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Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics

  • Markus LazarEmail author
  • Eleni Agiasofitou
  • Giacomo Po
Original Paper
  • 64 Downloads

Abstract

In this work, based on Eringen’s theory of nonlocal anisotropic elasticity, the three-dimensional nonlocal anisotropic elasticity of generalized Helmholtz type is developed. The derivation of a new three-dimensional nonlocal anisotropic kernel, which is the Green function of the three-dimensional anisotropic Helmholtz equation, enables to capture anisotropic length scale effects by means of a length scale tensor, which is a symmetric tensor of rank two. The derived nonlocal kernel function possesses up to six internal characteristic lengths on the Ångström-scale. The presented theory of nonlocal elasticity possesses the appropriate property to be a generalized continuum theory of Ångström-mechanics, since the range of its validity and applicability is up to the Ångström-scale. The connection between the theory of nonlocal anisotropic elasticity and lattice theory is established. The tensor function of nonlocal elastic moduli as well as the nonlocal kernel function is given in terms of the Hessian matrix in the lattice approach. In the framework of the considered theory, the modeling of dislocations in anisotropic materials taking into consideration anisotropic dislocation core effects is presented. Important dislocation key formulas, namely the anisotropic Peach–Koehler stress formula, the Peach–Koehler force and the anisotropic Blin’s formula, are derived. A major tool used in deriving the expression of anisotropic Blin’s formula is Kirchner’s so-called \({\varvec{F}}\)-tensor, which is here generalized toward nonlocal anisotropic elasticity. The main feature and advantage of the derived fields, compared with the corresponding ones in classical anisotropic elasticity, is that they are free of singularities. Numerical applications to straight dislocations in bcc Fe are given, revealing the ability and advantage of the considered theory to describe adequately nonsingular anisotropic stress and self-energy fields capturing the effects of anisotropy on the Ångström-scale.

Notes

Acknowledgements

The authors Markus Lazar and Eleni Agiasofitou gratefully acknowledge a grant from the Deutsche Forschungsgemeinschaft (Grant number La1974/4-1). Markus Lazar and Eleni Agiasofitou wish to thank Paul Steinmann for inspiring discussions. The author Giacomo Po acknowledges the support of the U.S. Department of Energy, Office of Fusion Energy through award number DE-FG02-03ER54708, the Air Force Office of Scientific Research (AFOSR) through award number FA9550-11-1-0282 and the National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation (CMMI), through award number 1563427 with UCLA.

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Authors and Affiliations

  1. 1.Department of PhysicsDarmstadt University of TechnologyDarmstadtGermany
  2. 2.Department of Mechanical and Aerospace EngineeringUniversity of California Los AngelesLos AngelesUSA
  3. 3.Department of Mechanical and Aerospace EngineeringUniversity of MiamiCoral GablesUSA

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