Three-dimensional analysis of doubly curved functionally graded magneto-electro-elastic shells

Three-dimensional (3D) solutions for the static analysis of doubly curved functionally graded (FG) magneto-electro-elastic shells are presented by an asymptotic approach. In the present formulation, the twenty-nine basic equations are firstly reduced to ten differential equations in terms of ten primary variables of elastic, electric and magnetic fields. After performing through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of two-dimensional (2D) governing equations for various order problems. These 2D governing equations are merely those derived in the classical shell theory (CST) based on the extended Love–Kirchhoffs' assumptions. Hence, the CST-type governing equations are derived as a first-order approximation to the 3D magneto-electro-elasticity. The leading-order solutions and higher-order corrections can be determined by treating the CST-type governing equations in a systematic and consistent way. The 3D solutions for the static analysis of doubly curved multilayered and FG magneto-electro-elastic shells are presented to demonstrate the performance of the present asymptotic formulation. The coupling magneto-electro-elastic effect on the structural behavior of the shells is studied.     

Some variational principles in elastic dielectric and elastic magnetic materials

The electroelastic and magnetoelastic analyses in the dielectric and magnetic materials should be studied together with their environments. From the first thermodynamic law a general variational theory is proposed. Some nonlinear variational principles with electric or magnetic Gibbs free energy and the complete governing equations, derived from these principles are given in this paper. The governing equations are complete and useful in engineering. It is noted that the virtual displacements do not only produce the variation of the strain, but also produce the variation of electric potential or magnetic potential. It is also noted that even in a small deformation the effect of the volume change on the electromagnetic field cannot be neglected. Just due to these two reasons the Maxwell stress can be naturally derived from the variational equation and it is unique. In these derivations new elctrostrictive and magnetostrictive coefficients are introduced and the anti-symmetric part of the stress can be naturally got. These principles are also useful for computational mechanics.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

On the micro-mechanical study of 1–3 type piezoelectric composites with semi-coupled thermo-electro-elastic effects

This paper deals with a two-dimensional generalized plane strain micro-mechanical model to simulate semi-coupled thermo-electro-elastic behavior of transversely polarized piezoelectric fibrous composites. The solution domain includes a representative volume element (RVE) consists of a long piezoelectric fiber surrounded by corresponding matrix in a square array arrangement. Fibers have orthotropic and/or transversely isotropic properties while are perfectly bonded to the isotropic matrix. In addition, the constituents are assumed to have linear thermo-electro-elastic behavior. The virtual form of equilibrium equations has been extended to cover the semi-coupled thermo-electro-elastic loading by using appropriate constitutive relations. The element-free Galerkin method is employed to discretize the governing equations in terms of three main primary variables including, displacements, electric potential and temperature. The performance of the present micro-mechanical study reveals close agreement compared with other techniques available in the literature. Based on the present study, ample results are addressed to provide an insight into the effects of the local fields, i.e. displacement, electric potential, electric field, and stress distributions within the RVE for the specific fiber volume fraction.     

Physical variational principle and thin plate theory in electro-magneto-elastic analysis

Under the assumption of the quasi-static electric and magnetic fields the electro-magneto-elastic analysis including medium and its environment is studied in this paper. The complete governing equations under the finite deformation can be derived from the physical variational principle. In the physical variational principle the variations of the electric potential and magnetic potential are divided into local variations and migratory variations. From the virtual change of the sum of the electromagnetic energy and the couple energy produced by the migratory variation we can get the electromagnetic force and in this case the virtual variation of the volume should be considered. It is also found that the Maxwell stress is directly related to the strain in a material with piezoelectric or piezomagnetic behavior for the finite deformation case. The thin plate theory in first order is derived from the general theory in this paper and the Maxwell stress is naturally included in the governing equations.     

Exact piezothermoelastic solution of simply-supported orthotropic circular cylindrical panel in cylindrical bending

Summary This work presents an exact piezothermoelastic solution of infinitely long, simply supported, cylindrically orthotropic, piezoelectric, radially polarised, circular cylindrical shell panel in cylindrical bending under thermal and electrostatic excitation. The general solution of the governing differential equations is obtained by separation of variables. The displacements, electric potential and temperature are expanded in appropriate Fourier series in the circumferential coordinate to satisfy the boundary conditions at the simply-supported longitudinal edges. The governing equations reduce to Euler-Cauchy type of ordinary differential equations. Their general solution involves six constants for each Fourier component. These are solved from the algebraic equations obtained by satisfying the boundary conditions at the lateral surfaces. The solution of the inverse problem of inferring the applied temperature field from the given measured distribution of electrical potential difference between the lateral surfaces of the shell has also been presented. Numerical results are presented for typical thermal and electrostatic loadings for various values of radius to thickness ratio.     

Internal energy variational principles and governing equations in electroelastic analysis

The first thermodynamic law contains a universal thermodynamic variational principle. The complete internal energy variational principle in the electroelastic analysis is not discussed in previous papers. In this paper this principle will be discussed. From this principle the simple complete governing equations can be deduced, and the Maxwell stress can be naturally derived from this variational principle. It is shown that the Maxwell stress has slightly different forms determined by using internal energy or electric Gibbs free energy variational principle, but substantially they are the same. In the second-order precision the Maxwell stress is uniquely determined, and its expression has the same form for all deformable and rigid dielectrics. The electroelastic analyses in the dielectric should be studied together with its environment, because the electric field exists in all materials except the ideal conductor. The complete governing equations under finite deformation in the initial configuration are also discussed.     

数值模拟离子强度敏感水凝胶的多场特性

发展了多物理模型来研究溶液中离子强度敏感水凝胶电-力-化学多场耦合的特性. 模 型的主要控制方程包括：计算水凝胶内外离子浓度分布的Nernst-Planck化学场方程；描述 膨胀变形的力学场方程和描述电场的泊松方程. 无网格有限云团法和牛顿迭代法用来数值离 散和求解控制方程. 通过对比多场耦合的响应, 包括胶的膨胀率和胶内外离子浓度和电势的分布，探讨了影响胶体变形的主要因素. 对数值模拟结果和实验结果进行了对照.     

考虑电场梯度的挠曲电纳米板弯曲性能分析

具有尺度依赖的挠曲电效应在器件的设计中扮演着越来越关键的角色, 研究人员在微纳米尺度多物理场分析中进行了大量工作. 基于考虑挠曲电和电场梯度效应的弹性介电材料非经典理论, 以二维纳米板为例, 通过理论建模, 分析纳米板在弯曲问题中的力?电耦合行为. 根据Mindlin假设给出板的位移场和电势场的一阶截断, 选取板的材料为立方晶体(m3m点群), 将广义三维本构方程代入到高阶应力、高阶偶应力、高阶电位移和高阶电四极矩的表达式中得到相应的二维本构方程, 利用弹性电介质变分原理得到板的控制方程和边界上的线积分等式, 分别将二维本构方程和边界上外法线的方向余弦代入, 得到板的高阶弯曲方程、高阶电势方程以及对应的四边简支边界条件. 利用四边简支矩形板的高阶弯曲方程、高阶电势方程和相应的边界条件, 根据Navier解理论, 求解纳米板的电势场, 重点分析电场梯度对板内一阶电势的影响. 数值计算结果表明: 电场梯度对纳米板中由挠曲电效应产生的一阶电势有削弱作用, 且材料参数g₁₁越大, 一阶电势受到的削弱越大; 同时电场梯度的存在消除了纳米板在受横向集中载荷作用时一阶电势的奇异性. 本文是对具有挠曲电效应和电场梯度效应的纳米板结构分析理论的一个扩展, 为微纳米尺度器件的结构设计提供参考.      

Exact electroelastic analysis of functionally graded piezoelectric shells

A paper focuses on implementation of the sampling surfaces (SaS) method for the three-dimensional (3D) exact solutions for functionally graded (FG) piezoelectric laminated shells. According to this method, we introduce inside the nth layer I_n not equally spaced SaS parallel to the middle surface of the shell and choose displacements and electric potentials of these surfaces as basic shell variables. Such choice of unknowns yields, first, a very compact form of governing equations of the FG piezoelectric shell formulation and, second, allows the use of strain–displacement equations, which exactly represent rigid-body motions of the shell in any convected curvilinear coordinate system. It is worth noting that the SaS are located inside each layer at Chebyshev polynomial nodes that leads to a uniform convergence of the SaS method. As a result, the SaS method can be applied efficiently to 3D exact solutions of electroelasticity for FG piezoelectric cross-ply and angle-ply shells with a specified accuracy by using a sufficient number of SaS.     

Indentation of a compressible soft electroactive half-space: Some theoretical aspects

We theoretically study the indentation response of a compressible soft electroactive material by a rigid punch. The half-space material is assumed to be initially subjected to a finite deformation and an electric biasing field. By adopting the linearized theory for incremental fields, which is established on the basis of a general nonlinear theory for electroelasticity, the appropriate equations governing the perturbed infinitesimal elastic and electric fields are derived particularly when the material is subjected to a uniform equibiaxial stretch and a uniform electric displacement. A general solution to the governing equations is presented, which is concisely expressed in terms of four quasi-harmonic functions. By adopting the potential theory method, exact contact solutions for three common perfectly conducting rigid indenters of flat-ended circular, conical and spherical geometries can be derived, and some explicit relations that are of practical importance are outlined.     

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates of transversely isotropic materials is analyzed based on linear three-dimensional theory of elasticity coupled with magnetic and electric fields. The transverse loads are expanded in Fourier-Bessel series and therefore can be arbitrarily distributed along the radial direction. The radial distributions of the displacements are assumed in combination of Fourier-Bessel series and polynomials as well as the electric potential and magnetic potential. If the material properties obey the exponential law along the thickness of the plate, two three-dimensional exact solutions for two unusual boundary conditions can be derived since they satisfy the governing equations and specified boundary conditions point by point. For simply supported or clamped boundary, the obtained solutions satisfy the governing equations exactly and the boundary conditions approximately. A layer wise model is also introduced to treat with the plates whose material property components vary independently and arbitrarily along the thickness of the plates. The numerical results are finally tabulated and plotted to demonstrate the presented method and agree well with those from finite element methods.     

Magnetoelectric effects in multiferroic laminated plates with imperfect interfaces

Two-dimensional(2D)equations for multiferroic(MF)laminated plates with imperfect interfaces are established in this paper.The interface between two adjacent sublayers,which are not perfectly bonded together,is modeled as a general spring-type layer.The mechanical displacements,and the electric and magnetic potentials of the two adjacent layers are assumed to be discontinuous at the interface.As an example,the influences of imperfect interfaces on the magnetoelectric(ME)coupling effects in an MF sandwich plate are investigated with the established 2D governing equations.Numerical results show that the imperfect interfaces have a significant impact on the ME coupling effects in MF laminated structures.     

Nonlinear Electroelastic Deformations

Electro-sensitive (ES) elastomers form a class of smart materials whose mechanical properties can be changed rapidly by the application of an electric field. These materials have attracted considerable interest recently because of their potential for providing relatively cheap and light replacements for mechanical devices, such as actuators, and also for the development of artificial muscles. In this paper we are concerned with a theoretical framework for the analysis of boundary-value problems that underpin the applications of the associated electromechanical interactions. We confine attention to the static situation and first summarize the governing equations for a solid material capable of large electroelastic deformations. The general constitutive laws for the Cauchy stress tensor and the electric field vectors for an isotropic electroelastic material are developed in a compact form following recent work by the authors. The equations are then applied, in the case of an incompressible material, to the solution of a number of representative boundary-value problems. Specifically, we consider the influence of a radial electric field on the azimuthal shear response of a thick-walled circular cylindrical tube, the extension and inflation characteristics of the same tube under either a radial or an axial electric field (or both fields combined), and the effect of a radial field on the deformation of an internally pressurized spherical shell.     

Asymptotic homogenization of three-dimensional thermoelectric composites

Thermoelectric composites are promising for high efficiency energy conversion between thermal flows and electric conduction, though their effective behaviors remain poorly understood due to nonlinear thermoelectric coupling. In this paper, we develop an asymptotic homogenization theory to analyze the effective behavior of three-dimensional (3D) thermoelectric composites, built on the observation that the equations governing microscopic field fluctuations in the composite are actually linear instead of nonlinear after separation of length scales. A set of solutions similar to Green's function method are used to construct the unit cell problem, and appropriate interfacial continuity conditions and boundary conditions are derived. The homogenized governing equations are then developed for thermoelectric composites, and they are further reduced for a special case wherein the heat flow and electric conduction in the composite remains one-dimensional (1D) at macroscopic scale, even though the composite itself is 3D in general. The general homogenization theory is implemented using finite element method, and a key constant in the constructed solutions is determined using the reformulated eigenvalue problem. The algorithm is validated, and is applied for a number of case studies for the effective behavior of thermoelectric composites.     

Continuum Electromechanical Theory for Nematic Continua with Application to Freedericksz Instability

Of interest in this work are nematic continua that exhibit electromechanical coupling. The first part of this paper presents a novel variational formulation with a potential energy depending on four independent variables (the displacement, director, specific polarization and electric displacement perturbation). Variations of the potential energy with respect to each one of these variables lead to the governing mechanical equilibrium and constitutive relations plus Maxwell’s equations.The proposed variational formulation is next applied to the study of bifurcation of an infinite layer of a nematic liquid crystal confined between two parallel plates and subjected to a uniform electric field perpendicular to these plates under full anchoring boundary conditions. As the electric field exceeds a critical value, the nematic directors which are initially parallel to the plates, rotate and tend to align with the electric field orientation. This phenomenon, termed in the literature as Freedericksz transition, is treated here as a bifurcation problem using a fully 2D formulation. It is shown that the solution corresponding to the lowest applied electric field, also termed the critical load, is uniform in the direction parallel to the plates and that the corresponding bifurcated path is stable near this critical load. This result holds for arbitrary positive constants of the Frank-Oseen energy (and values of electric susceptibility constants that allow bifurcation) and justifies the 1D treatment of the Freedericksz transition in 2D settings that is widely adopted in the liquid crystal literature. An asymptotic analysis of the supercritical, stable bifurcated equilibrium path about the critical load is also presented and compared with the exact bifurcated solution.     

Hyperbolic Models of Homogeneous Two-Fluid Mixtures

One derives the governing equations and the Rankine–Hugoniot conditions for a mixture of two miscible fluids using an extended form of Hamilton's principle of least action. The Lagrangian is constructed as the difference between the kinetic energy and a potential depending on the relative velocity of components. To obtain the governing equations and the jump conditions one uses two reference frames related with the Lagrangian coordinates of each component. Under some hypotheses on flow properties one proves the hyperbolicity of the governing system for small relative velocity of phases.     

Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network

In this paper a new approach is presented to reduce vibrations for one- and two-dimensional mechanical structures, as beam or thin plates, by means of several piezoelectric transducers shunted with a proper electric network system. The governing equations of the whole system are coupled to each other through the direct and converse piezoelectric effect. More in detail, the mechanical equations are expressed in accordance with the modal theory considering n vibration modes and the electrical equations reduce to the one-dimensional charge equation of electrostatics for each of n considered piezoelectric transducers. In this electromechanical system, a shunting electric device forms an electric subsystem working as multi degrees of freedom (dof’s) damped vibration absorber for the mechanical subsystem. Herein, it is introduced a proper transformation of the electric coordinates in order to approximate the governing equations for the whole shunted system with n uncoupled, single mode piezoelectric shunting systems that can be readily damped by the methods reported in literature. A further numerical optimisation problem on the spatial distribution of the piezoelectric elements allows to achieve a better performance. Numerical case studies of two relevant systems, a double clamped beam and a fully clamped plate, allow to take into account issues relative to the proposed approach. Laboratory experiments carried out in real time on a beam clamped at both ends consent to validate the proposed technique.     

压电材料反平面裂纹自相似扩展分析

通过对耦合的波动方程和方程解耦,用自模拟方法研究了压电材料中反平面裂纹的自相似扩展问题。研究表明：对反平面问题,介质内的耦合场与裂纹扩展速度有关,在裂纹尖端有r^-1/2阶的奇异性;动态应力强度因子与电位称载荷有关,与静态结论不同;电位移强度因子与机械载 荷无关,与静态结果的表达形式一致。     

Thermo-electro-magneto-mechanical bending behavior of size-dependent sandwich piezomagnetic nanoplates

Thermo-electro-magneto-mechanical bending analysis of a sandwich nanoplate is presented in this paper based on Kirchhoff’s plate theory and nonlocal theory. The sandwich nanoplate includes an elastic nano-core and two piezomagnetic face-sheets actuated by applied electric and magnetic potentials. The governing equations for the electro-magneto-mechanical bending are derived in terms of the displacement components and electric and magnetic potentials. Then, the problem is solved analytically by using Navier’s method. A parametric study is presented to show the effects of the nonlocal parameter, temperature rise, applied electric and magnetic potentials on the bending behaviors of sandwich nanoplates for simply-supported boundary conditions. As a main result of study, it is concluded that the deflection decreases as applied electric potential increases and applied magnetic potential decreases. In addition, the increase of nonlocal parameter leads to increase of deflection and maximum electric potential through the thickness direction.