功能梯度压电层/压电半空间结构中Love波的传播:一种改进的拉盖尔多项式方法

使用传统的拉盖尔多项式方法求解层状半空间结构时,存在因层间材料差异所造成的应力、电位移不连续的现象。为了克服此方法的不足,提出了一种改进的拉盖尔多项式方法,研究了功能梯度压电层状半空间中Love波的传播特性。与文献中应用WKB法得到的结果进行对比,验证了该方法的正确性。计算和分析了相应的频散曲线、应力和电位移分布曲线。结果表明:该方法能够避免因层间材料差异所造成的应力、电位移不连续现象的出现;高频Love波的应力和电位移主要分布在功能梯度压电层中速度较低的一侧。该研究为基于Love波传感器的设计与优化奠定了一定的理论基础。     

具有初应力的多层空心圆柱体中的导波

基于“增量变形力学”理论,研究了轴向初应力和径向初应力对多层空心圆柱体中导波传播特性的影响,应用Legendre多项式方法求解了耦合波动方程,并通过数值算例讨论了该方法的收敛性,分析了初应力对纵向波和扭转波的影响,数值分析结果表明初应力对两者的影响是非常不同的。此外,轴向初应力对频散曲线的影响与径向初应力的影响也非常不同。     

Analytical extraction of leaky modes in circular slab waveguides with arbitrary refractive index profile

Circular slab waveguides are conformally transformed into straight inhomogeneous waveguides, whereupon electromagnetic fields in the core are expanded in terms of Legendre polynomial basis functions. Thereafter, different analytical expression of electromagnetic fields in the cladding region, viz. Wentzel-Kramers-Brillouin solution, modified Airy function expansion, and the exact field solution for circular waveguides, i.e., Hankel function of complex order, are each matched to the polynomial expansion of the transverse electric field within the guide. This field matching process renders different boundary conditions to be satisfied by the set of orthogonal Legendre polynomial basis functions. In this fashion, the governing wave equation is converted into an algebraic and easy to solve eigenvalue problem, which is associated with a matrix whose elements are analytically given. Various numerical examples are presented and the accuracy of each of the abovementioned different boundary conditions is assessed. Furthermore, the computational efficiency and the convergence rate of the proposed method with increasing number of basis functions are briefly discussed.     

Longitudinal Legendre polynomial expansion of electromagnetic fields for analysis of arbitrary-shaped gratings

The Legendre polynomial expansion method (LPEM), which has been successfully applied to homogenous and longitudinally inhomogeneous gratings [J. Opt. Soc. Am. B24, 2676 (2007)], is now generalized for the efficient analysis of arbitrary-shaped surface relief gratings. The modulated region is cut into a few sufficiently thin arbitrary-shaped subgratings of equal spatial period, where electromagnetic field dependence is now smooth enough to be approximated by keeping fewer Legendre basis functions. The R-matrix propagation algorithm is then employed to match the Legendre polynomial expansions of the transverse electric and magnetic fields across the upper and lower interfaces of every slice. The proposed strategy then enhances the overall computational efficiency, reduces the required memory size, and permits the efficient study of arbitrary-shaped gratings. Here the rigorous approach is followed, and analytical formulas of the involved matrices are given.     

正交各向异性空心圆柱体中的纵向导波

基于三维线弹性理论,采用勒让德正交多项式展开法,推导了正交各向异性材料中纵向导波的耦合波动方程,并对耦合波动方程进行了数值求解。首先,为确定方法的适用性和准确性,利用Disperse软件求解各向同性空心圆柱体中纵向导波的频散曲线,并将其与勒让德正交多项式展开法求解结果相对比,二者结果完全一致。然后,讨论了勒让德正交多项式截止值对轴对称导波频散曲线收敛性的影响,并从数值计算的角度分析了产生影响的原因。最后,针对碳纤维缠绕的复合材料空心圆柱体,分别求解纵向、扭转和弯曲三种不同模态纵向导波的相速度频散曲线。计算了不同径厚比下的弯曲模态相速度频散曲线,分析径厚比的变化对频散曲线的影响。     

压电空心圆柱体中的周向SH波

基于三维线性压电弹性理论,采用一种正交多项式级数方法研究了轴向极化、电开路时正交各向异性压电空心圆柱中的周向SH波。把位移和电势展开成勒让德多项式级数,引入点确定的材料常数以解决边界条件,最后把问题的解简化为一个特征值问题。计算了不同径厚比下FZT-4管道周向SH的波频散曲线和电势分布。讨论了压电的影响。     

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.     

Integral equation analysis and optimization of 2D layered nanolithography masks by complex images Green's function technique in TM polarization

Analysis and optimization of diffraction effects in nanolithography through multilayered media with a fast and accurate field-theoretical approach is presented. The scattered field through an arbitrary two-dimensional (2D) mask pattern in multilayered media illuminated by a TM-polarized incident wave is determined by using an electric field integral equation formulation. In this formulation the electric field is represented in terms of complex images Green's functions. The method of moments is then employed to solve the resulting integral equation. In this way an accurate and computationally efficient approximate method is achieved. The accuracy of the proposed method is vindicated through comparison with direct numerical integration results. Moreover, the comparison is made between the results obtained by the proposed method and those obtained by the full-wave finite-element method. The ray tracing method is combined with the proposed method to describe the imaging process in the lithography. The simulated annealing algorithm is then employed to solve the inverse problem, i.e., to design an optimized mask pattern to improve the resolution. Two binary mask patterns under normal incident coherent illumination are designed by this method, where it is shown that the subresolution features improve the critical dimension significantly.     

Generalized thermoelastic waves in spherical curved plates without energy dissipation

In this article, the propagation of thermoelastic waves in orthotropic spherical curved plates subjected to stress-free, isothermal boundary conditions is investigated in the context of the Green–Naghdi (GN) generalized thermoelastic theory (without energy dissipation). The theoretical formulation is based on the linear GN thermoelastic theory. The coupled wave equation and heat conduction equation expressed by the displacement and temperature are obtained. By the Legendre orthogonal polynomial series expansion approach, the coupled controlling equations are solved. The convergence of the method is demonstrated through a numerical example. The dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Dispersion curves of the corresponding purely elastic spherical plate are also shown to analyze the influence of thermoelasticity on elastic modes. The displacement, temperature and stress distributions of both elastic modes and thermal modes are calculated to show their differences. A thermoelastic spherical plate with a different ratio of radius to thickness is considered to show the influence of the ratio on the characteristics of thermoelastic waves.     

纤维增强复合板中声弹Lamb波的波结构分析

基于线性三维弹性理论和“增量变形力学”理论,采用勒让德正交多项式展开法,推导了在水平和垂直方向施加初应力时,沿纤维增强复合板的非主对称轴方向传播的声弹Lamb波的波动方程,并对波动方程进行数值求解。为了验证方法的准确性,将该文方法求解的各向同性材料的相速度频散曲线与Disperse?软件的计算结果进行比较,两者吻合良好。以单层单向纤维增强复合材料板为例,计算了无初应力状态下的波结构应力曲线,并与应力自由边界初始条件的一致性进行了比较。研究了水平和垂直方向初应力效应对Lamb波频散曲线的影响。针对声弹效应较为敏感的Lamb波A₀模态,着重分析了初应力效应对波结构位移分布曲线的影响。     

代数双曲空间中拟Legendre基的应用

鉴于Legendre基等正交基在代数多项式空间中的广泛应用,论文在深入研究代数双曲空间的拟Legendre基性质的基础上,给出了其在反函数逼近和等距曲线逼近上的应用。利用多项式和双曲函数的混合多项式序列来逼近反函数,并通过实例证明给出方法的有效性;对基曲线的法矢曲线进行逼近,构造H-Bézier曲线的等距曲线的最佳逼近,这种方法直接求得逼近曲线的控制顶点,计算简单,截断误差小。     

Guided waves in functionally graded viscoelastic plates

In this paper, a dynamic solution for the propagating viscoelastic waves in functionally graded material (FGM) plates subjected to stress-free conditions is presented in the context of the Kelvin–Voigt viscoelastic theory. The FGM plate is composed of two orthotropic materials. The material properties are assumed to vary in the thickness direction according to a known variation law. The three obtained wave equations are divided into two groups, which control viscoelastic Lamb-like wave and viscoelastic SH wave, respectively. They are solved respectively by the Legendre orthogonal polynomial series approach. The validity of the method is confirmed through a comparison with the Lamb wave solution of a pure elastic FGM plate and a comparison with the SH wave solution of a viscoelastic homogeneous plate. The dispersion curves and attenuation curves for the graded and homogeneous viscoelastic plates are calculated to highlight their differences. The viscous effect on dispersion curves is shown. The influences of gradient variations are illustrated.     

基于Legendre正交多项式逼近法的结构可靠性分析

提出了结构可靠性分析的Legendre正交多项式逼近法。主要是基于数值逼近原理,以Legendre正交函数族做基,利用功能函数的高阶矩信息,通过计算功能函数概率密度函数的逼近表达式,然后根据工程结构可靠性的一般表达式来计算结构的失效概率,进行可靠性分析。通过数值检验,证明该方法可以很好地逼近各种经典理论分布曲线(正态分布、指数分布等6种经典分布)。文后给出了结构构件失效概率的实例计算,并和其它几种常用方法进行对比,进一步表明了Legendre正交多项式数值逼近法在结构可靠性分析理论上的正确性和实用性。     

Generalized thermoelastic waves in functionally graded plates without energy dissipation

In this paper, a dynamic solution of the propagating thermoelastic waves in functionally graded material (FGM) plate subjected to stress-free, isothermal boundary conditions is presented in the context of the Green–Naghdi (GN) generalized thermoelastic theory. The FGM plate is composed of two orthotropic materials. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The coupled wave equation and heat conduction equation are solved by the Legendre orthogonal polynomial series expansion approach. The convergency of the method is discussed through a numerical example. The dispersion curves of the inhomogeneous thermoelastic plate and the corresponding pure elastic plate are compared to show the characteristics of thermal modes and the influence of the thermoelasticity on elastic modes. The displacement, temperature and stress distributions of elastic modes and thermal modes are shown to discuss their differences. A plate with a different gradient variation is calculated to illustrate the influence of the gradient field on the wave characteristics.     

Influences of initial stresses on guided waves in functionally graded hollow cylinders

Limitations of the manufacturing technology result in the existence of initial stresses in functionally graded material (FGM) structures. In the context of the theory of “Mechanics of Incremental Deformations,” the guided wave characteristics in FGM hollow cylinders under initial stresses in the radial and axial directions are investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. The convergence of the method is discussed through numerical examples. It is found that the influences of initial stresses on the longitudinal waves and on the torsional waves are quite distinct, and that the influences of initial stresses in the axial direction are very different from those in the radial direction, both on the dispersion curves and on the displacement and stress distributions.     

平面应变轴对称压电圆柱壳的随机响应

提出求解随机激励轴对称压电圆柱壳响应的一种方法，并导出相应的解析表达式。首先给出压电圆柱壳在边界随机激励下的基本方程；然后通过位移与电势的变换，将随机激励变换到运动方程中；再利用Legendre多项式展开位移，应用Galerkin法化偏微分的运动方程为常微分方程组；最后根据随机振动理论，得到压电圆柱壳位移与加速度响应的均方值，由此可计算随机响应、分析有关因素的影响与机电耦合关系等。分析说明了存在的机电耦合项，及由此产生广义刚度的非对称性。     

Exact Solutions for the Stability and Free Vibration of Multilayered Functionally Graded Material Hollow Cylinders under Axial Compression

Exact three-dimensional stability and free vibration analyses of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under axial compression are presented. The material properties of each FGM layer are assumed to obey a power-law distribution of the volume fractions of constituents through the thickness coordinate. The Pagano method, which is based on the principle of virtual displacement and is conventionally used for the analysis of laminated composite structures, is modified to be feasible for the study of multilayered FGM cylinders, in which Reissner's mixed variational theorem, the successive approximation and transfer matrix methods, and the transformed real-valued solutions of the system equations are used. The present modified Pagano solutions for laminated composite cylinders are in excellent agreement with the exact 3D ones available in the literature, and those for sandwich FGM cylinders may be used as the benchmark solutions to assess the ones obtained using various two-dimensional theories and numerical models. The influence of some effects on the lowest critical load parameters of multilayered FGM cylinders and laminated composite ones is investigated, such as the derivation between using von Karman nonlinearity and full kinematic one, and the difference between using the uniform stress assumption and the uniform strain one. In addition, a parametric study with regard to some effects on the lowest frequency parameters of axially loaded, multilayered FGM cylinders is carried out, such as the magnitude of the applied compressive loads, the radius-to-thickness, length-to-radius and orthotropic ratios, and the material-property gradient index.     

黏弹性正交各向异性空心圆柱中纵向导波的传播

基于线性三维弹性理论和Kelvin-Voigt模型,采用勒让德正交多项式展开法推导了黏弹性正交各向异性空心圆柱中纵向导波的波动方程,数值求解了波动方程并阐述了相关方程的含义。首先计算了大径厚比下黏弹性管的相速度频散曲线和衰减曲线,并与已发表文献的结果进行了对比,验证了程序的正确性,并进一步计算了低阶纵向导波的位移分布和应力分布曲线,验证了方法的可靠性。然后利用方程的解耦特性,分别求解了不同径厚比、不同黏性常数下纵向模态和扭转模态的频散和衰减曲线,研究了径厚比和黏性常数效应对两种模态的影响。最后针对扭转模态导波,研究了材料相关黏弹性常数对其频散特性和衰减特性的影响。      

On a problem of the vibration of functionally graded conical shells with mixed boundary conditions

This paper presents a theoretical approach to solve vibration problems of functionally graded (FG) truncated conical shells under mixed boundary conditions. The material properties of FG shell are assumed to vary continuously through the thickness of the conical shell. The fundamental relations, motion and strain compatibility equations of FG truncated conical shells are derived by means of the Airy stress function method. Two cases of mixed boundary conditions are investigated. The basic equations are solved by using Galerkin method and fundamental cyclic frequencies of FG truncated conical shells are obtained. The results are compared and validated with the results available in the literature. The detailed parametric studies are carried out to investigate the influences of radius-to-thickness ratio, lengths-to-radius ratio, material composition and mixed boundary conditions on the fundamental cyclic frequencies of truncated conical shells.     

Multilayered piezomagnetic/piezoelectric composite with periodic interfacial Yoffe-type cracks under magnetic or electric field

The problem of periodic mode-III Yoffe-type cracks propagating subsonically along the interfaces in a multilayered piezomagnetic/piezoelectric composite under in-plane magnetic or electric field is studied. By means of periodic conditions, the analysis of the multilayered problem is simplified to a bilayer model with an interfacial Yoffe-type crack, which can be reduced to the Cauchy singular integration equation of the first kind, by utilizing the Fourier transform. The normalized dynamic stress intensity factor (NDSIF) can be obtained numerically. Results show that the NDSIF generally depends on the layer thickness ratio, crack moving speed, electric or magnetic loading as well as material properties. In regard to the curve monotony of the NDSIF versus the passive layer thickness, there generally exist three different cases distinguished by a parameter, θ, which depends on the crack moving speed as well as material mismatch parameters. Similar behavior has been reported for the periodic static cracks where the different monotonies are judged by the material mismatch parameter G (Wan et al. in Eng. Fract. Mech. 84:132–145, 2012). The present results reduce to those of periodic static cracks when considering the vanishing crack speed. As far as the curve monotony and the judging parameter are concerned, no matter what the crack moving speed is, the Yoffe-type crack problem is identical to the static crack problem when the piezomagnetic and piezoelectric layers share the same shear wave velocity. In addition, detailed analyses are also provided for NDSIF versus crack moving speed for different layer thickness ratio and material mismatch parameters. This study can be considered as an extension of the previous analysis of periodic static cracks problem and is expected together to provide some guidelines for the optimal design of a multilayered PM/PE composite.