2.4 Piezoelectric coefficients and constants

Piezoelectric coefficients and constants are crucial for understanding how materials convert mechanical energy to electrical energy and vice versa. These values help engineers design and optimize devices that harness piezoelectric effects for various applications.

Charge and voltage constants, stress constants, and coupling factors describe a material's ability to generate electricity from stress or deform under electric fields. Elastic and dielectric properties further characterize how piezoelectric materials behave under different conditions.

Piezoelectric Constants

Charge and Voltage Constants

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• Piezoelectric charge constant (d) measures the strain produced per unit of applied electric field
  • Expressed in meters per volt (m/V) or coulombs per newton (C/N)
  • Relates mechanical strain to applied electric field
  • Higher d values indicate greater piezoelectric effect (PZT ceramics typically have d33 values around 300-600 pC/N)
• Piezoelectric voltage constant (g) represents the electric field generated per unit of mechanical stress
  • Measured in volt-meters per newton (Vm/N)
  • Describes the ability of a material to generate voltage in response to applied stress
  • Materials with high g constants are suitable for sensor applications (quartz has a g11 value of about 50 x 10^-3 Vm/N)

Stress Constant and Coupling Factor

• Piezoelectric stress constant (e) relates the stress produced to the applied electric field
  • Expressed in newtons per coulomb (N/C) or coulombs per square meter (C/m^2)
  • Represents the material's ability to convert electrical energy into mechanical stress
  • Used in actuator design calculations (PZT ceramics can have e33 values around 15-25 C/m^2)
• Electromechanical coupling factor (k) quantifies the efficiency of energy conversion
  • Dimensionless parameter ranging from 0 to 1
  • Indicates how much of the input energy is converted to the desired form of energy
  • Higher k values suggest better energy conversion (PZT ceramics can have k33 values of 0.6-0.75)

Material Properties

Elastic Compliance and Stiffness

• Elastic compliance (s) measures the strain produced per unit of applied stress
  • Expressed in square meters per newton (m^2/N)
  • Inverse of elastic stiffness
  • Describes the material's ability to deform under stress (typical s11 values for PZT ceramics range from 10-16 x 10^-12 m^2/N)
• Elastic stiffness (c) represents the stress required to produce a unit strain
  • Measured in newtons per square meter (N/m^2)
  • Inverse of elastic compliance
  • Indicates the material's resistance to deformation (c11 values for PZT ceramics can be around 60-120 GPa)

Dielectric Properties

• Dielectric permittivity (ε) quantifies a material's ability to store electrical energy
  • Expressed in farads per meter (F/m)
  • Often reported as relative permittivity (εr) compared to vacuum permittivity
  • Higher values indicate greater charge storage capacity (PZT ceramics can have εr values of 1000-3000)
• Dielectric loss tangent (tan δ) measures the energy dissipation in the material
  • Dimensionless parameter
  • Lower values indicate less energy loss during electrical cycling
  • Important for high-frequency applications (typical tan δ values for PZT ceramics range from 0.01 to 0.03)

Mathematical Representation

Tensor Notation and Directional Properties

• Tensor notation uses subscripts to represent directions and properties
  • First subscript denotes the direction of applied stimulus
  • Second subscript indicates the direction of measured response
  • Examples: d31 represents strain in direction 3 due to electric field in direction 1
• Directional properties vary due to crystal structure anisotropy
  • Longitudinal effect: stimulus and response in same direction (d33, g33)
  • Transverse effect: stimulus and response in perpendicular directions (d31, g31)
  • Shear effect: involves rotational deformation (d15, g15)

Constitutive Equations and Matrix Form

• Constitutive equations describe the relationships between electrical and mechanical variables
  • Strain-charge form: $S = sT + dE$ and $D = dT + εE$
  • Stress-charge form: $T = cS - eE$ and $D = eS + εE$
  • S: strain, T: stress, E: electric field, D: electric displacement
• Matrix form simplifies complex tensor relationships
  • 6x6 matrices for elastic properties
  • 3x6 matrices for piezoelectric constants
  • 3x3 matrices for dielectric properties
• Voigt notation reduces 3D tensor to 2D matrix representation
  • Combines symmetrical tensor components
  • Simplifies calculations and material property representation