Point force in a plane in the context of fractional nonlocal elasticity

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Authors

  • Y. Povstenko Jan Dlugosz University in Czestochowa, Poland 0000-0002-7492-5394
  • T. Kyrylych Jan Dlugosz University in Czestochowa, Poland 0000-0003-1630-7958
  • A. Yatsko Koszalin University of Technology, Poland 0000-0002-3860-5318

Abstract

Solutions to the point load problems for elastic solids have different applications in geomechanics, contact mechanics, tribology as well as in modeling of lattice defects in crystals. Nonlocal elasticity assumes an integral constitutive equation for the stress tensor, takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit. Often, the nonlocal kernel of a stress constitutive equation is selected as the Green function of the Cauchy problem for appropriate partial differential equation. In this paper, we obtain the solution of elasticity problems for a point force in a plane and for a point load on the boundary of a half-plane in the context of the new theory of nonlocal elasticity in which the nonlocal modulus is the Green function of the Cauchy problem for the fractional diffusion equation with the Caputo derivative with respect to the nonlocality parameter and the fractional Riesz derivative with respect to spatial coordinates.

Keywords:

point force, Kelvin problem, Flamant problem, Cerruti problem, nonlocal elasticity, Caputo derivative, Riesz derivative

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