*(English)*Zbl 1195.74090

Summary: In this paper, the traction boundary element method (TBEM) and the method of fundamental solutions (MFS), formulated in the frequency domain, are used to evaluate the 3D scattered wave field generated by 2D empty cracks embedded in an elastic slab and a half-space. Both models overcome the thin-body difficulty posed when the classical BEM is applied.

The crack exhibits arbitrary cross section geometry and null thickness. In neither model are the horizontal formation surfaces discretized, since appropriate fundamental solutions are used to take them into consideration.

The TBEM models the crack as a single line. The singular and hypersingular integrals that arise during the TBEM model’s implementation are computed analytically, which overcomes one of the drawbacks of this formulation. The results provided by the proposed TBEM model are verified against responses provided by the classical BEM models derived for the case of an empty cylindrical circular cavity.

The MFS solution is approximated in terms of a linear combination of fundamental solutions, generated by a set of virtual sources simulating the scattered field produced by the crack, using a domain decomposition technique. To avoid singularities, these fictitious sources are not placed close to the crack, and the use of an enriched function to model the displacement jumps across the crack is unnecessary.

The performances of the proposed models are compared and their limitations are shown by solving the case of a C-shaped crack embedded in an elastic slab and a half-space domain.

The applicability of these formulations is illustrated by presenting snapshots from computer animations in the time domain for an elastic slab containing an S-shaped crack, after applying an inverse Fourier transformation to the frequency domain computations.

##### MSC:

74J25 | Inverse problems (waves in solid mechanics) |

74S15 | Boundary element methods in solid mechanics |

74S30 | Other numerical methods in solid mechanics |

74R10 | Brittle fracture |

65N80 | Fundamental solutions, Green’s function methods, etc. (BVP of PDE) |