A self-consistent approach to dynamic effective properties of a composite reinforced by distributed spherical particles

Summary A self-consistent approach to dynamic effective properties of a composite reinforced by randomly distributed spherical inclusions is studied. The coherent plane waves propagating through the particle-reinforced composite are of attenuation nature. It implies that there is an analogy between the particle-reinforced composite and the effective medium with complex-valued elastic constants from the viewpoints of wave propagation. A composite sphere consisting of the inclusion, the matrix and the interphase between them is assumed embedded in the effective medium. The effective wavenumbers of the coherent plane waves propagating through the particle-reinforced composite are obtained by the dynamic self-consistent conditions which require that the forward scattering amplitudes of such a composite sphere embedded in the effective medium are equal to zero. The dynamic effective properties (effective phase velocity, effective attenuation and effective elastic constants) obtained by the present dynamic self-consistent approach for SiC-Al composites are compared numerically with that obtained by the effective field approach at various volume concentrations. It is found that there is a good agreement between the two approaches at a relatively low frequency and low volume concentration but the numerical results deviate from each other at a relatively high frequency and high volume concentration.     

Scattering of antiplane shear waves in a fiber-reinforced composite medium with interfacial layers

Summary This paper deals with the scattering of antiplane shear waves in a metal matrix composite reinforced by fibers with interfacial layers. We assume same-size cylindrical inclusions and same-thickness interface layers with nonhomogeneous elastic properties. The effective complex wave numbers follow from the coherent wave equation which depends only upon the scattering amplitude of the single scattering problem. Effective elastic constants can be obtained from phase velocities of coherent waves. Numerical calculations for an SiC-fiber-reinforced Al composite are carried out, and the effect of interface properties on scattering cross section, phase velocity, attenuation of coherent plane wave, and effective elastic constant is shown graphically.     

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。     

球面波传播特性和围岩动态裂纹产生的数值试验研究

利用非线性动力有限元分析程序,对于半无限岩土介质中的洞室,研究了球面波在围岩中的传播特性和动态裂纹的产生过程,比较了球面波在毛洞和锚杆加固洞室围岩中的衰减规律和动态裂纹形态的差别。结果表明:围岩动态裂纹是自由面产生的反射拉伸波及其相互叠加以及加固区与未加固区相互错动剪切的综合作用造成的;球面波强度在毛洞围岩中的衰减速度比在加固洞室中的快;锚杆加固洞室的破坏程度比毛洞轻一些;将模拟所得加固洞室围岩破坏形态与相应模型试验的结果进行了比较,发现洞室破坏形态及规律较为一致,说明了数值分析结果是可信的。研究结果对球面波的认识和防护工程的加固设计具有较好的参考价值。     

碳纤维增强复合材料层合板Lamb波衰减特性研究

为提取适用于碳纤维增强复合材料层合板声发射故障诊断的模态信号,利用三维弹性理论及传递矩阵法获得Lamb波的频散曲线。以碳纤维增强复合材料层合板为研究对象搭建实验平台,改变断铅激励位置从而获得不同声发射信号。对采集的声发射信号进行小波尺度谱分析,结合频散曲线分离出不同模式的Lamb波,分别研究其不同频率的幅度及能量衰减特性。实验结果表明,较其它信号,低频率S0波幅度信号衰减速度较低,对碳纤维增强复合材料层合板的声发射故障诊断研究具有较大优势。     

Harmonic wave propagation in viscoelastic heterogeneous materials Part I: Dispersion and damping relations

An analytico-numerical method is presented to study the propagation of plane harmonic waves in infinite periodic linear viscoelastic media. Part I considers only the dispersion and attenuation of acoustical longitudinal and shear waves. To show the accuracy of the method, examples of plane harmonic wave propagation in an infinite homogeneous medium and in a periodic layered viscoelastic medium are presented. The method is then used to calculate the damping and dispersion relations for a fibre-reinforced viscoelastic composite material. The results show clearly the influence of materials' viscoelastic properties and heterogeneities on the propagation of plane harmonic waves through the media.     

Computational Prediction of Micro-crack Induced Ultrasound Attenuation in CFRP Composites

A computational study is presented of ultrasound attenuation arising from linear elastic scattering by matrix micro-cracking in carbon fiber reinforced polymer (CFRP) composites. The model considers ultrasound propagation in a unidirectionally reinforced laminate containing randomly distributed matrix micro-cracks, configured such that neither the cracked laminate nor wavefield display dependence on the spatial direction of fiber reinforcement. The resulting 2D scattering formulation is solved using the boundary element method (BEM). Scatter-induced ultrasound attenuation is computed using a fully-interacting scattering matrix, as well as approximate multiple scattering formulations. Through comparison to the fully-interacting computation, the validity of level 1 scattering (independent scattering approximation), level 2 scattering (single neighbor interaction), and level 3 scattering (two neighbor interaction) is noted as a function of micro-crack density. A single parameter estimation of attenuation dependence on micro-crack density is obtained by fitting exponential dependence to the fully-interacting scattering results. Additionally, this single parameter is shown to be extracted equally well by fitting over the limited validity range of the independent scattering formulation. When uniform crack morphology is assumed, the exponential fit to the independent scattering approximation yields a practical estimation of micro-crack induced attenuation over the full range of micro-crack density through consideration of forward scattering by a single micro-crack. Comparison of results to limited experimental data, and to an independent alternate computational approach, lends plausibility to the study’s conclusions.     

Laser-Generated Lamb Waves Propagation in Multilayered Plates Composed of Viscoelastic Fiber-reinforced Composite Materials

The propagation characteristics of laser-generated Lamb waves in multilayered fiber-reinforced composite plates with different fiber orientations and number of layers have been investigated quantitatively. Considering the viscoelasticity of the composite materials, we have set up finite element models for simulating the laser-generated Lamb waves in two types of the multilayered composite plates. In the first type, different fiber orientations are adopted. In the second one, different number of layers are considered. The results illustrate the occurrence of attenuation and dispersion, which is induced by the viscoelasticity and multilayer structure, respectively.     

Periodic Green's functions analysis of SAW and leaky SAWpropagation in a periodic system of electrodes on a piezoelectriccrystal

A method of periodic Green's functions with a propagation factor exp(iβx), unknown in advance, is used to calculate dispersion curves and attenuation coefficients for Rayleigh- and leaky- waves propagating in a periodic system of thin electrodes on a piezoelectric surface. To describe the charge distribution on the electrodes both a step approximation and Chebyshev polynomials are used, the last being more adequate in most cases. Numerically determined values of the Green's function are used and interpolated either linearly or using a modified variant of Ingebrigtsen's formula. Such basic parameters as stopband width, stopband center frequency, wave velocity and attenuation in the stopband are found. These parameters can be used in the coupling-of-modes (COM) analysis and design of SAW devices. The analysis includes bulk wave radiation and scattering. The dependence of the corresponding attenuation coefficient on frequency is determined. Results obtained allow the determination directly and properly of the COM parameters and the design of SAW devices having large number of electrodes most precisely and rapidly. Numerical results for Rayleigh waves on YZ-LiNbO₃ and leaky waves on 36°YX-LiTaO₃ substrates are presented     

Dynamic Effective Properties of Particle-Reinforced Composites with Viscoelastic Matrix

The frequency-dependent dynamic effective properties of the particle-reinforced composites with the viscoelastic matrix are studied. Several equations to predict the effective wavenumber of the coherent plane waves propagating through particle-reinforced composites are discussed and the equation given by Gubernatis, J.E., [‘Effects of microstructure on speed and attenuation of elastic waves in porous materials’, Wave Motion, 6, 1984, 579–589] based on the independent scattering approximation is used in this paper. The effective phase velocity, the effective attenuation and the effective elastic moduli are evaluated. Numerical calculations are carried out for two kinds of composites, namely, Lead-Epoxy and Glass-Epoxy and the numerical results show that the frequency-dependent dynamic effective properties are related to both the multiple scattering effects among the distributed particles and the viscous dissipative effects of the viscoelastic matrix. However, these effects in the composites with distributed heavy particles (lead) and light particles (glass) are of evidently different features.     

基于数值计算方法的CFRP层板二维细观形貌多孔隙超声散射衰减

针对超声波在含有多孔隙的复合材料中传播时,邻近孔隙超声散射波之间相互作用和散射衰减机制尚未澄清的问题,对孔隙率为7.47%的碳纤维增强复合材料（Carbon Fiber Reinforced Plastic,CFRP）采用时域有限差分方法进行数值计算,对比研究了CFRP层板中不同尺寸范围二维真实形貌孔隙及圆形孔隙对应的超声散射衰减系数。结果表明,对于横向尺寸m≤λ/8、λ/8_s普遍小于真实形貌孔隙的α_s。对于孔隙横向尺寸满足m<2λ/3的情况,大尺寸孔隙以及长条形孔隙的存在,整体上会使超声波散射衰减加重。     

钢纤维混凝土动态抗拉强度的实验研究

基于线弹性和一维应力波假定,采用Φ75mmSHPB对钢纤维体积率Vf分别为0、0.75%和1.50%的三种混凝土材料进行了一维杆层裂实验,考虑了应力波在混凝土材料内传播时的波形弥散效应和应力幅值衰减,通过计算应变片记录的应力信号确定了材料的动态抗拉强度。结果表明,钢纤维混凝土的动态抗拉强度受应变率和钢纤维体积率的影响,本文为测试脆性材料的动态抗拉强度提供了一种有效方法。基于微观扫描技术,对钢纤维增强机理进行了分析。     

Scattering of Thermal Waves and Non-steady Effective Thermal Conductivity of Unidirectional Fibrous Composites with Functionally Graded Interface

In this paper, a thermal wave method is applied to investigate the non-steady effective thermal conductivity of unidirectional fibrous composites with a functionally graded interface, and the analytical solution of the problem is obtained. The Fourier heat conduction law is applied to analyze the propagation of thermal waves in the fibrous composite. The scattering and refraction of thermal waves by a cylindrical fiber with an inhomogeneous interface layer in the matrix are analyzed, and the results of the single scattering problem are applied to the composite medium. The wave fields in different material layers are expressed by using the wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary conditions of the layers. The theory of Waterman and Truell is employed to obtain the effective propagating wave number and the non-steady effective thermal conductivity of composites. As an example, the effects of a graded interface on the effective thermal conductivity of composites are graphically illustrated and analyzed. Analysis shows that the non-steady effective thermal conductivity under higher frequencies is quite different from the steady thermal conductivity. In the region of intermediate and high frequencies, the effect of the properties of the interface on the effective thermal conductivity is greater. Comparisons with the steady thermal conductivity obtained from other methods are also presented.     

Broadband Laser-Ultrasonic Spectroscopy for Quantitative Characterization of Porosity Effect on Acoustic Attenuation and Phase Velocity in CFRP Laminates

This work aims at applying the method of broadband laser-ultrasonic spectroscopy for quantitative evaluation of the effect of isolated dispersed voids and additional extended interply delaminations on the acoustic attenuation and on the phase velocity in CFRP laminates. This method is based on the laser thermoelastic generation of broadband reference pulses of longitudinal ultrasonic waves in the specially designed source of ultrasound. The high-sensitivity piezoelectric transducer is used to detect these pulses propagating normal to the fiber plies in composite specimens. The laminate specimens investigated have different total porosity levels up to 10.5 % determined by the X-ray computer tomography. The resonance peak of the attenuation coefficient and the corresponding jump of the phase velocity are observed governed by the periodic layered structure of the specimens. The absolute maximum and the frequency bandwidth of the resonance attenuation peak depend on the total porosity level formed by the predominant type of imperfections, either of isolated spheroidal voids entrapped in epoxy layers or of extended interply delaminations. With an increase of the specimen’s total porosity dispersion of the phase velocity becomes noticeable in the low-frequency band before the resonance jump. The derived empirical relations between the total porosity level and the parameters of the frequency dependencies of the ultrasonic attenuation coefficient and of the phase velocity can be used for rapid quantitative characterization of the structure of CFRP laminates subject to different fabrication conditions.     

Propagation of electroacoustic axial shear waves in a piezoelectric medium reinforced by continuous fibers

The propagation of electroacoustic axial shear waves in a fiber reinforced piezocomposites is studied in which matrix and fibers consist of piezoelectric transversely isotropic materials with symmetry axes parallel to the fiber axes. The effective medium method self-consistent variant as developed by Sabina and Willis is used to obtain explicit equations for the complex wave vector and it is solved numerically. Its real part determines the effective wave velocity and the imaginary part the attenuation factor. Integral equations expressed via dynamic Green’s function kernels are set up. The central problem of the method is the axial shear electroacoustic wave scattering on one isolated fiber in the medium having the effective piezoelectric properties. It is solved approximately by the Galerkin type method. The obtained expressions for the effective wave velocity and attenuation factor cover not only the long-wave region but the intermediate wave and it is valid for long wavelenghts up to the diameter of the inclusion. Wave velocity and attenuation coefficient coincide with ones obtained earlier in some other way. Some numerical examples are presented for real materials.     

Scattering of elastic waves by spherical inclusions with applications to low frequency wave propagation in composites

Scattering of plane elastic waves by a spherical inclusion is considered. A unified method of solution is presented which treats compressional and shear incidence on a similar basis. Explicit results are given for Rayleigh scattering. We apply the results of the single scattering problem to the propagation of low frequency waves in a composite containing a dilute concentration of spherical inclusions. Explicit formulae are given for the effective wave speeds and attenuations when the inclusions are voids. Both the compressional and shear wave speeds decrease initially as a function of frequency.     

Finite element modeling of lamb wave propagation in anisotropic hybrid materials

A finite element approach for modeling of acoustic emission sources and signal propagation in hybrid multi-layered plates is presented. Modeling results are validated by Laser vibrometer measurements and comparison to calculated dispersion curves. We investigate hybrid plates as typically found in composite pressure vessels, composed of fiber reinforced polymers with arbitrary stacking sequences and attached metal or polymer materials. Hybrid plate thickness, the ratio between anisotropic and isotropic materials and material properties are varied. Lamb-wave propagation in a geometry representative of a pressure vessel is modeled. It is demonstrated, that acoustic emission sources in multi-layered structures can cause Lamb-waves superimposed by guided waves within the individual layers.     

Attenuation of a plane compressional wave by a random distribution of thin circular cracks

The problem of the diffraction of obliquely incident plane compressional waves by a thin circular crack located in an infinite isotropic elastic medium is considered. An analytic estimation of the scattering cross-section is obtained for the Rayleigh limit. The attenuation coefficient associated with a random distribution of such cracks is derived for not too closely-spaced cracks, even when weak scattering density conditions are not satisfied. It is found that, for long wave-lengths, the attenuation varies directly with the average of the sixth power of the radius and inversely with the fourth power of the wave-length, with a coefficient which depends on Poisson's ratio. Comparisons are made with known results for spherical cavities and with the attenuation we could calculate for scalar waves. Conclusions are drawn about attenuation measurements used as a non-destructive test to characterize crack distribution in a sample.     

Lamb waves in a thin isotropic layer between two anisotropic layers

Attenuative Lamb wave propagation in adhesively bonded anisotropic composite plates is introduced. The isotropic adhesive exhibits viscous behavior to stimulate the poor curing of the middle layer. Viscosity is assumed to vary linearly with frequency, implying that attenuation per wavelength is constant. Attenuation can be implemented in the analysis through modification of elastic properties of isotropic adhesive. The new properties become complex, but cause no further complications in the analysis. The characteristic equation is the same as that used for the elastic plate case, except that both real and imaginary parts of the wave number (i.e., the attenuation) must be computed. Based on the Lowe's solution in finding the complex roots of characteristic equation, the effect of longitudinal and shear attenuation coefficients of the middle adhesive layer on phase velocity dispersion curves and attenuation dispersion curves of Lamb waves propagating in bonded anisotropic composites is visualized numerically.     

基于三维弹性理论的碳纤维板中Lamb波建模

基于主动Lamb波的结构健康监测是目前复合材料结构损伤监测技术研究的热点之一,了解Lamb波的传播特性对进行可靠的损伤监测非常重要.本文结合经典三维弹性理论与Lamb波的运动位移方程,对碳纤维复合材料板中传播的Lamb波传播特性进行了建模研究,在此基础上推导了碳纤维板的相速度频散曲线,并讨论了Lamb波传播方向与坐标轴之间的夹角及碳纤维铺层方向对频散曲线的影响,建模结果证明了这种建模方法的正确性.