TY - JOUR
T1 - Detection of dislocations in a 2D anisotropic elastic medium
AU - Aspri, Andrea
AU - Beretta, Elena
AU - De Hoop, Maarten
AU - Mazzucato, Anna L.
N1 - Funding Information:
A. Aspri acknowledges the hospitality of the Department of Mathematics at NYU-Abu Dhabi. Part of this work was conducted while A. Mazzucato was on leave at NYU-Abu Dhabi. She is partially supported by the US National Science Foundation Grant DMS-1909103. M.V. de Hoop is supported by the Simons Foundation under the MATH + X program, the National Science Foundation under Grant DMS-1815143, and the corporate members of the Geo- Mathematical Imaging Group at Rice University USA.
Funding Information:
Acknowledgements. A. Aspri acknowledges the hospitality of the Department of Mathematics at NYU-Abu Dhabi. Part of this work was conducted while A. Mazzucato was on leave at NYU-Abu Dhabi. She is partially supported by the US National Science Foundation Grant DMS-1909103. M.V. de Hoop is supported by the Simons Foundation under the MATH + X program, the National Science Foundation under Grant DMS-1815143, and the corporate members of the Geo-Mathematical Imaging Group at Rice University USA.
Publisher Copyright:
© 2021 Universita degli Studi di Roma La Sapienza. All rights reserved.
PY - 2021
Y1 - 2021
N2 - 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.
AB - 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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M3 - Article
AN - SCOPUS:85104187708
SN - 1120-7183
VL - 42
SP - 183
EP - 195
JO - Rendiconti di Matematica e delle Sue Applicazioni
JF - Rendiconti di Matematica e delle Sue Applicazioni
IS - 3
ER -