An advanced finite element formulation for piezoelectric beam structures

Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.     

Exact 3D Saint Venant type solutions for piezoelectric d 15 shear-mode bi-morph and sandwich torsion actuation and sensing problems

This paper is concerned with the analysis of two types of piezoelectric torsion transducers using the d ₁₅-effect of mono-morph piezoelectric materials. The first problem is concerned with a bi-morph transducer made of two identical mono-morph straight rods, which are perfectly bonded to each other along their width; the polarization direction is parallel, but opposite in sign, such that the piezoelectric material parameter d ₁₅ has an opposite sign as well. The second problem sandwiches the bi-morph transducer between two identical elastic face layers. In both cases, the resulting transducer represents a torsion transducer. Therefore, we analyze the electromechanically coupled problem in the framework of Saint Venant’s torsion theory for straight rods taking into account the electrical problem as well. The results of our approach are compared to electromechanically coupled three-dimensional finite element computation and a very good agreement for the mechanical as well as the electrical entities is achieved, in particular for the rate-of-twist, the axial warping function, and the sensed voltage.     

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

In this paper, we present a non‐linear 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 non‐linear 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 in the electric potential in the thickness direction of the shell. Incorporating non‐linear 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 the 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 the resultant dielectric displacement fields. Accounting for material non‐linearities, the ferroelectric hysteresis phenomena are considered using the Preisach model. As a special aspect, the formulation includes temperature‐dependent effects and thus the change of the piezoelectric material parameters due to the temperature. This enables the element to describe temperature‐dependent hysteresis curves. Copyright © 2011 John Wiley & Sons, Ltd.     

Two-scale homogenization of electromechanically coupled boundary value problems

The contribution addresses a direct micro-macro transition procedure for electromechanically coupled boundary value problems. The two-scale homogenization approach is implemented into a so-called FE²-method which allows for the computation of macroscopic boundary value problems in consideration of microscopic representative volume elements. The resulting formulation is applicable to the computation of linear as well as nonlinear problems. In the present paper, linear piezoelectric as well as nonlinear electrostrictive material behavior are investigated, where the constitutive equations on the microscale are derived from suitable thermodynamic potentials. The proposed direct homogenization procedure can also be applied for the computation of effective elastic, piezoelectric, dielectric, and electrostrictive material properties.     

Geometrically non-linear analysis of composite structures with integrated piezoelectric sensors and actuators

This paper deals with the geometrically non-linear analysis of thin plate/shell laminated structures with embedded integrated piezoelectric actuators or sensors layers and/or patches. The motivation for the present developments is the lack of studies in the behavior of adaptive structures using geometrically non-linear models, where only very few published works were found in the open literature.

The model is based on the Kirchhoff classical laminated theory and can be applied to plate and shell adaptive structures with arbitrary shape, general mechanical and electrical loadings.

The finite element model is a non-conforming single layer triangular plate/shell element with 18 degrees of freedom for the generalized displacements and one electrical potential degree of freedom for each piezoelectric layer or patch.

An updated Lagrangian formulation associated to Newton–Raphson technique is used to solve incrementally and iteratively the equilibrium equations.

The model is applied in the solution of four illustrative cases, and the results are compared and discussed with alternative solutions when available.     

Optimal design of laminated piezocomposite energy harvesting devices considering stress constraints

Energy harvesting devices are smart structures capable of converting the mechanical energy (generally, in the form of vibrations) that would be wasted otherwise in the environment into usable electrical energy. Laminated piezoelectric plate and shell structures have been largely used in the design of these devices because of their large generation areas. The design of energy harvesting devices is complex, and they can be efficiently designed by using topology optimization methods (TOM). In this work, the design of laminated piezocomposite energy harvesting devices has been studied using TOM. The energy harvesting performance is improved by maximizing the effective electric power generated by the piezoelectric material, measured at a coupled electric resistor, when subjected to a harmonic excitation. However, harmonic vibrations generate mechanical stress distribution that, depending on the frequency and the amplitude of vibration, may lead to piezoceramic failure. This study advocates using a global stress constraint, which accounts for different failure criteria for different types of materials (isotropic, piezoelectric, and orthotropic). Thus, the electric power is maximized by optimally distributing piezoelectric material, by choosing its polarization sign, and by properly choosing the fiber angles of composite materials to satisfy the global stress constraint. In the TOM formulation, the Piezoelectric Material with Penalization and Polarization material model is applied to distribute piezoelectric material and to choose its polarization sign, and the Discrete Material Optimization method is applied to optimize the composite fiber orientation. The finite element method is adopted to model the structure with a piezoelectric multilayered shell element. Numerical examples are presented to illustrate the proposed methodology. Copyright © 2015 John Wiley & Sons, Ltd.     

Active control of forced vibrations in adaptive structures using a higher order model

This paper deals with a finite element formulation for active control of forced vibrations, including resonance, of thin plate/shell laminated structures with integrated piezoelectric layers, acting as sensors and actuators, based on third-order shear deformation theory. The finite element model is a single layer triangular nonconforming plate/shell element with 24 degrees of freedom for the generalized displacements, and one electrical potential degree of freedom for each piezoelectric element layer, which are surface bonded or embedded in the laminate.

The Newmark method is considered to calculate the dynamic response of the laminated structures, forced to vibrate in the first natural frequency. To achieve a mechanism of active control of the structure dynamic response, a feedback control algorithm is used, coupling the sensor and active piezoelectric layers. The model is applied in the solution of illustrative cases, and the results are presented and discussed.     

Coupled mechanics and finite element for non‐linear laminated piezoelectric shallow shells undergoing large displacements and rotations

A theoretical framework is presented for analysing the coupled non‐linear response of shallow doubly curved adaptive laminated piezoelectric shells undergoing large displacements and rotations. The formulated mechanics incorporate coupling between in‐plane and flexural stiffness terms due to geometric curvature, coupling between mechanical and electric fields, and encompass geometric non‐linearity effects due to large displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear co‐ordinates and are combined with the kinematic assumptions of a mixed‐field shear‐layerwise shell laminate theory. Based on the above formulation, a finite element methodology together with an incremental‐iterative technique, based on Newton–Raphson method is formulated. An eight‐node coupled non‐linear shell element is also developed. Various evaluation cases on laminated curved beams and cylindrical panels illustrate the capability of the shell finite element to predict the complex non‐linear behaviour of active shell structures including buckling, which is not captured by linear shell models. The numerical results also show the inherent capability of piezoelectric shell structures to actively induce large displacements through piezoelectric actuators, by jumping between multiple equilibrium states. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite Rotation Piezoelectric Exact Geometry Solid-Shell Element with Nine Degrees of Freedom per Node

This paper presents a robust non-linear piezoelectric exact geometry (EG) four-node solid-shell element based on the higher-order 9-parameter equivalent single-layer (ESL) theory, which permits one to utilize 3D constitutive equations. The term EG reflects the fact that coefficients of the first and second fundamental forms of the reference surface are taken exactly at each element node. The finite element formulation developed is based on a new concept of interpolation surfaces (I-surfaces) inside the shell body. We introduce three I-surfaces and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows us to represent the finite rotation piezoelectric higher-order EG solid-shell element formulation in a very compact form and to utilize in curvilinear reference surface coordinates the strain-displacement relationships, which are objective, that is, invariant under arbitrarily large rigid-body shell motions. To avoid shear and membrane locking and have no spurious zero energy modes, the assumed displacement-independent strain and stress resultant fields are introduced. In this connection, the Hu-Washizu variational equation is invoked. To implement the analytical integration throughout the element, the modified ANS method is applied. As a result, the present finite rotation piezoelectric EG solid-shell element formulation permits the use of coarse meshes and very large load increments.     

压电材料的电热函及机-电耦合分析

研究基于电热函的广义哈密顿原理在压电材料的机(力)-电耦合效应分析中的应用。作为应用实例,推导了两个简单压电材料结构的变形及应变致(strain induced)电场的解析解, 并与己有的解析解进行了比较并讨论了它们存在的若干问题。 计算结果表明,基于电热函的广义哈密顿原理是分析压电材料的机(力)-电耦合问题简单而又精确的模型。它既可准确地计算压电势能而又不必对二维问题和三维问题加以区分; 并且这种方法既可用于解析分析,也可作为压电复合材料有限元分析的数学基础。     

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three‐layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed‐ or open‐circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three‐layered smart beam with different EBC is proposed considering a fully coupled electro‐mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi‐static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC. Copyright © 2005 John Wiley & Sons, Ltd.     

A perturbation method for finite element modeling of piezoelectric vibrations in quartz plate resonators

When the piezoelectric stiffening matrix is added to the mechanical stiffness matrix of a finite element model, its sparse matrix structure is destroyed. A direct consequence of this loss in sparseness is a significant rise in memory and computational time requirements for the model. For weakly coupled piezoelectric materials, the matrix sparseness can be retained by a perturbation method which separates the mechanical eigenvalue solution from its piezoelectric effects. Using this approach, a perturbation and finite element scheme for weakly coupled piezoelectric vibrations in quartz plate resonators has been developed. Finite-element matrix equations were derived specifically for third-overtone thickness-shear, SC-cut quartz plate resonators with electrode platings. High-frequency piezoelectric plate equations were employed in the formulation of the finite element matrix equation. Results from the perturbation method compared well with the direct solution of the piezoelectric finite element equations. This method will result in significant savings in the computer memory and computational time. Resonance and antiresonance frequencies of a certain mode could be calculated easily by using the same eigenpair from the purely mechanical stiffness matrix. Numerical results for straight crested waves in a third overtone SC-cut quartz strip with and without electrodes are presented. The steady-state response to an electrical excitation is calculated.     

FEM analysis of electro-mechanical coupling effect of piezoelectric materials

Based on the mechanical and electrical equilibrium equations of piezoelectric materials, the minimum potential theory is presented by using the virtual work principle in this paper. A finite element method (FEM) formulation accounting for the electro-mechanical coupling effect of piezoelectric materials is given. Some problems in the numerical simulation are discussed and the extreme illness of the stiffness matrix is overcome by the dimension changing method. As a simple application, the response of an elliptical cavity in infinite media of piezoelectric materials is analyzed. Such a geometry leads to stress and electric field concentrations.     

Efficient coupled refined finite element for dynamic analysis of sandwich beams containing embedded shear-mode piezoelectric layers

An efficient and computationally low cost finite element (FE) model is developed for dynamic free and forced response of sandwich beams with embedded shear piezoelectric layers based on a coupled refined high-order global-local theory. Contrary to most of the available models, all of the kinematic and stress boundary conditions are ensured at the interfaces of the shear piezoelectric layers. Moreover, both the electrical-induced strains components and transverse flexibility are taken into account for the first time in the present theory. For validation of the proposed model, various free and forced vibration tests for thin and thick sandwich beams are carried out. For various electrical and mechanical boundary conditions, excellent agreement has been found between the results obtained from the proposed formulation with previously published and the coupled two-dimensional (2D) FE results.     

压电曲壳单元及其形状控制

板壳结构作为航空、航天工程中的主要工作元件,要承受多种环境荷载,而对形状变化非常敏感的机翼、天线等结构,有必要进行形状控制。推导了空间压电曲壳单元的有限元方程,采用约束方程法连接压电曲壳和主体结构,建立了整体结构的有限元分析模型,并基于等效应变原则验证了模型的正确性。在此基础上,利用最小二乘法对结构进行了形状控制,得到压电驱动器上电压的最优分布。算例表明:该文模型能提高计算精度和速度,达到形状控制的要求。     

Fully coupled non‐linear analysis of piezoelectric solids involving domain switching

Domain switching is the cause of significant non‐linearity in the response of piezoelectric materials to mechanical and electrical effects. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The constitutive equations are non‐linear. Moreover, due to the domain switching phenomenon, the resulting governing equations become highly non‐linear. The corresponding non‐linear finite element equations are derived and solved by using an incremental technique. The developed formulation is first verified against a number of benchmark problems for which a closed‐form solution exists. Next, a cantilever beam made of PZT‐4 is studied to evaluate the effect of domain switching on the overall force–displacement response of the beam. A number of interesting observations are made with respect to the extent of non‐linearity and its progressive spread as the load on the beam increases. Copyright © 2002 John Wiley & Sons, Ltd.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

A nine-node assumed strain finite element for large-deformation analysis of laminated shells

An application of the element-based Lagrangian formulation is described for large-deformation analysis of both single-layered and laminated shells. Natural co-ordinate-based stresses, strains and constitutive equations are used throughout the formulation of the present shell element which offers significant implementation advantages compared with the traditional Lagrangian formulation. In order to avoid locking phenomena, an assumed strain method has been employed with judicious selection of the sampling points. Three strictly successive finite rotations are used to represent the current orientation of the shell normal. The equivalent natural constitutive equation is derived using an explicit transformation scheme to consider the multi-layer effect of laminated structures. The arc-length control method is used to trace complex load-displacement paths. Several numerical analyses are presented and discussed in order to investigate the capabilities of the present shell element. © 1998 John Wiley & Sons, Ltd.     

Active control of functionally graded laminated cylindrical shells

An analytical method on active vibration control of smart FG laminated cylindrical shells with thin piezoelectric layers is presented based on Hamilton’s principle. The thin piezoelectric layers embedded on inner and outer surfaces of the smart FG laminated cylindrical shell act as distributed sensor and actuator, which are used to control vibration of the smart FG laminated cylindrical shell under thermal and mechanical loads. Here, the modal analysis technique and Newmark’s integration method are used to calculate the dynamic response of the smart FG laminated cylindrical shell with thin piezoelectric layers. Constant-gain negative velocity feedback approach is used for active vibration control with the structures subjected to impact, step and harmonic excitations. The influences of different piezoelectric materials (PZT-4, BaTiO₃ and PZT-5A) and various loading forms on the active vibration control are described in the numerical results.