Defining a Piezoelectric Material
A piezoelectric material responds to an electric potential gradient by straining, while stress causes an electric potential gradient in the material. This coupling between electric potential gradient and strain is the material's piezoelectric property. The material will also have a dielectric property so that an electrical charge exists when the material has a potential gradient. Piezoelectric material behavior is discussed in Piezoelectric analysis.
The mechanical properties of the material must be modeled by linear elasticity (Linear Elastic Behavior). The mechanical behavior can be defined by
in terms of the piezoelectric stress coefficient matrix, eφmij, or by
in terms of the piezoelectric strain coefficient matrix, dφmkl. The electrical behavior is defined by
where
- σij
-
is the mechanical stress tensor;
- εij
-
is the strain tensor;
- qi
-
is the electric “displacement” vector;
- DEijkl
-
is the material's elastic stiffness matrix defined at zero electrical potential gradient (short circuit condition);
- eφmij
-
is the material's piezoelectric stress coefficient matrix, defining the stress σij caused by the electrical potential gradient Em in a fully constrained material (it can also be interpreted as the electrical displacement qm caused by the applied strain εij at a zero electrical potential gradient);
- dφmkl
-
is the material's piezoelectric strain coefficient matrix, defining the strain εkl caused by the electrical potential gradient Em in an unconstrained material (an alternative interpretation is given later in this section);
- φ
-
is the electrical potential;
- Dφ(ε)ij
-
is the material's dielectric property, defining the relation between the electric displacement qi and the electric potential gradient Ej for a fully constrained material; and
- Ei
-
is the electrical potential gradient vector, -∂φ/∂xi.
The material's electrical and electro-mechanical coupling behaviors are, thus, defined by its dielectric property, Dφ(ε)ij, and its piezoelectric stress coefficient matrix, eφmij, or its piezoelectric strain coefficient matrix, dφmkl. These properties are defined as part of the material definition (Material Data Definition).