Re-polarization of elastic waves at a frictional contact interface—II. Incidence of a P or SV wave

This is Part II of a two-part paper which analyses the re-polarization of elastic waves at a frictional contact interface between two solids. The re-polarization of SH waves was solved in Part I by the use of the Fourier analysis. Here, in Part II, we consider the re-polarization of P or SV waves. It is assumed that the two solids are pressed together and, at the same time, loaded by anti-plane and in-plane shearing traction. If the incident wave is sufficiently strong, localized separation and slip may take place at the interface. As a result, the incident in-plane wave is re-polarized at the interface so that the anti-plane waves (SH waves) are induced. Using the method similar to that of Part I and considering the boundary conditions involving separation and slip, we manage to reduce the problem to a set of algebraic equations coupled with simple integral equations. An iterative method is developed based on the solution to the perfectly bonded interface. The locations and sizes of the separation and slip zones, the interface traction, the slip velocities, the global sliding velocities and the energy dissipation and partition are displayed for the case of two identical materials. It is found that the separation zones and the gaps are independent of the induced waves.     

一般各向异性单侧接触界面上波的反射和折射

研究简谐弹性波在一般各向异性介质单侧接触界面上的反射和折射问题．利用Fouier分析方法将非线性Coulomb摩擦接触边界波动问题化为一组代数方程．给出了确定局部分离、滑移和粘着区的思路和方法及各区域的解；讨论了出现界面局部分离和滑移的条件．对特定材料组合情况进行了详细数值计算，给出了界面力、相对滑移速度、张开位移、高频谐波的反射折射系数等特征参量；考察了平面和反平面波动的耦合及整体滑移等．其中关于高频谐波的结果可对已有实验结果给出很好的定性解释．在大多数情况下，即使对摩擦系数无穷大的粘滞接触界面，分离区端部也总是存在一个很小的滑移区。     

Numerical study of nonlinear interaction between a crack and elastic waves under an oblique incidence

A Finite Element (FE) model is proposed to study the interaction between in-plane elastic waves and a crack of different orientations. The crack is modeled by an interface of unilateral contact with Coulombs friction. These contact laws are modified to take into account a pre-stress _σ0${\sigma }_0$ that closes the crack. Using the FE model, it is possible to obtain the contact stresses during wave propagation. These contact stresses provide a better understanding of the coupling between the normal and tangential behavior under oblique incidence, and explain the generation of higher harmonics. This new approach is used to analyze the evolution of the higher harmonics obtained as a function of the angle of incidence, and also as a function of the excitation level. The pre-stress condition is a governing parameter that directly changes the nonlinear phenomenon at work at the interface and therefore the harmonic generation. The diffracted fields obtained by the nonlinear and linear models are also compared.     

Hysteretic linear and nonlinear acoustic responses from pressed interfaces

An analysis of the linear and nonlinear acoustic responses from an interface between rough surfaces in elastoplastic contact is presented as a model of the ultrasonic wave interactions with imperfect interfaces and closed cracks. A micromechanical elastoplastic contact model predicts the linear and second order interfacial stiffness from the topographic and mechanical properties of the contacting surfaces during a loading–unloading cycle. The effects of those surface properties on the linear and nonlinear reflection/transmission of elastic longitudinal waves are shown. The second order harmonic amplitudes of reflected/transmitted waves decrease by more than an order of magnitude during the transition from the elastic contact mode to the elastoplastic contact mode. It is observed that under specific loading histories the interface between smooth surfaces generates higher elastoplastic hysteresis in the interfacial stiffness and the acoustic nonlinearity than interfaces between rough surfaces. The results show that when plastic flow in the contacting asperities is significant, the acoustic nonlinearity is insensitive to the asperity peak distribution. A comparison with existing experimental data for the acoustic nonlinearity in the transmitted waves is also given with a discussion on its contact mechanical implication.     

Modelling of in-plane wave propagation in a plate using spectral element method and Kane–Mindlin theory with application to damage detection

This paper presents results of experimental and numerical analyses of in-plane waves propagating in a 5 mm-thick steel plate in the frequency range of 120–300 kHz. For such a thickness/frequency ratio, extensional waves reveal dispersive character. To model in-plane wave propagation taking into account the thickness-stretch effect, a novel 2D spectral element, based on the Kane–Mindlin theory, was formulated. An application of in-plane waves to damage detection is also discussed. Experimental investigations employing a laser vibrometer demonstrated that the position and length of a defect can precisely be identified by analysing reflected and diffracted waves.     

群速度控制格式及二维Riemann解

强激波和强接触间断的数值模拟一直是计算流体力学里一个富有挑战性的课题,它们是很多实际流动的基础。三阶迎风紧致格式是一种具有较高分辨率的高精度方法,但是在计算激波时仍有数值振荡产生。本文根据数值解的群速度特性,在三阶迎风紧致格式的基础上提出了一种群速度控制格式,使得能够正确模拟含有强激波和强接触间断的复杂流动。在此基础上构造了求解包含大压力比和密度比的二维界面问题的数值方法。计算结果表明,方法对激波和接触间断的分辨效果是令人满意的。     

Edge waves in asymmetrically laminated plates

We study edge waves propagating along the edge of an asymmetrically laminated elastic plate for which the out-of-plane component of displacement is coupled with the in-plane components. A Stroh-like formulation is used to show that such a plate can support at most two edge waves. An efficient method for computing the edge-wave speeds is proposed and explained through examples.     

Analytical extension of Finite Element solution for computing the nonlinear far field of ultrasonic waves scattered by a closed crack

Nonlinear scattering of ultrasonic waves by closed cracks subject to contact acoustic nonlinearity (CAN) is determined using a 2D Finite Element (FE) coupled with an analytical approach. The FE model, which includes unilateral contact with Coulomb friction to account for contact between crack faces, provides the near-field solution for the interaction between in-plane elastic waves and a crack of different orientations. The numerical solution is then analytically extended in the far-field based on a frequency domain near-to-far field transformation technique, yielding directivity patterns for all linear and nonlinear components of the scattered waves. The proposed method is demonstrated by application to two nonlinear acoustic problems in the case of tone-burst excitations: first, the scattering of higher harmonics resulting from the interaction with a closed crack of various orientations, and second, the scattering of the longitudinal wave resulting from the nonlinear interaction between two shear waves and a closed crack. The analysis of the directivity patterns enables us to identify the characteristics of the nonlinear scattering from a closed crack, which provides essential understanding in order to optimize and apply nonlinear acoustic NDT methods.     

垂直入射弹性波在压电体单侧接触界面上的传播特性

弹性波与压电材料接触界面的相互作用问题是工程应用中常见而复杂的问题,入射波足够强会引起界面出现滑移和分离,但滑移和分离的边界未知,边界条件具有非线性特性。通过Fourier分析,将混合边值问题的求解转化为非线性代数方程,利用软件通过迭代修正的方法进行了求解;给出3种状态边界的求解,分析入射波强度、外加应力及电场对界面状态的影响,并对高频谐波的特性进行分析,通过实例对理论推导进行验证,结果显示:入射波强度、外加荷载和电场的大小及摩擦因数均会影响到界面,通过改变这些条件可以控制界面状态,另外检测高频谐波的信号也可以反映界面状态。     

FORCED WAVE PROPAGATION IN VISCOELASTIC CABLE WITH SMALL SAG

Based on the linear viscoelastic differential constitutive law and cable structural model, the coupled longitudinal-transverse waves that propagate along a viscoelastic cable with small curvature is investigated. A mathematical model is presented that describes the three-dimensional nonlinear response of a viscoelastic cable. An asymptotic form of this model is obtained for the linear response of cables having small equilibrium curvature. The spectral relation governing the propagating waves is derived using transform methods. The spectral relation is employed in deriving a Green's function that is then used to construct solutions for in-plane response under distributed harmonic excitation. Analysis of forced response reveals the existence of two types of periodic waves that propagate through the cable, one characterizing extension-comprehensive deformation and the other characteristic transverse deformation. Project supported by the National Natural Science Foundation of China (No. 59635140).     

Experimental observation of stress waves propagating in laminated composites

The theoretical research on stress waves propagating in laminated composites has been reported by many authors. However, there has been little work on experimental studies of stress waves in those materials. This paper presents an experimental investigation on stress waves propagating parallel to the layers of a laminated composite. A sandwich laminated composite consisting of two aluminum facings and an epoxy core is used as a specimen. The stress wave in the specimen is observed by use of high-speed holographic interferometry with a pulsed laser. In order to obtain the relative fringe orders, the interference fringe pattern in the reconstructed image is treated as an image-processing system with a personal computer. For the calculation of the in-plane displacement, an approximate relative-fringe-order method is used. The in-plane displacements obtained at some sampling points on the surface are smoothed by using a spline function. Distributions of the in-plane displacement and the shear stress are then obtained quantitatively over the whole analyzed field.     

Effect of initial deformations on wave front propagation in anisotropic plates

The effects of large amplitudes and initial deformations on shock waves and acceleration waves propagating in fiber-reinforced laminated plates are investigated. Three cases are discussed, namely the large amplitude shock under initial in-plane deformations, small amplitude waves under in-plane deformations, and small amplitude waves propagating in a plate with large deflection. It is found that the in-plane force has a substantial effect on the transverse shear mode but little effects on other modes. The large initial deflection, however, is found to have considerable effects on all modes. A general procedure for constructing the wave surfaces is also presented.     

Dynamic analysis of suspended cables carrying moving oscillators

An efficient procedure for analyzing in-plane vibrations of flat-sag suspended cables carrying an array of moving oscillators with arbitrarily varying velocities is presented. The cable is modelled as a mono-dimensional elastic continuum, fully accounting for geometrical nonlinearities. By eliminating the horizontal displacement component through a standard condensation procedure, the nonlinear integro-differential equation governing vertical cable vibrations is derived. Due to the dynamic interaction at the contact points with the moving oscillators, such equation is coupled to the set of ordinary differential equations ruling the response of the travelling sub-systems. An improved series representation of vertical cable displacement is proposed, which allows to overcome the inability of the traditional Galerkin method to reproduce the kinks and abrupt changes of cable configuration at the interface with the moving sub-systems. Following the philosophy of the well-known “mode-acceleration” method, the convergence of the series expansion of cable response in terms of appropriate basis functions is improved through the introduction of the so-called “quasi-static” solution. Numerical results demonstrate that, despite the basis functions are continuous, the improved series enables to capture with very few terms the abrupt changes of cable profile at the contact points between the cable and the moving oscillators.     

Experimental contact pattern analysis for a gear-rack system

Gears perform their main task, namely load transmission, by means of very small contact areas originated by tooth interaction and thus, the analysis of phenomena occurring at the interface between mating teeth represents a critical issue in ensuring the optimal functioning of such devices. Nevertheless, while literature proposes a huge amount of numerical tooth contact analyses (TCA), a lack of experimental validation of such approaches is to be noted, since it is extremely difficult to inspect a contact interface which is, by its own nature, closed towards the outside world. One of the most promising techniques employed in investigating contact in metallic interfaces is based on the use of high frequency ultrasonic waves; their reflection from the interface (which is known to be related to contact conditions) can be graphically processed to build maps from which it is possible to assess geometrical features of the nominal contact area and, after a suitable calibration procedure, contact pressure distribution.     

层状地基任意形状刚性基础动力响应求解

提出了基于积分变换、对偶方程与精细积分算法求解多层地基任意形状刚性基础的动力刚度问题. 首先在频率波数域内圆柱坐标体系中利用圆形微元的对称与反对称特性建立多层地基中格林影响函数的波动方程,然后将应力和位移关系表示成对偶形式进行精细积分求解以提高计算精度和稳定性. 再将任意形状刚性基础与地基的交界面离散化为一系列圆形微元,利用格林影响函数建立其平动与转动动力刚度的矩阵方程. 该求解方法高效、准确并且计算稳定,适于任意复杂多层地基任意形状基础动力刚度的计算.      

Anti-symmetric waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface on the dispersive behavior of in-plane time-harmonic symmetric waves in a pre-stressed incompressible symmetric layered composite, was analyzed recently by Leungvichcharoen and Wijeyewickrema (2003). In the present paper the corresponding case for time harmonic anti-symmetric waves is considered. The bi-material composite consists of incompressible isotropic elastic materials. The imperfect interface is simulated by a shear-spring type resistance model, which can also accommodate the extreme cases of perfectly bonded and fully slipping interfaces. The dispersion relation is obtained by formulating the incremental boundary-value problem and using the propagator matrix technique. The dispersion relations for anti-symmetric and symmetric waves differ from each other only through the elements of the propagator matrix associated with the inner layer. The behavior of the dispersion curves for anti-symmetric waves is for the most part similar to that of symmetric waves at the low and high wavenumber limits. At the low wavenumber limit, depending on the pre-stress for perfectly bonded and imperfect interface cases, a finite phase speed may exist only for the fundamental mode while other higher modes have an infinite phase speed. However, for a fully slipping interface in the low wavenumber region it may be possible for both the fundamental mode and the next lowest mode to 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 the higher modes tend to the phase speeds of the surface wave or the interfacial wave or the limiting phase speed of the composite. The bifurcation equation obtained from the dispersion relation yields neutral curves that separate the stable and unstable regions associated with the fundamental mode or the next lowest mode. 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 imperfect interfaces on anti-symmetric waves is clearly evident in the numerical results.     

磁电弹复合材料和刚性导电导磁圆柱压头的二维微动接触分析

本文求解平面应变状态下磁电弹复合材料半平面和刚性导电导磁圆柱压头的二维微动接触问题。假设压头具有良好的导电导磁性,且表面电势和磁势是常数。微动接触依赖载荷的加载历史,所以首先求解单独的法向加载问题,然后在法向加载问题的基础上求解循环变化的切向加载问题。整个接触区可以分为内部的中心粘着区和两个外部的滑移区,其中滑移区满足Coulomb摩擦法则。利用Fourier积分变换,磁电弹半平面的微动接触问题将简化为耦合的Cauchy奇异积分方程组,然后数值离散为线性代数方程组,利用迭代法求解未知的粘着/滑移区尺寸、电荷分布、磁感应强度、法向接触压力和切向接触力。数值算例给出了摩擦系数、总电荷和总磁感应强度对各加载阶段的表面接触应力、电位移和磁感应强度的影响。     

Analysis of transient wave scattering from an inclusion with an unilateral smoothly contact interface by BEM

Summary A 2D time-domain Boundary Element Method (BEM) is applied to solve the problem of transient scattering of plane waves by an inclusion with a unilateral smooth contact interface. The incident wave is assumed strong enough so that localized separations take place along the interface. The present problem is indeed a nonlinear boundary value problem since the mixed boundary conditions involve unknown intervals (separation and contact regions). In order to determine the unknown intervals, an iterative technique is developed. As an example, we consider the scattering of plane waves by the cross section of a circular cylinder embedded in an infinite solid. Numerical results for the near field solutions are presented. The distortion of the response waves and the variation of the interface states are discussed. The financial support by the China National Natural Science Foundation under Grant No. 19872001 and No. 59878004 is gratefully acknowledged. The second author is also grateful to the support of the National Science Fund for Distinguished Young Scholars under Grant No. 10025211.     

Ultrasonic assessment of rough surface contact between solids from elastoplastic loading-unloading hysteresis cycle

An ultrasonic method to characterize the elastoplastic contact between two rough surfaces is presented. Ultrasonic experiments are performed on three different interfaces formed by aluminum surfaces with different levels of roughness. The frequency-dependent ultrasonic reflection coefficient from the interface is measured during loading and unloading cycles as a function of pressure, from which the ultrasonic interfacial contact stiffness is reconstructed by the least-squares inversion procedure. It is shown that one should distinguish between the ultrasonic (dynamic) interfacial stiffness and static interfacial stiffness for rough surfaces in elastoplastic contact (they are identical for purely elastic contact). It is shown that ultrasonic stiffness is associated with local unloading stiffness. An elastoplastic micromechanical model is used to describe the plasticity-induced hysteresis in the ultrasonically measured interfacial stiffness during loading-unloading cycles. The topographic parameters of the interface contact are reconstructed by matching the model-predicted results with the experimentally determined ultrasonic stiffness. Using these parameters the real area of contact, which is not directly measurable, is predicted during loading-unloading cycles using the model.