Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 50. no. 1. P. 78 – 91. ISSN 2079-6641

MATHEMATICS
https://doi.org/10.26117/2079-6641-2025-50-1-78-91
Research Article
Full text in English
MSC 45K05, 47G20

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Dedicated to the 80th anniversary of Academician AS RUz Sh. A. Alimov

Peridynamic Model of Vibrations in a Two-Dimensional Periodic Structure

A. V. Yuldasheva^{\ast}

Lomonosov Moscow State University, Tashkent Branch, 100060, Tashkent, Amir Temur st, 22, Uzbekistan

Abstract. Unlike classical continuum mechanics, where the linearized model is described by partial differential equations, the peridynamic model leads to an integro-differential equation with a non integrable kernel. The proposed method belongs to the category of nonlocal models, as particles separated by a finite distance can interact with each other. This allows the description of processes occurring in structures with cracks and discontinuities. Fracture is considered a natural result of deformation arising from the equation of motion and the constitutive model. Consequently, modeling crack growth in the peridynamic framework does not require additional data or equations that would be necessary in traditional fracture mechanics to determine crack initiation. The study examines a peridynamic model on a two-dimensional periodic structure related to graphene -a two dimensional allotropic form of carbon. It can be thought of as a single plane of layered graphite separated from the bulk crystal. Estimates suggest that graphene possesses high mechanical stiffness and record-breaking thermal conductivity. Its exceptionally high charge carrier mobility, which is the highest among all known materials (for the same thickness), makes it a promising material for various applications, particularly as a future foundation for nanoelectronics. The work investigates a hypersingular integro-differential equation describing oscillations in a two-dimensional periodic structure. A transformation has been found that allows the regularization of the singular integral operator involved in the equation. This made it possible to obtain a unique solution to the problem in the introduced Sobolev space.

Key words: Integro-differential equation, singular integral operator, peridynamics.

Received: 03.04.2025; Revised: 15.04.2025; Accepted: 17.04.2025; First online: 18.04.2025

For citation. Yuldasheva A. V. Peridynamic model of vibrations in a two-dimensional periodic structure. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 50: 1, 78-91. EDN: LRTAJN. https://doi.org/10.26117/2079-6641-2025-50-1-78-91.

Funding. The study was conducted without the support of foundations.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. The author participated in the writing of the article and is fully responsible for the submission of the final version of the article for publication.

^{\ast}Correspondence: E-mail: a_v_yuldasheva@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Yuldasheva A. V., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

References

  1. Alimov S. A., Cao Y., Ilhan O. A.On the problems of peridynamics with special convolution kernels. J. of Integral Equations and Applications, 2014. vol. 26, no. 3, pp. 301-321.
  2. Alimov S. A., Sheraliev S.On the solvability of the singular equation of peridynamics. Complex Variables and Elliptic Equations, 2019. vol. 64, no. 5, pp. 873-887 DOI: 10.1080/17476933.2019.1565406.
  3. Alimov S. A., Sheraliev S. N.On the solvability of hypersingular equation of peridynamics. Bulletin of National University of Uzbekistan, Mathematics and Natural Sciences, 2020. vol. 3, no. 4, pp. 278-298.
  4. Alimov Sh. A., Yuldasheva A. V.On the Solvability of the Peridynamic Equation with a Singular Kernel. Differential Equations, 2021. vol. 57, no. 3, pp. 353–365 DOI: 10.1134/S0012266121030083.
  5. Alimov Sh. A., Yuldasheva A. V.A Solvability of Singular Equations of Peridynamics on Two- Dimensional Periodic Structures. J Peridyn Nonlocal Model, 2023. vol. 5, no. 3, pp. 241–259 DOI:10.1007/s42102-021-00070-1.
  6. Du Q., Kamm J. R., Lehoucq R. B., Parks Michael L.A new approach for a nonlocal, nonlinear conservation law. SIAM J. Appl. Math., 2012. vol. 72, no. 1, pp. 464–487 DOI: 10.1137/110833233.
  7. Du Q., Zhou K. Mathematical analysis for the peridynamic nonlocal continuum theory. Mathematical Modelling and Numerical Analysis, 2011. vol. 45, no. 2, pp. 217-234 DOI: 10.1051/m2an/2010040.
  8. Emmrich E., Lehoucq R., Puhst D.Peridynamics: A Nonlocal Continuum Theory. Lecture Notes in Computational Science and Engineering, 2013. vol. 89, pp. 45-65.
  9. Gunzburger M., Lehoucq R.A nonlocal vector calculus with application to nonlocal boundary value problems. Multiscale Model. Simul., 2010. vol. 8, no. 5, pp. 1581-1598 DOI: 10.1137/090766607.
  10. Kantorovich L. V., Akilov G.P. Functional Analysis. Moscow: Nauka, 1977 (In Russian).
  11. Nikol’skiy S. M. Approximation of functions of several variables and imbedding theorems. New York: Springer-Verlag, 1971. 418 pp.
  12. Seleson P., Parks Michael L., Gunzburger M., Lehoucq R. B.Peridynamics as an upscaling of molecular dynamics. Multiscale Model. Simul., 2009. vol. 8, no. 1, pp. 204-207 DOI: 10.1137/09074807X.
  13. Silling S. A. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids, 2000. vol. 48, no. 1, pp. 175-209 DOI: 10.1016/S0022-5096(99)00029-0.
  14. Silling S. A., Bobaru F.Peridynamic modeling of membranes and fibers. International Journal of Non-Linear Mechanics, 2005. vol. 40, no. 1, pp. 395-409.
  15. Titchmarsh E. C. Eigenfunction expansions associated with second-order differential equations. Oxford: Clarendon Press, 1958. 436 pp.
  16. Watson G. N. A treatise on the theory of Bessel functions. Cambridge: Cambridge University Press, 1944. 814 pp.
  17. Zhou K., Du Q. Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal., 2010. vol. 48, no. 5, pp. 1759–1780 DOI: 10.1137/090781267.
  18. Yuldasheva A. V. Initial data problem for an equation related to a peridynamic model in a two dimensional domain. Vestnik KRAUNC. Fiz.-Mat. Nauki, 2021. vol. 37, no. 4, pp. 45–52 DOI: 10.26117/2079-6641-2021-37-4-45-52 (In Russian).

Information about the author

Yuldasheva Asal Viktorovna – D. Sci. (Phys. & Math.), professor, Lomonosov Moscow State University, Tashkent Branch, Tashkent, Uzbekistan, ORCID 0000-0001-6861-8331.

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