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Research Papers: Design Automation

# Design of an Isotropic Metamaterial With Constant Stiffness and Zero Poisson's Ratio Over Large Deformations

[+] Author and Article Information
A. Delissen

Department of Precision and Microsystems
Engineering,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands
e-mail: A.A.T.M.delissen@tudelft.nl

Department of Precision and Microsystems
Engineering,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands

L. A. Shaw

Department of Mechanical and Aerospace
Engineering,
University of California, Los Angeles,
420 Westwood Plaza,
Los Angeles, CA 90095
e-mail: lucas.shaw@engineering.ucla.edu

J. B. Hopkins

Department of Mechanical and Aerospace
Engineering,
University of California, Los Angeles,
420 Westwood Plaza,
Los Angeles, CA 90095
e-mail: hopkins@seas.ucla.edu

J. L. Herder

Department of Precision and Microsystems
Engineering,
Delft University of Technology,
Mekelweg 2,
Delft 2628 CD, The Netherlands
e-mail: J.L.Herder@tudelft.nl

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 4, 2018; final manuscript received August 1, 2018; published online September 7, 2018. Assoc. Editor: Carolyn Seepersad.

J. Mech. Des 140(11), 111405 (Sep 07, 2018) (10 pages) Paper No: MD-18-1188; doi: 10.1115/1.4041170 History: Received March 04, 2018; Revised August 01, 2018

## Abstract

A great deal of engineering effort is focused on changing mechanical material properties by creating microstructural architectures instead of modifying chemical composition. This results in meta-materials, which can exhibit properties not found in natural materials and can be tuned to the needs of the user. To change Poisson's ratio and Young's modulus, many current designs exploit mechanisms and hinges to obtain the desired behavior. However, this can lead to nonlinear material properties and anisotropy, especially for large strains. In this work, we propose a new material design that makes use of curved leaf springs in a planar lattice. First, analytical ideal springs are employed to establish sufficient conditions for linear elasticity, isotropy, and a zero Poisson's ratio. Additionally, Young's modulus is directly related to the spring stiffness. Second, a design method from the literature is employed to obtain a spring, closely matching the desired properties. Next, numerical simulations of larger lattices show that the expectations hold, and a feasible material design is presented with an in-plane Young's modulus error of only 2% and Poisson's ratio of $2.78×10−3$. These properties are isotropic and linear up to compressive and tensile strains of 0.12. The manufacturability and validity of the numerical model is shown by a prototype.

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## Figures

Fig. 1

(a) The unit cell with the springs indicated, two springs exist between nodes 2 and 4. Note that in the unloaded configuration, the springs would show a zero length. (b) The hexagonal grid shown with axial unit vectors. Some arbitrary axial coordinates are indicated. (c) Displacements and forces (arbitrarily chosen) in the unit cell. The displacement of node 1 is not shown for clarity.

Fig. 2

The spring needs to fit within the indicated feasible area (gray fill). The shape of the spring is defined by the dotted sections, which are positioned by control points (circles). These points are penalized when moving outside of the feasible area. The flexible springs are connected via rigid members to the nodes.

Fig. 3

Different sizes of (unrotated) RVE (a) a (I,J) = (1,1) RVE and (b) (I,J) = (2,3). Different boundaries are indicated. All the rigid members are black and the springs are drawn in between. (c) A general RVE with periodic boundary conditions shown.

Fig. 4

(a) The final spring design with the spring indicated in the middle, within the design area (dashed). The unit cell length is l and the spring thickness is to scale. (b) The deformed spring into different directions, to the maximum displacement uend.

Fig. 5

Spring characteristics in different displacement directions, indicated by different colors, corresponding to those in Fig. 4(b). The dashed line and circles are the desired values at the sample points used for optimization. Positive axial displacements are those to the right in Fig. 4(b) and vice versa negative to the left. (a) The axial force profile, (b) the transverse force profile, and (c) the axial force error.

Fig. 10

(a) The measured stress–strain curve, compared with the desired analytical curve. (b) The transverse strain–strain curve, with the analytical approximation and linear trend-lines indicated. Note that the slope of this line is equal to Poisson's ratio (with a minus sign). The thick colored lines are the averaged results of each sample.

Fig. 6

The lattice of (a) I,J=(1,1) and (b) I,J=(3,3) strained in extension. The undeformed lattice is shown in gray on the background.

Fig. 7

The responses in terms of (a) stress in the x-directionσx and (b) transverse strain εy. The maximum and minimum strain limits are indicated (dotted), as is the analytical estimation (dashed). Different orientations are given in different colors.

Fig. 8

Over a strain range, the material properties are shown: (a) the modulus of elasticity E and (b) Poisson's ratio ν. Different colors indicate different lattice orientations. The maximum and minimum strain limits as well as the expected properties are indicated.

Fig. 9

(a) The 0deg and (b) the 30deg lattice, loaded in extension. (c) Video tracking with the observed features indicated. The middle lines are used to calculate strain in the x-direction, the lines just inside the sample to normalize and correct for camera misalignments, and the lines at the top and bottom edges to determine the strain in the y-direction.

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