Research Papers

Compressive Response of Pyramidal Lattices Embedded in Foams

[+] Author and Article Information
C. I. Hammetter

Department of Mechanical Engineering,
University of California,
Santa Barbara, CA 93106

F. W. Zok

Department of Materials,
University of California,
Santa Barbara, CA 93106

Select simulations on the lattices alone were performed using both the explicit and implicit versions of the Abaqus code. These simulations were used to confirm that the results from the explicit code had indeed converged. For simulations of the lattice/foam composites, the implicit code usually led to convergence problems because of “pinching” of thin foam elements located between the buckled struts and the rigid compression platens and the consequent excessive distortion of these elements. As a result, the explicit code was used exclusively for these simulations.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 12, 2012; final manuscript received April 13, 2013; accepted manuscript posted May 7, 2013; published online August 22, 2013. Assoc. Editor: Daining Fang.

J. Appl. Mech 81(1), 011006 (Aug 22, 2013) (11 pages) Paper No: JAM-12-1553; doi: 10.1115/1.4024408 History: Received December 12, 2012; Revised April 13, 2013; Accepted May 07, 2013

Recent endeavors to combine the desirable energy-absorption characteristics of stochastic foams with the comparatively high strengths of pyramidal lattices have shown promise for creating composites that outperform their constituents alone under compressive loading. Herein we employ numerical and analytical models to identify both the mechanisms by which synergistic behavior is obtained in such composites and the constituent mass fractions that yield maximum benefits. We find that the loading boundary conditions play a crucial role. When, for instance, composites are loaded between plates that are well bonded to the composites, their specific strengths invariably exceed those predicted by a rule-of-mixtures; however, these strengths can always be improved through an optimized lattice of equivalent mass. In contrast, when the composites are loaded between frictionless plates, their specific strengths exceed not only rule-of-mixtures predictions but, in many cases, also that of any mass-equivalent pyramidal lattice alone subject to the same (frictionless) conditions. The origin of this behavior is found to arise from foam-stabilization of lattice bending and splaying: deformation modes that govern strength in the absence of foam. In essence, the foam causes a transition from bend-dominated to stretch-dominated behavior in the lattice.

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Fig. 1

Unit cell of single-layered pyramidal lattice

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Fig. 2

Two-dimensional representations of a multilayered pyramidal lattice undergoing (a) uniaxial compressive straining, (b) uniaxial compressive stressing, and (c) pure bending. The deformation is stretch dominated only in (a).

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Fig. 3

Compressive stress–strain curves for stochastic foams, computed from Eqs. (4)–(8)

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Fig. 4

Effects of boundary conditions and relative density on the compressive response of lattices alone

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Fig. 5

Variations in lattice strength with relative density obtained from both the FE simulations and the analytical models

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Fig. 6

Effects of composite relative density and lattice mass fraction on the compressive response of lattice/foam composites and of corresponding neat foams and lattices for bonded boundary conditions

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Fig. 7

Effects of lattice mass fraction on composite compressive strengths from FEA and analytical models for the bonded boundary conditions

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Fig. 8

Effects of boundary conditions and mass fraction on the compressive response of composites with 10% relative density. (Compare results with those in Fig. 6(b).)

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Fig. 9

Effects of lattice mass fraction on composite compressive strengths from FEA and analytical models for the frictionless conditions. Predicted strengths for strut buckling in the composite lie above the range of strengths plotted here (see Fig. 7(b)).

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Fig. 10

FEA results showing effects of lattice mass fraction and compressive strain on the apparent Poisson's ratio for the composite and lattice

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Fig. 11

Deformed meshes with contours of equivalent plastic strain at two macroscopic strains (points A and B in Fig. 8) for a composite with ρ¯c = 0.1 and m = 0.4

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Fig. 12

Deformed meshes of a lattice (ρ¯L = 0.02) with contours of equivalent plastic strain for the bonded case and for the frictionless case, alone and filled with foam. The deformed meshes labeled C and D correspond to the points in Fig. 8.

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Fig. 13

Effects of boundary conditions and lattice mass fraction on the compressive response of composites, all consisting of a lattice with 10% relative density and foams with relative densities varying between 5 and 15%. Also shown for comparison are the results for the lattices and foams alone at pertinent relative densities.

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Fig. 14

Compressive strengths obtained from FEA and the analytical models for filled and unfilled lattices (each with ρ¯L = 0.1) for each of the three boundary conditions

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Fig. 15

Load-displacement response of a strut embedded in foam and displaced laterally, perpendicular to its axis

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Fig. 16

Failure maps for two boundary conditions

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Fig. 17

Schematic of loads acting on a single strut embedded in foam during plastic hinging




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