On non‐linear behaviour of spherical shallow shells bonded with piezoelectric actuators by the differential quadrature element method (DQEM)

Int. J. Numer. Meth. Engng

2005

This paper is concerned with a geometrically nonlinear solid shell element to analyze piezoelectric structures. The finite element formulation is based on a variational principle of the Hu-Washizu type and includes six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. The element has 8 nodes with 4 nodal degrees of freedoms, 3 displacements and the electric potential. A bilinear distribution through the thickness of the independent electric field is assumed to fulfill the electric charge conservation law in bending dominated situations exactly. The presented finite shell element is able to model arbitrary curved shell structures and incorporates a 3D-material law. A geometrically nonlinear theory allows large deformations and includes stability problems. Linear and nonlinear numerical examples demonstrate the ability of the proposed model to analyze piezoelectric devices.

Section: Introductionmentioning

confidence: 99%

“…All of the above cited piezoelectric shell formulations except [12] assume a geometrically linear theory for the shell kinematics. In the theoretical contributions of Tzou [21,22] it is pointed out that nonlinear characteristics can significantly influence the performance of piezoelectric structures and systems.…”

Section: Introductionmentioning

confidence: 99%

A geometrically non-linear piezoelectric solid shell element based on a mixed multi-field variational formulation

Int. J. Numer. Meth. Engng

2005

“…The variation of the generalized geometric vector δε g can be written as δε g = [δε 11 , δε 22 , 2δε 12 , δκ 11 , δκ 22 , 2δκ with…”

Section: Variational Formulationmentioning

confidence: 99%

“…References [1,2,3], and classical formulations, which model the shell by a reference surface, see e.g. References [4,5,6,7,8,9,10,11]. Some of these element formulations are restricted to shallow shell structures, [5,6,9,11], where the initial shell curvature is assumed to be small.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Schulz

Numerical Meth Engineering

2011

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.

An advanced finite element formulation for piezoelectric shell structures

Legner

Numerical Meth Engineering

2013

SUMMARYPiezoelectric materials are increasingly employed throughout the field of sensor and actuator applications. The physical description leads to a boundary value problem with electromechanically coupled differential equations that can be solved approximatively, for example, by the FEM. As the structures usually are quite thin, piezoelectric shell elements are very well suited for this task. In piezoelectric materials, the mechanical and the electrical fields are coupled by the constitutive relations. Especially for problems dominated by bending, this results in incompatibilities when using element formulations where the mechanical and the electrical DOFs are interpolated with lowest order functions. As a consequence, parasitic approximations and incorrect computation results occur. The present work proposes a concept to avoid these incompatible approximation spaces and the occurring computation errors when dealing with thin structures. The element is based on a mixed variational formulation and uses six mechanical and two electrical nodal DOFs. It employs the Reissner–Mindlin kinematics, considers strains throughout the thickness, and allows for 3D‐electromechanical constitutive relations. Numerical examples show that the element formulation is electromechanically consistent and enables to analyze piezoelectric shell structures without parasitic approximations for all typical load cases. Copyright © 2013 John Wiley & Sons, Ltd.