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Research Papers

Elastic Boundary Layers in Two-Dimensional Isotropic Lattices

[+] Author and Article Information
A. Srikantha Phani

Department of Mechanical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, UKspa21@bath.ac.uk

Norman A. Fleck

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UKnaf1@eng.cam.ac.uk

J. Appl. Mech 75(2), 021020 (Feb 27, 2008) (8 pages) doi:10.1115/1.2775503 History: Received February 21, 2007; Revised June 15, 2007; Published February 27, 2008

The phenomenon of elastic boundary layers under quasistatic loading is investigated using the Floquet–Bloch formalism for two-dimensional, isotropic, periodic lattices. The elastic boundary layer is a region of localized elastic deformation, confined to the free edge of a lattice. Boundary layer phenomena in three isotropic lattice topologies are investigated: the semiregular Kagome lattice, the regular hexagonal lattice, and the regular fully triangulated lattice. The boundary layer depth is on the order of the strut length for the hexagonal and the fully triangulated lattices. For the Kagome lattice, the depth of boundary layer scales inversely with the relative density. Thus, the boundary layer in a Kagome lattice of low relative density spans many cells.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 12

Boundary layer in a fully triangulated lattice parallel to the x1 direction for ρ¯=0.05.

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Figure 1

The two-dimensional isotropic lattices considered in the present study: (a) semiregular Kagome lattice, (b) regular hexagonal lattice, and (c) regular fully triangulated lattice

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Figure 4

Unit cells for (a) semiregular Kagome lattice, (b) regular hexagonal lattice, and (c) fully triangulated lattice. The axis of reflective symmetry is shown as a dashed line. Joints on the boundary are labeled numerically.

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Figure 5

Boundary layers in a Kagome lattice parallel to the x2 direction for ρ¯=0.05: (a) eigenvector 1, (b) eigenvector 2, (c) eigenvector 3, and (d) eigenvector 4

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Figure 6

Elastic boundary layers in a Kagome lattice subjected to macroscopic uniaxial tension along the x2 direction and subjected to simple shear

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Figure 7

The attenuation of each of the four eigenvectors versus ρ¯ in a Kagome lattice

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Figure 8

Boundary layer in a Kagome lattice parallel to the x1 direction for ρ¯=0.05

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Figure 9

Boundary layer in a hexagonal lattice parallel to the x2 direction, for ρ¯=0.05

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Figure 10

Boundary layers in a hexagonal lattice parallel to the x1 direction for ρ¯=0.05: (a) eigenvector 1, (b) eigenvector 2, (c) eigenvector 3, and (d) eigenvector 4

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Figure 11

Boundary layers in a fully triangulated lattice parallel to the x2 direction for ρ¯=0.05: (a) eigenvector 1, (b) eigenvector 2, (c) eigenvector 3, and (d) eigenvector 4

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Figure 2

Deformed mesh of Kagome lattice (ρ¯=10%) revealing a boundary layer at the sides of the specimen for (a) uniaxial tension and (b) simple shear. Based on the work of Fleck and Qiu (1).

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Figure 3

A unit cell for a two-dimensional periodic structure showing the degrees of freedom shared with the neighboring unit cells and the coordinate system employed

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