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Research Papers

Qualitative Analysis and Conceptual Design of Planar Metamaterials With Negative Poisson's Ratio

[+] Author and Article Information
Sree Kalyan Patiballa

Mem. ASME
Department of Industrial and
Enterprise Systems Engineering,
University of Illinois Urbana-Champaign,
Urbana, IL 61801
e-mail: patibal2@illinois.edu

Girish Krishnan

Mem. ASME
Department of Industrial and
Enterprise Systems Engineering,
University of Illinois Urbana-Champaign,
Urbana, IL 61801
e-mail: gkrishna@illinois.edu

1Corresponding author.

Manuscript received September 23, 2017; final manuscript received December 11, 2017; published online February 5, 2018. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 10(2), 021006 (Feb 05, 2018) (10 pages) Paper No: JMR-17-1321; doi: 10.1115/1.4038977 History: Received September 23, 2017; Revised December 11, 2017

This paper presents a new mechanics-based framework for the qualitative analysis and conceptual design of mechanical metamaterials, and specifically materials exhibiting auxetic behavior. The methodology is inspired by recent advances in the insightful synthesis of compliant mechanisms by visualizing a kinetostatic field of forces that flow through the mechanism geometry. The framework relates load flow in the members of the microstructure to the global material properties, thereby enabling a novel synthesis technique for auxetic microstructures. This understanding is used to qualitatively classify auxetic materials into two classes, namely, high-shear and low-shear microstructures. The ability to achieve additional attributes such as isotropy is shown to be related to the qualitative class that the microstructure belongs.

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Figures

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Fig. 1

Different load cases for SEBH: (a) prescribed horizontal strain, (b) prescribed vertical strain, (c) prescribed shear strain, and (d) prescribed bi-axial strain

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Fig. 2

Definition of load flow using the concept of transferred force in a displacement amplifying compliant inverter. (a) Force applied at i produces the same deformation at point j as a transferred force ftrj acting at j. (b) The deformed profile of the mechanism. (c) The transferred force in TC sets is maximally decoupled from other members.

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Fig. 3

A compliant dyad as a TC set

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Fig. 4

Output displacement of a dyad building block: (a) semicircular band capturing the possible directions of displacement and (b) actual displacement determined by the intersection of the semicircular band and the freedom direction of the constraint beam

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Fig. 5

(a) Microstructure geometry for auxetic material [25], (b) corresponding quarter cell geometry used for analysis, and (c) corresponding deformation of the microstructure geometry, (d) microstructure geometry exhibiting positive Poisson's ratio [13,15], (e) corresponding quarter cell geometry for analysis, and (f) corresponding deformation of the microstructure geometry

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Fig. 6

Load flow analysis in evaluating X direction stiffness: (a) boundary condition and applied strain for X axis, (b) substituting boundary conditions with beam constraints, and (c) load flow analysis on individual TC sets. Similarly, (d)–(f) analyzes load flow for evaluating stiffness in the Y direction.

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Fig. 7

Load flow analysis in evaluating bulk modulus and Poisson's ratio: (a) boundary condition and applied strain, (b) substituting boundary conditions with beam constraints, (c) load flow analysis on individual TC sets for applying X strain alone, and (d) load flow analysis on individual TC sets for applying Y strain alone

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Fig. 8

Load flow analysis in evaluating shear modulus: (a) boundary condition and applied strain, (b) substituting boundary conditions with beam flexures, (c) load flow analysis on individual TC sets for applying X strain alone, and (d) load flow analysis on individual TC sets for applying Y strain alone

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Fig. 9

Load flow analysis in evaluating shear modulus of a microstructure in Figs. 5(c) and 5(d): (a) boundary condition and applied strain for shear modulus analysis, (b) substituting boundary conditions with beam constraints, (c) load flow analysis on individual TC sets for applying X strain alone, (d) load flow analysis on individual TC sets for applying Y strain alone, and (e)–(h) load flow analysis to evaluate bulk modulus

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Fig. 10

Steps involved in the design of auxetic microstructure using the load flow framework

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Fig. 11

Fabricated prototype of the auxetic microstructure from Fig. 10(f): (a) undeformed microstructure and (b) a positive X direction displacement at the right face shows a clear positive Y direction deformation in the top face

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Fig. 12

Variation of Poisson's ratio with strain (a) shows for νxy and (b) shows for νyx

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Fig. 13

Design of alternative microstructure topologies from load flow-based framework and their deformations demonstrating auxetic behavior

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Fig. 14

Qualitative analysis of shear modulus using load flow visualization: (a)–(d) auxetic design with low shear modulus and (e)–(h) auxetic design with high shear modulus

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Fig. 15

(a) is the initial geometry, (b) is the optimized quarter cell geometry with isotropy, and (c) is the full microstructure geometry with isotropy

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