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REVIEW ARTICLES

Lattice models in micromechanics

[+] Author and Article Information
Martin Ostoja-Starzewski

Department of Mechanical Engineering, McGill University, 817 Sherbrooke St West, Montréal, Québec, Canada H3A 2K6; martin.ostoja@mcgill.ca

Appl. Mech. Rev 55(1), 35-60 (Jan 01, 2002) (26 pages) doi:10.1115/1.1432990 History:
Copyright © 2002 by ASME
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Figures

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A 1D chain of particles of lattice spacing s, connected by axial springs (thin lines)
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a) A 1D chain of dumbbell particles (vertical rigid bars) of X-braced girder geometry, pin-connected by axial springs (thin lines); and b) the shear and curvature modes of a single bay
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Planar lattices and their repeating elements (after Noor and Nemeth 4)
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A wire rope of constant helix angle
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a) A hexagonal lattice with three different choices of unit cell; b) a square lattice with a square unit cell; and c) a triangular lattice with hexagonal unit cell
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a) Unit cell of a triangular lattice model; α(1),[[ellipsis]],α(6) are the normal spring constants, β(1),[[ellipsis]],β(6) are the angular spring constants; in the isotropic Kirkwood model α(b)(b+3) and β(b)(b+3),b=1,2,3; and b) details of the angular spring model.
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a) A triple honeycomb lattice made of three different spring types α, β, and γ belonging, respectively to three sublattices A, B, and C; and b) A 42×42 unit cell of a triangular lattice of hexagonal pixels, with 11 pixel diameter circular inclusions centered on pixels and randomly placed with periodic boundary conditions; after Snyder et al. 19
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a) Parameter plane: aspect ratio of inclusions and the contrast; b) spring network as a basis for resolution of round disks, ellipses, pixels, and needles in the parameter plane; and c) another interpretation of the parameter plane: from pixels to needles
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A functionally graded matrix-inclusion composite with 47.2% volume fraction of black phase is partitioned into 8×8 subdomains, corresponding to a 64-processor parallel computer
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Scanned image of a very thin polycrystal aluminum sheet. All the grain boundaries are orthogonal to the plane of the sheet
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The lattice geometry (a); curvature and internal loads in a single beam element (b)
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a) A decrease in pore sizes (from left to right): from large holes (slender beams), through a lattice of stubby beams, to a plate perforated with small holes; shown at porosities p=10%, 50%, and 90%-from right to left; b) The effective Young’s modulus Eeff, normalized by the beam’s Young’s modulus E(b), as a function of p for: 1) the central-force lattice, 2) the Timoshenko beams lattice, 3) the Bernoulli-Euler beams lattice, 4) the Cox model, and 5) the effective medium theory for a perforated plate; and c) the effective Poisson’s ratio νeff as a function of p, models (1–5) shown
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Two structures, I and II, resulting in a lattice with local (nearest neighbor) and nonlocal (second neighbor) interactions. Note that the structure II consists of three triangular networks having separate sets of nodes, and that all these nodes coincide with the nodes of structure I.
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A perspective view of a triangular geometry lattice, showing the relevant internal loads in a beam cross section
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A triangular lattice with 71 edges and 37 vertices; it is generically rigid.
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Samples of a) a planar Poisson line field and b) a finite fiber field generated from (5.4) with a1=1 and all other ai’s equal zero. Test windows of size L×L are considered. c) Deformation of a network of (b), with 195 fibers at a preference in the x1-direction at δ=2 with originally straight fibers, with fiber bending present, subjected to axial strain ε11=1%.d) The same network, with fiber bending almost absent, subjected to axial strain ε11=1%. All displacements in c) and d) are magnified by a factor 8 for clarity. Figure d) shows large, mechanism-type motions of the network including those of some fibers which spring outside the original domain of the network.
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a) A cluster of five grains showing the lines of interactions; b) a discrete element model showing the normal force, the shear force, and the moment exerted by grains 2 and 3 each onto the grain 1; c) a most general model showing the same grain-grain interactions as before but augmented by an internal, angular spring constant ka; and d) a simplified model adopted in this paper, showing only normal (kn) and angular (ka) effects.
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Substitutional a) versus topological disorder b) of a hard-core Delaunay network
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A periodic Poisson-Delaunay network with 200 vertices

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