A finite element formulation for piezoelectric shell structures considering geometrical and material non‐linearities

Abstract

In this paper we present a nonlinear finite element formulation for piezoelectric shell structures. Based on a mixed multi field variational formulation, an electro-mechanical coupled shell element is developed considering geometrically and materially nonlinear behavior of ferroelectric ceramics. The mixed formulation includes the independent fields of displacements, electric potential, strains, electric field, stresses, and dielectric displacements. Besides the mechanical degrees of freedom, the shell counts only one electrical degree of freedom. This is the difference of the electric potential in thickness direction of the shell. Incorporating nonlinear kinematic assumptions, structures with large deformations and stability problems can be analyzed. According to a Reissner-Mindlin theory, the shell element accounts for constant transversal shear strains. The formulation incorporates a three-dimensional transversal isotropic material law, thus the kinematic in thickness direction of the shell is considered. The normal zero stress condition and the normal zero dielectric displacement condition of shells are enforced by the independent resultant stress and resultant dielectric displacement fields. Accounting for material nonlinearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperaturedependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature dependent hysteresis curves.

Computational homogenization of fibrous piezoelectric materials

“…In this section we describe the kinematics assumed for the macroscale piezoelectric shell, Fig. 3, following [27], which in turn is largely based on Naghdi's theory for the mechanical part [33,36]. For more details, see the original paper [27].…”

“…3, following [27], which in turn is largely based on Naghdi's theory for the mechanical part [33,36]. For more details, see the original paper [27]. The Green-Lagrange strains and the electric field are derived in convective coordinates.…”

“…This is in bending dominated situations not correct [27]. Herein, a linear approximation is adopted, which is sufficient to pass the out-of-plane bending patch test [27].…”

“…where 0 33 , 1 33 are the constant and linear components of the thickness strain, while E 0 3 , E 1 3 represent the constant and linear parts of the electric field in the thickness direction [27]. The relation between the Green-Lagrange strains and the independent shell strains can be stated in compact form by introducing the matrix A such as:…”

Numerical homogenization of piezoelectric textiles with electrospun fibers for energy harvesting

“…A few of these concern electromechanically coupled problems such as in the case of piezoelectricity [16]. Moreover, although several macroscale formulations were developed for piezoelectric shell elements [17][18][19], computational homogenization of shells is only recently receiving major attention [20,21]. To the best of our knowledge, such approaches have not yet been proposed for piezoelectric shells.…”