基于群速度失配的超声兰姆波二次谐波的时域测量方法

考虑到发射和接收换能器对超声兰姆波时域二次谐波信号所带来的不可避免的影响,提出一种基于基频与二倍频兰姆波群速度失配的超声兰姆波二次谐波的时域测量方法。当基频与二倍频超声兰姆波的相速度匹配而群速度失配时,在超声兰姆波传播过程中所发生的二次谐波信号,在时域上可与源于斜劈换能器的二次谐波信号相分离。采用仅源自于基频兰姆波的时域二次谐波的积分振幅,定量描述兰姆波二次谐波的发生效率。以铝板中传播的兰姆波为例,给出了时域二次谐波的具体测量过程。本文提出的测量方法放宽了超声兰姆波二次谐波的测量条件,且扣除了换能器对二次谐波信号所带来的影响,所测得的二次谐波信号完全来自于基频兰姆波时域信号的二次谐波发生效应。      

一种评价固体板材表面性质的非线性超声兰姆波方法

在基频与二倍频兰姆波相速度匹配但群速度失配的条件下, 通过选择适当的兰姆波二次谐波时域信号的测量起止时间, 可完全扣除换能器对二次谐波积分振幅测量所带来的影响。本文提出采用兰姆波二次谐波的积分振幅作为评价参量, 以实现对板材表面性质的准确评价。当板材表面性质发生改变时, 原本在理想表面条件下成立的基频与二倍频兰姆波相速度匹配的条件不再严格满足, 这将显著地影响到兰姆波的二次谐波发生效率, 相应的二次谐波积分振幅随表面性质的改变也将发生非常敏感的单调变化。实验结果表明,利用扣除换能器影响之后所测得的兰姆波二次谐波的积分振幅,可对板材表面性质的变化情况进行准确评价。     

Second Harmonic Generation of Lamb Wave in Numerical Perspective

The influences of phase and group velocity matching on cumulative second harmonic generation of Lamb waves are investigated in numerical perspective.Finite element simulations of nonlinear Lamb wave propagation are performed for Lamb wave mode pairs with exact and approximate phase velocity matching,with and without group velocity matching,respectively.The evolution of time-domain second harmonic Lamb waves is analyzed with the propagation distance.The amplitudes of primary and second harmonic waves are calculated to characterize the acoustic nonlinearity.The results verify that phase velocity matching is necessary for generation of the cumulative second harmonic Lamb wave in numerical perspective,while group velocity matching is demonstrated to not be a necessary condition.     

圆管结构中周向导波非线性效应的模式展开分析

在二阶微扰近似条件下, 采用导波模式展开分析方法研究了圆管结构中周向导波的非线性效应. 伴随基频周向导波传播所发生的二次谐波, 可视为由一系列二倍频周向导波模式叠加而成. 从动量定理出发, 结合柱坐标系下非线性应力张量及其散度的数学表达式, 针对圆管中某一基频周向导波模式, 推导出相应的二倍频应力张量及二倍频彻体驱动力的数学表达式, 建立了确定二倍频周向导波模式展开系数的控制方程, 得到了伴随基频周向导波传播所发生的二次谐波声场的形式解. 理论分析和数值计算表明, 当构成二次谐波声场的某一二倍频周向导波模式与基频周向导波的相速度匹配时, 该二倍频周向导波模式的位移振幅表现出随传播周向角积累增长的性质; 当两者的相速度失配时, 二倍频周向导波的振幅随传播周向角表现出“拍”效应.     

固体板中SH板波非线性效应的实验观察

采用微扰近似和导波的模式展开分析方法,从理论上简要分析了SH板波的二次谐波发生效应;尽管在无限大固体介质中单个切变波的二次谐波发生效应非常微弱,但在一定条件下由两个切变波构成的SH板波可具有强烈的非线性效应;本文的主要工作就是对此结论加以实验验证。试制了激发SH板波的切变波斜劈换能器和接收二次谐波信号的液体斜劈换能器,建立了非线性SH板波的实验研究系统;通过详细的理论分析和对比实验研究,阐明了在一定条件下实验观察到的显著二次谐波信号来源于SH板波传播过程中的强烈非线性效应。此外,针对不同的SH板波传播距离,在远场条件下分别测量了相应的二次谐波幅频曲线;在基频SH板波与二倍频对称兰姆波相速度相等所对应的频率值附近,分析了二次谐波的振幅随传播距离的变化关系,结果证明在一定条件下SH板波的二次谐波振幅可随传播距离积累增长,即SH板波可具有强烈的非线性效应。      

General solution of cumulative second harmonic by Lamb wave propagation in a solid plate

A straightforward approach has been developed for the general solution of cumulative second harmonic by Lamb wave propagation in a solid plate. The present analyses of second-harmonic generation by Lamb waves focus on the cases where the phase velocity of the fundamental Lamb wave is exactly or approximately equal to that of the double frequency Lamb wave （DFLW）. Based on the general solution obtained, the numerical analyses show that the cumulative second-harmonic fields are associated with the position of excitation source and the difference between the phase velocity of the fundamental Lamb wave and that of the dominant DFLW component.     

Cumulative second-harmonic generation of Lamb waves propagating in a two-layered solid plate

The physical process of cumulative second-harmonic generation of Lamb waves propagating in a two-layered solid plate is presented by using the second-order perturbation and the technique of nonlinear reflection of acoustic waves at an interface. In general, the cumulative second-harmonic generation of a dispersive guided wave propagation does not occur. However, the present paper shows that the second-harmonic of Lamb wave propagation arising from the nonlinear interaction of the partial bulk acoustic waves and the restriction of the three boundaries of the solid plates does have a cumulative growth effect if some conditions are satisfied. Through boundary condition and initial condition of excitation, the analytical expression of cumulative second-harmonic of Lamb waves propagation is determined. Numerical results show the cumulative effect of Lamb waves on second-harmonic field patterns.     

Influence of Change in Inner Layer Thickness of Composite Circular Tube on Second-Harmonic Generation by Primary Circumferential Ultrasonic Guided Wave Propagation

The influence of change in inner layer thickness of a composite circular tube is investigated on second-harmonic generation(SHG) by primary circumferential ultrasonic guided wave(CUGW) propagation. Within a secondorder perturbation approximation, the nonlinear effect of primary CUGW propagation is treated as a second-order perturbation to its linear response. It is found that change in inner layer thickness of the composite circular tube will influence the efficiency of SHG by primary CUGW propagation in several aspects. In particular, with change in inner layer thickness, the phase velocity matching condition that is originally satisfied for the primary and double-frequency CUGW mode pair selected may no longer be satisfied. This will remarkably influence the efficiency of SHG by primary CUGW propagation. Theoretical analyses and numerical results show that the effect of SHG by primary CUGW propagation is very sensitive to change in inner layer thickness, and it can be used to accurately monitor a minor change in inner layer thickness of the composite circular tube.     

Enhancement effect of cumulative second-harmonic generation by closed propagation feature of circumferential guided waves

On the basis of second-order perturbation approximate and modal expansion approach,we investigate the enhancement effect of cumulative second-harmonic generation(SHG) of circumferential guided waves(CGWs) in a circular tube,which is inherently induced by the closed propagation feature of CGWs.An appropriate mode pair of primary-and doublefrequency CGWs satisfying the phase velocity matching and nonzero energy flux is selected to ensure that the second harmonic generated by primary CGW propagation can accumulate along the circumference.Using a coherent superposition of multi-waves,a model of unidirectional CGW propagation is established for analyzing the enhancement effect of cumulative SHG of primary CGW mode selected.The theoretical analyses and numerical simulations performed directly demonstrate that the second harmonic generated does have a cumulative effect along the circumferential direction and the closed propagation feature of CGWs does enhance the magnitude of cumulative second harmonic generated.Potential applications of the enhancement effect of cumulative SHG of CGWs are considered and discussed.The theoretical analysis and numerical simulation perspective presented here yield an insight previously unavailable into the physical mechanism of the enhancement effect of cumulative SHG by closed propagation feature of CGWs in a circular tube.     

圆管结构中的非线性周向导波*

采用将二阶微扰近似与模式展开分析相结合的方法,可从理论上有效求解圆管结构中的非线性周向导波问题。通过数值计算和有限元仿真发现,基频与二倍频周向导波模式的相速度匹配程度,可显著影响二倍频周向导波模式随传播周向角的积累增长程度。针对基频与二倍频周向导波模式的相速度和群速度均相匹配的情形,通过实验研究发现,周向导波确可存在强烈的非线性效应,且周向导波的二次谐波发生效应可对圆管早期损伤状态做出敏感的响应。文中给出的有关结果,可为进一步开展非线性周向导波的研究工作奠定理论和实验基础。     

Numerical Perspective of Second-Harmonic Generation of Circumferential Guided Wave Propagation in a Circular Tube

The effect of second-harmonic generation(SHG) by primary(fundamental) circumferential guided wave(CGW)propagation is investigated from a numerical standpoint.To enable that the second harmonic of the primary CGW mode can accumulate along the circumferential direction,an appropriate mode pair of primary and double frequency CGWs is chosen.Finite element simulations and evaluations of nonlinear CGW propagation are analyzed for the selected CGW mode pair.The numerical simulations performed directly demonstrate that the response of SHG is completely generated by the desired primary CGW mode that satisfies the condition of phase velocity matching at a specific driving frequency,and that the second harmonic of the primary CGW mode does have a cumulative effect with circumferential angles.The numerical perspective obtained yields an insight into the complicated physical process of SHG of primary CGW propagation unavailable previously.     

非线性兰姆波在厚度缓慢变化和衰减下的特性分析*

该文运用解析的方式推导了考虑声波衰减时兰姆波二次谐波的累积和传播规律,并用半解析方式将该理论推广到缓慢变厚度板的情况。由于色散特性,兰姆波二次谐波和基频波相速度不匹配,传播通常会产生拍频效应,使得二次谐波的振幅沿着传播距离周期性的归零。当考虑声波衰减或板的厚度缓慢变化的情况时,拍频效应将不再严格地被满足。二次谐波的振幅依然会沿着传播距离而振荡,但不会归零。该研究可以用于分析如何高效地激发和接收兰姆波的二次谐波,表征和评估不同厚度变化的结构中的微观结构损伤。     

Study of second-harmonic generation of shear horizontal modes using modal analysis

In this work a modal analysis was employed to study generation of the second harmonics of shear horizontal (SH) modes in a solid plate. Under second order perturbation, second-harmonic generation will occur accompanying SH mode propagation due to the bulk elastic non-linearity of plate material. In solid plate the total second-harmonic fields of a SH mode propagation are regarded as sum of the fields of a series of double frequency Lamb modes (DFLMs). The contribution of each DFLM component to the total second-harmonic fields is dependent of the difference of phase velocities of the corresponding DFLM and SH mode. The analysis results show that the DFLM field component may have cumulative growth effect once its phase velocity exactly or approximately equals that of a SH mode. It is also found that the fields of the total second harmonics of a SH mode are only symmetrical. The examples of field distributions of several DFLMs on the plate surface are considered.     

Analysis of Second-Harmonic Generation of Low-Frequency Dilatational Lamb Waves in a Two-Layered Composite Plate

We analyze the effect of second-harmonic generation(SHG) of primary Lamb wave propagation in a two-layered composite plate, and then investigate the influence of interfacial properties on the said effect at low frequency. It is found that changes in the interfacial properties essentially affect the dispersion relation and then the maximum cumulative distance of the double-frequency Lamb wave generated. This will remarkably influence the efficiency of SHG. To overcome the complications arising from the inherent dispersion and multimode natures in analyzing the SHG effect of Lamb waves, the present work focuses on the analysis of the SHG effect of low-frequency dilatational Lamb wave propagation. Both the numerical analysis and finite element simulation indicate that the SHG effect of low-frequency dilatational Lamb wave propagation is found to be much more sensitive to changes in the interfacial properties than primary Lamb waves. The potential of using the SHG effect of low-frequency dilatational Lamb waves to characterize a minor change in the interfacial properties is analyzed.     

Finite-amplitude waves in isotropic elastic plates

The propagation of finite-amplitude waves in a homogeneous, isotropic, stress-free elastic plate is investigated theoretically. Geometric and weak material non-linearities are included, and perturbation is used to obtain solutions of the non-linear equations of motion for harmonic generation in the waveguide. Solutions for the second-harmonic, sum, and difference-frequency components are obtained via modal decomposition. Ordinary differential equations for the modal amplitudes in the expansion of the second-order solution are obtained using a reciprocity relation. There are no restrictions on the modes or frequencies of the primary waves. Two conditions for internal resonance are quantified: phase matching, and transfer of power from the primary to the secondary wave.     

Characterization of surface properties of a solid plate using nonlinear Lamb wave approach

A nonlinear Lamb wave approach is presented for characterizing the surface properties of a solid plate. This characterization approach is useful for some practical situations where ultrasonic transducers cannot touch the surfaces to be inspected, e.g. the inside surfaces of sealed vessels. In this paper, the influences of changes in the surface properties of a solid plate on the effect of second-harmonic generation by Lamb wave propagation were analyzed. A surface coating with the different properties was used to simulate changes in the surface properties of a solid plate. When the areas and thicknesses of coatings on the surface of a given solid plate changed, the amplitude-frequency curves both of the fundamental waves and the second harmonics by Lamb wave propagation were measured under the condition that Lamb waves had a strong nonlinearity. It was found that changes in the surface properties might clearly affect the efficiency of second-harmonic generation by Lamb wave propagation. The Stress Wave Factors (SWFs) in acousto-ultrasonic technique were used for reference, and the definitions of the SWFs of Lamb waves were introduced. The preliminary experimental results showed that the second-harmonic SWF of Lamb wave propagation could effectively be used to characterize changes in the surface properties of the given solid plate.     

单轴晶体Ⅰ类相位匹配时的转换效率分析

 在考虑了二次谐波的吸收对转换效率影响的情况下,由光在介质中的波动方程,推导出负单轴晶体Ⅰ类相位匹配时平面二次谐波的耦合波方程和二次谐波转换效率的理论公式。在此基础上,再考虑走离效应的影响,给出了二次谐波转换效率的理论公式。根据这些公式,以负单轴晶体K₂Al₂B₂O₇（KABO）为例,对二次谐波的转换效率进行了数值计算。结果表明：考虑二次谐波的吸收时,存在倍频晶体长度的最佳值,当倍频晶体取该值时,二次谐波的转换效率最高;走离效应的存在使二次谐波的转换效率降低。     

超声Lamb波二次谐波发生效率分析与模式选择

在导波模式展开分析方法的基础上,提出激发效率参量来定量表征超声Lamb波积累二次谐波的发生效率。以P92钢板为例,理论计算得到了与频散曲线对应的理论激发效率参量分布图谱,从图谱中选择理论激发效率参量大小不同的两种基频Lamb波模式:纵波型S1模式和交点型A2/S2模式,分别测量这两种基频Lamb波模式在钢板中传播时产生的二次谐波信号。理论计算和实验测量结果表明,这两种基频Lamb波模式的理论和实验激发效率参量的比值基本一致,且激发效率参量较大的纵波型S1模式能激发出效率更高的二次谐波信号。研究结果表明激发效率参量可以有效的用于Lamb波二次谐波发生效率的表征及模式选择。      

分层半空间表面非线性瑞利波的激发*

针对非线性瑞利波在均匀分层半空间结构中的激发和传播规律进行研究。根据摄动理论和模态分解将分层半空间结构中瑞利波的二次谐波声场表示为二倍频瑞利波模式的线性组合,经由互易关系得到各模式的展开系数表达式。对不同分层半空间结构中瑞利波二次谐波的激发和传播特性进行讨论,结果表明相速度匹配的瑞利波模式其二次谐波分量随传播距离线性增长,非匹配模式的二次谐波分量则沿传播方向周期震荡传播。此外,文中定义非线性参数表征瑞利波模式产生的非线性程度,这有利于选择出具有明显非线性效应的匹配点,为后续检测工作提供理论依据,具有指导意义。     

Ultrasonic measurement of the elastic properties of benzoyl glycine single crystals

Certain organic crystals are found to possess high non-linear optical coefficients, often one to two orders of magnitude higher than those of the well-known inorganic non-linear optical materials. Benzoyl glycine is one such crystal whose optical second-harmonic generation efficiency is much higher than that of potassium dihydrogen phosphate. Single crystals of benzoyl glycine are grown by solvent evaporation technique usingN, N-dimethyl formamide as the solvent. All the nine second-order elastic stiffness constants of this orthorhombic crystal are determined from ultrasonic wave velocity measurements employing the pulse echo overlap technique. The anisotropy of elastic wave propagation in this crystal is demonstrated by plotting the phase velocity, slowness, Young’s modulus and linear compressibility surfaces along symmetry planes. The volume compressibility, bulk modulus and relevant Poisson’s ratios are also determined. Variation of the diagonal elastic stiffness constants with temperature over a limited range are measured and reported.