Hooke's Law and Dielectrics |
What is a constitutive equation? For mechanical problems, a constitutive equation describes
how a material strains when it is stressed, or vice-versa. Constitutive equations exist
also for electrical problems; they describe how charge moves in a (dielectric) material when it
is subjected to a voltage, or vice-versa.
Engineers are already familiar with the most common mechanical constitutive equation that applies for everyday metals and plastics. This equation is known as Hooke's Law and is written as: |
In words, this equation states: Strain = Compliance × Stress.
However, since piezoelectric materials are concerned with electrical properties too, we must also consider the constitutive equation for common dielectrics: |
In words, this equation states: ChargeDensity = Permittivity × ElectricField. |
Coupled Equation |
Piezoelectric materials combine these two seemingly dissimilar constitutive equations into one coupled equation, written as: |
The piezoelectric coupling terms are in the matrix
d.
In order to describe or model piezoelectric materials, one must have knowledge about the material's mechanical properties (compliance or stiffness), its electrical properties (permittivity), and its piezoelectric coupling properties. |
Matrix Subscript Definitions |
The subscripts in piezoelectric constitutive equations have very important meanings.
They describe the conditions under which the material property data was measured.
For example, the subscript E on the compliance matrix s_{E} means that the compliance data was measured under at least a constant, and preferably a zero, electric field. Likewise, the subscript T on the permittivity matrix e_{T} means that the permittivity data was measured under at least a constant, and preferably a zero, stress field. |