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Next: Three-dimensional scattering of scalar Up: Numerical examples Previous: One penny-shaped crack

Many penny-shaped cracks

We now consider a multi-crack problem. Namely, we compute crack opening displacements and the stresses within the body having many cracks in its interior. Analyses of this type have applications in the fracture mechanics of rock-like materials.

Specifically, we consider an infinite elastic body which contains an array of 8 $\times$ 8 $\times$ 8 (=512) penny-shaped cracks of the same radius a0. The centres of these cracks are located regularly at the interval of 4 a0 in all the coordinate directions. The directions of these cracks are taken random. The whole system is subjected to a uniform remote tension of the magnitude p0. Hence the function $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })$ which appears in (3.100) is written as $\mbox{\boldmath$\space t $ }^{\infty}(\mbox{\boldmath$\space x $ })=\mbox{\bold...
...oldmath$\space x $ })\mbox{\boldmath$\space n $ }(\mbox{\boldmath$\space x $ })$ where

$\displaystyle \mbox{\boldmath$ \sigma $ }^{\infty}(\mbox{\boldmath$ x $ }) =
\l...
...\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & p_0
\end{array}\right) .$     (3.113)

Again, the material constants are chosen as $\lambda=\mu$.

Fig.3.34 shows the mesh for 512 cracks (total DOF=241,152), where each crack is discretised with the 471 DOF mesh shown in Fig.3.29. In Fig.3.35 we display the computed crack opening displacements by superposing the non-dimensional crack opening displacement $\mu\mbox{\boldmath$\space \phi $ }/(a_0 p_0)$ on the non-dimensional position vector $\mbox{\boldmath$\space x $ }/a_0$. The required CPU time for computing the crack opening displacements was 20,854 (sec), and the number of iterations needed for the convergence of GMRES was 10. We next show the results of the stress computation. We take a surface $\Pi$, shown in Fig.3.36, which has no intersection with the cracks. Observation points are taken to be the 200$\times$200 lattice points located on $\Pi$ at an equal interval. Fig.3.37 shows the distribution of the second invariant of the stress deviator defined by

\begin{eqnarray*}J_2 = \frac{1}{2} \sigma'_{ij} \sigma'_{ij}, \quad
\sigma'_{ij} = \sigma_{ij} - \frac{1}{3} \delta_{ij} \sigma_{kk}
\end{eqnarray*}


computed on these observation points. The additional CPU time required for the calculation of stress components at these 40,000 points was 477 (sec) after the computation of the crack opening displacements. Finally, Fig.3.38 shows the crack opening displacements for a case similar to the one in Fig.3.35 except that the radius of each crack is also varied randomly between a0/2 and a0. The CPU time was almost the same as in the constant radius case.
  
Figure 3.29: Mesh and crack opening displacement
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY...
...}\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY1b.eps,scale=1.2}\end{center}\end{figure}


  
Figure 3.30: Crack opening displacement vs distance from the centre of the crack
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY2.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.31: Stress $\sigma_{33}$ vs distance from the crack tip
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY3.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.32: CPU time (sec) vs number of unknowns
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY4.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.33: CPU time (sec) per iteration vs number of unknowns
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY5.eps,scale=1.0}\end{center}\end{figure}


  
Figure 3.34: Mesh for 512 cracks (DOF=241,152)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY6.ps,scale=0.8}\end{center}\end{figure}


  
Figure 3.35: Crack opening displacements of 512 cracks (DOF=241,152)
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY7.ps,scale=0.8}\end{center}\end{figure}


  
Figure 3.36: Surface $\Pi$
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY8.eps,scale=1}\end{center}\end{figure}


  
Figure 3.37: J2 on surface $\Pi$
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY9.eps,scale=1}\end{center}\end{figure}


  
Figure 3.38: Crack opening displacements of 512 cracks with random radii (DOF=241,152).
\begin{figure}
\begin{center}
\leavevmode
\epsfile{file=FIG/ES3D_GALERKIN/NMEKXY10.ps,scale=0.8}\end{center}\end{figure}


next up previous contents
Next: Three-dimensional scattering of scalar Up: Numerical examples Previous: One penny-shaped crack
Ken-ichi Yoshida
2001-07-28