1 Theory of dielectric elastomers Zhigang Suo Harvard University 11:15 am – 12:00 noon, 11 January 2010, Monday PhD Winter School Dielectric Elastomer Transducers Monte Verita, Ascona, Switzerland, January

2 Dielectric elastomer Compliant Electrode Dielectric Elastomer Reference StateCurrent State Pelrine, Kornbluh, Pei, Joseph High-speed electrically actuated elastomers with strain greater than 100%. Science 287, 836 (2000).

3 Parallel-plate capacitor P P battery force electrode vacuum electrode Electric field Electric displacement field stress field  0, permittivity of vacuum Maxwell stress

Maxwell stress Electrostriction Trouble with Maxwell stress in dielectrics In general,  varies with deformation. In general, E 2 dependence has no special significance. Wrong sign? Our complaints:

5 An atom in an electric field Hydrogen atom External electric field displaces positive and negative charges somewhat. Polarization: Induce more charge on the electrodes. Deformation: Distort the shape of the electron cloud. battery

6 A dipole in an electric field Polar molecules External electric field reorients dipoles. Polarization: Induce more charge on the electrodes. Deformation: Distort the shape of the sample. battery

7 Field equations in vacuum, Maxwell (1873) A field of forces maintain equilibrium of a field of charges Electrostatic field P P Maxwell stress

8 Include Maxwell stress in force balance h “Free-body” diagram

9 James Clerk Maxwell ( ) “I have not been able to make the next step, namely, to account by mechanical considerations for these stresses in the dielectric. I therefore leave the theory at this point…” A Treatise on Electricity & Magnetism (1873), Article 111

10 Trouble with electric force in dielectrics In a vacuum, external force is needed to maintain equilibrium of charges +Q In a solid dielectric, force between charges is NOT an operational concept PP

11 The Feynman Lectures on Physics Volume II, p.10-8 (1964) “It is a difficult matter, generally speaking, to make a unique distinction between the electrical forces and mechanical forces due to solid material itself. Fortunately, no one ever really needs to know the answer to the question proposed. He may sometimes want to know how much strain there is going to be in a solid, and that can be worked out. But it is much more complicated than the simple result we got for liquids.”

12 All troubles are gone if we use measurable quantities Reference State Current State equilibrate elastomer and loads equations of state divide by volume name quantities Nominal True Suo, Zhao, Greene,J. Mech. Phys. Solids 56, 467 (2008)

13 Dielectric constant is insensitive to stretch Kofod, Sommer-Larsen, Kornbluh, Pelrine Journal of Intelligent Material Systems and Structures 14, 787 (2003). VHB 4910

14 Elasticity Polarization Ideal dielectric elastomer Dielectric behavior is liquid-like, unaffected by deformation. For an ideal dielectric elastomer, electromechanical coupling is purely a geometric effect: incompressibility

15 Ideal dielectric elastomer In terms of nominal quantities In terms of true quantities

16 Maxwell stress represented in three ways Compliant Electrode Dielectric Elastomer Reference State Current State For incompressible material, the 3 states of stress give the same deformation Uniaxial stress biaxial stresstriaxial stress

Maxwell stress Electrostriction

18 Non-ideal dielectric elastomer Dielectric constant Area ratio Wissler, Mazza, Sens. Actuators, A 138, 384 (2007). deformation affects dielectric constant

19 Quasi-linear dielectric elastomer Zhao, Suo, JAP 104, (2008)

20 The nominal vs. the true Reference StateCurrent State Battery does work True electric field and true electric displacement are NOT work-conjugate Nominal electric field and nominal electric displacement are work-conjugate Battery does work

21 Extend the theory to general loads

22 Ideal dielectric elastomer Reference State Current State Incompressibility Elastic energy density Equations of state

23 Neo-Hookean model Elastic energy density Equations of state Incompressibility

24 Extend the theory to inhomogeneous field

25 A field of markers: stretch Reference stateCurrent state Xx(X, t) X+dX x(X+dX, t) L l

26 A field of batteries: electric field X x(X, t) X+dX x(X+dX, t) ground Reference stateCurrent state L l

27 3D inhomogeneous field Need to specify a material model Condition of equilibrium

28 PDEs Suo, Zhao, Greene, J. Mech. Phys. Solids 56, 467 (2008) Toupin (1956), Eringen (1963), Tiersten (1971), Goulbourne, Mockensturm and Frecker (2005), Dorfmann & Ogden (2005), McMeeking & Landis (2005) …

29 The nominal vs the true Reference StateCurrent State

30 Stretch Polarization Ideal dielectric elastomers Zhao, Hong, Suo, Physical Review B 76, (2007) Liquid-like dielectric behavior, unaffected by deformation Ideal electromechanical coupling is purely a geometric effect:

31 Conditions of thermodynamic equilibrium Finite element method Solve for Legendre transformation

32 States of equilibrium Simple LayerRing Sphere Zhao and Suo, APL 93, (2008)

33 Oscillating Balloon Zhao and Suo, APL 93, (2008) Zhu, Cai, Suo, Excitation by sinusoidal voltage Oscillation of the balloon

34 Summary Ideal dielectric elastomer: dielectric behavior is liquid-like. Only for ideal dielectric elastomer, the Maxwell stress is valid. A more general model can represent both the Maxwell stress and electrostriction. Inhomogeneous, time-dependent field can also be modeled.