Detection of dislocations in a 2D anisotropic elastic medium

Andrea Aspri, Elena Beretta, Maarten De Hoop, Anna L. Mazzucato

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)183-195
Number of pages13
JournalRendiconti di Matematica e delle Sue Applicazioni
Volume42
Issue number3
StatePublished - 2021

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Modeling and Simulation
  • Geometry and Topology
  • Fluid Flow and Transfer Processes
  • Discrete Mathematics and Combinatorics
  • Computational Mathematics
  • Applied Mathematics

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