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Research Papers

Tuning Elastic Waves in Soft Phononic Crystal Cylinders Via Large Deformation and Electromechanical Coupling

[+] Author and Article Information
Bin Wu, Weijian Zhou, Ronghao Bao

Department of Engineering Mechanics,
Zhejiang University,
Hangzhou 310027, China

Weiqiu Chen

Department of Engineering Mechanics,
Zhejiang University,
Hangzhou 310027, China;
State Key Laboratory of Fluid Power
and Mechatronic Systems,
Zhejiang University,
Hangzhou 310027, China;
Key Laboratory of Soft Machines and
Smart Devices of Zhejiang Province,
Zhejiang University,
Hangzhou 310027, China;
Soft Matter Research Center,
Zhejiang University,
Hangzhou 310027, China
e-mail: chenwq@zju.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received November 18, 2017; final manuscript received December 13, 2017; published online January 4, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(3), 031004 (Jan 04, 2018) (16 pages) Paper No: JAM-17-1643; doi: 10.1115/1.4038770 History: Received November 18, 2017; Revised December 13, 2017

Soft electroactive materials can undergo large deformation subjected to either mechanical or electrical stimulus, and hence, they can be excellent candidates for designing extremely flexible and adaptive structures and devices. This paper proposes a simple one-dimensional soft phononic crystal (PC) cylinder made of dielectric elastomer (DE) to show how large deformation and electric field can be used jointly to tune the longitudinal waves propagating in the PC. A series of soft electrodes, which are mechanically negligible, are placed periodically along the DE cylinder, and hence, the material can be regarded as uniform in the undeformed state. This is also the case for the uniformly prestretched state induced by a static axial force only. The effective periodicity of the structure is then achieved through two loading paths, i.e., by maintaining the longitudinal stretch and applying an electric voltage over any two neighboring electrodes or by holding the axial force and applying the voltage. All physical field variables for both configurations can be determined exactly based on the nonlinear theory of electroelasticity. An infinitesimal wave motion is further superimposed on the predeformed configurations, and the corresponding dispersion equations are derived analytically by invoking the linearized theory for incremental motions. Numerical examples are finally considered to show the tunability of wave propagation behavior in the soft PC cylinder. The outstanding performance regarding the band gap (BG) property of the proposed soft dielectric PC is clearly demonstrated by comparing with the conventional design adopting the hard piezoelectric material. One particular point that should be emphasized is that soft dielectric PCs are susceptible to various kinds of failure (buckling, electromechanical instability (EMI), electric breakdown (EB), etc.), imposing corresponding limits on the external stimuli. This has been carefully examined for the present soft PC cylinder such that the applied electric voltage is always assumed to be less than the critical voltage except for one case, in which we illustrate that the snap-through instability of the axially free PC cylinder made of a generalized Gent material may be used to efficiently trigger a sharp transition in the BGs.

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Figures

Grahic Jump Location
Fig. 1

Diagram of a soft DE cylinder with mechanically negligible electrodes bonded onto the two end (top and bottom) surfaces: (a) undeformed configuration, along with the reference coordinates (R,  Θ,  Z) and (b) deformed configuration induced by an axial electric voltage V and an axial force N, along with the current coordinates (r,  θ,  z)

Grahic Jump Location
Fig. 2

Geometry and actuation of an infinitely long PC made of identical DE subcylinders separated by mechanically negligible insulating glue layers: (a) undeformed configuration, (b) prestretched configuration induced by an axial force N only, (c) path A configuration: fixed prestretch combined with a periodically applied electric voltage V, and (d) path B configuration: fixed axial force combined with a periodically applied electric voltage V

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Fig. 3

Variations of prestretches λ1pre and λ3pre as functions of the dimensionless resultant axial force N¯ in the DE phononic cylinder in the absence of the electric voltage for both neo-Hookean and Gent models

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Fig. 4

Dimensionless critical voltages V¯c corresponding to α=1 and dimensionless EB voltages V¯EB versus the axial prestretch λ3 in the DE phononic cylinder (path A) for both neo-Hookean and Gent models

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Fig. 5

Variations of the initial radial stretch λ1 ((a) and (d)), the axial stress σ¯33 ((b) and (e)), and the axial force N¯ ((c) and (f)) as functions of the dimensionless electric voltage V¯ in the DE phononic cylinder (path A) at different axial prestretches for both neo-Hookean ((a)–(c)) and Gent ((d)–(f)) models

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Fig. 6

Dispersion diagrams of the incremental longitudinal waves in the DE phononic cylinder (path A) for the neo-Hookean model at different electric voltages and three different axial prestretches: (a) λ3=1, (b) λ3=1.5, and (c) λ3=3

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Fig. 7

Frequency limits of the first ((a) and (b)) and second ((c) and (d)) Bragg BGs versus the dimensionless electric voltage V¯ in the DE phononic cylinder (path A) at different axial prestretches for both neo-Hookean ((a) and (c)) and Gent ((b) and (d)) models

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Fig. 8

Frequency limits of the first Bragg BG versus the axial prestretch λ3 in the DE phononic cylinder (path A) at different electric voltages for both neo-Hookean (a) and Gent (b) models

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Fig. 9

Comparison of the long wavelength limit (43) with the exact solution (39) of the incremental longitudinal waves in the DE phononic cylinder (path A) at different axial prestretches and electric voltages for the neo-Hookean model

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Fig. 10

Variations of the radial stretch ratio λ1/λ1pre and the axial stretch ratio λ3/λ3pre as functions of the dimensionless electric voltage V¯ in the DE phononic cylinder (path B) at different axial forces for the neo-Hookean model, where λ1pre and λ3pre are the prestretches induced by the axial force only

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Fig. 11

Dimensionless electric voltage V¯ versus the radial stretch λ1 in the DE phononic cylinder (path B) for both neo-Hookean and Gent models at three different axial forces: (a) N¯=0, (b) N¯=2.5, and (c) N¯=5

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Fig. 12

Normalized critical voltages V¯c corresponding to α=1 and the normalized EB voltages V¯EB versus the axial force N¯ in the DE phononic cylinder (path B) for both neo-Hookean and Gent models

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Fig. 13

Dispersion diagrams of the incremental longitudinal waves in the DE phononic cylinder (path B) for the neo-Hookean model at different electric voltages and two different axial forces: (a) N¯=0 and (b) N¯=2.5

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Fig. 14

Frequency limits of the first Bragg BG versus the dimensionless electric voltage V¯ in the DE phononic cylinder (path B) at different axial forces for both neo-Hookean (a) and Gent (b) models

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Fig. 15

Frequency limits of the first Bragg BG versus the axial force N¯ in the DE phononic cylinder (path B) at different electric voltages for both neo-Hookean (a) and Gent (b) models

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Fig. 16

(a) Nonlinear response of the radial stretch λ1 to the dimensionless electric voltage V¯ in the axially free DE phononic cylinder for the neo-Hookean and Gent models. The snap-through transitions associated with the Gent model only are denoted by the blue-dashed arrows. (b) The frequency limits of the first Bragg BG versus the dimensionless electric voltage V¯ in the axially free DE phononic cylinder for the Gent model.

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