Abstract
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.
Original language | English (US) |
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Pages (from-to) | 183-195 |
Number of pages | 13 |
Journal | Rendiconti di Matematica e delle Sue Applicazioni |
Volume | 42 |
Issue number | 3 |
State | Published - 2021 |
Keywords
- Anisotropic
- Dislocations
- Elasticity
- Inverse problem
- Uniqueness
- Well-posedness
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
- Modeling and Simulation
- Geometry and Topology
- Fluid Flow and Transfer Processes
- Discrete Mathematics and Combinatorics
- Computational Mathematics
- Applied Mathematics