Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves

The use of nonlinear ultrasonic waves has been accepted as a potential technique to characterize the state of material micro-structure in solids. The typical nonlinear phenomenon is generation of second harmonics. Second harmonic generation of ultrasonic waves propagation has been vigorously studied for tracking material micro-damages in unbounded media and plate-like waveguides. However, there are few studies of launching second harmonic guided wave propagation in tube-like structures. Considering that second harmonics could provide useful information sensitive for material degradation condition, this research aims at developing a procedure for detecting second harmonics of ultrasonic guided wave in an isotropic pipe. The second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe is investigated. Flexible polyvinylidene fluoride (PVDF) comb transducers are used to measure fundamental wave and second harmonic one. Experimental results show that nonlinear parameters increase monotonically with propagation distance. This work experimentally verifies that the second harmonics of guided waves in pipe have the cumulative effect with propagation distance. The proposed procedure is applied to assessing thermal fatigue damage indicated by nonlinearity in an aluminum pipe. The experimental observation verifies that nonlinear guided waves can be used to assess damage levels in early thermal fatigue state by correlating them with the acoustic nonlinearity.     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性,同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应,利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析,在不同的条件下,该方程在相平面上存在同宿轨道或异宿轨道, 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法,对该非线性方程进行 求解,得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解,讨论了这些解存 在的必要条件,这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程,从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

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.     

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.     

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.     

Three kinds of nonlinear dispersive waves in elastic rods with finite deformation

On the basis of classical linear theory on longitudinal, torsional and flexural waves in thin elastic rods, and taking finite deformation and dispersive effects into consideration, three kinds of nonlinear evolution equations are derived. Qualitative analysis of three kinds of nonlinear equations are presented. It is shown that these equations have homoclinic or heteroclinic orbits on the phase plane, corresponding to solitary wave or shock wave solutions, respectively. Based on the principle of homogeneous balance, these equations are solved with the Jacobi elliptic function expansion method. Results show that existence of solitary wave solution and shock wave solution is possible under certain conditions. These conclusions are consistent with qualitative analysis.     

Second-harmonic generation in an infinite layered structure with nonlinear spring-type interfaces

The second-harmonic generation characteristics in the elastic wave propagation across an infinite layered structure consisting of identical linear elastic layers and nonlinear spring-type interlayer interfaces are analyzed theoretically. The interlayer interfaces are assumed to have identical linear interfacial stiffness but can have different quadratic nonlinearity parameters. Using a perturbation approach and the transfer-matrix method, an explicit analytical expression is derived for the second-harmonic amplitude when the layered structure is impinged by a monochromatic fundamental wave. The analysis shows that the second-harmonic generation behavior depends significantly on the fundamental frequency reflecting the band structure of the layered structure. Two special cases are discussed in order to demonstrate such dependence, i.e., the second-harmonic generation by a single nonlinear interface as well as by multiple consecutive nonlinear interfaces. In particular, when the second-harmonic generation occurs at multiple consecutive nonlinear interfaces, the cumulative growth of the second-harmonic amplitude with distance is only expected in certain frequency ranges where the fundamental as well as the double frequencies belong to the pass bands of the layered structure. Furthermore, a remarkable increase of the second-harmonic amplitude is found near the terminating edge of pass bands. Approximate expressions for the low-frequency range are also obtained, which show the cumulative growth of the second-harmonic amplitude with quadratic frequency dependence.     

Lagrangian description of transport equations for shock waves in three-dimensional elastic solids

A set of transport equations for the growth or decay of theamplitudes of shock waves along an arbitrary propagation directionin three-dimensional nonlinear elastic solids is derived using theLagrangian coordinates.The transport equations obtained showthat the time derivative of the amplitude of a shock wave alongany propagation ray depends on (i) an unknown quantity immediatelybehind the shock wave,(ii) the two principal curvatures of theshock surface,(iii) the gradient taken on the shock surface ofthe normal shock wave speed and (iv) the inhomogeneous term.whichis related to the motion ahead of the shock surface.vanisheswhen the motion ahead of the shock surface is uniform.Severalchoices of the propagation vector are given for which the tran-sport equations can be simplified.Some universal relations,which relate the time derivatives of various jump quantities toeach other but which do not depend on the constitutive equationsof the material,are also presented.     

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.     

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     

Simulation seismic wave propagation in topographic structures using asymmetric staggered grids

A new 3 D finite- difference ( FD ) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures. The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise. Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme, The method had no interpolation between the fine and coarse grids. All grids were computed at the same spatial iteration. Complicated geometrical structures like rough submarine interface, fault and nonplanar interfaces were treated with fine irregular grids. Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time, at the same time, has satisfactory stability and accuracy.     

Nonlinear flexural waves in fluid–filled elastic channels

Nonlinear waves on liquid sheets between thin infinite elastic plates are studied analytically and numerically. Linear and nonlinear models are used for the elastic plates coupled to the Euler equations for the fluid. One-dimensional time-dependent equations are derived based on a long-wavelength approximation. Inertia of the elastic plates is neglected, so linear perturbations are stable. Symmetric and mixed-mode travelling waves are found with the linear plate model and symmetric travelling waves are found for the nonlinear case. Numerical simulations are employed to study the evolution in time of initial disturbances and to compare the different models used. Nonlinear effects are found to decrease the travelling wave speed compared with linear models. At sufficiently large amplitude of initial disturbances, higher order temporal oscillations induced by nonlinearity can lead to thickness of the liquid sheet approaching zero.     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

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.     

The nonlinear analysis of elastic wave of piezoelectric crystal plate with perturbation method

When investigating or designing acoustic wave sensors, the behavior of piezoelectric devices is supposed to be linear. However, if the sensors are subjected to a strong elastic field, the amplitude of the elastic strain induced in the piezoelectric material is so large that the nonlinearity, which affects the stability and performance of the piezoelectric sensors, can no longer be ignored. In this paper, we perform a theoretical analysis on nonlinear anti-symmetric thickness vibration of thin-film acoustic wave resonators made from quartz. Using Green’s identity, under the usual approximation of neglecting higher time harmonics, a perturbation analysis is performed from which the resonator frequency–amplitude (A–F) relation is obtained. Numerical calculations are made. Furthermore, the validity of the method is examined.     

有限变形弹性杆中的非线性波及周期解

利用Ham ilton变分原理,导出了计及有限变形和横向Possion效应的弹性杆中非线性纵向波动方程.利用Jacob i椭圆正弦函数展开和第三类Jacob i椭圆函数展开法,对该方程和截断的非线性方程进行求解,得到了非线性波动方程的两类准确周期解及相应的孤波解和冲击波解,讨论了这些解存在的必要条件.     

Shock waves in saturated thermoelastic porous media

The objective of this paper is to develop and present the macroscopic motion equations for the fluid and the solid matrix, in the case of a saturated porous medium, in the form of coupled, nonlinear wave equations for the fluid and solid velocities. The nonlinearity in the equations enables the generation of shock waves. The complete set of equations required for determining phase velocities in the case of a thermoelastic solid matrix, includes also the energy balance equation for the porous medium as a whole, as well as mass balance equations for the two phase. In the special case of a rigid solid matrix, the wave after an abrupt change in pressure propagates only through the fluid.     

Primary wave transmission in systems of elastic rods with granular interfaces

We study analytically and numerically primary pulse transmission in one dimensional systems of identical linearly elastic non-dispersive rods separated by identical homogeneous granular layers composed of n beads. The beads interact elastically through a strongly (essentially) nonlinear Hertzian contact law. The main challenge in studying pulse transmission in such strongly nonlinear media is to analyze the ‘basic problem’, namely, the dynamical response of a single intermediate granular layer, confined from both ends by barely touching linear elastic rods subject to impulsive excitation of the left rod. The analysis of the basic problem is carried out under two basic assumptions; namely, of sufficiently small duration of the shock excitation applied to the first layer of the system, and of sufficiently small mass of each bead in the granular interface compared to the mass of each rod. In fact, the smallness of the mass of the bead defines the small parameter in the asymptotic analysis of this problem. Both assumptions are reasonable from the point of view of practical applications. In the analysis we focus only in primary pulse propagation, by neglecting secondary pulse reflections caused by wave scattering at each granular interface and considering only the transmission of the main (primary) pulse across the interface to the neighboring elastic rod. Two types of shock excitations are considered. The first corresponds to fixed time duration (but still much smaller compared to the characteristic time of pulse propagation through the length of each rod), whereas the second type corresponds to a pulse duration that depends on the small parameter of the problem. The influence of the number of beads of the granular interface on the primary wave transmission is studied, and it is shown that at granular interfaces with a relatively low number of beads fast time scale oscillations are excited with increasing amplitudes with increasing number of beads. For a larger number of beads, primary pulse transmission is by means of solitary wave trains resulting from the dispersion of the original shock pulse; in that case fast oscillations result due to interference phenomena caused by the scattering of the main pulse at the boundary of the interface. Considering a periodic system of rods we demonstrate significant reduction of the primary pulse when transmitted through a sequence of granular interfaces. This result highlights the efficacy of applying granular interfaces for passive shock mitigation in layered elastic media.     

NONLINEAR WAVES AND PERIODIC SOLUTION IN FINITE DEFORMATION ELASTIC ROD

A nonlinear wave equation of elastic rod taking account of finite deformation, transverse inertia and shearing strain is derived by means of the Hamilton principle in this paper. Nonlinear wave equation and truncated nonlinear wave equation are solved by the Jacobi elliptic sine function expansion and the third kind of Jacobi elliptic function expansion method. The exact periodic solutions of these nonlinear equations are obtained, including the shock wave solution and the solitary wave solution. The necessary condition of exact periodic solutions, shock solution and solitary solution existence is discussed.