0
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

G. Radaelli

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

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

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 2.78×103. 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.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Lee, J.-H. , Singer, J. P. , and Thomas, E. L. , 2012, “Micro-/Nanostructured Mechanical Metamaterials,” Adv. Mater., 24(36), pp. 4782–4810. [CrossRef] [PubMed]
Zadpoor, A. A. , 2016, “Mechanical Meta-Materials,” Mater. Horiz., 3(5), pp. 371–381. [CrossRef]
Lakes, R. S. , 1987, “Foam Structures With a Negative Poisson's Ratio,” Science, 235(4792), pp. 1038–1040. [CrossRef] [PubMed]
Lakes, R. S. , 1993, “Design Considerations for Materials With Negative Poisson's Ratios,” ASME J. Mech. Des., 115(4), p. 696. [CrossRef]
Milton, G. W. , and Cherkaev, A. V. , 1995, “Which Elasticity Tensors are Realizable?,” ASME J. Eng. Mater. Technol., 117(4), p. 483. [CrossRef]
Grima, J. N. , and Evans, K. E. , 2000, “Auxetic Behavior From Rotating Squares,” J. Mater. Sci. Lett., 19(17), pp. 1563–1565. [CrossRef]
Milton, G. W. , 2013, “Complete Characterization of the Macroscopic Deformations of Periodic Unimode Metamaterials of Rigid Bars and Pivots,” J. Mech. Phys. Solids, 61(7), pp. 1543–1560. [CrossRef]
Bu¨ckmann, T. , Schittny, R. , Thiel, M. , Kadic, M. , Milton, G. W. , and Wegener, M. , 2014, “On Three-Dimensional Dilational Elastic Metamaterials,” New J. Phys., 16(3), p. 33032. [CrossRef]
Kadic, M. , Bückmann, T. , Stenger, N. , Thiel, M. , and Wegener, M. , 2012, “On the Practicability of Pentamode Mechanical Metamaterials,” Appl. Phys. Lett., 100(19), p. 191901. [CrossRef]
Bertoldi, K. , Reis, P. M. , Willshaw, S. , and Mullin, T. , 2010, “Negative Poisson's Ratio Behavior Induced by an Elastic Instability,” Adv. Mater., 22(3), pp. 361–366. [CrossRef] [PubMed]
Overvelde, J. T. B. , Shan, S. , and Bertoldi, K. , 2012, “Compaction Through Buckling in 2D Periodic, Soft and Porous Structures: Effect of Pore Shape,” Adv. Mater., 24(17), pp. 2337–2342. [CrossRef] [PubMed]
Florijn, B. , Coulais, C. , and van Hecke, M. , 2014, “Programmable Mechanical Metamaterials,” Phys. Rev. Lett., 113(17), p. 175503. [CrossRef] [PubMed]
Shan, S. , Kang, S. H. , Wang, P. , Qu, C. , Shian, S. , Chen, E. R. , and Bertoldi, K. , 2014, “Harnessing Multiple Folding Mechanisms in Soft Periodic Structures for Tunable Control of Elastic Waves,” Adv. Funct. Mater., 24(31), pp. 4935–4942. [CrossRef]
Shan, S. , Kang, S. H. , Zhao, Z. , Fang, L. , and Bertoldi, K. , 2015, “Design of Planar Isotropic Negative Poisson's Ratio Structures,” Extreme Mech. Lett., 4, pp. 96–102. [CrossRef]
Grima, J. N. , Mizzi, L. , Azzopardi, K. M. , and Gatt, R. , 2016, “Auxetic Perforated Mechanical Metamaterials With Randomly Oriented Cuts,” Adv. Mater., 28(2), pp. 385–389. [CrossRef] [PubMed]
Larsen, U. D. , Sigmund, O. , and Bouwstra, S. , 1996, “Design and Fabrication of Compliant Micromechanisms and Structures With Negative Poisson's Ratio,” IEEE Ninth International Workshop on Micro Electromechanical Systems, San Diego, CA, Feb. 11–15, pp. 365–371.
Lira, C. , Scarpa, F. , Olszewska, M. , and Celuch, M. , 2009, “The SILICOMB Cellular Structure: Mechanical and Dielectric Properties,” Phys. Status Solidi, 246(9), pp. 2055–2062. [CrossRef]
Grima, J. N. , Caruana-Gauci, R. , Attard, D. , and Gatt, R. , 2012, “Three-Dimensional Cellular Structures With Negative Poisson's Ratio and Negative Compressibility Properties,” Proc. R. Soc. A, 468(2146), pp. 3121–3138. [CrossRef]
Bu¨ckmann, T. , Stenger, N. , Kadic, M. , Kaschke, J. , Frölich, A. , Kennerknecht, T. , Eberl, C. , Thiel, M. , and Wegener, M. , 2012, “Tailored 3D Mechanical Metamaterials Made by Dip-in Direct-Laser-Writing Optical Lithography,” Adv. Mater., 24(20), pp. 2710–2714. [CrossRef] [PubMed]
Grima, J. N. , Oliveri, L. , Attard, D. , Ellul, B. , Gatt, R. , Cicala, G. , and Recca, G. , 2010, “Hexagonal Honeycombs With Zero Poisson's Ratios and Enhanced Stiffness,” Adv. Eng. Mater., 12(9), pp. 855–862. [CrossRef]
Olympio, K. R. , and Gandhi, F. , 2010, “Zero Poisson's Ratio Cellular Honeycombs for Flex Skins Undergoing One-Dimensional Morphing,” J. Intell. Mater. Syst. Struct., 21(17), pp. 1737–1753. [CrossRef]
Soman, P. , Fozdar, D. Y. , Lee, J. W. , Phadke, A. , Varghese, S. , and Chen, S. , 2012, “A Three-Dimensional Polymer Scaffolding Material Exhibiting a Zero Poisson's Ratio,” Soft Matter, 8(18), pp. 4946–4951. [CrossRef] [PubMed]
Silva, S. P. , Sabino, M. A. , Fernandes, E. M. , Correlo, V. M. , Boesel, L. F. , and Reis, R. L. , 2005, “Cork: Properties, Capabilities and Applications,” Int. Mater. Rev., 50(6), pp. 345–365. [CrossRef]
Wang, A.-J. , and McDowell, D. L. , 2004, “In-Plane Stiffness and Yield Strength of Periodic Metal Honeycombs,” J. Eng. Mater. Technol., 126(2), pp. 137–156. [CrossRef]
Zhu, H. X. , Fan, T. X. , and Zhang, D. , 2015, “Composite Materials With Enhanced Dimensionless Young's Modulus and Desired Poisson's Ratio,” Sci. Rep., 5(1), p. 14103. [CrossRef] [PubMed]
Lee, W. , Kang, D.-Y. , Song, J. , Moon, J. H. , and Kim, D. , 2016, “Controlled Unusual Stiffness of Mechanical Metamaterials,” Sci. Rep., 6, p. 20312. [CrossRef] [PubMed]
Wang, F. , Sigmund, O. , and Jensen, J. S. , 2014, “Design of Materials With Prescribed Nonlinear Properties,” J. Mech. Phys. Solids, 69, pp. 156–174. [CrossRef]
Almgren, R. F. , 1985, “An Isotropic Three-Dimensional Structure With Poisson's Ratio = −1,” J. Elast., 15(4), pp. 427–430. [CrossRef]
Cabras, L. , and Brun, M. , 2014, “Auxetic Two-Dimensional Lattices With Poisson's Ratio Arbitrarily Close to −1,” Proc. R. Soc. A, 470(2172), p. 20140538. [CrossRef]
Christensen, R. M. , 1987, “Sufficient Symmetry Conditions for Isotropy of the Elastic Moduli Tensor,” ASME J. Appl. Mech., 54(4), pp. 772–777.
Trease, B. P. , Moon, Y.-M. , and Kota, S. , 2005, “Design of Large-Displacement Compliant Joints,” ASME J. Mech. Des., 127(4), pp. 788–798.
Jutte, C. V. , and Kota, S. , 2008, “Design of Nonlinear Springs for Prescribed Load-Displacement Functions,” ASME J. Mech. Des., 130(8), p. 081403. [CrossRef]
Rahman, M. U. , and Zhou, H. , 2014, “Design of Constant Force Compliant Mechanisms,” Int. J. Eng. Res. Technol., 3(7), pp. 14–19. https://www.ijert.org/phocadownload/V3I7/IJERTV3IS070028.pdf
Saxena, A. , 2005, “Topology Design of Large Displacement Compliant Mechanisms With Multiple Materials and Multiple Output Ports,” Struct. Multidiscip. Optim., 30(6), pp. 477–490. [CrossRef]
Lahuerta, R. D. , Nigro, P. S. B. , Sim, E. T. , Pimenta, P. M. , and Silva, N. , 2014, “Design of Compliant Mechanism Considering Large Deformation Using Topology Optimization Method,” 13th International Symposium on Multiscale, Multifunctional and Functionally Graded Materials (MM&FGM), pp. 1–8.
Liu, L. , Xing, J. , Yang, Q. , and Luo, Y. , 2017, “Design of Large-Displacement Compliant Mechanisms by Topology Optimization Incorporating Modified Additive Hyperelasticity Technique,” Math. Probl. Eng., 2017, pp. 1–11.
Jagla, E. A. , and Dalvit, D. A. R. , 1991, “Null Length Springs: Some Curious Properties,” Am. J. Phys., 59(5), pp. 434–436. [CrossRef]
Herder, J. L. , 2001, “Energy-Free Systems: Theory, Conception, and Design of Statically Balanced Spring Mechanisms,” Delft University of Technology, Delft, The Netherlands.
Delissen, A. A. T. M. , Radaelli, G. , and Herder, J. L. , 2017, “Design and Optimization of a General Planar Zero Free Length Spring,” Mech. Mach. Theory, 117, pp. 56–77. [CrossRef]
de Payrebrune, K. M. , and O'Reilly, O. M. , 2017, “On the Development of Rod-Based Models for Pneumatically Actuated Soft Robot Arms: A Five-Parameter Constitutive Relation,” Int. J. Solids Struct., 120, pp. 226–235. [CrossRef]
Syam, W. P. , Jianwei, W. , Zhao, B. , Maskery, I. , Elmadih, W. , and Leach, R. , 2017, “Design and Analysis of Strut-Based Lattice Structures for Vibration Isolation,” Precis. Eng., 52, pp. 494–506.
Nagy, A. P. , 2011, “Isogeometric Design Optimisation,” Delft University of Technology, Delft, The Netherlands.
Hill, R. , 1963, “Elastic Properties of Reinforced Solids: Some Theoretical Principles,” J. Mech. Phys. Solids, 11(5), pp. 357–372. [CrossRef]
Smit, R. J. M. , Brekelmans, W. A. M. , and Meijer, H. E. H. , 1998, “Prediction of the Mechanical Behavior of Nonlinear Heterogeneous Systems by Multi-Level Finite Element Modeling,” Comput. Methods Appl. Mech. Eng., 155(1–2), pp. 181–192. [CrossRef]
Stratasys Inc., 2007, “P430 ABS Material Properties,” Stratasys Inc., Eden Prairie, MN.
Hernandez, R. , Slaughter, D. , Whaley, D. , Tate, J. , and Asiabanpour, B. , 2016, “Analyzing the Tensile, Compressive, and Flexural Properties of 3D Printed ABS P430 Plastic Based on Printing Orientation Using Fused Deposition Modeling,” 27th Annual International Solid Freeform Fabrication Symposium, Austin, TX, pp. 939–950.

Figures

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In