We consider a model for elastic dislocations in geophysics. We model a portion of the Earth’s crust as a bounded, inhomogeneous elastic body with a buried fault surface, along which slip occurs. We prove well-posedness of the resulting mixed-boundary-value-transmission problem, assuming only bounded elastic moduli. We establish uniqueness in the inverse problem of determin- ing the fault surface and the slip from a unique measurement of the displacement on an open patch at the surface, assuming in addition that the Earth’s crust is an isotropic, layered medium with Lamé coefficients piecewise Lipschitz on a known partition and that the fault surface satisfies certain geo- metric conditions. These results substantially extend those of the authors in [Arch. Ration. Mech. Anal. 236, 71–111 (2020)].
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Detection of dislocations in a 2D anisotropic elastic medium
We study a model of dislocations in two-dimensional elastic media. In this model, the displacement satisfies the system of linear elasticity with mixed displacement-traction homogeneous boundary conditions in the complement of an open curve in a bounded planar domain, and has a specified jump, the slip, across the curve, while the traction is continuous there. The stiffness tensor is allowed to be anisotropic and inhomogeneous. We prove well-posedness of the direct problem in a variational setting, assuming the coefficients are Lipschitz continuous. Using unique continuation arguments, we then establish uniqueness in the inverse problem of determining the dislocation curve and the slip from a single measurement of the displacement on an open patch of the traction-free part of the boundary. Uniqueness holds when the elasticity operators admits a suitable decomposition and the curve satisfies additional geometric assumptions. This work complements the results in Arch. Ration. Mech. Anal., 236(1):71-111, (2020), and in Preprint arXiv:2004.00321, which concern three-dimensional isotropic elastic media.
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- Award ID(s):
- 1909103
- NSF-PAR ID:
- 10249404
- Editor(s):
- Mascia, Corrado; Terracina, Andrea; Tesei, Alberto
- Date Published:
- Journal Name:
- Rendiconti di matematica e delle sue applicazioni
- Volume:
- 42
- Issue:
- 3-4
- ISSN:
- 2532-3350
- Page Range / eLocation ID:
- 183-195
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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