Plane Problem of the Theory of Elasticity on the Identification of Nodal Points of a Quadrature Inclusion

A. V. Kaptsov; Капцов А. В.; E. I. Shifrin; Шифрин Е. И.

doi:10.31857/S0572329923600147

Plane Problem of the Theory of Elasticity on the Identification of Nodal Points of a Quadrature Inclusion

Authors: Kaptsov A.V.¹, Shifrin E.I.¹
Affiliations:
1. Ishlinsky Institute for Problems in Mechanics RAS
Issue: No 6 (2023)
Pages: 47-68
Section: Articles
URL: https://rjmseer.com/1026-3519/article/view/672642
DOI: https://doi.org/10.31857/S0572329923600147
EDN: https://elibrary.ru/XOMURK
ID: 672642

Cite item

Full Text

Open Access
Restricted Access

Access granted
Restricted Access

Subscription or Fee Access

Abstract

The problem of detection and identification of an elastic inclusion in an isotropic, linearly elastic plane is considered. It is assumed that constant stresses are set at infinity. It is also assumed that on some closed curve containing an inclusion inside, the acting forces and displacements are known quantities. For the case of quadrature domain occupied by the inclusion, a method for identifying its nodal points has been developed. The developed method is based on the application of the principle of reciprocity. Numerical examples are given.

Keywords

elasticity, plane problem, quadrature domain, nodal points, inverse problem

About the authors

A. V. Kaptsov

Ishlinsky Institute for Problems in Mechanics RAS

Email: shifrin@ipmnet.ru
119526, Moscow, Russia

E. I. Shifrin

Ishlinsky Institute for Problems in Mechanics RAS

Author for correspondence.
Email: shifrin@ipmnet.ru
119526, Moscow, Russia

References

Andrieux S., Ben Abda A. Identification of planar cracks by complete overdetermined data: inversion formulae // Inverse Probl. 1996. V. 12. P. 553–563.
Andrieux S., Ben Abda A., Bui H. Reciprocity principle and crack identification // Inverse Probl. 1999. V. 15. P. 59–65.
Goldstein R.V., Shifrin E.I., Shushpannikov P.S. Application of invariant integrals to the problems of defect identification // Int. J. Fract. 2007. V. 147. P. 45–54. https://doi.org/10.1007/978-1-4020-6929-1_6
Шифрин Е.И. Идентификация эллипсоидального дефекта в упругом теле по результатам одного испытания на одноосное растяжение (сжатие) // Изв. РАН. МТТ. 2010. № 3. С. 131–142.
Shifrin E.I., Shushpannikov P.S. Identification of a spheroidal defect in an elastic solid using a reciprocity gap functional // Inverse Probl. 2010. V. 26. 055001. https://doi.org/10.1088/0266-5611/26/5/055001
Shifrin E.I., Shushpannikov P.S. Identification of an ellipsoidal defect in an elastic solid using boundary measurements // Int. J. Solids Struct. 2011. V. 48. № 7–8. P. 1154–1163. https://doi.org/10.1016/j.ijsolstr.2010.12.016
Shifrin E.I., Shushpannikov P.S. Identification of small well-separated defects in an isotropic elastic body using boundary measurements // Int. J. Solids Struct. 2013. V. 50. № 22–23. P. 3707–3716. https://doi.org/ 10.1016/j.ijsolstr.2013.07.009
Shifrin E.I., Kaptsov A.V. Identification of multiple cracks in 2D elasticity by means of the reciprocity principle and cluster analysis // Inverse Probl. 2018. V. 34. 015009. https://doi.org/10.1088/1361-6420/aa9d74
Davis P.J. The Schwarz function and its applications. The Carus Mathematical Monographs 17. Mathematical Association of America. 1974.
Aharonov D., Shapiro H.S. Domains on which analytic functions satisfy quadrature identities // J. d’Analyse Math. 1976. V. 30. P. 39–73.
Gustafsson B. Quadrature identities and the Schottky double // Acta Appl. Math. 1983. V. 1. P. 209–240.
Bell S.R. Quadrature domains and kernel function zipping // Arkiv mat. 2005. V. 43. P. 271–287.
Bell S.R. Density of quadrature domains in one and several complex variables // Complex Var. Elliptic Equ. 2009. V. 54. № 3–4. P. 165–171.
Ameur Y., Helmer M., Tellander F. On the uniqueness problem for quadrature domains // Comput. Methods Funct. Theory. 2021. V. 21. P. 473–504. https://doi.org/10.1007/s40315-021-00373-w
Мусхелишвили Н.И. Некоторые основные задачи математической теории упругости. М.: Наука, 1966. 708 с.
Golub G.H., Milanfar P., Varah J. A stable numerical method for inverting shape from moments // SIAM J. Sci. Comput. 1999. V. 21. № 4. P. 1222–1243. https://doi.org/10.1137/S1064827597328315
El Badia A., Ha-Duong T. An inverse source problem in potential analysis // Inverse Probl. 2000. V. 16. № 3. P. 651–663. https://doi.org/10.1088/0266-5611/16/3/308
Kang H., Lee H. Identification of simple poles via boundary measurements and an application of EIT // Inverse Probl. 2004. V. 20. № 6. P. 1853–1863. https://doi.org/10.1088/0266-5611/20/6/010
Lee S.-Y., Makarov N.G. Topology of quadrature domains // J. Am. Math. Soc. 2016. V. 29. P. 333–369.
Шифрин Е.И., Капцов А.В. Идентификация узловых точек упругого включения в упругой плоскости // Доклады РАН. Математика, информатика, процессы управления. 2023. Т. 509. С. 77–82. https://doi.org/10.31857/S268695432370011X