Communications in Mathematical Sciences

Volume 13 (2015)

Number 5

Multiscale analysis of linearized peridynamics

Pages: 1193 – 1218

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n5.a6

Authors

Tadele Mengesha (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.; and Department of Mathematics, University of Tennessee, Knoxville, Tenn., U.S.A.)

Qiang Du (Department of Mathematics, Pennsylvania State University, University Park, Penn., U.S.A.; and Department of Applied Physics and Applied Mathematics, Columbia University, New York, N.Y., U.S.A.)

Abstract

In this paper, we study the asymptotic behavior of a state-based multiscale heterogeneous peridynamic model. The model involves nonlocal interaction forces with highly oscillatory perturbations representing the presence of heterogeneities on a finer spatial length scale. The two-scale convergence theory is established for a steady state variational problem associated with the multiscale linear model. We also examine the regularity of the limit nonlocal equation and present the strong approximation to the solution of the peridynamic model via a suitably scaled two-scale limit.

Keywords

multiscale analysis, peridynamics, nonlocal equations, elasticity, Navier equation, homogenization, heterogeneous materials, two-scale convergence

2010 Mathematics Subject Classification

45F99, 45P05, 74E05, 74H10, 74Q05

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Published 22 April 2015