Constitutive relations for piezoelectric benders under various boundary conditions

In this article, the constitutive relations of three types of piezoelectric bender-the unimorph bender, bimorph bender and triple-layer bender-are derived based on beam theory under the quasi-static equilibrium condition. The relation coefficients are expressed in terms of the geometrical and material properties of the benders. Constitutive relations are derived for a range boundary conditions (fixed-free, fixed-roll and fixed-simply supported) under conditions allowing different lengths of the piezoelectric and elastic layers. The complicated constitutive relations can be easily formulated and verified by using the function for handling symbolic calculations. The derived constitutive relations may assist in the determination of the optimal values of geometrical parameters in the design of actuators and sensors for particular applications.