Piezoelectric Properties

• E-field due to stress --> Direct effect
  • Dipole created = polarisation= E-field
• D=P under short curcuit conditions (E=0)
• Einstein convention : add contributions of different components together
• voltage coeffcient: E-field in response to applied stress in open circuit conditions

Converse effect: electromechanical actuators

• strain due to E-field 
• 1 component of E is applied using electrodes on 1 side
• improve movement of actuators by stacking layers OR introducing Mechanical constraints --> bending action
• Bimorph/ uni morph
• direct and converse effects are thermodynamically equivalent
• use voigt notation to deal with matrices
• 3 direction is the poling direction
• 1 and 2 directions are equivalent 

Uniaxial coefficient d33 (disc)

• poled along x-axis , with stress along same direction
• change in polaristaion
• spark generators

Transverse coefficient d31(patch sensor): sheet poled perpendicular to sheet, stress parallel to the plane

Hydrostatic coeffcient dh

• disc poled along axis, hydrostatic stress
• underwater sensors

Hydrostatic figure of merit

• use to determine/select best piezoelectric material for sensor
• need high voltage output to cancel out effects of the connecting cable capacitance

Combined loading

• can get combined loading of electrical and mechanical conditions
• stress should be held constant if measuring permittivity as it may effect polarisation

Blocking force

• Stress required to reduce strain back to zero after an E-field has been applied
• Large for uniaxial mode devices
• reduced to <1N for bending 

Stack actuators

• large blocking force (~ 5kN)
• Total displacemnet = sum of displacement
• small displacement (~ 100 microns)

• Can relate the free (X=0) and clamped permittivity using the (x =0) coupling coefficient
• permittvity can be reduced by ~50% in clamped state relative to free state
• The compliance in open circuit (D=0) is less than in short circuit 
• can use the coupling coefficient to measure electromechanial energy conversion [mechanical --> electrcal]
• kp: planar (radial mode of vibration)
• kt: thickness mode

Resonators

• frequencies determined by the specimens dimensions and wave velocity
• high frequencies obtained in thickness mode (has smaller dimensions )
• flexural vibrations occur at lower frequencies

Nonlinear properties: at high field amplitudes linear relation breaks down

• above a threshold field (E_t) there is a gradual increase in relative permittivity with field amplitude
• P-E and x-E curves
  • at E>Ec nonlinear and hysteric behaviour
  • E<Ec: cycling around remnant state less prominence of non-linerarity
• x-E: at lower field can get positive or negative of E-field relative to remnant polaristaion
• dielectric permittivity: 3 regions - E<Et: permittvity ~ constant
  • E< Ec: Rayleigh behaviour 
  • E> Ec: sharp increase in permittivity due to large scale ferroelectric switching 
• P-E loop: observe as 2 parabolic functions

Converse effect: change in piezoelectric coefficient with E is linear;under high field x>> x under linear piezoelectric relation [where d is constant]

Direct effect

• d33 increases with increase pressure (linerarly)
• Charge density -X loop similar to x -E 

Origin of non-lineariry

• Transition of a domain wall across pinning defects
• Domain switching
  • Added mechanism for re-orientation of dipoles 
  • ease is dependent on the applied E-field

Relaxor 

• x-E inherently non-linear as have quadratic relation
• electrostrictive behaviour (ferroelectric) strongest near to peak of relative permittvity
• strong dependence on temeperature
• relaxor: transition to normal ferroelectric behavior upon cooling
• displacement decreases rapidly with increasing temperature