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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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References

Pelrine, R. , Kornbluh, R. , Pei, Q.-B. , and Joseph, J. , 2000, “ High-Speed Electrically Actuated Elastomers With Strain Greater Than 100%,” Science, 287(5454), pp. 836–839. [CrossRef] [PubMed]
Bar-Cohen, Y. , 2004, Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges, Vol. 136, SPIE Press, Bellingham, WA.
Rudykh, S. , Bhattacharya, K. , and deBotton, G. , 2012, “ Snap-Through Actuation of Thick-Wall Electroactive Balloons,” Int. J. Nonlinear Mech., 47(2), pp. 206–209. [CrossRef]
Li, T. , Keplinger, C. , Baumgartner, R. , Bauer, S. , Yang, W. , and Suo, Z. , 2013, “ Giant Voltage-Induced Deformation in Dielectric Elastomers Near the Verge of Snap-Through Instability,” J. Mech. Phys. Solids, 61(2), pp. 611–628. [CrossRef]
McKay, T. , O'Brien, B. , Calius, E. , and Anderson, I. , 2010, “ An Integrated, Self-Priming Dielectric Elastomer Generator,” Appl. Phys. Lett., 97(6), p. 062911. [CrossRef]
Kornbluh, R. D. , Pelrine, R. , Prahlad, H. , Wong-Foy, A. , McCoy, B. , Kim, S. , Eckerle, J. , and Low, T. , 2012, “ From Boots to Buoys: Promises and Challenges of Dielectric Elastomer Energy Harvesting,” Electroactivity in Polymeric Materials, Springer, Berlin, pp. 67–93.
Rudykh, S. , and Boyce, M. , 2014, “ Transforming Wave Propagation in Layered Media Via Instability-Induced Interfacial Wrinkling,” Phys. Rev. Lett., 112(3), p. 034301. [CrossRef] [PubMed]
Galich, P. I. , and Rudykh, S. , 2015, “ Influence of Stiffening on Elastic Wave Propagation in Extremely Deformed Soft Matter: From Nearly Incompressible to Auxetic Materials,” Extreme Mech. Lett., 4, pp. 156–161. [CrossRef]
Galich, P. I. , and Rudykh, S. , 2015, “ Comment on “Disentangling Longitudinal and Shear Elastic Waves by neo-Hookean Soft Devices” [Appl. Phys. Lett., 106, 161903 (2015)],” Appl. Phys. Lett., 107(5), p. 056101. [CrossRef]
Galich, P. I. , Slesarenko, V. , and Rudykh, S. , 2017, “ Shear Wave Propagation in Finitely Deformed 3D Fiber-Reinforced Composites,” Int. J. Solids Struct., 110–111, pp. 294–304. [CrossRef]
Gei, M. , Roccabianca, S. , and Bacca, M. , 2011, “ Controlling Bandgap in Electroactive Polymer-Based Structures,” IEEE/ASME Trans. Mechatronics, 16(1), pp. 102–107. [CrossRef]
Galich, P. I. , and Rudykh, S. , 2016, “ Manipulating Pressure and Shear Elastic Waves in Dielectric Elastomers Via External Electric Stimuli,” Int. J. Solids Struct., 91, pp. 18–25. [CrossRef]
Wu, B. , Su, Y. , Chen, W. , and Zhang, C. , 2017, “ On Guided Circumferential Waves in Soft Electroactive Tubes Under Radially Inhomogeneous Biasing Fields,” J. Mech. Phys. Solids, 99, pp. 116–145. [CrossRef]
Yang, W.-P. , and Chen, L.-W. , 2008, “ The Tunable Acoustic Band Gaps of Two-Dimensional Phononic Crystals With a Dielectric Elastomer Cylindrical Actuator,” Smart Mater. Struct., 17(1), p. 015011. [CrossRef]
Celli, P. , Gonella, S. , Tajeddini, V. , Muliana, A. , Ahmed, S. , and Ounaies, Z. , 2017, “ Wave Control Through Soft Microstructural Curling: Bandgap Shifting, Reconfigurable Anisotropy and Switchable Chirality,” Smart Mater. Struct., 26(3), p. 035001. [CrossRef]
Toupin, R. A. , 1956, “ The Elastic Dielectric,” Arch. Ration. Mech. Anal., 5, pp. 849–915.
Dorfmann, A. , and Ogden, R. W. , 2005, “ Nonlinear Electroelasticity,” Acta. Mech., 174(3–4), pp. 167–183. [CrossRef]
McMeeking, R. M. , and Landis, C. M. , 2005, “ Electrostatic Forces and Stored Energy for Deformable Dielectric Materials,” ASME J. Appl. Mech., 72(4), pp. 581–590. [CrossRef]
Suo, Z. , Zhao, X. , and Greene, W. H. , 2008, “ A Nonlinear Field Theory of Deformable Dielectrics,” J. Mech. Phys. Solids, 56(2), pp. 467–486. [CrossRef]
Cohen, N. , Dayal, K. , and deBotton, G. , 2016, “ Electroelasticity of Polymer Networks,” J. Mech. Phys. Solids, 92, pp. 105–126. [CrossRef]
deBotton, G. , Tevet-Deree, L. , and Socolsky, E. A. , 2007, “ Electroactive Heterogeneous Polymers: Analysis and Applications to Laminated Composites,” Mech. Adv. Mater. Struct., 14(1), pp. 13–22. [CrossRef]
Tian, L. , Tevet-Deree, L. , deBotton, G. , and Bhattacharya, K. , 2012, “ Dielectric Elastomer Composites,” J. Mech. Phys. Solids, 60(1), pp. 181–198. [CrossRef]
Rudykh, S. , Lewinstein, A. , Uner, G. , and deBotton, G. , 2013, “ Analysis of Microstructural Induced Enhancement of Electromechanical Coupling in Soft Dielectrics,” Appl. Phys. Lett., 102(15), p. 151905. [CrossRef]
Rudykh, S. , and deBotton, G. , 2011, “ Stability of Anisotropic Electroactive Polymers With Application to Layered Media,” Z. Angew. Math. Phys., 62(6), pp. 1131–1142. [CrossRef]
Bertoldi, K. , and Gei, M. , 2011, “ Instabilities in Multilayered Soft Dielectrics,” J. Mech. Phys. Solids, 59(1), pp. 18–42. [CrossRef]
Rudykh, S. , Bhattacharya, K. , and deBotton, G. , 2014, “ Multiscale Instabilities in Soft Heterogeneous Dielectric Elastomers,” Proc. R. Soc. A, 470(2162), p. 20130618. [CrossRef]
Abu-Salih, S. , 2017, “ Analytical Study of Electromechanical Buckling of a Micro Spherical Elastic Film on a Compliant Substrate—Part I: Formulation and Linear Buckling of Periodic Patterns,” Int. J. Solids Struct., 109, pp. 180–188. [CrossRef]
Goshkoderia, A. , and Rudykh, S. , 2017, “ Electromechanical Macroscopic Instabilities in Soft Dielectric Elastomer Composites With Periodic Microstructures,” Eur. J. Mech. A, 65, pp. 243–256. [CrossRef]
Dorfmann, A. , and Ogden, R. W. , 2010, “ Electroelastic Waves in a Finitely Deformed Electroactive Material,” IMA J. Appl. Math., 75(4), pp. 603–636. [CrossRef]
Shmuel, G. , and deBotton, G. , 2012, “ Band-Gaps in Electrostatically Controlled Dielectric Laminates Subjected to Incremental Shear Motions,” J. Mech. Phys. Solids, 60(11), pp. 1970–1981. [CrossRef]
Kolle, M. , Lethbridge, A. , Kreysing, M. , Baumberg, J. , Aizenberg, J. , and Vukusic, P. , 2013, “ Bio-Inspired Band-Gap Tunable Elastic Optical Multilayer Fibers,” Adv. Mater., 25(15), pp. 2239–2245. [CrossRef] [PubMed]
Rudykh, S. , Ortiz, C. , and Boyce, M. , 2015, “ Flexibility and Protection by Design: Imbricated Hybrid Microstructures of Bio-Inspired Armor,” Soft Matter, 11(13), pp. 2547–2554. [CrossRef] [PubMed]
Slesarenko, V. , and Rudykh, S. , 2016, “ Harnessing Viscoelasticity and Instabilities for Tuning Wavy Patterns in Soft Layered Composites,” Soft Matter, 12(16), pp. 3677–3682. [CrossRef] [PubMed]
Rytov, S. , 1956, “ Acoustical Properties of a Thinly Laminated Medium,” Sov. Phys. Acoust., 2, pp. 68–80.
Shmuel, G. , and deBotton, G. , 2017, “ Corrigendum to ‘Band-Gaps in Electrostatically Controlled Dielectric Laminates Subjected to Incremental Shear Motions’ [J. Mech. Phys. Solids, 60 (2012) 1970–1981],” J. Mech. Phys. Solids, 105, pp. 21–24. [CrossRef]
Galich, P. I. , Fang, N. X. , Boyce, M. C. , and Rudykh, S. , 2017, “ Elastic Wave Propagation in Finitely Deformed Layered Materials,” J. Mech. Phys. Solids, 98, pp. 390–410. [CrossRef]
Spinelli, S. A. , and Lopez-Pamies, O. , 2015, “ Some Simple Explicit Results for the Elastic Dielectric Properties and Stability of Layered Composites,” Int. J. Eng. Sci., 88, pp. 15–28. [CrossRef]
Zhao, X. , Hong, W. , and Suo, Z. , 2007, “ Electromechanical Hysteresis and Coexistent States in Dielectric Elastomers,” Phys. Rev. B, 76(13), p. 134113. [CrossRef]
Musgrave, M. , 1970, Crystal Acoustics: Introduction to the Study of Elastic Waves and Vibrations in Crystals, Holden-Day, San Francisco, CA.
Nayfeh, A. H. , 1995, Wave Propagation in Layered Anisotropic Media: With Applications to Composites, Elsevier Science, New York.
Langenberg, K. J. , Marklein, R. , and Mayer, K. , 2010, “ Energy vs. Group Velocity for Elastic Waves in Homogeneous Anisotropic Solid Media,” IEEE URSI International Symposium on Electromagnetic Theory (EMTS), Berlin, Aug. 16–19, pp. 733–736.
Tiersten, H. F. , 1963, “ Thickness Vibrations of Piezoelectric Plates,” J. Acoust. Soc. Am., 35(1), pp. 53–58. [CrossRef]
Shmuel, G. , and Band, R. , 2016, “ Universality of the Frequency Spectrum of Laminates,” J. Mech. Phys. Solids, 92, pp. 127–136. [CrossRef]
Arruda, E. M. , and Boyce, M. C. , 1993, “ A Three-Dimensional Constitutive Model for the Large Stretch Behavior of Rubber Elastic Materials,” J. Mech. Phys. Solids, 41(2), pp. 389–412. [CrossRef]
Gent, A. N. , 1996, “ A New Constitutive Relation for Rubber,” Rubber Chem. Technol., 69(1), pp. 59–61. [CrossRef]
Babaee, S. , Wang, P. , and Bertoldi, K. , 2015, “ Three-Dimensional Adaptive Soft Phononic Crystals,” J. Appl. Phys., 117(24), p. 244903. [CrossRef]

Figures

Grahic Jump Location
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.

Grahic Jump Location
Fig. 2

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

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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.

Grahic Jump Location
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⌣.

Grahic Jump Location
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.

Grahic Jump Location
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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