A nonlinear field theory of deformable dielectrics Zhigang Suo - PowerPoint PPT Presentation

A nonlinear field theory of deformable dielectrics Zhigang Suo Harvard University Work with X. Zhao, W. Greene, Wei Hong, J. Zhou Talk 1 in Session 21-2-2, 10:00 am - 12:00 pm , Wednesday, 6 June 2007, McMat 2007, Austin, Texas Slides are available at http://imechanica.org/node/635 1

Dielectric elastomer actuators •Electromechanical coupling •Large deformation, light weight, low cost… •Soft robots, artificial muscles… Dielectric Φ A Elastomer + Q a L l − Q Compliant Electrode Reference State Current State Pelrine, Kornbluh, Pei, Joseph 2 High-speed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000).

Electromechanical instability Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283. Φ Φ + Q l − Q Q thick thin Coexistent states: flat and wrinkled Plante, Dubowsky, Int. J. Solids and Structures 43 , 7727 (2006). 3

Trouble with electric force = − Φ i = E D i q i , i , = ε +Q +Q D E i i In a vacuum, F = qE force between charges can be measured i i Define electric field by E = F/Q +Q +Q Historical work Recent work •Toupin (1956) •Dorfmann, Ogden (2005) •Eringen (1963) •Landis, McMeeking (2005) •Tiersten (1971) • Suo, Zhao, Greene (2007) In a solid, …… …… force between charges is NOT an operational concept 4

Maxwell stress in vacuum (1864?) = − Φ i = = ε E D i q D E i , i , 0 i i ( ) = = = − F qE D E D E D E i i j , j i j i j i , j , j ε ( ) i = − Φ = = ε = 0 E E D E E E E E i , j , ij j , i j i , j 0 j j , i k k , 2 ε ⎛ ⎞ P = ε − δ ⎜ ⎟ 0 F E E E E 0 i j i k k ij ⎝ ⎠ 2 + Q , j ε σ = 2 0 2 E ε − σ = ε − δ Q 0 E E E E ij 0 j i k k ij 2 5 P

Trouble with Maxwell stress in dielectrics Maxwell said that he didn’t make progress with dielectrics, but his qualms have not prevented people from using his name anyway… - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + - + + + + + + + + + Maxwell stress ε Electrostriction σ = − 2 2 E 33 •In dielectrics, electric force is not an operational concept. • ε varies with deformation in general. •Why E 2 dependence? •How about the sign of the Maxwell stress? 6 In solid, Maxwell stress has NO theoretical basis

All troubles are gone if we focus on measurable quantities P δ Work done by the weight l Φ δ Work done by the battery Q electrode A system of elastic conductors and dielectrics dielectric ( ) is specified by a free-energy function U , l Q δ Q Φ δ = δ + Φ δ U P l Q ground ( ) ( ) P ∂ ∂ δ ( ) U l , Q ( ) U l , Q l = Φ = P l , Q , l , Q ∂ ∂ l Q A transducer ( ) U , l Q is measurable 7 Suo, Zhao, Greene JMPS, in press. http://imechanica.org/node/635

Intensive quantities Reference State Current State Intensive Quantities λ = l / L Φ = + A s P / A Q a ~ L = Φ = Φ l ( E / l ) E / L − Q ~ = D = ( Q / a ) D Q / A P ( ) ( ) ( ) δλ δ = δ λ = P l sA L AL s Work done by the weight ( ) ( ) ( ) ~ ~ ~ ~ Φ δ = δ = δ Work done by the battery Q E L D A AL E D ( ) ~ λ Free energy per unit volume W , D = ∂ / λ ∂ ⎧ s W ~ δ ~ δ = δλ + δ = δ + Φ δ ⎨ W s E D U P l Q ~ ~ 8 = ∂ ∂ ⎩ E W / D

Ideal dielectric elastomer Decouple elastic and dielectric energy ( ) ( ) ( ) ~ λ = λ + W , D W W D 0 1 2 Q Q ~ ( ) D = = = λ = Linear dielectric liquid D D W D λ λ ε 1 a A 2 1 2 ( ) ( ) ⎡ ⎤ ( ) 1 1 11 = µ − + − + − + Arruda-Boyce elastomer: 2 3 W I 3 I 9 I 27 ... ⎢ ⎥ 0 ⎣ 2 ⎦ 2 20 N 1050 N µ : small-strain shear modulus N: number of rigid units between neighboring crosslinks = λ + λ + λ 2 2 2 9 I 1 2 3

Hysteresis and coexistent states Φ + Q l − Q ~ = Φ = Φ ( E / l ) E / L ~ = D = ( Q / a ) D Q / A 10 Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283.

Equilibrium & Stability Free energy of the system ( ) ~ = λ λ − λ − λ − Φ G L L L W , , D P L P L Q 1 2 3 1 2 1 1 1 2 2 2 λ λ Elastomer weights battery 3 L 3 L 3 3 λ λ ⎛ ⎞ ⎛ ⎞ δ ∂ ∂ ∂ ⎛ ⎞ 1 L 1 L G W W W ~ ~ ⎜ ⎟ ⎜ ⎟ = − δλ + − δλ + − δ ⎜ ⎟ 1 1 Φ Φ s s E D ⎜ ⎟ ⎜ ⎟ ~ ∂ λ ∂ λ ∂ 1 1 2 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ L L L D 1 2 3 1 2 λ λ Q Q 2 L 2 L ∂ ∂ ∂ 2 2 2 1 W 1 W 1 W 2 2 ~ + δλ + δλ + δ 2 2 2 D ~ ∂ λ ∂ λ ∂ 1 2 2 2 2 2 2 2 D 1 2 ∂ ∂ ∂ 2 2 2 W W W ~ ~ + δλ δλ + δλ δ + δλ δ D D ~ ~ ∂ λ ∂ λ ∂ λ ∂ ∂ λ ∂ 1 2 1 2 D D 1 2 1 2 P P P P 1 1 2 2 ∂ ∂ ∂ W W ~ W = = = s Equilibrium state s E ~ ∂ λ ∂ λ ∂ 1 2 D 1 2 Equilibrium state becomes unstable when the Hessian ceases to be positive-definite , ( ) = det 0 11 H ,

Pre-stresses enhance actuation λ λ 3 L 3 L 3 3 λ λ 1 L 1 L 1 1 Φ Φ λ λ Q Q 2 L 2 L 2 2 P P P P 1 1 2 2 Experiment: Pelrine, Kornbluh, Pei, Joseph Science 287, 836 (2000). Theory: Zhao, Suo http://imechanica.org/node/1456 12

Inhomogeneous field ( ) ⎛ ⎞ ∂ ξ ∂ ∂ 2 ~ x ( ) x X , t ∫ ∫ ∫ ~ ~ = ⎜ − ρ ⎟ ξ + ξ = i i i s dV b dV t dA ⎜ ⎟ A field of weights F X , t ∂ ∂ ∂ iK i i i i 2 iK ⎝ ⎠ X t X K K ( ) ∂ Φ ⎛ ⎞ ∂ η ~ ( ) X , t ~ = − A field of batteries ∫ ∫ ∫ ~ ~ ⎜ ⎟ − = η + η ω E , t X D dV q dV dA ⎜ ⎟ ∂ K ∂ K X ⎝ ⎠ X K K ~ δ ~ δ = δ + W s F D E Material laws iK iK K K ( ) ( ) ~ ~ ( ) ( ) ∂ ∂ , ~ W F , D ~ ~ W F D = = E F , D s F , D ~ ∂ K ∂ iK F D iK K •Linear PDEs •Nonlinear material laws δ Q Φ 13 Suo, Zhao, Greene P JMPS, in press. http://imechanica.org/node/635 δ l

Finite element method Thick State Transition Thin State Thin State Transition Thick State 14 Zhou, Hong, Zhao, Zhang, Suo http://imechanica.org/node/1447

Summary • A nonlinear field theory. No Maxwell stress. No electric body force. No polarization vector. • Electromechanical instability. • Hysteresis and coexistent states. • Finite element method. These slides are available at http://imechanica.org/node/635. iMechanica get together. Wednesday, 5.45pm-9:00pm, Room 2.120. Beer, snacks… 15