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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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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(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