Dispersion effects of extensional waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface, on time-harmonic extensional wave propagation in a pre-stressed symmetric layered composite is considered. The bimaterial composite consists of incompressible isotropic elastic materials. The shear spring type resistance model employed to simulate the imperfect interface can accommodate the extreme cases of perfect bonding and a fully slipping interface. The dispersion relation obtained by formulating the incremental boundary-value problem and the use of the propagator matrix technique, is analyzed at the low and high wavenumber limits. For the perfectly bonded and imperfect interface cases in the low wavenumber region, only the fundamental mode has a finite phase speed, while other higher modes have an infinite phase speed when the dimensionless wavenumber approaches zero. However, for the fully slipping interface in the low wavenumber region, both the fundamental mode and the next lowest mode have finite phase speeds. In the high wavenumber region, when the dimensionless wavenumber tends to infinity, the phase speeds of the fundamental mode and the higher modes depend on the phase speeds of the surface and interfacial waves and on the limiting phase speed of the composite. An expression to determine the cut-off frequencies is obtained from the dispersion relation. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of the imperfect interface is clearly evident in the numerical results.     

Wave propagation along a non-principal direction in a compressible pre-stressed elastic layer

The dispersive behavior of small amplitude waves propagating along a non-principal direction in a pre-stressed, compressible elastic layer is considered. One of the principal axes of stretch is normal to the elastic layer and the direction of propagation makes an angle θ with one of the in-plane principal axes. The dispersion relations which relate wave speed and wavenumber are obtained for both symmetric and anti-symmetric motions by formulating the incremental boundary value problem for a general strain energy function. The behavior of the dispersion curves for symmetric waves is for the most part similar to that of the anti-symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress and propagation angle, it may be possible for both the fundamental mode and the next lowest mode to have finite phase speeds, while other higher modes have an infinite phase speed. At the high wavenumber limit, the phase speeds of the fundamental mode and the higher modes tend to the Rayleigh surface wave speed and the limiting wave speeds of the layer, respectively. Numerical results are presented for a Blatz–Ko material and the effect of the propagation angle is clearly illustrated.     

Time-harmonic wave propagation in a pre-stressed compressible elastic bi-material laminate

The dispersive behaviour of time-harmonic waves propagating along a principal direction in a perfectly bonded pre-stressed compressible elastic bi-material laminate is considered. The dispersion relation which relates wave speed and wavenumber is obtained by formulating the incremental boundary value problem and the use of the propagator matrix technique. At the low wavenumber limit, depending on the pre-stress, both the fundamental mode and the next lowest mode may have finite phase speeds. For the higher modes which have infinite phase speeds in the low wavenumber region, an expression to determine the cut-off frequencies is obtained. At the high wavenumber limit, the phase speeds of the fundamental mode and higher modes tend to phase speeds of the surface wave, the interfacial wave or the limiting phase speed of the composite. For numerical examples, either a two-parameter compressible neo-Hookean material or a two-parameter compressible Varga material is assumed.     

Wave propagation analysis in 2D nonlinear hexagonal periodic networks based on second order gradient nonlinear constitutive models

We analyze the propagation of nonlinear waves in homogenized periodic nonlinear hexagonal networks, considering successively 1D and 2D situations. Wave analysis is performed on the basis of the construction of the effective strain energy density of periodic hexagonal lattices in the nonlinear regime. The obtained second order gradient nonlinear continuum has two propagation modes: an evanescent subsonic mode that disappears after a certain wavenumber and a supersonic mode characterized by an increase of the frequency with the wavenumber. For a weak nonlinearity, a supersonic mode occurs and the dispersion curves lie above the linear dispersion curve (v_p =v_p0). For a higher nonlinearity, the wave changes from a supersonic to an evanescent subsonic mode at s=0.7 and the dispersion curves drops below the linear case and vanish for certain values of the wavenumber. An important decrease in the frequency occurs for both subsonic and supersonic modes when the lattice becomes auxetic, and the longitudinal and shear modes become very close to each other. The influence of the lattice geometrical parameters of the lattice on the dispersion relations is analyzed.     

Dispersion relations for SH wave in magneto-electro-elastic heterostructures

In the present work the dispersion relations of stationary SH waves in a heteroestructure with magneto-electro-elastic properties have been obtained. The calculations were done taking into consideration the symmetry of the system and separating the solutions in symmetric and anti-symmetric parts. Different limit cases are presented. The dispersion curves and amplitudes of vibration are shown for different configurations.     

正交各向异性板中非主轴方向的Lamb波

基于线性三维弹性理论,采用勒让德正交多项式展开法,推导了波沿正交各向异性材料非主轴方向传播时的Lamb波耦合波动方程,并对耦合波动方程进行了数值求解。为验证该方法的适用性和正确性,首先将此方法应用于各向同性材料,并与已知的数据结果进行了比较;然后以单向纤维增强复合材料为例,计算了耦合Lamb波沿不同的非主轴方向传播时的相速度频散曲线,并分别研究了传播方向改变时低阶模态Lamb波和高阶模态Lamb波频散特性的变化。最后,针对潜在用于各向异性复合材料结构健康监测的耦合Lamb波低阶模态,给出了其在不同传播方向时的相速度分布和群速度分布。同时,结合低阶模态Lamb波的位移分布特性和材料的各向异性特点,阐释了S0模态对波的传播方向变化最为敏感的原因。     

Symmetric and anti-symmetric Rayleigh–Lamb modes in sinusoidally corrugated waveguides: An analytical approach

The wave propagation analysis in corrugated waveguides is considered in this paper. Elastic wave propagation in a two-dimensional periodically corrugated plate is studied here analytically. The dispersion equation is obtained by applying the traction free boundary conditions. Solution of the dispersion equation gives both symmetric and anti-symmetric modes. In a periodically corrugated waveguide all possible spectral order of wave numbers are considered for the analytical solution. It has been observed that the truncation of the spectral order influences the results. Truncation number depends on the degree of corrugation and the frequency of the wave. Usually increasing frequency requires increasing number of terms in the series solution, or in other words, a higher truncation number. For different degrees of corrugation the Rayleigh–Lamb symmetric and anti-symmetric modes are investigated for their non-propagating ‘stop bands’ and propagating ‘pass bands’. To generate the dispersion equation for corrugated plates with a wide range of the degree of corrugation, appropriate truncation of the spectral orders has to be considered. Analytical results are given for three different degrees of corrugation in three plates. Resonance of symmetric and anti-symmetric modes in these plates, their ‘cut-off’, ‘cut-on’, ‘branch-point’, ‘change-place’, ‘mode conversion’ and ‘pinch points’ at various frequencies are also studied.     

Propagation behaviour of guided waves in tapered sandwich structures and debonding identification using time reversal

Fundamental symmetric and anti-symmetric Lamb wave modes are characterised in terms of their velocity and magnitude variation as they encounter a gradual change in the thickness of a composite sandwich plate with a high density foam core (Dyvinicell HP100). Wave mode selection is optimised to enhance the sensitivity of guided waves for debonding inspection of tapered sandwich structures, and the conversion in wave modes after interaction with the taper is also studied. A taper change in the sandwiched structure can induce a clear change in the wave magnitude when the guided waves are excited at relatively low frequency (below around 150 kHz), while there is less effect on the velocity of the propagating waves. The effect of the taper on the signal magnitude and velocity is reduced when the excitation frequency is increased beyond 200 kHz. An imaging algorithm based on time reversal is developed to detect multiple debonding in tapered composite sandwich panels, using guided waves at different excitation frequencies ranging from 150 to 200 kHz, from an active sensor network. Correlation coefficients between the original and reconstructed time reversal signals are calculated to define a damage index for individual sensing paths, which are subsequently used in a fusion process to identify the presence of multiple debonded areas.     

气液两相流压力波传播速度研究

将双流体模型用于绝热无相的管道气液两相流,依据小扰动线化分析原理,导出了压力波波数Ｋ方程通过对不同空隙率下肉体上压力波小随角频率变化的计算,研究了虚拟质量力和狭义相间阻力对压力波波速及其人色散性的影响。对泡状流和弹状流压力波波速的计算结果与前人的测量结果作了比较,两者符合良好。     

Wave propagation in layer with two preferred directions

This article is concerned with overall or macroscopic properties of a composite material with no distinction made between the fibres and the matrix which they are embedded in. All the properties with dimensions larger than the fibre diameter and spacing are regarded as averaged over a volume of material. The systems of particular interest here are in the fibre reinforced composites with the fibres being very much stiffer and stronger than the matrix.Laminated plates of fibre-reinforced material are often fabricated from prepreg tapes, laid up according to some specific arrangement of fibre orientation and then bonded together. An angle-ply laminate is formed by alternating plies so that the families in adjacent laminas are inclined by angle ϕ and −ϕ to given direction alternately. The process of fabricating a multilayered plate of this material gives rise to a laminate in which the plies are separated by resin rich layer, and when this layer is thin enough that its thickness is negligible it may be regarded as plate reinforced by two families of fibres. Problems shall be considered in three dimensions, but attention shall be restricted to linear elasticity theory. The plate under consideration is reinforced by two mechanically equivalent families of fibres, but with no other preferred directions, so that it is locally orthotropic with respect to the plane of the fibres and to the two planes that orthogonally bisect the fibres.In this article linear elastic stress–strain relation is employed to derive dispersion curves for plane harmonic waves propagating in a plate of finite thickness but of infinite lateral extent. Attention is restricted to waves propagating in the plane parallel to stress free plate faces where waves travelling at any angle to one of the families of very strong fibres are examined. The dispersion equations, relating the phase velocity to the wavelength, are obtained. The fundamental modes are examined for symmetric as well as for anti-symmetric deformations. This leads to full understanding of displacement field as well as stress field.     

Theoretical modeling of guided wave propagation in a sandwich plate subjected to transient surface excitations

A fast and efficient two-dimensional (2D) semi-analytical model based on global matrix method is developed to study the general characteristics of guided wave propagation in a honeycomb composite sandwich structure (HCSS) subjected to time-dependent transient surface excitations. The HCSS used in this study has an extremely lightweight and thick nomex honeycomb core, which is sandwiched between two thin graphite woven composite skins. The homogenized material properties of the skin and the core are considered to be elastic and quasi-isotropic in nature. Far-field time history of surface displacements are calculated for vertical and horizontal tone-burst surface excitations that are representative of thickness and radial mode of vibrations of piezoelectric transducers, respectively. Results are compared with those obtained from Finite element modeling (FEM) in LS-DYNA showing good agreement. Wavelet transform is then performed on the time-domain signal to obtain the group velocities of the propagating modes for their accurate identification on the basis of the theoretical dispersion curve. It is found that the response signal is dominated by the first anti-symmetric mode for a vertical excitation, whereas, the signal characteristics are multimodal in nature with dominating higher order symmetric and anti-symmetric modes for a horizontal excitation. The model is expected to be helpful for appropriate guided wave mode tuning and rapid analysis of data for experimental detection of disbonds in these novel structures.     

Leaky and non-leaky behaviours of guided waves in CF/EP sandwich structures

The aim of this study is to investigate the leaky and non-leaky behaviours of guided waves, between the composite skin and the core in CF/EP sandwich structures, focusing on the fundamental symmetric like and anti-symmetric like guided wave modes and Rayleigh waves. In investigating the core effect on the guided wave propagation different types of cores are used, namely Nomex honeycomb (HRH 10 1/8-3) 10 and 20 mm in thickness and foam (Divinycell^®  PVC). The behaviour of the guided wave modes is characterised and the conversion mechanism to the Rayleigh wave is investigated. Further, leaky and non-leaky behaviours of guided waves upon interacting with debonded areas are explored, where the ability of guided waves to identify debonding of different sizes was assessed. Finite element analysis simulations are presented to support the experimental analysis, where propagation of ultrasonic waves and their interaction with debonded areas are quantitatively examined.     

撑杆式预应力钢压杆屈曲形态函数研究

基于撑杆式预应力钢压杆正对称和反对称两个基本屈曲模态, 给出了相应的 屈曲形态函数, 通过3个典型算例并变化关键参数着重研究了模拟曲线项数, 同时 给出了不对称屈曲的形态函数. 研究表明, 取形态函 数模拟项数为预应力钢压杆节间数, 可以很好满足工程精度.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

粘性可压混合层时间稳定性对称紧致差分求解

基于可压扰动方程组的一阶改型 ,将高精度对称紧致格式引入边值法数值线性稳定性分析。对所获非线性离散特征值问题给出了一个通用形式二阶迭代局部算法 ,实现了时间模式和空间模式的统一求解 ,并将扰动特征值及其特征函数同时得到。据此分析了可压平面自由混合层时间稳定性 ,涉及二维 /三维扰动波、粘性 /无粘扰动波、第一 /第二模态、特征函数、伪特征值谱等。研究表明 ,压缩性效应和粘性效应对最不稳定扰动波数和增长率呈相似的减抑作用 ;在 Mc=1附近 ,从高波数段开始 ,粘性效应可强化二维不稳定扰动波由第一模态向第二模态的过渡     

Elastic waves guided by a material interface

The propagation of interfacial small-amplitude waves along a rectilinear thin film separating two pre-stressed, incompressible, elastic media is addressed. The film is modelled as a material surface possessing its own mass density and normal and flexural stiffnesses. It is shown that these features induce dispersion as the obtained secular equations are polynomials of the second degree in the wavenumber when bending stiffness is absent (membrane-like interface), and of the fourth degree otherwise (plate-like interface). In both case, beyond the modified Stoneley mode, a bending mode for the interface, an additional propagating wave can exist, with amplitude polarized along the interface (extensional mode). The associated bifurcation problem is analyzed with focus on the effects of compressive residual forces at the interface. The buckling strain of a compressed metal layer embedded in an elastomeric medium is computed also with an exact approach, to provide the range of validity of the proposed simplified model of material interface.     

Surface wave transference in a piezoelectric cylinder coated with reinforced material

The present study deals with the propagation of a polarized shear horizontal(SH)wave in a pre-stressed piezoelectric cylinder circumscribed by a self-reinforced cylinder.The interface of the two media is assumed mechanically imperfect.For obtaining the dispersion relation,the mathematical formulation has been developed and solved by an analytical treatment.The effects of various parameters,i.e.,the thickness ratio,the imperfect interface,the initial stress,the reinforcement,and the piezoelectric and dielectric constants,on the dispersion curve are observed prominently.The dispersion curves for different modes have been also plotted.The consequences of the study may be used for achieving optimum efficiency of acoustic wave devices.     

WAVE PROPAGATION IN PIEZOELECTRIC/PIEZOMAGNETIC LAYERED PERIODIC COMPOSITES

This paper is concerned with the dynamic behaviors of wave propagation in layered periodic composites consisting of piezoelectric and piezomagnetic phases. The dispersion relations of Lamb waves axe derived. Dispersion curves and displacement fields are calculated with different piezoelectric volume fractions. Numerical results for BaTiO3/CoFe2O4 composites show that the dispersion curves resemble the symmetric Lamb waves in a plate. Exchange between the longitudinal （i.e. thickness） mode and coupled mode takes place at the crossover point between dispersion curves of the first two branches. With the increase of BaTiO3 volume fraction, the crossover point appears at a lower wave number and wave velocity is higher. These findings are useful for magnetoelectric transducer applications.     

高频纵向导波在钢杆中传播特性的研究

计算了钢杆中纵向轴对称导波模态的衰减频散曲线和群 速度频散曲线. 分析了1~3MHz范围内高频纵向轴对称超声导波在钢杆中的传播特 性. 理论分析表明，各高阶纵向模态都存在一个衰减最小值，在此衰减最小值所对应频率下 的高阶模态能传播较远距离，可用于钢杆导波检测. 建立实验系统，采用轴对称同端激 励接收的方法，根据第1次端面回波做出群速度和端面回波幅值随频率变化曲线，实验结 果与理论分析基本吻合. 表明考虑材料衰减特性的钢杆频散曲线可以作为实验指导依据.