Advanced finite element formulations for modeling thin piezoelectric structures

Abstract

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense.Some numerical examples will demonstrate the capability of the presented finite element formulation.

A computational framework for the analysis of linear piezoelectric beams using hp-FEM

“…The coefficients used in the analytical solution presented in Section 6 are as follows 21 4 À a 14 a 18 4 a 1 ; …”

“…In the case of quadratic electric potential distribution, a diagonal electric permittivity tensor and piezoelectric component P 133 ¼ 0, which is the case for a major class of piezoceramics [31] and, specifically, the ones used in piezoelectric beam literature [6,10,5,1,3,8,9,21], the three differential Eq. (57) take the form,…”

“…Along the same lines, Koutsawa [23] attempts to solve the problem of static piezoelectric beams by using higher order displacement theories for beams. This paper aims to present a simple three-dimensional finite element framework for linear piezoelectric beams, derived from first principles, in order to bridge the gap between existing simplified lumped or distributed parameter models [1,6,[8][9][10]12,13] and the most sophisticated nonlinear warping beam models [19][20][21][22][23]. In the process, interesting new physical magnitudes, such as the coupled shear or the coupled bending/torsional moment introduced as a result of an electric displacement, will naturally arise.…”

“…Wagner and co-workers [19][20][21] introduce a sophisticatedthree-dimensional beam finite element model with linear and nonlinear strain measures, including hysteresis, using a six field variational formulation and assuming a quadratic electric potential distribution across the cross sectional area. The work is restricted to static analysis only and requires a preprocessing stage to compute the warping patterns by solving a two-dimensional boundary value problem, using a separate finite element discretisation.…”

“…As it is common in piezoelectric beam literature[1,19,21], for the electrical part we have postulated similar Dirichlet and Neumann boundary conditions to those of the mechanical problem i.e. all electric variables are zero at the fixed end and the derivatives are specified at the free end in a coupled fashion.…”