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Research Papers

Three-Dimensional-Printed Multistable Mechanical Metamaterials With a Deterministic Deformation Sequence

[+] Author and Article Information
Kaikai Che

G.W.W. School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: kche@gatech.edu

Chao Yuan

G.W.W. School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332;
State Key Laboratory for Strength and
Vibration of Mechanical Structures,
Department of Engineering Mechanics,
School of Aerospace Engineering,
Xi'an Jiaotong University,
Shaanxi, Xi'an 710049, China
e-mail: chao.yuan@me.gatech.edu

Jiangtao Wu

G.W.W. School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: jiangtaowu@gatech.edu

H. Jerry Qi

G.W.W. School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: qih@me.gatech.edu

Julien Meaud

G.W.W. School of Mechanical Engineering,
Georgia Institute of Technology,
Atlanta, GA 30332
e-mail: julien.meaud@me.gatech.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 29, 2016; final manuscript received September 10, 2016; published online October 14, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(1), 011004 (Oct 14, 2016) (10 pages) Paper No: JAM-16-1381; doi: 10.1115/1.4034706 History: Received July 29, 2016; Revised September 10, 2016

Multistable mechanical metamaterials are materials that have multiple stable configurations. The geometrical changes caused by the transition of the metamaterial from one stable state to another, could be exploited to obtain multifunctional and programmable materials. As the stimulus amplitude is varied, a multistable metamaterial goes through a sequence of stable configurations. However, this sequence (which we will call the deformation sequence) is unpredictable if the metamaterial consists of identical unit cells. This paper proposes to use small variations in the unit cell geometry to obtain a deterministic deformation sequence for one type of multistable metamaterial that consists of bistable unit cells. Based on an analytical model for a single unit cell and on the minimization of the total strain energy, a rigorous theoretical model is proposed to analyze the nonlinear mechanics of this type of metamaterials and to inform the designs. The proposed theoretical model is able to accurately predict the deformation sequence and the stress–strain curves that are observed in the finite-element simulations with an elastic constitutive model. A deterministic deformation sequence that matches the sequence predicted by the theory and finite-element simulations is obtained in experiments with 3D-printed samples. Furthermore, an excellent quantitative agreement between simulations and experiments is obtained once a viscoelastic constitutive model is introduced in the finite-element model.

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Figures

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Fig. 1

Theoretical model for 1D multistable metamaterials. (a) An example of a 1D multistable metamaterial with three rows, each with 3 unit cells. (b) Geometrical parameters of a single unit cell. (c) Simplified clamped–clamped beam model of the unit cell (a transverse force, f, is applied at the midpoint of the beam). (d) and (e) Comparisons of normalized force and normalized strain energy versus normalized deformation curves between theory and finite-element model for a single unit cell without third-mode imperfection (a3 = 0). (f) and (g) Comparisons of the normalized force and normalized strain energy versus normalized deformation curves between theory and finite-element model for a single unit cell without third-mode imperfection (a3 = 0.1). The geometrical parameters have the following values: l/h=30, h/t=10, w=15t, H=1.5t, and T = 30t.

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Fig. 2

(a) Multiple unit cells in series and (b) schematics of the theoretical model with multiple unit cells in series

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Fig. 3

Contour plot of normalized total strain energy for two identical unit cells in series as a function of the normalized deformations d¯tot and d¯1 (* denotes the local maximum), the numerical values on the contour lines correspond to the values of the normalized strain energy U¯tot. The solid line corresponds to one of the possible deformation paths during loading, the dashed line corresponds to the other possible path.

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Fig. 4

Theoretical analysis of the effect of varying the thickness from row to row on the deformation path. (a) Normalized force versus deformation curves for single unit cells of different thicknesses. (b) Normalized strain energy versus deformation curves for single unit cells of different thicknesses. In (a) and (b), solid lines correspond to the theoretical model and dashed lines to FEA simulations. (c) Contours of the total strain energy of a metamaterial with two unit cells with different thicknesses. The deformation path obtained using the theoretical model is shown for loading (solid line) and unloading (dashed line). (d) Comparison between the deformation paths of the metamaterial obtained using the theoretical model and FEA simulations. (e) Normalized force versus normalized total deformation (for loading only).

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Fig. 5

Theoretical analysis of the effect of mode shape imperfection on the deformation path. (a) Metamaterial with two unit cells in series: unit cell 2 has no mode shape imperfection (a3,2 = 0), unit cell 1 has a third-mode shape imperfection (a3,1 = 0.1). (b) and (c) Comparison curves of normalized force versus deformation and normalized strain energy versus deformation for single unit cell 2 with a3 = 0 and unit cell 1 with a3 = 0.1 (solid lines are theoretical results, dashed lines are results from FEA). (d) Contour plot of total strain energy for the metamaterial with two unit cells (shown in a). The deformation path obtained using the theoretical model is shown for loading (solid line) and unloading (dashed line). (e) Comparison between the deformation paths of the metamaterial (shown in a) obtained using the theoretical model and FEA simulations. (f) Normalized force versus normalized total deformation (for loading only).

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Fig. 6

Experimental validation of the method of thickness variation to obtain a deterministic deformation sequence (metamaterial A). (a) 5 × 5 multistable metamaterial with unit cells of thickness t varies from row to row. (b) The customized compression fixture. (c) and (d) Snapshots of the multistable architecture at different values of the effective strain ϵ ((c) experiment, (d) FEA).

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Fig. 7

Experimental validation of the method of mode shape imperfection to obtain a deterministic deformation sequence (metamaterial B). (a) 5 × 5 multistable metamaterial with unit cells of mode imperfection size a3 varies from row to row. (b) and (c) Snapshots of the multistable architecture at different values of the effective strain ϵ ((b) Experiment, (c) FEA).

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Fig. 8

Quantitative analysis of the deformation sequence. (a) Metamaterial A (see Fig. 6(a)). (b) Metamaterial B (see Fig. 7(a)). d¯i is the normalized deformation of row i defined as Eq. (11). The dashed black vertical lines are corresponding to the effective strains of the snapshots in Figs. 6 and 7.

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Fig. 9

Stress versus strain curves for metamaterial A (a and b) and for metamaterial B (c and d). In (a) and (c), experimental data are compared to FEA with a viscoelastic constitutive model. In (b) and (d), the theoretical model is compared to FEA simulations using an elastic constitutive model.

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Fig. 10

Comparison of relaxation Young's modulus between experiment and fit using generalized standard linear solid

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