Constitutive model for third harmonic generation in elastic solids

In this article, we present a new constitutive model for studying ultrasonic third harmonic generation in elastic solids. The model is hyperelastic in nature with two parameters characterizing the linear elastic material response and two other parameters characterizing the nonlinear response. The limiting response of the model as the nonlinearity parameters tend to zero is shown to be the well-known St Venant–Kirchhoff model. Also, the symmetric response of the model in tension and compression and its role in third harmonic generation is shown. Numerical simulations are carried out to study third harmonic generation in materials characterized by the proposed constitutive model. Predicted third harmonic guided wave generation reveals an increasing third harmonic content with increasing nonlinearity. On the other hand, the second harmonics are independent of the nonlinearity parameters and are generated due to the geometric nonlinearity. The feasibility of determining the nonlinearity parameters from third harmonic measurements is qualitatively discussed.     

Nonlinear Lamb waves in plate/shell structures

鉴于常规超声检测技术对分布式材料细微损伤和接触类结构损伤的检测效果不佳,近年来非线性超声技术逐渐引起广泛关注.超声波在板壳结构中通常以兰姆波的形式进行传播,然而由于兰姆波的频散及多模特性,使得非线性兰姆波的理论和实验研究进展缓慢.本文从经典非线性理论出发,总结了源于材料固有非线性诱发的非线性兰姆波的理论和实验两个方面的研究进展,并综述了兰姆波的二次谐波发生效应在材料损伤评价方面的若干应用;从接触声非线性理论出发,讨论了目前由于接触类结构损伤诱发的非线性兰姆波的研究现状.最后展望了非线性兰姆波的未来研究重点及发展趋势.     

Interesting effects in harmonic generation by plane elastic waves

The harmonics of plane longitudinal and trans-verse waves in nonlinear elastic solids with up to cubic nonlinearity in a one-dimensional setting are investigated in this paper. It is shown that due to quadratic nonlinearity, a transverse wave generates a second longitudinal harmonic. This propagates with the velocity of transverse waves, as well as resonant transverse first and third harmonics due to the cubic and quadratic nonlinearities. A longitudinal wave generates a resonant longitudinal second harmonic, as well as first and third harmonics with amplitudes that increase linearly and quadratically with distance propagated. In a second investigation, incidence from the linear side of a pri-mary wave on an interface between a linear and a nonlinear elastic solid is considered. The incident wave crosses the interface and generates a harmonic with interface conditions that are equilibrated by compensatory waves propagating in two directions away from the interface. The back-propagated compensatory wave provides information on the nonlinear elastic constants of the material behind the interface. It is shown that the amplitudes of the compensatory waves can be increased by mixing two incident longitudinal waves of appropriate frequencies.     

Quadratically Nonlinear Cylindrical Hyperelastic Waves: Primary Analysis of Evolution

The propagation and interaction of hyperelastic cylindrical waves are studied. Nonlinearity is introduced by means of the Murnaghan potential and corresponds to the quadratic nonlinearity of all basic relationships. To analyze wave propagation, an asymptotic representation of the Hankel function of the first order and first kind is used. The second-order analytical solution of the nonlinear wave equation is similar to that for a plane longitudinal wave and is the sum of the first and second harmonics, with the difference that the amplitudes of cylindrical harmonics decrease with the distance traveled by the wave. A primary computer analysis of the distortion of the initial wave profile is carried out for six classical hyperelastic materials. The transformation of the first harmonic of a cylindrical wave into the second one is demonstrated numerically. Three ways of allowing for nonlinearities are compared __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 73–82, July 2005.     

Second-harmonic generation of two-dimensional elastic wave propagation in an infinite layered structure with nonlinear spring-type interfaces

The two-dimensional elastic wave propagation in an infinite layered structure with nonlinear interlayer interfaces is analyzed theoretically to investigate the second-harmonic generation due to interfacial nonlinearity. The structure consists of identical isotropic linear elastic layers that are bonded to each other by spring-type interfaces possessing identical linear normal and shear stiffnesses but different quadratic nonlinearity parameters. Explicit analytical expressions are derived for the second-harmonic amplitudes when a single monochromatic Bloch mode propagates in the structure in arbitrary directions by applying the transfer-matrix approach and the Bloch theorem to the governing equations linearized by a perturbation method. The second-harmonic generation by a single nonlinear interface and by multiple consecutive nonlinear interfaces are shown to be profoundly influenced by the band structure of the layered structure, the fundamental Bloch wave mode, and its propagation direction. In particular, the second harmonics generated at multiple consecutive interfaces are found to grow cumulatively with the propagation distance when the phase matching occurs between the Bloch modes at the fundamental and double frequencies.     

管道中具有累积效应的二阶谐纵向导波生成点的数值验证

利用非线性导波检测早期损伤的潜在优势,在于其对微裂纹或以微结构变化的形式表现出来的材料性能退化的敏感性。而长距离传播能力使得振幅具有累积效应的非线性二阶谐纵向导波成为检测管道的理想方法。本文提出了在均匀各向同性的应力自由弹性固体管道中,产生轴对称二阶谐纵向导波的方案,该方案使用同频零阶纵向导波作为激励波。根据相速度匹配条件得到与此方案相对应的可能的谐导波生成点后,利用另一条件即非零功率流对生成点进行了数值验证,最终确定了在管道中产生二阶谐纵向导波的生成点。     

Nonlinear surface acoustic waves on an elastic solid of general anisotropy

The effect of elastic nonlinearity on the propagation of Rayleigh waves in an anisotropic elastic solid is considered. A nonlinear integro-differential equation is derived for a quantity which is related to the Fourier transform of the displacement component on the surface. The variation of this quantity along the surface accounts for the slow modulation of the wave through formation and depletion of the different harmonics. Explicit results are given for harmonic generation in an initially sinusoidal wave and for parametric amplification of a weak signal by a pump wave of twice its frequency.     

Plane Waves in Cubically Nonlinear Elastic Media

The theory of plane waves in nonlinear materials described by the Murnaghan potential is proposed. The theory takes into account both the classical quadratic nonlinearity and the cubic nonlinearity of the basic wave equations. Some new opportunities for the wave interaction analysis are commented on: in addition to the second harmonics, a longitudinal plane wave generates the third one, a transverse plane wave generates the third harmonics, and horizontally and vertically polarized transverse plane waves jointly generate new waves     

Rayleigh wave in a quadratic nonlinear elastic half-space (Murnaghan model)

The nonlinear equations that underlie the analysis of classical Rayleigh waves are derived for the two-dimensional case of nonlinear elastic deformation described by the Murnaghan model. In addition to the case of presence of both geometrical and physical nonlinearities, two special cases are considered where one only type of nonlinearity is taken into account. It is shown that unlike the one-dimensional problems for plane waves where only three types of nonlinear interaction should be allowed for, the two-dimensional problems should include 24 types of nonlinear interaction. In the case of geometrical nonlinearity alone, a preliminary analysis of the nonlinear equations is carried out. Second-order equations are derived. The second approximation includes the second harmonics of the wave itself and its attenuating amplitude and is nonlinearly dependent on the initial amplitude of the Rayleigh wave and linearly increasing with the distance traveled by the wave     

金属板结构疲劳损伤二倍频兰姆波检测方法研究

针对重大基础设施安全运行需要,本文进行了金属板结构疲劳损伤非线性兰姆波检测方法研究。基于兰姆波二次谐波产生条件,确定了产生二次谐波积累增长效应的两种兰姆波模态及对应的激励频率。通过有限元仿真,研究了材料性能改变对兰姆波非线性效应的影响,证明了二倍频兰姆波非线性系数对材料性能退化表征的有效性。在此基础上,开展了金属板结构疲劳损伤非线性兰姆波检测实验研究。将极性反转方法应用于疲劳试件检测实验中,有效提高了检测信号中二倍频兰姆波的幅值和信噪比。实验结果表明,两种兰姆波模态对的二次谐波非线性系数均随疲劳损伤增加呈线性增长趋势,但基频S(0,2)模态和二倍频S(0,4)模态对对疲劳损伤检测的灵敏度更高,更适合金属板结构疲劳损伤检测。     

材料力学性能退化的超声无损检测与评价

材料性能退化总是伴随着某种形式的材料非线性力学行为, 从而引起超声波传播的非线性, 即高频谐波的产生.基于此,材料和结构的超声无损检测与主人技术发展成起来.首先介绍固体介质内的非线性超声波动方程的基础, 并综述了利用超声波传播的非线性特性对结构材料和粘结面的力学性能退化进行无损检测与评价的研究进展.之后对材料在疲劳、拉伸以及蠕变载荷作用下, 其力学性能退化进行超声无损检测与评价的试验研究进行了介绍;综述了超声波传播非线性的机理研究, 以及利用超声波对粘结面的粘结强度及其力学性能退化评估所开展的研究. 最后指出了今后该领域需要进一步研究的问题.      

钢轨轨底垂直振动模式导波检测技术的实验研究

现有铁路钢轨超声探伤车技术无法检测线路钢轨轨底缺陷,给铁路运输安全带来很大隐患.本文开展了钢轨轨底超声导波传播特性和垂直振动模式导波检测技术研究,采用半解析有限元方法分析了我国60型钢轨轨底的各振动模态导波频散曲线和波结构.应用模态锤技术对自由状态钢轨轨底垂直振动模态导波传播特性进行了实验测量,结果表明,在0～100kHz频率范围内,钢轨轨底垂直振动模态优势模式与厚度为14 mm板中的A0模式兰姆波具有等效性.进一步研究了激励频率、激励脉冲周数、传播距离对轨底垂直振动模态导波传播的影响,设计了导波斜探头,选择合适的参数在钢轨轨底激励出垂直振动模态导波并检测出了轨底的人工缺陷.本文的研究结果为线路钢轨轨底的导波检测技术奠定了一定的基础.     

Modeling nonlinear Rayleigh wave fields generated by angle beam wedge transducers—A theoretical study

Nonlinear Rayleigh wave fields generated by an angle beam wedge transducer are modeled in this study. The calculated area sound sources underneath the wedge are used to model the fundamental Rayleigh sound fields on the specimen surface, which are more accurate than the previously used line sources with uniform or Gaussian amplitude distributions. A general two-dimensional nonlinear Rayleigh wave equation without parabolic approximation is introduced and the solutions are obtained using the quasilinear theory. The second harmonic Rayleigh wave due to material nonlinearity is given in an integral expression with these fundamental Rayleigh waves radiated by the wedge transmitter acting as a forcing function. Multi-Gaussian beam (MGB) models are employed to simplify these integral solutions and to extract the diffraction and attenuation correction terms explicitly. The effect of nonlinearity of generating sources on the second harmonic Rayleigh wave fields is taken into consideration; simulation results show that it will affect the magnitude and diffraction correction of the second harmonic waves in the region close to the Rayleigh wave sound sources. This research provides a theoretical improvement to alleviate the experimental restriction on analyzing the effects of diffraction, attenuation and source nonlinearity when using angle beam wedge transducers as transmitters.     

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.     

Nonlinear elastic wave propagation in a phononic material with periodic solid–solid contact interface

Phononic materials enable enhanced dynamic properties, and offer the ability to engineer the material response. In this work we study the wave propagation in such a structure when introduced with nonlinearity. Our system is comprised of pre-compressed material with periodic solid–solid contacts, which exhibit a quadratic nonlinearity for small displacements. We suggest a new approach to modeling this system, where we discretize the unit cell in order to derive an approximate analytical solution using a perturbation method, which we are then able to easily validate numerically. With these methods, we study the band structure in the system and the second harmonic generation originating from the nonlinearity. We qualitatively analyze the second harmonic response of the system in terms of the single-crack response with linear band structure considerations. Significant band structure manipulation by changing system parameters is demonstrated, including possible in-situ tuning. The system also exhibits effective frequency doubling, i.e. the transmitted wave is primarily comprised of the second harmonic wave, for a certain range of frequencies. We demonstrate very high robustness to disorder in the system, in terms of band structure and second harmonic generation. These results have possible applications as frequency-converting devices, tunable engineered materials, and in non-destructive evaluation.     

Generation of the second,fourth, eighth,and subsequent harmonics by a quadratic nonlinear hyperelastic longitudinal plane wave

The perturbation (small-parameter) method is used to obtain the first three approximations for the problem of a harmonic longitudinal plane wave propagating in a quadratic nonlinear hyperelastic material described by the classical Murnaghan model. The subsequent approximations are discussed. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that the third approximation corrects the prediction of the evolution of the initial wave profile. Which of the harmonics dominates depends on the distance traveled by the wave: the second harmonic is generated first, then it transforms into the fourth harmonic, and finally, as the distance increases, the eighth harmonic shows     

有限单元法在超声导波检测技术中的应用

超声导波检测技术具有对波导结构中的缺陷进行远距离无损检测的能力,多年来一直是无损检测领域关注的热点之一.有限单元法具有对各种复杂动力学问题进行计算的能力,已成为超声导波检测技术研究的重要工具.本文结合超声导波检测技术研究领域中的热点问题,对相关的有限单元法进行了简要综述.介绍了有限单元法的发展及其在多物理场耦合机制下导波的激励与接收、线弹性和黏弹性结构中导波的传播特性、非线性超声导波等多个方面的应用研究情况. 最后,基于超声导波检测技术研究趋势展望了相关有限单元法的未来研究重点和发展方向.      

Experimental and Numerical Investigation of Second Harmonic Generation by Creep Induced Micro-voids

Background: Identification of damage at an early stage is crucial for critical structural components. Objective: Creep induced micro-voids in heat treated polycrystalline pure copper are experimentally and numerically characterized. Methods: This is accomplished by the use of non-linear ultrasonic waves. Numerically, the study is carried out on an elastic material with randomly located micro-voids. The finite difference in time domain method is used. Results: Experimentally it is found that material damage due to micro-voids that are developed before 40% creep life may effectively be detected with nonlinear ultrasonic waves in the low power regime. Increase in second order nonlinearity parameter with percentage creep life is observed along with the corresponding increase in micro-void concentration. Optical metallography and micro-hardness measurements were used to corroborate the experimental results. For the numerical studies, void dimensions selected for study are much lower than the probing wave length. Concentration of micro-voids is varied from 0.01% to 3%. The second order nonlinearity parameter showed a slight increase in the initial stages and an abrupt increase with higher micro-void concentration in simulations. This agrees with experimental trends. Conclusion: The feasibility of detecting micro-voids in the early stages of creep using nonlinear ultrasonic waves is experimentally demonstrated and a simple model with voids is numerically studied to show the sensitivity of the second order nonlinearity parameter to micro voids.

     

Generation of the second, fourth, and eighth harmonics by a hyperelastic longitudinal planewave: numerical simulation

The generation of the first, second, fourth, and eighth harmonics of a harmonic longitudinal plane wave propagating in quadratic nonlinear materials described by the classical Murnaghan model is numerically modeled. This situation was also analyzed analytically. The materials are fibrous composites with micro- and nanoscale structure. Using such materials introduces additional real restrictions on the main parameters and allows simulating various real situations: from the generation of all harmonics for feasible distances and times to the generation of the second harmonics for feasible distances and times. The contribution of each approximation to the overall wave pattern is analyzed. It is shown that these approximations affect the prediction of the evolution of the initial wave profile: first there is a tendency to the generation of the second harmonic which then transforms into a tendency to the generation of the fourth and eight harmonics