Research Papers

Kirigami Auxetic Pyramidal Core: Mechanical Properties and Wave Propagation Analysis in Damped Lattice

[+] Author and Article Information
Fabrizio Scarpa

Advanced Composites Centre for Innovation and Science (ACCIS),
University of Bristol,
BS8 1TR Bristol, UK
e-mail: f.scarpa@bristol.ac.uk

Manuel Collet

e-mail: morvan.ouisse@univ-fcomte.fr
Institut FEMTO-ST,
Département Mécanique Appliquée,
UMR CNRS 6174,
Besançon 25000,France

Kazuya Saito

Institute of Industrial Science (IIS),
University of Tokyo,
Tokyo 153-8505, Japan
e-mail: saito-k@iis.u-tokyo.ac.jp

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 13, 2012; final manuscript received May 4, 2013; published online June 6, 2013. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 135(4), 041001 (Jun 06, 2013) (11 pages) Paper No: VIB-12-1100; doi: 10.1115/1.4024433 History: Received April 13, 2012; Revised May 04, 2013

The work describes the manufacturing, mechanical properties, and wave propagation characteristics of a pyramidal lattice made exhibiting an auxetic (negative Poisson's ratio) behavior. Contrary to similar lattice tessellations produced using metal cores, the pyramidal lattice described in this work is manufactured using a kirigami (origami plus cutting pattern) technique, which can be applied to a large variety of thermoset and thermoplastic composites. Due to the particular geometry created through this manufacturing technique, the kirigami pyramidal lattice shows an inversion between in-plane and out-of-plane mechanical properties compared to classical honeycomb configurations. Long wavelength approximations are used to calculate the slowness curves, showing unusual zero-curvature phononic properties in the transverse plane. A novel 2D wave propagation technique based on Bloch waves for damped structures is also applied to evaluate the dispersion behavior of composite (Kevlar/epoxy) lattices with intrinsic hysteretic loss. The 2D wave propagation analysis shows evanescence directivity at different frequency bandwidths and complex modal behavior due to unusual deformation mechanism of the lattice.

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Queheillalt, D. T., and Wadley, H. N., 2005, “Pyramidal Lattice Truss Structures With Hollow Trusses,” Mater. Sci. Eng., A, 397, pp. 132–137. [CrossRef]
Queheillalt, D. T., and Wadley, H. N., 2009, “Titanium Alloy Lattice Truss Structures,” Mater. Des., 30, pp. 1966–1975. [CrossRef]
Pingle, S. M., Fleck, N. A., Deshpande, V. S., and Wadley, H. N. G., 2011, “Collapse Mechanism Maps for a Hollow Pyramidal Lattice,” Proc. R. Soc. London, Ser. A, 467(2128), pp. 985–1011. [CrossRef]
Wadley, H. N., 2006, “Multifunctional Periodic Cellular Metals,” Philos. Trans. R. Soc. London, Ser. A, 364(1838), pp. 31–68. [CrossRef]
Alderson, A., and Alderson, K. L., 2007, “Auxetic Materials,” Proc. Inst. Mech. Eng., Part G, 221, pp. 565–575. [CrossRef]
Gonella, S., and Ruzzene, M., 2008, “Analysis of In-Plane Wave Propagation in Hexagonal and Re-Entrant Lattices,” J. Sound Vib., 312, pp. 125–139. [CrossRef]
Spadoni, A., and Ruzzene, M., 2007, “Numerical and Experimental Analysis of the Static Compliance of Chiral Truss Core Airfoils,” J. Mech. Mater. Struct., 2, pp. 965–981. [CrossRef]
Tee, K. F., Spadoni, A., Scarpa, F., and Ruzzene, M., 2010, “Wave Propagation in Auxetic Tetrachiral Honeycombs,” ASME J. Vibr. Acoust., 132, p. 031007. [CrossRef]
Bettini, P., Airoldi, A., Sala, G., Landro, L. D., Ruzzene, M., and Spadoni, A., 2010, “Composite Chiral Structures for Morphing Airfoils: Numerical Analyses and Development of a Manufacturing Process,” Composites, Part B, 41, pp. 133–147. [CrossRef]
Martin, J., Heyder-Bruckner, J. J., Remillat, C. D. L., Scarpa, F., Potter, K., and Ruzzene, M., 2008, “The Hexachiral Prismatic Wingbox Concept,” Phys. Status Solidi B, 245, pp. 570–577. [CrossRef]
Grima, J. N., Williams, J. J., Gatt, R., and Evans, K. E., 2005, “Modelling of Auxetic Networked Polymers Built From Calix[4]Arene Building Blocks,” Mol. Simul., 31(13), pp. 907–913. [CrossRef]
Coluci, V. R., Hall, L. J., Kozlov, M. E., Zhang, M., Dantas, S. O., Galvão, D. S., and Baughman, R. H., 2008, “Modeling the Auxetic Transition for Carbon Nanotube Sheets,” Phys. Rev. B, 78, p. 115408. [CrossRef]
Hall, L. J., Coluci, V. R., Galvão, D. S., Kozlov, M. E., Zhang, M., Dantas, S. O., and Baughman, R. H., 2008, “Sign Change of Poisson's Ratio for Carbon Nanotube Sheets,” Science, 320(5875), pp. 504–507. [CrossRef] [PubMed]
Nojima, T., and Saito, K., 2006, “Development of Newly Designed Ultra-Light Core Structures,” JSME Int. J., Ser. A, 49, pp. 38–42. [CrossRef]
Saito, K., Scarpa, F., and Neville, R., 2011, “Origami Composite Auxetic Honeycomb,” 16th International Conference on Composite Structures (ICCS16), Porto, Portugul, June 28–30, Paper No. 117.
Saito, K., Agnese, F., and Scarpa, F., 2011, “A Cellular Kirigami Morphing Wingbox Concept,” J. Intell. Mater. Syst. Struct., 22(9), pp. 935–944. [CrossRef]
Norris, A. N., Shuvalov, A. L., and Kutsenko, A. A., “Analytical Formulation of Three-Dimensional Dynamic Homogenization for Periodic Elastic Systems,” Proc. R. Soc. London, Ser. A, 468, pp. 1629–1651. [CrossRef]
Srivastava, A., and Nemat-Nasser, S., 2012, “Overall Dynamic Properties of Three-Dimensional Periodic Elastic Composites,” Proc. R. Soc. London, Ser. A, 268, pp. 269–287. [CrossRef]
Willis, J. R., 2012, “Effective Constitutive Relations for Waves in Composites and Metamaterials,” Proc. R. Soc. London, Ser. A, 467, pp. 1865–1879. [CrossRef]
Wolfe, J. P., 1998, Imaging Phonons, Cambridge University, Cambridge, UK.
Hussein, M. I., 2009, “Theory of Damped Bloch Waves in Elastic Media,” Phys. Rev. B, 80, p. 212301. [CrossRef]
Hussein, M. I., and Frazier, M. J., 2010, “Band Structure of Phononic Crystals With General Damping,” J. Appl. Phys., 108(9), p. 093506. [CrossRef]
Moiseyenko, R. P., and Laude, V., 2011, “Material Loss Influence on the Complex Band Structure and Group Velocity in Phononic Crystals,” Phys. Rev. B, 83, p. 064301. [CrossRef]
Farzbod, F., and Leamy, M. J., 2011, “Analysis of Bloch's Method in Structures With Energy Dissipation,” ASME J. Vibr. Acoust., 133, p. 051010. [CrossRef]
Bornert, M., Bretheau, T., and Gilormini, P., 2001, Homogénéisation en Mécanique des Matériaux 1, Hermes Science Europe, Stanmore, UK.
Odegard, G., 2004, “Constitutive Modeling of Piezoelectric Polymer Composites,” Acta Mater., 52(18), pp. 5315–5330. [CrossRef]
Nayfeh, A. H., 1995, Wave Propagation in Layered Anisotropic Media With Applications to Composites, North Holland-Elsevier Science, Amsterdam.
Wang, L., and Ryne, K. G., 2007, “Existence of Extraordinary Zero-Curvature Slowness Curve in Anisotropic Elastic Media,” J. Acoust. Soc. Am., 122(4), pp. 1873–1875. [CrossRef] [PubMed]
Allaire, G., and Congas, C., 1998, “Bloch Waves Homogenization and Spectral Asymptotic Analysis,” J. Math. Pures Appl., 77, pp. 153–208.
Ichchou, M. N., Akrout, S., and Mencik, J., 2007, “Guided Waves Group and Energy Velocities Via Finite Elements,” J. Sound Vib., 305(4–5), pp. 931–944. [CrossRef]
Mencik, J., and Ichchou, M., 2005, “Multi-Mode Propagation and Diffusion in Structures Through Finite Elements,” Eur. J. Mech. A/Solids, 24(5), pp. 877–898. [CrossRef]
Houillon, L., Ichchou, M., and Jezequel, L., 2005, “Wave Motion in Thin-Walled Structures,” J. Sound Vib., 281(3–5), pp. 483–507. [CrossRef]
Collet, M., Ouisse, M., Ruzzene, M., and Ichchou, M., 2011, “A Floquet-Bloch Decomposition of the Elastodynamical Equations: Application to Bi-dimensional Wave's Dispersion Computation of Damped Mechanical System,” Int. J. Solids Struct., 48, pp. 2837–2848. [CrossRef]
Collet, M., Cunefare, K., and Ichchou, N., 2009, “Wave Motion Optimization in Periodically Distributed Shunted Piezocomposite Beam Structures,” J. Intell. Mater. Syst. Struct., 20(7), pp. 787–808. [CrossRef]
Gavrić,, L., 1995, “Computation of Propagative Waves in Free Rail Using a Finite Element Technique,” J. Sound Vib., 185, pp. 531–543. [CrossRef]
Maysenhölder, W., 1994, Körperschall-energie Grundlagen zur Berechnung von Energiedichten und Intensitäten, Wissenschaftliche Verlagsgesellschaft, Stuttgart, Germany.
Berthelot, J.-M., Assarar, M., Sefrani, Y., and Mahi, A. E., 2008, “Damping Analysis of Composite Materials and Structures,” Compos. Struct., 85(3), pp. 189–204. [CrossRef]
Lehoucq, R., Sorensen, D., and Yang, C., 1998, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems With Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, PA.
Schenk, O., and Gärtner, K., 2004, “Solving Unsymmetric Sparse Systems of Linear Equations With PARDISO,” FGCS, Future Gener. Comput. Syst., 20(3), pp. 475–487. [CrossRef]
Scarpa, F., Panayiotou, P., and Tomlinson, G., 2000, “Numerical and Experimental Uniaxial Loading on In-Plane Auxetic Honeycombs,” J. Strain Anal. Eng. Des., 35(5), pp. 383–388. [CrossRef]
Gibson, L. J., and Ashby, M. F., 1982, “The Mechanics of Three-Dimensional Cellular Materials,” Proc. R. Soc. London, Ser. A, 382(1782), pp. 43–59. [CrossRef]


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Fig. 3

(a) RVE of the pyramidal kirigami core with its nondimensional geometry parameters; (b) FE RVE with α = 1,β = 0.05,δ = 6,θ = 20 deg

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Fig. 4

Generic 3D periodic cells

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Fig. 2

Steps for the manufacturing for the kirigami auxetic pyramidal lattice core

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Fig. 1

Production of pyramidal lattice core using perforation/rolling/folding technique on metals (from Ref. [4])

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Fig. 7

Distribution of the Poisson's ratios (a) νxy and (b) νxz versus the angle θ for different δ parameters and α = 1

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Fig. 8

(a) Nondimensional in-plane and transverse Young's modulus versus the internal angle θ for different aspect ratios δ; (b) similar parametric curves for nondimensional in-plane shear and transverse modulus. For all calculations, α = 1.

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Fig. 10

Slowness curves in the xy plane for α = 1.0,θ = 20 deg

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Fig. 12

Dispersion curves of propagative modes of the studied system (imaginary part of λn(ω) or real part of kn(ω))

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Fig. 5

Layout of the damped kirigami lattice unit cell considered for the 2D wave propagation study

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Fig. 6

Unrefined (a) and refined (b) mesh cases

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Fig. 15

Directivity using the evanescence index in Eq. (23) saturated at unit value for (a) undamped and (b) damped lattice

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Fig. 13

Dispersion curves of propagative modes of the studied system (imaginary part of λn(ω) or real part of kn(ω))

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Fig. 14

Propagative wave numbers of damped system (Im(λn(ω)) for different angles

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Fig. 9

Slowness curve in the xz plane for α = 1,θ = 20 deg

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Fig. 11

Dispersion curves of all modes of the studied system (imaginary part of λn(ω) or real part of kn(ω))




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