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Boundary Integrals for Data Reconstruction on an Elastostatic Crack

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Abstract

The elastostatic Cauchy problem of fracture mechanics is studied in a two-dimensional bounded domain containing a crack. Given Cauchy data on the boundary of the domain, the displacement and normal stress (traction) are reconstructed on the crack. The reconstruction is done by reducing the original problem, via the elastostatic potential, to a system of integral equations to be solved for densities over the boundary of the domain and the crack. Discretization is carried out by the Nyström method using quadrature formulas adjusted for singularities manifesting at the endpoints of the crack. Tikhonov regularization is applied for the stable solution of the discretized system. The results of numerical experiments for different input data and parameters are given showing that relevant physical quantities on the crack can be stably reconstructed.

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Data Availability Statement

In the numerical simulations of the submitted work, synthetic data is used. Formulas to generate the data are stated within the work itself.

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Acknowledgements

The authors are grateful for valuable suggestions and careful reading by the anonymous referees.

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Authors and Affiliations

  1. Faculty of Applied Mathematics and Informatics, Ivan Franko National University of Lviv, Lviv, 79000, Ukraine

    Roman Chapko & Mariia Vlasiuk

  2. Mathematics, ITN, Campus Norrköping, Linköping University, 601 74, Norrköping, Sweden

    B. Tomas Johansson

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  1. Roman Chapko
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  2. B. Tomas Johansson
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  3. Mariia Vlasiuk
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Correspondence to Roman Chapko.

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Chapko, R., Johansson, B.T. & Vlasiuk, M. Boundary Integrals for Data Reconstruction on an Elastostatic Crack. Int. J. Appl. Comput. Math 8, 40 (2022). https://doi.org/10.1007/s40819-021-01232-x

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