Electric Charge / Deformation and Polarization Matt Pharr ES 241 5/21/09

Electric Charge 10 protons, 10 electrons Net charge = = 0 Total charge is conserved A B SI Units: 1 Coulomb = = * elementary charges A B Q A = 0 +Q B = 0 Q net = 0 Q A = +4 +Q B = -4 Q net = 0

F’(Q’) F’’(Q’’) F’(Q’) F’’(Q’’) Capacitor Φ SI Units: 1 Volt = 1.602e-19 Joule

Measurement of Electric Potential I R V? Current measured with galvanometer Ohm’s Law: V = IR gives the potential

A Capacitor, a Weight, and a Battery Mechanical work Electric work In equilibrium

F(l,Q) and Stress Experimental Relation Recall: oil Φ l a +Q -Q Electric field Electric displacement Stress field Maxwell Stress

Deformable Dielectrics Reference StateCurrent State Nominal electric field Nominal electric displacement Nominal stress Stretch Nominal free- energy density L

Definition of Stress Nominal stress No weight, no stress??? Analogous to thermal expansion Stress-free deformation Small α Very stiff σ σ Stress generated due to constraint How is there deformation due to voltage change?

3D Homogeneous Deformation

Field Theory Recovers Maxwell Stresses in a Vacuum Electric energy per current volume Recall P Maxwell Stresses

Ideal Dielectric Elastomers Elastomer Structure Incompressibility Stretching ization Polar-

Electrostriction Well below extension limit Low cross-link density Polarization unaffected by deformation Close to extension limit High cross-link density Deformation affects polarization

Deformation Affects Polarization A model: quasi-linear dielectrics Ideal dielectric elastomer Quasi-linear dielectrics

Pull-in Instability Experimental Observation for oil This can lead to electrical breakdown

Pull-in Instability Exercise: Find critical electric field for instability subject to a biaxial force in the plane of membrane Assume ideal dielectric elastomer and incompressibility Choose a free energy of stretching function: Neo-Hookean law

Pull-in Instability For equal biaxial stress, s 1 = s 2 = s and λ 1 = λ 2 = λ Combining these two equations gives the following In equilibrium

Pull-in Instability reaches a peak when If s/μ = 0 If s/μ = 1 Larger stretch before breakdown

Solder Bumps e-e-

Multiple forces Chemical potential Electric current Package Warpage Temperature gradient Multiple phases Solder: Relation to Class Ideas from Paper Covered in Class Kinetic laws – chemical potential, diffusion flux Principle of virtual work – work conjugates Traction Deformation Rate Eulerian vs. Lagrangian

Backup Slides

Maxwell stress in vacuum (1873) P P A field of forces needed to maintain equilibrium of a field of charges Electrostatic field

Include Maxwell stress in a free-body diagram h “Free-body” diagram

Work-conjugate, or not Reference State Current 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