# Piezoelectric beam

## On a concept to avoid electromechanical locking in a piezoelectric finite beam element

*D. Legner, J. Wackerfuß, S. Klinkel, W. Wagner*Within the development process of piezoelectric sensor and actuator applications, numerical tools are essential. These tools solve approximately the electromechanical boundary value problems received from physical descriptions. Since many piezoelectric components are generally thin rod like structures, a piezoelectric finite beam element is presented in this contribution which can be used effectively to analyze a wide range of piezoelectric devices. The mechanical strains and the electric field are coupled by the constitutive relations. This results in incompatibilities within the numerical approximations in case of bending dominated problems, which can be compared to those causing the well-known shear locking in purely mechanical problems. This effect occurs in standard finite element formulations, where the mechanical and electrical degrees of freedom are derived from standard kinematic assumptions.

The present element formulation is based on the classical Timoshenko beam theory, extended by a piezoelectric part. Each finite element node contains eight degrees of freedom: three displacements, three rotations and the electrical potential on the top and the bottom of the beam. The related fields are interpolated with linear approximation functions. In order to overcome the described problem of incompatible approximation spaces, a mixed multi-field variational approach is introduced, wherein six independent fields are considered. These are the displacements, strains, stresses, electric potential, electric field and the dielectric displacements. It allows for 3D constitutive equations and approximations of the strains and the electric field independent of the linear interpolation functions. Throughout the beam's cross section, a set of special polynomial approaches for strains and electric field components is proposed. This leads to well-balanced approximation functions regarding coupling of electrical and mechanical fields. Thus, electromechanical locking does not occur as is demonstrated by means of several numerical examples.

## Publication

Legner, D., Wackerfuß, J., Klinkel, S., Wagner, W.: On a concept to avoid electromechanical locking in a piezoelectric finite beam element. PAMM Proceedings in Applied Mathematics and Mechanics, Volume 10, Issue 1, pp. 185–186, 2010

Legner, D., Wackerfuß, J., Klinkel, S. & Wagner, W. 2013, 'An advanced finite element formulation for piezoelectric beam structures', Computational Mechanics, Volume 52, S. 1331-1349.

This project has been realized in cooperation with Legner, D., Wagner, W., both affiliated to the Karlsruhe Institute of Technology, and Klinkel S. of the Technische Universität Kaiserslautern.