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.     

A finite difference method for elastic wave scattering by a planar crack with contacting faces

This paper presents a finite difference time-domain technique for 2D problems of elastic wave scattering by cracks with interacting faces. The proposed technique introduces cracks into the finite difference model using a set of split computational nodes. The split-node pair is bound together when the crack is closed while the nodes move freely when open, thereby a unilateral contact condition is considered. The development of the open/close status is determined by solving the equation of motion so as to yield a non-negative crack opening displacement. To check validity of the proposed scheme, 1D and 2D scattering problems for which exact solutions are known are solved numerically. The 1D problem demonstrates accuracy and stability of the scheme in the presence of the crack-face interaction. The 2D problem, in which the crack-face interaction is not considered, shows that the proposed scheme can properly reproduce the stress singularity at the tip of the crack. Finally, scattered fields from cracks with interacting faces are investigated assuming a stick and a frictionless contact conditions. In particular, the directivity and higher-harmonics are investigated in conjunction with the pre-stress since those are the basic information required for a successful ultrasonic testing of closed cracks.     

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.     

Sphere scattering of nonlinearly interacting acoustic waves

Sphere scattering of the field of nonlinearly interacting plane acoustic waves when the sphere is located in the region of nonlinear interaction between the primary pumping waves of a parametric antenna is considered. An analytic expression for the secondary field pressure at the difference frequency is obtained. This expression describes the process of nonlinear interaction of the incident and scattered waves. The secondary-field total pressure components, which characterize the interaction between the incident plane waves and scattered spherical waves are analyzed. The numerical results and experimental data are given.Taganrog. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 4–12, March–April, 1995.     

Plasmonic cloaking and scattering cancelation for electromagnetic and acoustic waves

One popular approach to cloaking objects from electromagnetic waves at moderately long wavelengths is the scattering cancelation technique. This mechanism is based on the use of a single homogeneous thin layer to cover an object of interest in order to provide scattering suppression in a given frequency band. This approach has also been recently extended to acoustic waves. This paper provides an investigation of the physical nature of scattering cancelation by a uniform thin layer for both electromagnetic and acoustic waves in inviscid fluids. Two distinct scattering cancelation regions are obtained within the available parameter space: a non-resonant plasmonic cloaking region and an anti-resonant cloaking region, which are identified and compared in both the electromagnetic and acoustic domains. Although both types of operations allow for the suppression of the dominant scattering orders, the resulting internal fields and physical functionality of the cloaks present distinct differences between the two domains. We discuss analogies and differences between these functionalities and their implications in electromagnetic and acoustic cloaking problems, with an insight into their practical implementation.     

Off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid sphere

In this paper, the off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid immovable (fixed) sphere is investigated. It is shown here that shifting the sphere off the axis of wave propagation induces a dependence of the scattering on the azimuthal angle. Theoretical expressions for the incident and scattered field from a rigid immovable sphere are derived. The near- and far-field acoustic scattering fields are expressed using partial wave series involving the spherical harmonics, the scattering coefficients of the sphere, the half-conical angle of the wave number components of the beam, its order and the beam-shape coefficients. The scattering coefficients of the sphere and the 3D scattering directivity plots in the near- and far-field regions are evaluated using a numerical integration procedure. The calculations indicate that the scattering directivity patterns near the sphere and in the far-field are strongly dependent upon the position of the sphere facing the incident high-order Bessel vortex beam.     

线性与非线性流固耦合动力学数值方法的进展及应用

本文综述了线性与非线性流固耦合问题数值方法的进展及工程应用。讨论了四种数值分析方法：(1)混合有限元–子结构–子区域数值模型,以求解有限域线性流固耦合问题,如流体晃动,声腔–结构耦合,流体中的压力波,化工容器的地震响应,坝水耦合等;(2)混合有限元–边界元数值模型,以求解涉及无限域的线性流固耦合问题,如大型浮体承受飞机降落冲击,船舰的炮击回应等;(3)混合有限元–有限差分(体积)数值模型,以求解不涉及破浪和两相分离的非线性流固耦合问题;(4)混合有限元–光滑粒子数值模型,以求解涉及破浪和两相分离的非线性流固耦合问题。文中推荐分区迭代求解过程,以便应用现有的固体及流体求解器,于毎一时间步长分别求解固体及流体的方程,通过耦合迭代收敛,向前推进以达问题求解。文中选用的工程应用例子包含气–液–壳三相耦合,液化天然气船水晃动,人体步行冲击引起的声腔–建筑结构耦合,大型浮体承受飞机降落冲击的瞬态动力回应,涉及破浪和两相分离的气–翼耦合及结构于水上降落的冲击。数值分析结果与可用的实验或计算结果作了比较,以说明所述方法的精度及工程应用价值。文中列出了基于流固耦合的波能采积装置模型,以应用线性系统的共振及非线性系统的周期解原理,有效地采积波能。本文列出了231篇参考文献,以便读者进一步研讨所感兴趣方法。     

Developments of numerical methods for linear and nonlinear fluid-solid interaction dynamics with applications

本文综述了线性与非线性流固耦合问题数值方法的进展及工程应用. 讨论了四种数值分析方法: (1) 混合有限元-子结构-子区域数值模型, 以求解有限域线性流固耦合问题, 如流体晃动, 声腔-结构耦合, 流体中的压力波, 化工容器的地震响应,坝水耦合等; (2) 混合有限元-边界元数值模型, 以求解涉及无限域的线性流固耦合问题, 如大型浮体承受飞机降落冲击, 船舰的炮击回应等; (3) 混合有限元-有限差分(体积) 数值模型, 以求解不涉及破浪和两相分离的非线性流固耦合问题; (4) 混合有限元-光滑粒子数值模型, 以求解涉及破浪和两相分离的非线性流固耦合问题. 文中推荐分区迭代求解过程, 以便应用现有的固体及流体求解器, 于毎一时间步长分别求解固体及流体的方程, 通过耦合迭代收敛, 向前推进以达问题求解. 文中选用的工程应用例子包含气-液-壳三相耦合, 液化天然气船水晃动, 人体步行冲击引起的声腔-建筑结构耦合, 大型浮体承受飞机降落冲击的瞬态动力回应, 涉及破浪和两相分离的气-翼耦合及结构于水上降落的冲击. 数值分析结果与可用的实验或计算结果作了比较, 以说明所述方法的精度及工程应用价值. 文中列出了基于流固耦合的波能采积装置模型, 以应用线性系统的共振及非线性系统的周期解原理, 有效地采积波能. 本文列出了231 篇参考文献, 以便读者进一步研讨所感兴趣方法.     

基于互补理论和扩展有限元法的摩擦接触裂纹应力强度因子计算

从虚功方程出发,结合扩展有限元离散技术与接触条件的非线性互补表述,建立了摩擦接触裂纹问题的扩展有限元非线性互补模型,将不等式接触条件转化为非线性互补类的非光滑方程组,并采用基于广义导数的非光滑阻尼牛顿法求解方程组,无需引入任何额外人工变量以及迭代求解。以含中心倾斜裂纹平板和边裂纹平板为例,运用相互作用积分法计算摩擦接触裂纹的应力强度因子,将其结果与理论解进行对比分析,该方法都能给出精确的计算结果;基于扩展有限元方法对单轴压缩作用下倾斜裂纹扩展过程进行了数值模拟,计算结果表明,受压裂纹数值结果与实验结果比较吻合,从而验证了本文方法的有效性与正确性。     

结构中微裂纹与超声波的混频非线性作用数值仿真研究

针对结构中微裂纹检测难题,本文对结构中微裂纹与超声波的混频非线性作用进行了数值仿真研究。基于经典非线性理论,得到了两列超声纵波相互作用产生混频效应的理论条件。通过有限元仿真,研究了两列纵波与微裂纹相互作用产生混频的条件,并分析了界面处静应力、摩擦系数和裂纹方向对混频效应的影响。研究发现,超声波与微裂纹相互作用产生混频非线性效应的发生条件仍符合经典非线性理论下的混频产生条件。裂纹界面处施加的静应力对差频横波幅值有明显影响;当施加静应力与无裂纹模型得到的最大应力值接近时,混频非线性效应最强;裂纹界面的摩擦系数对超声波的混频非线性效应影响较小;透射差频横波传播方向与经典非线性理论预测的理论差频分量方向基本一致,且几乎不受裂纹方向变化的影响,而反射差频横波的传播方向随裂纹方向的改变而有所不同。本文研究工作为微裂纹检出及方向识别做了有益探索。     

A numerical model for the nonlinear interaction of elastic waves with cracks

Microstructures such as cracks and microfractures play a significant role in the nonlinear interactions of elastic waves, but the precise mechanism of why and how this works is less clear. Here we simulate wave propagation to understand these mechanisms, complementing existing theoretical and experimental works.We implement two models, one of homogeneous nonlinear elasticity and one of perturbations to cracks, and then use these models to improve our understanding of the relative importance of cracks and intrinsic nonlinearity. We find, by modeling the perturbations in the speed of a low-amplitude P-wave caused by the propagation of a large-amplitude S-wave that the nonlinear interactions of P- and S-waves with cracks are significant when the particle motion is aligned with the normal to the crack face, resulting in a larger magnitude crack dilation. This improves our understanding of the relationship between microstructure orientations and nonlinear wave interactions to allow for better characterization of fractures for analyzing processes including earthquake response, reservoir properties, and non-destructive testing.     

SCATTERING OF ANTI-PLANE SHEAR WAVES BY A SINGLE CRACK IN AN UNBOUNDED TRANSVERSELY ISOTROPIC ELECTRO-MAGNETO-ELASTIC MEDIUM

A theoretical treatment of the scattering of anti-plane shear (SH) waves is provided by a single crack in an unbounded transversely isotropic electro-magneto-elastic medium. Based on the differential equations of equilibrium, electric displacement and magnetic induction intensity differential equations, the governing equations for SH waves were obtained. By means of a linear transform, the governing equations were reduced to one Helmholtz and two Laplace equations. The Cauchy singular integral equations were gained by making use of Fourier transform and adopting electro-magneto imperme ableboundary conditions. The closed form expression for the resulting stress intensity factor at the crack was achieved by solving the appropriate singular integral equations using Chebyshev polynomial. Typical examples are provided to show the loading frequency upon the local stress fields around the crack tips. The study reveals the importance of the electro-magneto-mechanical coupling terms upon the resulting dynamic stress intensity factor.     

On diffraction of acoustic and electric waves in piezoelectric medium by an absorbent half-plane electrode

Diffraction of incident acoustic and incident electric waves in a transversally isotropic piezoelectric medium at the boundary of a half-plane absorbent electrode is systematically investigated using the quasi-hyperbolic approximation. The electrode is assumed to be very thin so that its thickness and stiffness can be neglected. By exact inversion, the explicit expressions for the scattering waves are obtained. A closed form solution is obtained by applying Laplace transformations and the Wiener–Hopf technique. By means of the Cagniard–de Hoop method a detailed investigation of the structure of the electro-acoustic wave is conducted. The mode conversion between electric and acoustic waves, the effect of electro-acoustic head wave, the Bleustein–Gulyaev surface wave and the structure of the wave in terms of the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in detail. It is shown that in piezoelectric materials, absorbent electrodes are neither completely opaque nor completely transparent to electric and acoustic waves. The dynamic field intensity factors at the tip of the electrode are functions of the angle of incidence and time; they are derived explicitly and discussed through a detailed numerical analysis.     

Diffraction of weak shock waves by deformed bodies submerged in a liquid

Conclusion On the basis of an analysis of theoretical and experimental data obtained up to now by various investigators, we can note the following major advances in the field of the interaction of shock waves with barriers submerged in a liquid:Exact solutions have been obtained for problems in the diffraction of acoustic shock waves by rigid and stationary bodies of specified shape (plates, wedges, cones, parabolic, elliptical, and circular cylinders, spheres, paraboloids of revolution); approximate schemes have been worked out for estimating hydrodynamic loads, making it possible to investigate various stages of the interaction of shock waves with elastic shells of revolution and solid bodies; studies have been conducted in the exact formulation of the interaction of plane (spherical) nonstationary waves with elastic barriers (unbounded plates, plates in a screen, infinitely long thin-walled and thick-walled cylindrical shells, closed thin-walled and thick-walled spherical shells); an exact solution has been found for the internal problems in the case of cavities (circular and elliptical cylinders, spheres, spheroids) and elastic shells of revolution (infinitely long cylindrical and closed spherical shells); methods have been worked out for the approximate determination of the parameters of objects (elastic thin-walled infinitely long cylindrical and closed spherical shells) from reflected echo signals; estimates have been given for the influence of the structural characteristics of an object (support, concentrated masses), the nonlinear properties of interacting media, cavitation in liquid, and plastic deformations in the barrier material on the process of hydrodynamic interaction.We should also mention the main lines of further investigation and the problems which require solution: designing new experimental apparatus and measuring complexes for studying the nonstationary behavior of deformed bodies and structures in a liquid; solution of problems in diffraction by oonical and cylindrical shells of finite length, and by compound structures of complicated form in which account is taken of the structural characteristics and the internal elements; calculation of three-layer and multilayer shells acted upon by shock waves, taking account of the transverse compression of the filler; construction of more exact schemes (models) for the nonlinear and cavitation-type interaction of waves with barriers; development of numerical and combined methods for the solution of the problems in hydroelasticity.Mechanics Institute, Moscow State University. Translated from Prikladnaya Mekhanika, Vol. 16, No. 5, pp. 3–11, May, 1980.     

Acoustomechanical constitutive theory for soft materials

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by com-bining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momen-tum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and ana-lyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.     

A return toward equilibrium in a 2D Rayleigh–Taylor instability for compressible fluids with a multidomain adaptive Chebyshev method

We numerically simulate a single-mode Rayleigh–Taylor instability between compressible miscible fluids with a highly accurate self-adaptive pseudospectral Chebyshev multidomain method in two two-dimensional boxes at small aspect ratios. The simulations are started from rest and pursued until the return toward mechanical equilibrium of the mixing. Four regimes—linear and weakly nonlinear, nonlinear steady bubble rise, return toward equilibrium, and finally a system of acoustic waves—can be identified. We show that this one-dimensional system of stationary acoustic waves is damped by the physical viscosity. This provides a reference solution.      

Wave propagation in transversely isotropic cylinders

Various approaches have been used for model1ing problems dealing with interaction of acoustic/elastic waves with transversely isotropic cylinders. The authors developed the first mathematical model for the scattering of acoustic waves from transversely isotropic cylinders [Honarvar, F., Sinclair, A.N., 1996. Acoustic wave scattering from transversely isotropic cylinders. Journal of the Acoustical Society of America 100, 57–63.]. In the current paper, this model is used for derivation of the frequency equations of longitudinal and flexural wave propagation in free transversely isotropic cylinders. Consistency of this model with the physics of the problem is demonstrated and a systematic solution to the corresponding equations is developed. Numerical results obtained for a number of transversely isotopic cylinders are used for verification of the mathematical model.     

A combined analytical and numerical analysis of the flow-acoustic coupling in a cavity-pipe system

The generation of sound by flow through a closed, cylindrical cavity (expansion chamber) accommodated with a long tailpipe is investigated analytically and numerically. The sound generation is due to self-sustained flow oscillations in the cavity. These oscillations may, in turn, generate standing (resonant) acoustic waves in the tailpipe. The main interest of the paper is in the interaction between these two sound sources. An analytical, approximate solution of the acoustic part of the problem is obtained via the method of matched asymptotic expansions. The sound-generating flow is represented by a discrete vortex method, based on axisymmetric vortex rings. It is demonstrated through numerical examples that inclusion of acoustic feedback from the tailpipe is essential for a good representation of the sound characteristics.     

Solution of acoustic scattering problems by means of second kind integral equations

In the present paper, the problem of scattering of acoustic waves from a fluid inclusion in two dimensions is solved numerically. The boundary conditions are those of continuous pressure and normal displacement. First, the problem in the frequency domain is reduced to a pair of second kind Fredholm integral equations on the boundary of the scatterer. Then, the integral equations are discretized by means of the Nyström algorithm, and the resulting linear systems are solved iteratively. Finally, the time domain solution is obtained from a sequence of frequency domain values. The integral equations of the present paper possess a very simple physical interpretation which guarantees the stability of their numerical solution and rapid convergence of the iterative solver. The resulting algorithm is an efficient tool for solving relatively large scale two-dimensional scattering problems.     

二维含裂纹结构与声耦合问题研究

应用分形有限元方法结合边界元方法研究了二维含裂纹结构和声耦合问题.采用二级分形有限元方法对含裂纹的弹性结构体进行离散处理,这样可以使得自由度数大大地减少;无限大外域声场的计算使用边界元方法,可以自动满足无穷远辐射条件.数值仿真算例结果表明:结构声耦合系统的共振频率随着裂纹深度的增加而下降;裂纹附近的声场所受的影响较为明显.