A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications

Klinkel, Sven; Wagner, Werner

doi:10.1016/j.compstruc.2007.05.032

Cited by 56 publications

(40 citation statements)

References 33 publications

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“…A mixed solid shell element formulation based on a Hu-Washizu principle is introduced, details see [1]. As nodal degrees of freedom three displacements and the electric potential are assumed.…”

Section: Finite Element Formulationmentioning

confidence: 99%

A solid shell finite element formulation to simulate the behaviour of thin dielectric elastomer structures

Zwecker

Klinkel

2011

Proc Appl Math and Mech

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To analyse the behaviour of thin structures of dielectric elastomer (DE) material a solid shell finite element is presented. The main characteristics of DEs are a non-linear hyper elastic behaviour, the quasi-incompressibility, and the ability to transform electric energy into mechanical work. Applying a voltage to thin DE structures may produce large elongation strains of 120-380%. These large strains, the efficient electro-mechanical coupling, and the light weight make DEs very attractive for the usage in actuators. Thus, there is a need for detailed research. With respect to the electro-mechanical coupling a constitutive model is presented. An electric stress tensor and a total stress tensor are introduced by considering the electrical body force and couple in the balance of linear momentum and angular momentum, respectively. The governing equations are derived and embedded in the solid shell formulation. The element formulation is based on a Hu-Washizu mixed variational principle using six independent fields: displacements, electric potential, strains, electric field, mechanical stresses, and dielectric displacements. It allows large deformations and accounts for physical nonlinearities to capture two of the main characteristics of DEs. The shell element could be applied for the modelling of arbitrary curved thin structures. The ability of the present element formulation is demonstrated in several examples.

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Section: Finite Element Formulationmentioning

confidence: 99%

A solid shell finite element formulation to simulate the behaviour of thin dielectric elastomer structures

Zwecker

Klinkel

2011

Proc Appl Math and Mech

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“…In these theories, a priori assumptions concerning the mechanical displacements and the electrical field in certain directions are made to reduce the three-dimensional continuum to a lower-order one. With respect to nonlinear formulations for piezoelectric plates and shells, we refer to the rich literature, e.g., Zheng et al [8], Tan and Vu-Quoc [9], Klinkel and Wagner [10], Marinković et al [11] and Lentzen et al [12]. In this paper, we develop a novel hybrid approach to the modeling of the nonlinear behavior of piezoelectric shells, which is here presented in a holistic form for the first time; the approach is illustrated by two numerical examples.…”

Section: Introductionmentioning

confidence: 99%

Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells

2017

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We present a novel multistage hybrid asymptotic-direct approach to the modeling of the nonlinear behavior of thin shells with piezoelectric patches or layers, which is formulated in a holistic form for the first time in this paper. The key points of the approach are as follows: (1) the asymptotic reduction in the threedimensional linear theory of piezoelasticity for a thin plate; (2) a direct approach to geometrically nonlinear piezoelectric shells as material surfaces, which is justified and completed by demanding the mathematical equivalence of its linearized form with the asymptotic solution for a plate; and (3) the numerical analysis by means of an FE scheme based on the developed model of the reduced electromechanically coupled continuum. Our approach is illustrated by examples of static and steady-state analysis and verified with three-dimensional solutions computed with the commercially available FE code ABAQUS as well as by comparison with results reported in the literature.

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“…References [5,10,12,13,14] point out that geometrically nonlinear characteristics can significantly influence the performance of piezoelectric systems, especially for the sensor usage. A geometrically nonlinear theory that incorporates large rotations is presented in References [1,5,6,12]. A common assumption in piezoelectric models is that the electric field is constant through the thickness.…”

Section: Introductionmentioning

confidence: 99%

“…One can distinguish between solid shell elements, see e.g. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Schulz

Klinkel

Wagner

2011

Numerical Meth Engineering

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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.

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A piezoelectric solid shell element based on a mixed variational formulation for geometrically linear and nonlinear applications

Cited by 56 publications

References 33 publications

A solid shell finite element formulation to simulate the behaviour of thin dielectric elastomer structures

A solid shell finite element formulation to simulate the behaviour of thin dielectric elastomer structures

Hybrid asymptotic–direct approach to finite deformations of electromechanically coupled piezoelectric shells

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

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