非线性表面波的二阶微扰解及特性分析

曾胜洋; 贾璐; 张书增; 李雄兵; 王猛

doi:10.7498/aps.71.20212445

摘要

为解决非线性声表面波的求解难题, 本文从二阶非线性各向同性介质的超弹性本构方程出发, 采用位移势函数法, 建立二维表面波的非线性势函数控制方程; 通过微扰法推导非线性表面波的准线性解和绝对非线性系数, 讨论表面波二次谐波解的主要组成部分; 并建立模拟非线性表面波传播的有限元模型, 位移幅值的仿真结果与理论符合良好, 验证了本文非线性表面波理论的准确性. 根据微扰解的数值结果, 探讨了非线性表面波的传播以及非线性系数的特性, 结果表明: 表面波二次谐波由累积项及非累积项组成, 前者与表面波纵波分量自相互作用相关, 但当初始条件和传播距离相同时, 该部分谐波幅值比纯纵波的二次谐波幅值大; 此外, 纵波和表面波的非线性系数存在正比关系, 该比例关系由材料的二阶弹性系数确定. 本文探究的非线性表面波的传播特性及其绝对非线性系数的定义表达式, 对指导非线性表面波的实际应用具有一定意义.

关键词:

Abstract

The properties of ultrasonic nonlinear surface wave in the quasilinear region are investigated. In this work the governing equation of particle displacement potential is employed for surface wave in isotropic elastic solid with quadratic nonlinearity. Then, the quasilinear solution of the nonlinear surface wave is obtained by the perturbation method, and the absolute nonlinear parameter of the surface wave is derived. Subsequently, the main components of the second harmonic surface wave solution are discussed. A finite element model for the propagating nonlinear surface wave is developed, and simulation results of the nonlinear surface wave displacements agree well with the theoretical solutions, which indicates that the proposed theory is effective. Finally, the properties of wave propagation and the characteristic of the nonlinear parameter for the surface wave are analyzed based on the theoretical solutions. It is found that the second harmonic surface wave consists of cumulative and non-cumulative displacement terms. The cumulative displacement term is related to the self-interaction of the longitudinal wave component of the surface wave. However, its amplitude is larger than that of the pure longitudinal wave when the initial excitation conditions and propagation distances are the same. The nonlinear parameters for surface and longitudinal waves are related to each other, and an explicit relationship is found, which can be determined by the second-order elastic coefficients of the material. The propagation properties of nonlinear surface waves and the measurement method of absolute nonlinear parameters are also discussed, which will benefit the practical application of nonlinear surface waves.

Keywords:

作者及机构信息

1.
中南大学交通运输工程学院, 长沙　410075

2.
清华大学航天航空学院, 应用力学教育部重点实验室, 北京　100084

3.
清华大学, 柔性电子技术研究中心, 北京　100084

通信作者: 贾璐, jia-lu@mail.tsinghua.edu.cn ; 张书增, sz_zhang@csu.edu.cn ;

基金项目: 国家自然科学基金(批准号: 51805554)和中南大学中央高校基本科研业务费(批准号: 2021zzts0175)资助的课题.

Authors and contacts

1.
School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China

2.
AML, School of Aerospace Engineering, Tsinghua University, Beijing 100084, China

3.
Center for Flexible Electronics Technology, Tsinghua University, Beijing 100084, China

Corresponding author: Jia Lu, jia-lu@mail.tsinghua.edu.cn ; Zhang Shu-Zeng, sz_zhang@csu.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 51805554) and the Fundamental Research Fund for the Central Universities of Central South University, China (Grant No. 2021zzts0175).

文章全文 : translate this paragraph

参考文献

[1]	周正干, 刘斯明 2011 机械工程学报 47 2 Zhou Z G, Liu S M 2011 J. Mech. Eng. 47 2
[2]	Huang Y, Wang X, Gong X, Wu H, Zhang D, Zhang D 2020 Sci. Rep. 10 1
[3]	Matlack K H, Kim J Y, Jacobs L J, Qu J 2015 J. Nondestr. Eval. 34 273 Google Scholar
[4]	陈海霞, 林书玉 2021 70 114302 Google Scholar Chen H X, Lin S Y 2021 Acta Phys. Sin. 70 114302 Google Scholar
[5]	Zhang S, Li X, Chen C, Jeong H, Xu G 2019 J. Nondestr. Eval. 38 88 Google Scholar
[6]	Liu Y, He A, Liu J, Mao Y, Liu X 2020 Chin. Phys. B 29 054301 Google Scholar
[7]	Li W, Lan Z, Hu N, Deng M 2021 Ultrasonics 113 106356 Google Scholar
[8]	Zhang J, Xuan F Z, Yang F 2013 J. NonCryst. Solids 378 101 Google Scholar
[9]	Guo S, Lei Z, Mirshekarloo M S, Chen S, Yi F C, Zheng Z W, Shen Z, Liu H, Yao K 2016 Mater. Sci. Eng. A 669 41 Google Scholar
[10]	Kim G, Park S, Kim J Y, Kurtis K E, Hayes N W, Jacobs L J 2018 Constr. Build. Mater. 186 1114 Google Scholar
[11]	Torello D, Selby N, Kim J Y, Qu J, Jacobs L 2017 Ultrasonics 81 107 Google Scholar
[12]	张世功, 吴先梅, 张碧星, 安志武 2016 65 104301 Google Scholar Zhang S G, Wu X M, Zhang B X, An Z W 2016 Acta Phys. Sin. 65 104301 Google Scholar
[13]	Kube C M, Argulles A P 2017 J. Acoust. Soc. Am. 142 EL224 Google Scholar
[14]	Wang X, Gong X, Qin C, Zhang D, Wu H, Zhang D 2019 Mech. Syst. Sig. Process. 130 790 Google Scholar
[15]	Xu L, Wang K, Su Y, He Y, Yang J, Yuan S, Su Z 2021 Ultrasonics 118 106578
[16]	Zabolotskaya E A 1992 J. Acoust. Soc. Am. 91 2569 Google Scholar
[17]	Herrmann J, Kim J, Jacobs L J, Qu J, Littles J W, Savage M F 2006 J. Appl. Phys. 99 124913 Google Scholar
[18]	Masurkar F, Tse P 2020 Ultrasonics 108 106036 Google Scholar
[19]	Landau L D, Lifshitz E M 1986 Theory of Elasticity (Oxford: Pergamon Press) pp95–118
[20]	Rose J L 2014 Ultrasonic Guided Waves in Solid Media (New York: Cambridge University Press) pp108–114
[21]	Kundu T 2019 Nonlinear Ultrasonic and Vibro-acoustical Techniques for Nondestructive Evaluation (Gewerbestrasse: Springer Press) pp229–231
[22]	Jia L, Yan S, Zhang B, Huang J 2020 J. Acoust. Soc. Am. 148 EL289 Google Scholar
[23]	Norris A 1991 J. Elasticity 25 247 Google Scholar
[24]	Nagy P B, Qu J, Jacobs L J 2013 J. Acoust. Soc. Am. 134 1760 Google Scholar
[25]	Morlock M B, Kim J, Jacobs L J, Qu J 2015 J. Acoust. Soc. Am. 137 281 Google Scholar
[26]	Muir D D. 2009 Ph. D. Dissertation (Atlanta: Georgia Institute of Technology)
[27]	Zhong B, Zhu J 2021 Appl. Phys. Lett. 118 261903 Google Scholar
[28]	Stobbe D M 2005 M. S. Thesis (Atlanta: Georgia Institute of Technology)
[29]	de Araújo Freitas V L, de Albuquerque V H C, de Macedo Silva E, Silva A A, Tavares J M R 2010 Mater. Sci. Eng. A 527 4431 Google Scholar
[30]	Shui Y, Solodov I Y 1988 J. Appl. Phys. 64 6155 Google Scholar
[31]	Shull D J, Kim E E, Hamilton M F, Zabolotskaya E A 1995 J. Acoust. Soc. Am. 97 2126 Google Scholar

施引文献

图 1 表面波传播坐标系

Fig. 1. The coordinate for the propagation of the surface wave

下载: 全尺寸图片幻灯片

图 2 非线性表面波二维有限元模型示意图

Fig. 2. Schematic diagram of two-dimensional (2D) finite element model for nonlinear surface wave.

下载: 全尺寸图片幻灯片

图 3 非线性纵波二维有限元模型示意图

Fig. 3. Schematic diagram of 2D finite element model for the nonlinear longitudinal wave.

下载: 全尺寸图片幻灯片

图 4 纵波仿真结果图　(a)传播距离15 mm处探针的信号图; (b) 基波和二次谐波幅值随距离变化图, 蓝色点是仿真结果, 黄色实线是线性拟合结果

Fig. 4. Simulation results for the nonlinear longitudinal wave: (a) Typical signal at 15 mm propagation distance; (b) plots of fundamental wave and second harmonic amplitude versus propagation distance, where blue points denote the simulation results, and the yellow solid line denotes the fitting line.

下载: 全尺寸图片幻灯片

图 5 表面波仿真结果　(a) 传播距离15 mm处探针的信号图; (b) 基波和二次谐波幅值随距离变化图, 蓝色点是仿真结果, 黄色实线是拟合结果

Fig. 5. Simulation results for the nonlinear surface wave: (a) Typical signal at 15 mm propagation distance; (b) plots of fundamental wave and second harmonic amplitude versus propagation distance, where blue points denote the simulation results, and the yellow solid line denotes the fitting line.

下载: 全尺寸图片幻灯片

表 1 纯铝和三种铝合金的密度和弹性常数

Table 1. Densities and elastic coefficients of pure aluminum and three aluminum alloys.

材料	$ {\rho _{\text{0}}} $/ (kg·m³)	$\lambda $/ GPa	$ \mu $/ GPa	$A$/ GPa	$B$/ GPa	$C$/ GPa
Al^[24]	2700	51	26	–350	–155	–95
Al2024^[26]	2780	56.9	27.7	–305	–146	31
Al6061^[27]	2705	59.5	26.1	–337	–129.5	16.5
Al7075^[28]	2800	54.9	26.5	–351.2	–149.4	–102.8

下载: 导出CSV

表 2 初始条件和传播距离对面外分量和纵波二次谐波幅值的影响

Table 2. The effects of initial conditions and propagation distance on the amplitudes of out-of-plane component and the second harmonic longitudinal wave.

序号	传播距离r/m	频率f/MHz	基波幅值${A_1}$/nm	二次谐波幅值
序号	传播距离r/m	频率f/MHz	基波幅值${A_1}$/nm	${A_2}$/nm	$A_2^{\text{C}}$/nm	$u_{\text{l}}^{\left( 2 \right)}$/nm
1	0.2	1	1	2.91×10^–3	3.05×10^–5	3.80×10^–4
2	0.2	0.2	1	1.16×10^–4	6.11×10^–6	1.52×10^–5
3	0.2	1	5	7.28×10^–2	7.62×10^–4	9.48×10^–3
4	0.5	1	1	2.91×10^–2	1.22×10^–4	3.80×10^–3

下载: 导出CSV

表 3 表1中4种材料的非线性系数

Table 3. Nonlinear parameters of four materials listed in Table 1.

材料	${\beta _{\text{l}}}$	${\beta _{\text{r}}}$	${\bar \beta _{\text{r}}}$	$ {\bar \beta _{11}} $	${\delta _{\text{R}}}$
Al	14.670	24.672	4.740	1.166	22.191
Al2024	9.680	15.799	3.044	0.674	13.892
Al6061	9.695	14.753	2.860	0.733	14.444
Al7075	13.723	22.271	4.293	1.024	21.227

下载: 导出CSV

map

[1]	周正干, 刘斯明 2011 机械工程学报 47 2 Zhou Z G, Liu S M 2011 J. Mech. Eng. 47 2
[2]	Huang Y, Wang X, Gong X, Wu H, Zhang D, Zhang D 2020 Sci. Rep. 10 1
[3]	Matlack K H, Kim J Y, Jacobs L J, Qu J 2015 J. Nondestr. Eval. 34 273 Google Scholar
[4]	陈海霞, 林书玉 2021 70 114302 Google Scholar Chen H X, Lin S Y 2021 Acta Phys. Sin. 70 114302 Google Scholar
[5]	Zhang S, Li X, Chen C, Jeong H, Xu G 2019 J. Nondestr. Eval. 38 88 Google Scholar
[6]	Liu Y, He A, Liu J, Mao Y, Liu X 2020 Chin. Phys. B 29 054301 Google Scholar
[7]	Li W, Lan Z, Hu N, Deng M 2021 Ultrasonics 113 106356 Google Scholar
[8]	Zhang J, Xuan F Z, Yang F 2013 J. NonCryst. Solids 378 101 Google Scholar
[9]	Guo S, Lei Z, Mirshekarloo M S, Chen S, Yi F C, Zheng Z W, Shen Z, Liu H, Yao K 2016 Mater. Sci. Eng. A 669 41 Google Scholar
[10]	Kim G, Park S, Kim J Y, Kurtis K E, Hayes N W, Jacobs L J 2018 Constr. Build. Mater. 186 1114 Google Scholar
[11]	Torello D, Selby N, Kim J Y, Qu J, Jacobs L 2017 Ultrasonics 81 107 Google Scholar
[12]	张世功, 吴先梅, 张碧星, 安志武 2016 65 104301 Google Scholar Zhang S G, Wu X M, Zhang B X, An Z W 2016 Acta Phys. Sin. 65 104301 Google Scholar
[13]	Kube C M, Argulles A P 2017 J. Acoust. Soc. Am. 142 EL224 Google Scholar
[14]	Wang X, Gong X, Qin C, Zhang D, Wu H, Zhang D 2019 Mech. Syst. Sig. Process. 130 790 Google Scholar
[15]	Xu L, Wang K, Su Y, He Y, Yang J, Yuan S, Su Z 2021 Ultrasonics 118 106578
[16]	Zabolotskaya E A 1992 J. Acoust. Soc. Am. 91 2569 Google Scholar
[17]	Herrmann J, Kim J, Jacobs L J, Qu J, Littles J W, Savage M F 2006 J. Appl. Phys. 99 124913 Google Scholar
[18]	Masurkar F, Tse P 2020 Ultrasonics 108 106036 Google Scholar
[19]	Landau L D, Lifshitz E M 1986 Theory of Elasticity (Oxford: Pergamon Press) pp95–118
[20]	Rose J L 2014 Ultrasonic Guided Waves in Solid Media (New York: Cambridge University Press) pp108–114
[21]	Kundu T 2019 Nonlinear Ultrasonic and Vibro-acoustical Techniques for Nondestructive Evaluation (Gewerbestrasse: Springer Press) pp229–231
[22]	Jia L, Yan S, Zhang B, Huang J 2020 J. Acoust. Soc. Am. 148 EL289 Google Scholar
[23]	Norris A 1991 J. Elasticity 25 247 Google Scholar
[24]	Nagy P B, Qu J, Jacobs L J 2013 J. Acoust. Soc. Am. 134 1760 Google Scholar
[25]	Morlock M B, Kim J, Jacobs L J, Qu J 2015 J. Acoust. Soc. Am. 137 281 Google Scholar
[26]	Muir D D. 2009 Ph. D. Dissertation (Atlanta: Georgia Institute of Technology)
[27]	Zhong B, Zhu J 2021 Appl. Phys. Lett. 118 261903 Google Scholar
[28]	Stobbe D M 2005 M. S. Thesis (Atlanta: Georgia Institute of Technology)
[29]	de Araújo Freitas V L, de Albuquerque V H C, de Macedo Silva E, Silva A A, Tavares J M R 2010 Mater. Sci. Eng. A 527 4431 Google Scholar
[30]	Shui Y, Solodov I Y 1988 J. Appl. Phys. 64 6155 Google Scholar
[31]	Shull D J, Kim E E, Hamilton M F, Zabolotskaya E A 1995 J. Acoust. Soc. Am. 97 2126 Google Scholar

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