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TECHNICAL PAPERS

# Lattice Green’s Functions in Nonlinear Analysis of Defects

[+] Author and Article Information
S. Haq

Faculty of Engineering Sciences, GIK Institute, Topi 23640, NWFP Pakistan

A. B. Movchan

Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, UKabm@liv.ac.uk

G. J. Rodin

Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX 78712

We are grateful to Proffessor Wolfgang Wendland for bringing this reference to our attention.

J. Appl. Mech 74(4), 686-690 (Aug 20, 2006) (5 pages) doi:10.1115/1.2710795 History: Received November 11, 2004; Revised August 20, 2006

## Abstract

A method for analyzing problems involving defects in lattices is presented. Special attention is paid to problems in which the lattice containing the defect is infinite, and the response in a finite zone adjacent to the defect is nonlinear. It is shown that lattice Green’s functions allow one to reduce such problems to algebraic problems whose size is comparable to that of the nonlinear zone. The proposed method is similar to a hybrid finite-boundary element method in which the interior nonlinear region is treated with a finite element method and the exterior linear region is treated with a boundary element method. Method details are explained using an anti-plane deformation model problem involving a cylindrical vacancy.

###### FIGURES IN THIS ARTICLE
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Copyright © 2007 by American Society of Mechanical Engineers
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## Figures

Figure 1

Deformation of the lattice in the vicinity of the vacancy

Figure 2

Node and link sets involved in the analysis. The set Γ−, formed by the nodes separating the linear and nonlinear zones, is denoted by the squares containing minus sign. The sets Γ+ and C+ are denoted by the squares containing the plus and × signs, respectively. The union of these two sets is the first layer of the nodes in the linear zone. The set Γ, formed by the exterior nodes of a finite lattice, is denoted by black squares. All nodes inside Γ− and Γ− itself form the set Ω−; the remaining nodes form the set Ω¯+ and Ω+=Ω¯+\Γ. The links bounded by Γ− and Γ+ are denoted by Υ0, the links bounded by Γ+ are denoted by Υ−, and the links bounded by Γ+∪C+, and Γ are denoted by Υ+.

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