四边固支功能梯度板中波的传播

摘要：考虑剪切变形和转动惯性的影响,采用一阶剪切变形板理论和小应变的应变－位移关系,利用Hamilton原理推得运动控制方程,并应用特征值方程得到频散方程。给出了波在功能梯度板中传播的频散,相速度和群速度随波数变化的曲线,分析了不同的功能梯度材料指数对波传播的影响规律。     

Elastodynamic Behavior of Waves in Thermo-Microstretch Elastic Plate Bordered with Layers of Inviscid Liquid

The elastodynamic behavior of waves in a thermo-microstretch elastic homogeneous isotropic plate bordered with layers of inviscid liquid on both sides subjected to stress-free thermally insulated and isothermal conditions is investigated in the context of Lord and Shulman and Green and Lindsay theories of thermoelasticity. The mathematical model has been simplified by using the Helmholtz decomposition technique, and the frequency equations for different mechanical situations are obtained and discussed. The special cases such as short wavelength waves and regions of the secular equations are also discussed. Finally, the numerical solution is carried out for a magnesium crystal composite material plate bordered with water. The dispersion curves, attenuation coefficients, amplitudes of dilatation, microrotation, microstretch, and temperature distribution for the symmetric and skew-symmetric wave modes are presented graphically.     

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.     

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。     

On the propagation of Lamb waves in viscothermoelastic plates under fluid loadings

In this paper the propagation of plane and circular crested viscothermoelastic waves in a homogeneous isotropic, Kelvin-Voigt type viscoelastic thermally conducting, plate sandwiched between inviscid liquid layers is investigated in the context of classical and non-classical theories of thermoelasticity. The secular equations for the symmetric and skew-symmetric modes of plane and circular crested waves are derived in closed form and isolated mathematical conditions. It is noticed that the motion for both the plane and cylindrical waves in plates is governed by Rayleigh-Lamb-type secular equations. The secular equations for thin plate and short wave length waves are also obtained and discussed. The results in the absence of fluid loading, coupled and uncoupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. The dispersion curves, attenuation profiles and specific loss in case of symmetric and skew-symmetric wave modes are also presented graphically for a polymethyl methacrylate material plate under fluid loadings. The effect of dissipation due to viscosity is noticed to be quite significant and clearly visible from various curves in the graphs.     

Wave propagation and transient response of a FGM plate under a point impact load based on higher-order shear deformation theory

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load are presented. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Based on the higher-order shear deformation theory and considering the effect of the rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Then, using the dispersion relation and integral transforms, exact integral solutions for the functionally graded plate under a point impact load are obtained. The transient response curves of the functionally graded plates are plotted and the influence of volume fraction distributions on transient response of functionally graded plates is analyzed. Finally, the solutions of the higher-order shear deformation theory and the first-order shear deformation theory are studied.     

Analysis of weakly guiding cladded acoustic fibers of hexagonal crystal symmetry

An approximate analysis of the weakly guiding cladded acoustic fiber consisting of a core and infinite thick cladding both with hexagonal crystal symmetry is presented. The crystalline Z axis is parallel to the fiber axis. Weak guidance conditions require that the stiffness constants of the core are slightly less than those of the cladding, and the density of the core and cladding are almost the same. Approximate dispersion and cutoff equations are derived for all pure guided modes. Dispersion curves of several lower-order guided torsional, radial-axial and flexural modes are evaluated using both exact and approximate formulas and they are in good agreement. A simple approximate dispersion equation for leaky (longitudinal-type) modes is also obtained.     

非均匀复合材料板中剪切波传播的研究

基于弹性界层中弹性波干涉理论,采用有效介质法,研究了剪切波在非均匀、纤维随机分布复合材料板中的传播,得到了非均匀弹性介质内的有效波数。通过满足弹性有界层的上、下边界条件,得到了非均匀界层中的频散方程。作为特例,绘出了不同参数下板中的前四阶频散曲线。可以看出,非均匀弹性有界层中的频散曲线和均匀界层中的有很大不同。最后分析了纤维和基体特性比、纤维的体积份数以及板厚与纤维半径比对频散曲线的影响。     

Dispersion of Thermoelastic Waves in a Plate With and Without Energy Dissipation

In this paper, the dispersion and energy dissipation of thermoelastic plane harmonic waves in a thin plate bounded by insulated traction-free surfaces is studied on the basis of three generalized theories of thermoelasticity. The frequency equations corresponding to the symmetric and antisymmetric modes of vibration of the plate are obtained. Some limiting and particular cases of the frequency equations are then discussed. Results obtained in three theories of generalized thermoelasticity are compared. The results for the coupled thermoelasticity can be obtained as particular cases of the results by setting thermal relaxations times equal to zero. Numerical evaluations relating to the lower modes of the symmetric and antisymmetric waves are presented for an aluminum alloy plate.     

Thermoelastic Lamb waves in a transversely isotropic plate bordered with layers of inviscid liquid

Analysis for the propagation of thermoelastic waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides, is investigated in the context of coupled theory of thermoelasticity. Secular equations for homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and anti-symmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. It is shown that the purely transverse motion (SH mode), which is not affected by thermal variations, gets decoupled from rest of the motion of wave propagation and occurs along an in-plane axis of symmetry. The special cases, such as short wavelength waves and thin plate waves of the secular equations are also discussed. The secular equations for leaky Lamb waves are also obtained and deduced. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic plate of zinc material bordered with water. The dispersion curves for symmetric and anti-symmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The theory and numerical computations are found to be in close agreement.     

Guided thermoelastic waves in functionally graded plates with two relaxation times

Functionally graded material (FGM) is a promising heat insulation material. Wave propagation in FGM structures has received much attention for the purpose of non-destructive testing and evaluation. Few literatures dealt with the thermoelastic wave in FGM structures although the thermal effect would cause attenuations of elastic waves. In this paper, guided thermoelastic waves in FGM plates subjected to stress-free, isothermal boundary conditions are investigated in the context of the Green–Lindsay (GL) generalized thermoelastic theories (with two relaxation times). Coupled wave equations and heat conduction equation are solved by the Legendre polynomial approach. Dispersion curves for a pure elastic graded plate are calculated to make a comparison with the published data. For the thermoelastic graded plate, dispersion curves of thermal modes and elastic modes are illustrated simultaneously. Attenuation curves for graded plates with different relaxation times are compared. The influences of different material gradient shapes are discussed. Two homogeneous thermoelastic plates with different volume fractions are obtained to show their differences from graded plates. Finally, thermoelastic wave dispersion curves for a homogeneous plate and a graded plate are calculated in the context of the classical coupled thermoelastic theory (CT) to show its differences and similarities to the generalized theory.     

间隙波在功能梯度压电板和压电半空间结构中的传播

研究了间隙波在功能梯度压电板和压电半空间结构中的传播性质.功能梯度压电板的材料性能沿x2方向呈指数变化,首先推导了间隙波传播时的解析解,利用界面条件得到了间隙波的频散方程,基于推导的频散方程,结合数值算例分析了功能梯度压电材料的梯度、压电层厚度以及材料性能对间隙波相速度的影响,研究结果对功能梯度压电材料的覆层结构在声波器件中的应用具有重要的参考价值.     

Lamb wave propagation in the functionally graded piezoelectric–piezomagnetic material plate

The propagation behaviour of Lamb waves in the functionally graded piezoelectric–piezomagnetic material plate with material parameters varying continuously along the thickness direction is investigated in this paper. The power series technique is employed to solve these variable coefficient ordinary differential equations. Dispersion equations are given for different boundary conditions. In numerical examples, the influence of the variation of each parameter on dispersion curves and cut-off frequency in electrically and magnetically open cases is discussed in detail. Results show that the elastic parameters and density varying along the thickness direction obviously influence the variation of phase velocity. Some variations in electric and magnetic parameters also affect the phase velocity but the influence is too small, while others almost cannot affect the dispersion curves. Cut-off frequency is closely related to two elastic parameters and to density, whereas other parameters almost cannot influence it. All the results can provide theoretical guidance not only for the analysis and design of a magnetoelectric transducer using functionally graded materials, but also for ultrasonic nondestructive evaluation.     

Effect of uncertainty in material properties on wave propagation characteristics of nanorod embedded in elastic medium

The effect of uncertainty in material properties on wave propagation characteristics of nanorod embedded in an elastic medium is investigated by developing a nonlocal nanorod model with uncertainties. Considering limited experimental data, uncertain-but-bounded variables are employed to quantify the uncertain material properties in this paper. According to the nonlocal elasticity theory, the governing equations are derived by applying the Hamilton’s principle. An iterative algorithm based interval analysis method is presented to evaluate the lower and upper bounds of the wave dispersion curves. Simultaneously, the presented method is verified by comparing with Monte-Carlo simulation. Furthermore, combined effects of material uncertainties and various parameters such as nonlocal scale, elastic medium and lateral inertia on wave dispersion characteristics of nanorod are studied in detail. Numerical results not only make further understanding of wave propagation characteristics of nanostructures with uncertain material properties, but also provide significant guidance for the reliability and robust design of the next generation of nanodevices.     

Hydroelastic Behavior of a Floating Ring-Shaped Plate

The interaction between incident surface water waves and floating elastic plate is studied. This paper considers the diffraction of plane incident waves on a floating flexible ring-shaped plate and its response to the incident waves. An analytic and numerical study of the hydroelastic behavior of the plate is presented. An integro-differential equation is derived for the problem and an algorithm of its numerical solution is proposed. The representation of the solution as a series of Hankel functions is the key ingredient of the approach. The problem is first formulated. The main integro-differential equation is derived on the basis of the Laplace equation and thin-plate theory. The free-surface elevation, plate deflection and Green’s function are expressed in polar coordinates as superpositions of Hankel and Bessel functions, respectively. These expressions are used in a further analysis of the integro-differential equation. The problem is solved for two cases of water depth infinite and finite. For the coefficients in the case of infinite depth a set of algebraic equations is obtained, yielding an approximate solution. Then a solution is obtained for the general and most interesting case of finite water depth analogously in the seventh section. The exact solution might be approximated by taking into account a finite number of the roots of the plate dispersion relation. Also, the influence of the plate’s motion on wave propagation in the open water field and within the gap of the ring is studied. Numerical results are presented for illustrative purposes.     

Small amplitude vibration of viscoelastic plates

Summary The equations of incompressible linear viscoelasticity are averaged over the thickness of a thin plate. The basic equations governing the extensional and flexural motion of the plate are obtained when displacements are assumed to be linear across the thickness of the plate. A dispersion relation governing the propagation of flexural waves is obtained which incorporates characterizing parameters of the material. Numerical results are presented for the decay of waves in a semi-infinite plate which is excited harmonically at one boundary. The results show how characterizing parameters affect the decay of waves.With 2 Figures     

Wave propagation in an asymmetric fibre-reinforced laminated plate. Part II

Summary The propagation of waves in a two-ply laminated plate is examined. Each lamina is composed of the same transversely isotropic, homogeneous, elastic material with the direction of transverse isotropy lying in the plane of the lamina. The two laminae are perfectly bonded to each other in such a way that their respective axes of transverse isotropy are orthogonal. The dispersion equation is derived for waves propagating at an arbitrary angle in the plane of the plate. The long and short wavelength behaviour is examined analytically and subsequently the equation is solved numerically. Results are plotted for the fundamental mode and first two harmonics. In order to gain further understanding of these dispersion curves in certain regions, the stress variation through the plate is examined.The corresponding dispersion curves are obtained for the ideal material which is assumed to be inextensible in the direction of transverse isotropy.With 12 Figures     

Circular crested waves in anisotropic thermoelastic plates bordered with inviscid liquid

The propagation of circularly crested waves in a homogeneous, transversely isotropic, thermally conducting plate bordered with layers of inviscid liquid or half space of inviscid liquid on both sides is investigated in the context of conventional coupled thermoelasticity, Lord-Shulman and Green-Lindsay theories of thermoelasticity. Secular equations for circular homogeneous transversely isotropic plate in closed form and isolated mathematical conditions for symmetric and antisymmetric wave modes in completely separate terms are derived. The results for isotropic materials and uncoupled theories of thermoelasticity have been obtained as particular cases. The special cases such as short wavelength waves, thin plate waves and leaky Lamb waves of the secular equation are also deduced and discussed. The amplitudes of displacement components and temperature change have also been computed and studied. Finally, the numerical solution is carried out for transversely isotropic circular plate of cobalt material bordered with water. The dispersion curves for symmetric and antisymmetric wave modes, attenuation coefficient and amplitudes of displacement and temperature change in case of fundamental symmetric (S₀) and skew symmetric (A₀) modes are presented in order to illustrate and compare the theoretical results. The analytical and numerical results are found to be in close agreement.     

磁电弹功能梯度材料板中的波动特性

假设材料常数和电、 磁常数沿板厚方向呈梯度变化, 将电磁功能梯度材料板沿厚度方向划分为层单元, 建立单元的控制方程, 然后根据单元之间的连续条件将单元控制方程装配成系统的控制方程, 求解控制方程, 得到不同模态波的弥散曲线, 分析了电磁参数对波的弥散特性的影响, 同时研究了材料梯度对弹性位移、 静电势和静磁势分布的影响, 并对磁电功能梯度材料中波的六个特征波面进行了分析。计算结果表明材料梯度引起弹性位移、 静电势、 静磁势的分布集中于材料参数梯度减少的方向, 材料为正交各向异性弹性材料。      

Dispersion relations and modes of wave propagation in inclusion-reinforced composite plates

This paper is intended to examine the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness on the dispersion relations and modes of wave propagation in inclusion-reinforced composite plates. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Mori–Tanaka mean-field theory is used to predict the effective elastic moduli of the composite plate explicitly. The effective elastic moduli are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior. The resulting moduli are then used to determine the dispersion relations and the modal patterns of Lamb waves using the dynamic stiffness matrix method. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions are symmetrical or antisymmetric about the midplane of the plate. Classifying the wave type in an anisotropic plate is not as simple as that in an isotropic plate, and has not received proper attention in the literature. The wave types and orders are identified by analyzing the dispersion curves and inspecting the calculated modal patterns, and the results indicate that the Lamb waves in an orthotropic composite plate can also be classified as either symmetric or antisymmetric waves. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns. Propagation speed is generally increased with the aspect ratio, e.g., using longer fibers generally results in a higher propagation speed.