Exploring Tunable Phononic Crystals using Dielectric Elastomers

Transcription

1 Exploring Tunable Phononic Crystals using Dielectric Elastomers Penn State CAV Spring Workshop 2017 April 26, 2017 Michael Jandron Naval Undersea Warfare Center, Newport, RI David Henann Brown University, Providence, RI Approved for Public Release // Distribution is Unlimited 1

2 2 Consider these disturbances Environmental (acoustical) Natural (structural) Military (acoustical) Household (acoustical) Entertainment (acoustical and structural)

3 3 Phononic Crystals & Metamaterials Negative refraction Economou, University of Crete, Greece Waveguide of phonons Laude, CNRS, France Acoustic cloaking Norris, A. N., Proc. of the Royal Society A, 2008 Filtering Sound Current work Focusing Sound Lin, J. Appl. Phys. (2012)

4 4 Phononic Crystals Basic Principles Time space domain Frequency wavevector domain 1 st Bragg Plane 2 nd Bragg Plane 3 rd Bragg Plane

5 5 Phononic Crystals Basic Principles Consider homogeneous solid Consider 50/50 Steel in Epoxy 2 In-Plane Polarization Anti-Plane Polarization G X G X

6 6 2D Phononic Crystals In-plane shear response Anti-plane shear response 50 Displacement Transmissibility 50 Displacement Transmissibility Reduced Frequency Reduced Frequency

7 7 2D Phononic Crystals Efficient to explore in the frequency domain All directions of propagation explored In-plane shear response Anti-plane shear response Normalized Frequency db Wavevector 50 Normalized Frequency db Wavevector

8 8 2D Phononic Crystals Wavevector Wavevector

9 9 Past Research on Phononic Crystals Sigalas and Economou, JSV 1992 Kushwaha et. al., PRL 1993, PRB 1994 Bertoldi and Boyce, PRB 2008 Shim et. al., IJSS 2015 From linear elastics in mid 1990 s to present interest soft elastomers

10 10 Dielectric Elastomers Electric insulators made up of molecules that polarize in response to an electric field Roentgen (1880) Sacerdote (1899) Eguchi (1925) Toupin (1956, 1963) Pelrine, et. al. (1998, 2000) Kornbluh, et. al. (2000) Bar-Cohen, et. al. (2001, 2002) Carpi, et. al. (2005, 2008, 2009) Popularized in late 90s McMeeking and Landis (2005, 2007) Dorfmann, et. al. (2005, 2010) Suo, et. al. (2008) Bustamante, et. al. (2009) Henann, et. al. (2013)

11 11 Dielectric Elastomers Capri et. al., IEEE Trans. Mech 2010 Wang et. al. J. Appl. Mech Kornbluh et. al., Proc. SPIE 7976, 2011 Keplinger et. al., Science 2013 Rosset et. al., EPFL-LMTS

12 PCs of Dielectric Elastomers Unactuated Unactuated Actuated Michael Jandron, Naval Undersea Warfare Center Actuated 12

13 13 Modeling of PCs of Dielectric Elastomers (1) Considering an infinitely periodic material (2) Electric field in dielectric elastomer causes finite deformation Band-gap change? Tunability?

14 14 Two Step Solution Approach Step 1: Solve for non-linear, quasi-static, electro-mechanical response Increasing electric field Step 2: Solve for linear wave propagation Increasing frequency

15 Consider the square phononic crystal Both matrix and inclusions are dielectric elastomers Subject to an Anti-plane electric field* Michael Jandron, Naval Undersea Warfare Center *Referential Miller index representation 15

16 16 Anti-plane actuation (C) (D) (A) (B)

17 Consider the square phononic crystal Both matrix and inclusions are dielectric elastomers Subject to an In-plane electric field* Michael Jandron, Naval Undersea Warfare Center *Referential Miller index representation 17

18 18 In-plane actuation

19 19 Consider the hexagonal phononic crystal Both matrix and inclusions are dielectric elastomers Subject to an In-plane electric field* *Referential Miller direction

20 20 In-plane actuation

21 21 Research Questions What characteristics show the most promising tunability? Can we simulate electromechanical instabilities accurately? Can we harness the instabilities for enhanced tunability?

22 22 Future Work Constitutive model updates Viscoelastic additions Reese (1997) Bergstrom (1998) Linder (2011) Toyjanova (2014) Hossain(2012, 2015) Wang (2016) Blue = Load Control Red = Voltage Control Force 3M VHB Hossain(2015) Stretch Explore shapes/symmetries/prestretching Chirality, Holes, Fluid-filled Chirality Holes/Conductors? Vibration diodes Interesting tunability? Fluid-filled? Lucklum (2009) Harnessing electrocavitation instability Electromechanically triggered deformation Thin plates resist deformation until they buckle Build a phononic crystal of these? Interesting tunability? Wang et. al. (2012) G.I. Taylor (1966) Florijn et. al. (2016)

23 23 Conclusions Summary: 1. Dielectric elastomers enable electrically tunable phononic crystals. 2. Band-gaps are strongly influenced by an electric field. 3. Electro-mechanical instabilities enable enhanced tunability. Further Research Questions: 1. Role of viscoelasticity and damping? 2. Can we develop a predictive model of more electro-mechanical instabilities? 3. How can these instabilities be used for band-gap tunability?

24 Backup 24

25 25 Step 1 Kinematics Kinematics: Deformation gradient Volume ratio Left Cauchy-Green tensor McMeeking and Landis (2005, 2007), Dorfmann, Ogden, and Bustamante (2005, 2006, 2009) Suo, Zhao, and Greene (2008), Henann (2013)

26 26 Step 1 Electrical Quantities Kinematics: Deformation gradient Volume ratio Left Cauchy-Green tensor Electric potential and field: Electric potential Referential electric field Spatial electric field McMeeking and Landis (2005, 2007), Dorfmann, Ogden, and Bustamante (2005, 2006, 2009) Suo, Zhao, and Greene (2008), Henann (2013)

27 27 Step 1 Constitutive Relations & Balance Equations Cauchy Stress: Gent rubber elasticity Bulk response Electrostatic stress Electric Displacement: is the ground-state (long-term) shear modulus is the bulk modulus is a Gent material parameter denoting the stiffening of polymer chains is the electric permittivity of material PDEs for Electrostatics Equilibrium: Gauss s law: Solved in Abaqus: Henann, Chester, and Bertoldi, JMPS (2013)

28 Verification: Match analytic for homogeneous deformation Michael Jandron, Naval Undersea Warfare Center 28 Now we can use charge control to get through pull-in instability 1.5 I m =1 I m =5 I m =7 Exact Abaqus 1 I m =10 I m = Henann, Chester, and Bertoldi, JMPS (2013)

29 In Abaqus we make use of a 3D RVE Representative Volume Element (RVE) Superscript denotes degree of freedom Subscript denotes node Assumed infinitely periodic in 3D (Tvergard 1982, Socrate 1995, Danielsson 2002, Parsons 2006) Michael Jandron, Naval Undersea Warfare Center 29

30 30 Step 2 Linear, dynamic governing equations Introduce spatial increments: Gradient quantities: Which leads to the spatially linearized constitutive equations: Where the material tangents are defined as: Linearized governing equations become:

31 31 Step 2 Linear, dynamic governing equations Consider displacement and electric potential increments corresponding to a homogeneous plane wave: Defining acoustic quantities wave vector angular frequency and constitute the mode shape. Inserting into the linearized governing equations we obtain the eigenvalue problem electromechanical acoustic tensor

32 32 Step 2 Linear, dynamic governing equations With the acoustic quantities are where is the standard acoustic tensor. The eigenvalue problem then reduces to Beyond influencing the initial state of deformation, the electrical loading does not affect wave propagation in the ideal dielectric elastomer. This eigenvalue problem is also solved in Abaqus for given states of nonlinear deformation.

33 33 Bloch-Floquet Constraints Bloch-Floquet constraints allow us to impose an arbitrary bloch wavevector on a periodic domain where is the bloch wavevector, and is the vector between two points in the constraint For Abaqus decompose into real and imaginary parts Real Imaginary For Abaqus must solve constraints as part of system Bloch, F. Z. Physik (1929)

34 34 Mapping of Wavevectors But what wavevectors do we need? D Referential Spatial R Given referential lattice vectors we can compute the referential reciprocal vectors Then, given the deformation gradient we can map the referential to spatial: Then the boundary of the Irreducible Brillouin Zone (IBZ) is traced to define the wavevector in spatial coordinates.

35 Verification: Compare Abaqus to PWE for anti-plane actuation homogeneous deformation Plane Wave Expansion (PWE) is a popular analytical method to compute dispersion relations for phononic crystals (Kushwaha 1993, 1994) we use this for verification In-plane shear response Anti-plane shear response Representative Volume Element Irreducible Brillouin Zone (IBZ) Black = Abaqus Red = PWE Michael Jandron, Naval Undersea Warfare Center 35

36 36 Consider the hexagonal phononic crystal Both matrix and inclusions are dielectric elastomers Subject to an In-plane electric field* *Referential Miller direction

37 37 In-plane actuation

38 38 Halve mismatch Effect of mismatch Double mismatch

39 39 Effect of and mismatch Similar deformation but very different band-gaps

40 40 Effect of mismatch