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Smart materials and structures play an important role for sensor and actuator applications. For the simulation
of such systems it is essential to predict the material and system behavior as precisely as possible. A reliable
simulation may provide an easier, faster and cheaper development of such devices. In a wide range of technical
applications piezoelectric sensors and actuators typically have a shell-like structure. This motivates the present
contribution to deal with the consistent approximation of a piezoelectric shell formulation. A physical description
leads to a system of electromechanical differential equations. Due to the constitutive relations the strains and the
electric field are coupled. In case of bending dominated problems incompatible approximation functions of these
fields cause incorrect results. This effect occurs in standard finite element formulations, where the mechanical
and electrical degrees of freedom are interpolated with lowest order functions. The formulation presented in
this paper is based on the classical Reissner-Mindlin shell theory extended by a piezoelectric part. The shell
element has four nodes and bilinear interpolation functions. The eight degrees of freedom per node are three
displacements, three rotations and the electric potential on top and bottom of the shell. The finite shell element
incorporates a 3D-material law and is able to model arbitrary curved shell geometries of piezoelectric devices. In
order to overcome the described problem of incompatible approximation spaces a mixed multi-field variational
approach is introduced. Six independent fields are employed. These are the displacement, strain, stress, electric
potential, dielectric displacement and the electric field. It allows for approximations of the electric field and the
strains independent of the bilinear interpolation functions. A quadratic approach for the shear strains and the
corresponding electric field is proposed through the shell thickness. This leads to well balanced approximation
functions regarding coupling of electrical and mechanical fields. The numerical results are confirmed by analytical
considerations and an example illustrates the more precise results of the present formulation in contrast to
standard elements.
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