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arXiv · 1805.08817

Solvability of nonlocal systems related to peridynamics

Moritz Kassmann·Tadele Mengesha·James Scott
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Abstract

In this work, we study the Dirichlet problem associated with a strongly coupled system of nonlocal equations. The system of equations comes from a linearization of a model of peridynamics, a nonlocal model of elasticity. It is a nonlocal analogue of the Navier-Lam\'e system of classical elasticity. The leading operator is an integro-differential operator characterized by a distinctive matrix kernel which is used to couple differences of components of a vector field. The paper's main contributions are proving well-posedness of the system of equations and demonstrating optimal local Sobolev regularity of solutions. We apply Hilbert space techniques for well-posedness. The result holds for systems associated with kernels that give rise to non-symmetric bilinear forms. The regularity result holds for systems with symmetric kernels that may be supported only on a cone. For some specific kernels associated energy spaces are shown to coincide with standard fractional Sobolev spaces.

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Moritz Kassmann, Tadele Mengesha, James Scott. 2018-05-22. Solvability of nonlocal systems related to peridynamics. https://arxiv.org/abs/1805.08817

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