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Research Papers

Scaling of the Elastic Behavior of Two-Dimensional Topologically Interlocked Materials Under Transverse Loading

[+] Author and Article Information
T. Siegmund

e-mail: siegmund@purdue.edu

R. J. Cipra

School of Mechanical Engineering,
585 Purdue Mall,
Purdue University,
West Lafayette, IN 47907

J. S. Bolton

School of Mechanical Engineering,
Ray W. Herrick Laboratories,
140 South Martin Jischke Drive,
Purdue University,
West Lafayette, IN 47907

lCorresponding author.

Manuscript received April 8, 2013; final manuscript received June 21, 2013; accepted manuscript posted July 1, 2013; published online September 19, 2013. Editor: Yonggang Huang.

J. Appl. Mech 81(3), 031011 (Sep 19, 2013) (9 pages) Paper No: JAM-13-1152; doi: 10.1115/1.4024907 History: Received April 08, 2013; Accepted June 01, 2013; Revised June 21, 2013

Topologically interlocked materials (TIMs) are a class of 2D mechanical crystals made by a structured assembly of an array of polyhedral elements. The monolayer assembly can resist transverse forces in the absence of adhesive interaction between the unit elements. The mechanical properties of the system emerge as a combination of deformation of the individual unit elements and their contact interaction. The present study presents scaling laws relating the mechanical stiffness of monolayered TIMs to the system characteristic dimensions. The concept of thrust line analysis is employed to obtain the scaling laws, and model predictions are validated using finite element simulations as virtual experiments. Scaling law powers were found to closely resemble those of classical plate theory despite the distinctly different underlying mechanics and theory of TIM deformation.

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Figures

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Fig. 1

(a) Assembly process of tetrahedra unit elements leading to topological interlocking. (b) Example of a TIM assembly with 7 × 7 unit elements.

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Fig. 2

(a) TIMs with varying span, L0, but constant unit element edge length, a0. (b) TIMs with constant span, but varying unit element edge lengths. (c) TIMs with varying span but constant number of tetrahedra in each direction, N, and hence constant aspect ratio, a0/L0.

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Fig. 3

(a) A representative TIM and the abutments supporting the TIM. (b) Two characteristic cross sections of the TIM.

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Fig. 4

A schematic of a representative load carrying cross section of a TIM monolayer (a) undeformed state, and, (b) deformed state including the lines of thrust

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Fig. 5

Computationally predicted F − δ results for TIM assemblies corresponding to assemblies of Fig. 2: (a) a = const and N increasing; (b) L0/a0 = const and N increasing, and, (c) N = const and a varying

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Fig. 6

Compressive principal stress vectors, σP3, at displacement δ/a0 = 0.04 from simulations with slip suppressed. The 7 × 7 assembly (a) top view and (b) section view (shown for plane, P1).

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Fig. 7

Compressive principal stress vectors, σP3, at displacement δ/a0 = 0.04 from simulations with slip suppressed. The 19 × 19 assembly (a) top view and (b) section view (shown for plane, P1).

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Fig. 8

Stiffness scaling analysis results for assembled TIMs obtained from FEA for δ/a0 = 0.04, and from analytical model, Eq. (17). Shown are results for effect on stiffness of TIM span, Fig. 2(a).

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Fig. 9

Stiffness scaling analysis results for assembled TIMs obtained from FEA for δ/a0 = 0.04, and from analytical model, Eq. (17). Shown are results for effect on stiffness of unit element edge length, Fig. 2(b).

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Fig. 10

Stiffness scaling analysis results for assembled TIMs for constant a0/L0, Fig. 2(c), obtained from FEA for δ/a0 = 0.04, and analytical model, Eq. (18)

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Fig. 11

(a) Dependence of the TIM stiffness on the coefficient of friction for the reference 7 × 7 assembly; (b) section view of the assemblies for suppressed slip (μ = 100.00) and with slip (μ = 0.3)

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