbox{\boldmath$ y $ }) dS , \quad \mbox{\boldmath$ x $ }\in D,$     (3.7)

where $\mbox{\boldmath$\space u $ }^{\infty}$ is a solution of the equation of elastostatics in the whole space. Physically, we can interpret $\mbox{\boldmath$\space u $ }^{\infty}$ as the ``no crack solution''. Applying the traction operator $T_{ik}(\mbox{\boldmath$\space x $ })$ defined as

\begin{eqnarray*}T_{ik}(\mbox{\boldmath$ x $ }) := C_{ijkl} n_j(\mbox{\boldmath$ x $ }) \frac{\partial}{\partialx_l},
\end{eqnarray*}


on both sides of (3.7), we obtain
 
$\displaystyle t_i(\mbox{\boldmath$ x $ }) = t^{\infty}_i(\mbox{\boldmath$ x $ }...
...$ y $ })
\phi_j(\mbox{\boldmath$ y $ }) dS , \quad \mbox{\boldmath$ x $ }\in D,$     (3.8)

where ti and $t_i^{\infty}$ are traction vectors associated with ui and $u_i^{\infty}$, respectively. Finally, letting $\mbox{\boldmath$\space x $ }$ approach S in (3.8) and noting that S is free from traction, we have the following hypersingular integral equation
 
$\displaystyle - t^{\infty}_i(\mbox{\boldmath$ x $ })=
\pfint_{S} T_{ik}(\mbox{\...
...$ y $ })
\phi_j(\mbox{\boldmath$ y $ }) dS , \quad \mbox{\boldmath$ x $ }\in S,$     (3.9)

where $\pfint_{}$   stands for the finite part of a divergent integral. Hypersingular integral equations of this type are the basic equations for crack problems considered in this thesis.
next up previous contents
Next: A note on integral Up: Crack Previous: Notation
Ken-ichi Yoshida
2001-07-28