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Research Papers

The Imperfection Sensitivity of Isotropic Two-Dimensional Elastic Lattices

[+] Author and Article Information
Digby D. Symons, Norman A. Fleck

Engineering Department,  Cambridge University, Trumpington Street, Cambridge CB2 1PZ, UK

J. Appl. Mech 75(5), 051011 (Jul 17, 2008) (8 pages) doi:10.1115/1.2913044 History: Received September 28, 2007; Revised October 11, 2007; Published July 17, 2008

The imperfection sensitivity of the effective elastic properties is numerically explored for three planar isotropic lattices: fully triangulated, the Kagome grid, and the hexagonal honeycomb. Each lattice comprises rigid-jointed, elastic Euler–Bernoulli beams, which can both stretch and bend. The imperfections are in the form of missing bars, misplaced nodes, and wavy cell walls. Their effect on the macroscopic bulk and shear moduli is numerically investigated by considering a unit cell containing randomly distributed imperfections, and with periodic boundary conditions imposed. The triangulated and Kagome lattices have sufficiently high nodal connectivities that they are stiff, stretching dominated structures in their perfect state. In contrast, the perfect hexagonal honeycomb, with a low nodal connectivity of 3, is stretching dominated under pure hydrostatic loading but is bending dominated when the loading involves a deviatoric component. The high connectivity of the triangulated lattice confers imperfection insensitivity: Its stiffness is relatively insensitive to missing bars or to dispersed nodal positions. In contrast, the moduli of the Kagome lattice are degraded by these imperfections. The bulk modulus of the hexagonal lattice is extremely sensitive to imperfections, whereas the shear modulus is almost unaffected. At any given value of relative density and level of imperfection (in the form of missing bars or dispersed nodal positions), the Kagome lattice has a stiffness intermediate between that of the triangulated lattice and the hexagonal honeycomb. It is argued that the imperfections within the Kagome lattice switch the deformation mode from stretching to a combination of stretching and bending. Cell-wall waviness degrades the moduli of all three lattices where the behavior of the perfect structure is stretching dominated. Since the shear response of the perfect hexagonal honeycomb is by bar bending, the introduction of bar waviness has a negligible effect on the effective shear modulus.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Perfect geometries of three planar grids: (a) triangular, (b) Kagome, and (c) Hexagonal

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Figure 2

Planar grids with f=0.1 (10%) missing bars: (a) triangular, (b) Kagome, and (c) hexagonal

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Figure 3

Convergence of bulk modulus of planar Kagome with increasing size of unit cell; (a) f=0.01 (1%) missing bars and (b) f=0.1 (10%) missing bars (means of 20 simulations plotted, error bars represent ±1 standard deviation)

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Figure 4

Planar Kagome grids with f=0.01 (1%) missing bars: (a) equibiaxial strain t∕l=0.05; (b) equibiaxial strain t∕l=0.02

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Figure 5

Sensitivity of bulk modulus of planar grids to missing bars: (a) for varying relative density ρ¯ with f=0 and f=0.1 (10%) missing bars; (b) for varying proportion of missing bars f with fixed cell-wall thickness t∕l=0.02

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Figure 6

Sensitivity of shear modulus of planar grids to missing bars: (a) for varying relative density with f=0 and f=0.1 missing bars; (b) for cell-wall thickness t∕l=0.02

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Figure 7

Planar grids with stochastic nodal dispersion amplitude a=0.3: (a) triangular, (b) Kagome, and (c) hexagonal

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Figure 8

Sensitivity of bulk modulus of planar grids to stochastic nodal dispersion: (a) for varying relative density with dispersion amplitude a=0 and a=0.5; (b) for varying dispersion amplitude with fixed cell-wall thickness t∕l=0.02

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Figure 9

Sensitivity of shear modulus of planar grids to stochastic nodal dispersion: (a) for varying relative density with dispersion amplitude a=0 and a=0.5; (b) for varying dispersion amplitude with fixed cell-wall thickness t∕l=0.02

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Figure 10

(a) Definition of bar waviness; (b) sensitivity of bulk and shear moduli of triangular, Kagome, and hexagonal planar grids to bar waviness

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