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Research Papers

Shear Wave Propagation and Band Gaps in Finitely Deformed Dielectric Elastomer Laminates: Long Wave Estimates and Exact Solution

[+] Author and Article Information
Pavel I. Galich

Department of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel

Stephan Rudykh

Department of Aerospace Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel
e-mail: rudykh@technion.ac.il

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 11, 2017; final manuscript received June 17, 2017; published online July 7, 2017. Assoc. Editor: Daining Fang.

J. Appl. Mech 84(9), 091002 (Jul 07, 2017) (12 pages) Paper No: JAM-17-1199; doi: 10.1115/1.4037159 History: Received April 11, 2017; Revised June 17, 2017

We analyze small amplitude shear waves (SWs) propagating in dielectric elastomer (DE) laminates subjected to finite deformations and electrostatic excitations. First, we derive long wave estimates for phase and group velocities of the shear waves propagating in any direction in DE laminates subjected to any homogenous deformation in the presence of an electric filed. To this end, we utilize a micromechanics-based energy potential for layered media with incompressible phases described by neo-Hookean ideal DE model. The long wave estimates reveal the significant influence of electric field on the shear wave propagation. However, there exists a configuration, for which electric field does not influence shear waves directly, and can only alter the shear waves through deformation. We study this specific configuration in detail, and derive an exact solution for the steady-state small amplitude waves propagating in the direction perpendicular to the finitely deformed DE layers subjected to electrostatic excitation. In agreement with the long wave estimate, the exact dispersion relation and the corresponding shear wave band gaps (SBGs)—forbidden frequency regions—are not influenced by electric field. However, SBGs in DE laminates with highly nonlinear electroelastic phases still can be manipulated by electric field through electrostatically induced deformation. In particular, SBGs in DE laminates with electroelastic Gent phases widen and shift toward higher frequencies under application of an electric field perpendicular to the layers. However, in laminates with neo-Hookean ideal DE phases, SBGs are not influenced either by electric field or by deformation. This is due to the competing mechanisms of two governing factors: changes in geometry and material properties induced by deformation. In this particular case, these two competing factors entirely cancel each other.

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Figures

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Fig. 1

Schematic representation of the undeformed (a) and subjected to the electromechanical load (b) periodic layered material with alternating phases a and b. A unit cell (c); (e1,e2,e3) is the orthonormal basis.

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Fig. 2

Induced stretch as function of dimensionless Lagrangian (a) and Eulerian (b) electric fields

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Fig. 3

The phase velocities of shear waves (59)(61) as functions of the dimensionless electric displacement for laminates with v(a)=0.2 and μ(a)/μ(b)=5. The phase velocities are normalized by the corresponding values in the absence of electric field.

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Fig. 4

Slowness curves for the out-of-plane (a)–(c) and in-plane (d)–(f) shear waves propagating in the DE laminates with different compositions subjected to electric field perpendicular to the layers. Scale is 0.4 per division, and slowness is normalized by μ⌣/ρ¯. Note that the horizontal and vertical axes with the corresponding labels n1/c¯ and n2/c¯ serve for showing the principal directions and physical quantity presented on the polar plot only.

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Fig. 5

Energy curves for the out-of-plane (a)–(c) and in-plane (d)–(f) shear waves propagating in the DE laminates with different compositions subjected to electric field perpendicular to the layers. Scale is 0.4 per division, where group velocity is normalized by ρ¯/μ⌣. Note that the horizontal and vertical lines with the corresponding labels (n1v) and (n2v) serve for showing the principal directions and physical quantity presented on the polar plot only.

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Fig. 6

Shear wave band gaps as functions of dimensionless Lagrangian electric displacement for waves propagating perpendicular to the layers. The band gap structures are true for any contrast in electric permittivities ε(a)(b) between the layers. The locking parameters for Gent phases are Jm(a)=Jm(b)=0.5. The densities of the layers are identical, i.e., ρ(a)(b) = 1. Frequency is normalized as fn=(ωH/2π)ρ¯/μ⌣.

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Fig. 7

Comparison of the exact solution for long waves (59), dispersion relation (92), and results reported by Shmuel and deBotton [30,35] for the shear waves propagating perpendicular to the layers in the laminates with incompressible ideal DE neo-Hookean phases subjected to electric field perpendicular to layers, namely DL=1.27. The laminate is made of VHB-4910 and ELASTOSIL RT-625: v(a) = 0.5, μ(a)(b) = 1.19, ε(a)(b) = 1.74, and ρ(a)(b) = 0.94, where μ(b) = 342 kPa, ε(b) = 2.7, and ρ(b) = 1020 kg/m3.

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Fig. 8

Comparison of the exact solution for long waves (59), dispersion relation (92), and dispersion relation by Shmuel and deBotton [35] for the waves propagating perpendicular to the layers in the laminates with incompressible neo-Hookean phases subjected to electric field perpendicular to layers, namely, DL=0.37, v(a)=0.8, μ(a)/μ(b)=ε(a)/ε(b)=20, and ρ(a)/ρ(b)=1

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