A nonlinear field theory of deformable dielectrics

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… A L Dielectric Elastomer a Q Q  l Compliant Electrode Reference State Current State Pelrine, Kornbluh, Pei, Joseph High-speed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000). 2 Electromechanical instability Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283.  Q l  Q thick thin Q Coexistent states: flat and wrinkled Plante, Dubowsky, Int. J. Solids and Structures 43, 7727 (2006). 3 Trouble with electric force Ei   ,i +Q +Q Di ,i  q Di  Ei Fi  qEi In a vacuum, force between charges can be measured Define electric field by E = F/Q +Q +Q In a solid, force between charges is NOT an operational concept Historical work •Toupin (1956) •Eringen (1963) •Tiersten (1971) …… Recent work •Dorfmann, Ogden (2005) •Landis, McMeeking (2005) •Suo, Zhao, Greene (2007) …… 4 Maxwell stress in vacuum (1864?) Ei   ,i Di ,i  q Di   0 Ei Fi  qEi  D j , j Ei  D j Ei , j  D j Ei , j D j Ei , j   0 E j E j ,i  0 2 Ek Ek ,i Ei , j   ,ij  E j ,i 0   Fi    0 E j Ei  Ek Ek ij  2  , j  ij   0 E j Ei  0 2 Ek Ek  ij P Q  0 2 E2 Q P 5 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   33   E 2 2 -----------+ - +++++++++ Electrostriction •In dielectrics, electric force is not an operational concept. •  varies with deformation in general. •Why E2 dependence? •How about the sign of the Maxwell stress? In solid, Maxwell stress has NO theoretical basis 6 All troubles are gone if we focus on measurable quantities Work done by the weight Pl Work done by the battery Q electrode dielectri c Q  ground A system of elastic conductors and dielectrics is specified by a free-energy function U l , Q  U  Pl  Q P l Pl , Q   U l , Q  , l l , Q   U l , Q  Q A transducer Suo, Zhao, Greene JMPS, in press. http://imechanica.org/node/635 U l , Q  is measurable 7 Intensive quantities Reference State Current State  Intensive Quantities  l/L A L l a Q Q s  P/ A ~ E  /L ~ D Q/ A (E   / l) ( D  Q / a) P Work done by the weight Pl  sA L    AL s Work done by the battery Q  EL  DA   AL ED ~   ~  ~ ~ ~ Free energy per unit volume W  , D   s  W /  8 ~ ~  E  W / D U  Pl  Q ~ ~ W  s  ED Ideal dielectric elastomer Decouple elastic and dielectric energy ~ W , D  W0    W1 D   Linear dielectric liquid D2 W1 D   2 D Q Q ~   D a 12 A Arruda-Boyce elastomer: 1 11 1  W0    I  3  I2 9  I 3  27  ... 20 N 1050 N 2 2      : small-strain shear modulus N: number of rigid units between neighboring crosslinks 2 2 I  1  2  3 2 9 Hysteresis and coexistent states Q l  Q ~ E  /L ~ D Q/ A (E   / l) ( D  Q / a) 10 Zhao, X, Hong, W., Suo, Z., 2007. http://imechanica.org/node/1283. Equilibrium & Stability Free energy of the system ~ G  L1L2 L3W 1, 2 , D  P1L1  P22 L2  Q 1   3 L3 Elastomer 1 L1 G weights battery  2 L2 Q  W   W   W ~  ~   s1 1    s2 2   ~  E D     L1 L2 L3  1  D     2  1  2W 2 1  2W 2 1  2W ~ 2  1  2  ~ D 2 2 1 2 2 2 D 2 2  2W  2W ~  2W ~  12  1D  2D ~ ~ 1 2 1 D 2 D P2 P1 W 1 Equilibrium state s1  s2  W  2 ~ W E ~ D , Equilibrium state becomes unstable when the Hessian ceases to be positive-definite det H   0 , 11 Pre-stresses enhance actuation 3 L3 1 L1  2 L2 Q P2 P1 Experiment: Pelrine, Kornbluh, Pei, Joseph Science 287, 836 (2000). Theory: Zhao, Suo http://imechanica.org/node/1456 12 Inhomogeneous field xi X, t  A field of weights FiK X, t   X K  ~ ~  2 xi  i siK dV    bi   2 i dV   ~i dA ti  X K   t        X K   ~ ~ ~  DK dV  q dV  dA   ~ A field of batteries E X, t     X, t  K X K Material laws W  siK FiK  DKEK siK ~ ~   ~ ~ W F, D F, D  FiK   ~ ~ ~ W F, D E K F, D  ~ DK     •Linear PDEs •Nonlinear material laws Q  Suo, Zhao, Greene JMPS, in press. http://imechanica.org/node/635 P 13 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