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Research Papers

# Pyramidal Lattice Structures for High Strength and Energy Absorption

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

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

F. W. Zok

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

Manuscript received June 22, 2012; final manuscript received October 15, 2012; accepted manuscript posted October 22, 2012; published online May 16, 2013. Assoc. Editor: Vikram Deshpande.

J. Appl. Mech 80(4), 041015 (May 16, 2013) (11 pages) Paper No: JAM-12-1255; doi: 10.1115/1.4007865 History: Received June 22, 2012; Revised October 15, 2012; Accepted October 22, 2012

## Abstract

Recent developments in directed photocuring of polymers have enabled fabrication of periodic lattice structures with highly tailorable geometries. The present study addresses the mechanics of compressive deformation of such structures with emphasis on the effects of strut slenderness $L/D$, strut inclination angle $θ$, and number of repeat lattice layers $N$. We present analytic models and finite element calculations for a broad parameter space and identify designs that yield desirable combinations of specific strength and energy absorption. The optimal designs (those for which crushing occurs at nearly constant compressive stress) are found to be those in which there is only one pyramidal layer, the inclination angle is of intermediate value ($θ$ = 50 deg) and the strut slenderness ratio falls below a critical value, typically $L/D=4$. The performance of near-optimal structures is attributable to the balance between two competing processes during plastic deformation: (i) geometric hardening associated with lateral expansion of the nodes and the struts, and (ii) geometric softening arising from the corresponding reduction in strut angle. Comparisons with stochastic foams show that the lattice structures can be designed to attain levels of energy absorption not possible by foams (by factors of 3–5 on a mass basis), albeit at higher stress levels than those required for crushing foams.

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## Figures

Fig. 1

Schematics of pyramidal lattice structures with varying N

Fig. 2

Comparison of analytic estimates of relative density ρ¯*, Eq. (4), and ρ¯, Eq. (5), with (essentially exact) results from the finite element models

Fig. 3

Schematic of cross-sections midnode through a second-order (N = 2) pyramidal lattice structure

Fig. 4

Analytic predictions of lattice stress-strain response for various strut inclination angles

Fig. 5

Compressive response of the two geometric models, with single and multiple cells (θ = 61 deg, ρ¯= 0.2, N = 2). Inset schematics show the nature of the geometries.

Fig. 6

Effects of the number of layers on the compressive response of lattice structures with ρ¯ = 0.065 and θ = 60 deg. The normalizing stress, σb is that required for elastic strut buckling, using K = 0.5.

Fig. 7

Compressive response of lattice structures with N = 2

Fig. 8

Bottom: representative compressive stress-strain curves of lattice structures with ρ¯ = 0.25, showing effects of θ and N. Top: the lattice structures at various stages of deformation. Contours represent equivalent plastic strain (0–0.2).

Fig. 9

Comparisons of compressive strengths of two-layer lattices obtained from FEA with those of the analytic models as well as that for a stochastic foam

Fig. 10

Failure modes observed in FE simulations for the two-layer lattice. Dashed lines are the mode boundaries given by Eqs. (18), (19), and (20).

Fig. 11

Compressive response of lattice structures with N = 1

Fig. 12

Comparisons of compressive strengths of single-layer lattices obtained from FEA with those of the analytic models as well as that for stochastic foam

Fig. 13

Failure modes observed in FE simulations for the single-layer lattice. Dashed lines are the mode boundaries given by Eqs. (18), (19), and (21).

Fig. 14

Energy absorption for the lattice structures with θ = 40 deg, (a) on a volumetric basis and (b) on a mass basis, for both N = 1 and N = 2

Fig. 15

Energy absorption for lattices with θ = 60 deg, (a) on a volumetric basis and (b) on a mass basis, for both N = 1 and N = 2

Fig. 16

Comparison of energy absorption performance of single-layer lattices (N = 1) for various strut inclination angles with that of a stochastic foam, all with the same relative density

Fig. A1

Schematic of the changes in node geometry associated with local plastic deformation

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