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On Solvability of One Singular Equation of Peridynamics

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Abstract

In the classical theory of solid mechanics, the behavior of solids is described by partial differential equations (PDE) through Newton’s second law of motion. However, when spontaneous cracks and fractures exist, such PDE models are inadequate to characterize the discontinuities of physical quantities such as the displacement field. Recently, a peridynamic continuum model was proposed which only involves the integration over the differences of the displacement field. A linearized peridynamic model can be described by the integro-differential equation with initial values. In this paper, we study the well-posedness and regularity of a linearized peridynamic model with singular kernel. The novelty of the paper is that the singular kernel is represented as the Laplacian of a regular function. This let to convert the model to an operator valued Volterra integral equation. Then the existence and regularity of the solution of the peridynamics problem are established through the study of the Volterra integral equation.

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REFERENCES

  1. S. A. Silling, ‘‘Reformulation of elasticity theory for discontinuities and long-range forces,’’ J. Mech. Phys. Solids 48, 175–209 (2000).

    Article  MathSciNet  Google Scholar 

  2. E. Askari, F. Bobaru, R. B. Lehoucq, M. L. Parks, S. A. Silling, and O. Weckner, ‘‘Peridynamics for multiscale materials modeling,’’ J. Phys.: Conf. Ser. 125, 012078 (2008).

    Google Scholar 

  3. S. A. Silling and R. B. Lehoucq, ‘‘Convergence of peridynamics to classical elasticity theory,’’ J. Elasticity 93, 13–37 (2008).

    Article  MathSciNet  Google Scholar 

  4. S. A. Silling and R. B. Lehoucq, ‘‘Peridynamic theory of solid mechanics,’’ Adv. Appl. Mech. 44, 73–168 (2010).

    Article  Google Scholar 

  5. R. B. Lehoucq and S. A. Silling, ‘‘Force flux and the peridynamic stress tensor,’’ J. Mech. Phys. Solids 56, 1566–1577 (2008).

    Article  MathSciNet  Google Scholar 

  6. E. Emmrich and O. Weckner, ‘‘Analysis and numerical approximation of an integrodifferential equation modelling non-local effects in linear elasticity,’’ Math. Mech. Solids 12, 363–384 (2007).

    Article  MathSciNet  Google Scholar 

  7. E. Emmrich and O. Weckner, ‘‘On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity,’’ Commun. Math. Sci. 5, 851–864 (2007).

    Article  MathSciNet  Google Scholar 

  8. Q. Du, J. R. Kamm, R. B. Lehoucq, and M. L. Parks, ‘‘A new approach for a nonlocal, nonlinear conservation law,’’ SIAM J. Appl. Math. 72, 464–487 (2012).

    Article  MathSciNet  Google Scholar 

  9. P. Seleson, M. L. Parks, M. Gunzburger, and R. B. Lehoucq, ‘‘Peridynamics as an upscaling of molecular dynamics,’’ Multiscale Model. Simul. 8, 204–227 (2009).

    Article  MathSciNet  Google Scholar 

  10. E. Emmrich, R. Lehoucq, and D. Puhst,‘‘Peridynamics: A nonlocal continuum theory,’’ Lect. Notes Comput. Sci. Eng. 89, 45–65 (2013).

  11. K. Zhou and Q. Du, ‘‘Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions,’’ SIAM J. Numer. Anal. 48, 1759–1780 (2010).

    Article  MathSciNet  Google Scholar 

  12. Sh. A. Alimov, Y. Cao and O. A. Ilhan, ‘‘On the problems of peridynamics with special convolution kernels,’’ J. Integral Equat. Appl. 26, 301–321 (2014).

    Article  MathSciNet  Google Scholar 

  13. Sh. A. Alimov and Sh. Sheraliev, ‘‘On the solvability of the singular equation of peridynamics,’’ Complex Variables Ellipt. Equat. 5 (64), 873–887 (2019).

    MATH  Google Scholar 

  14. A. Yuldasheva, ‘‘The linear peridinamic model in Elasticity Theory,’’ Lob. J. Math. 41 (1),137–144 (2020).

    Google Scholar 

  15. S. Albeverio and Sh. A. Alimov, ‘‘On some integral equations in Hilbert space with an application to the theory of elasticity,’’ Integral Equat. Operator Theory 35, 153–168 (2006).

    Article  MathSciNet  Google Scholar 

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ACKNOWLEDGMENTS

We are grateful to the reviewers for careful reading of the manuscript and helpful remarks.

Funding

This work was supported by Academy of Sciences Republic of Uzbekistan OT-F4-(88) and OT-F4-(36/32).

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Authors and Affiliations

  1. Tashkent Branch, Moscow State University, 100060, Tashkent, Uzbekistan

    A. V. Yuldasheva

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  1. A. V. Yuldasheva
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Correspondence to A. V. Yuldasheva.

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(Submitted by E. K. Lipachev)

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Yuldasheva, A.V. On Solvability of One Singular Equation of Peridynamics. Lobachevskii J Math 41, 1131–1136 (2020). https://doi.org/10.1134/S1995080220060190

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  • DOI: https://doi.org/10.1134/S1995080220060190

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