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Acta Mechanica
Published in: Acta Mechanica 2/2020

28-11-2019 | Original Paper

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

Authors: Markus Lazar, Eleni Agiasofitou, Giacomo Po

Published in: Acta Mechanica | Issue 2/2020

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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.
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Appendix
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Footnotes
1
A consequence of the axiom of material invariance [18].
 
2
In the case of a finite domain, a modified nonlocal kernel has to be used (see [32, 72]).
 
3
The Fourier transform and its inverse are, respectively, defined as (see, e.g.,  [81]):
$$\begin{aligned} {\hat{f}}({\varvec{k}})=\int _{{\mathbb {R}}^3} f({\varvec{x}})\, \text {e}^{-\text {i}{\varvec{k}}\cdot {\varvec{x}}}{\mathrm {d}}V, \qquad f({\varvec{x}})=\frac{1}{(2\pi )^3}\int _{{\mathbb {R}}^3} {\hat{f}}({\varvec{k}})\, \text {e}^{\text {i}{\varvec{k}}\cdot {\varvec{x}}}\, \text {d}{\hat{V}}. \end{aligned}$$
 
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Metadata
Title
Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics
Authors
Markus Lazar
Eleni Agiasofitou
Giacomo Po
Publication date
28-11-2019
Publisher
Springer Vienna
Published in
Acta Mechanica / Issue 2/2020
Print ISSN: 0001-5970
Electronic ISSN: 1619-6937
DOI
https://doi.org/10.1007/s00707-019-02552-2

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