搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

引用本文:
Citation:

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

Second-order perturbation solution and analysis of nonlinear surface waves

Zeng Sheng-Yang, Jia Lu, Zhang Shu-Zeng, Li Xiong-Bing, Wang Meng
PDF
HTML
导出引用
  • 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.
    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.
      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.
      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).
    [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 273Google Scholar

    [4]

    陈海霞, 林书玉 2021 物理学报 70 114302Google Scholar

    Chen H X, Lin S Y 2021 Acta Phys. Sin. 70 114302Google Scholar

    [5]

    Zhang S, Li X, Chen C, Jeong H, Xu G 2019 J. Nondestr. Eval. 38 88Google Scholar

    [6]

    Liu Y, He A, Liu J, Mao Y, Liu X 2020 Chin. Phys. B 29 054301Google Scholar

    [7]

    Li W, Lan Z, Hu N, Deng M 2021 Ultrasonics 113 106356Google Scholar

    [8]

    Zhang J, Xuan F Z, Yang F 2013 J. NonCryst. Solids 378 101Google 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 41Google Scholar

    [10]

    Kim G, Park S, Kim J Y, Kurtis K E, Hayes N W, Jacobs L J 2018 Constr. Build. Mater. 186 1114Google Scholar

    [11]

    Torello D, Selby N, Kim J Y, Qu J, Jacobs L 2017 Ultrasonics 81 107Google Scholar

    [12]

    张世功, 吴先梅, 张碧星, 安志武 2016 物理学报 65 104301Google Scholar

    Zhang S G, Wu X M, Zhang B X, An Z W 2016 Acta Phys. Sin. 65 104301Google Scholar

    [13]

    Kube C M, Argulles A P 2017 J. Acoust. Soc. Am. 142 EL224Google Scholar

    [14]

    Wang X, Gong X, Qin C, Zhang D, Wu H, Zhang D 2019 Mech. Syst. Sig. Process. 130 790Google 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 2569Google Scholar

    [17]

    Herrmann J, Kim J, Jacobs L J, Qu J, Littles J W, Savage M F 2006 J. Appl. Phys. 99 124913Google Scholar

    [18]

    Masurkar F, Tse P 2020 Ultrasonics 108 106036Google 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 EL289Google Scholar

    [23]

    Norris A 1991 J. Elasticity 25 247Google Scholar

    [24]

    Nagy P B, Qu J, Jacobs L J 2013 J. Acoust. Soc. Am. 134 1760Google Scholar

    [25]

    Morlock M B, Kim J, Jacobs L J, Qu J 2015 J. Acoust. Soc. Am. 137 281Google Scholar

    [26]

    Muir D D. 2009 Ph. D. Dissertation (Atlanta: Georgia Institute of Technology)

    [27]

    Zhong B, Zhu J 2021 Appl. Phys. Lett. 118 261903Google 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 4431Google Scholar

    [30]

    Shui Y, Solodov I Y 1988 J. Appl. Phys. 64 6155Google Scholar

    [31]

    Shull D J, Kim E E, Hamilton M F, Zabolotskaya E A 1995 J. Acoust. Soc. Am. 97 2126Google Scholar

  • 图 1  表面波传播坐标系

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

    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.

    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.

    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.

    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.

    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]27005126–350–155–95
    Al2024[26]278056.927.7–305–14631
    Al6061[27]270559.526.1–337–129.516.5
    Al7075[28]280054.926.5–351.2–149.4–102.8
    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
    ${A_2}$/nm$A_2^{\text{C}}$/nm$u_{\text{l}}^{\left( 2 \right)}$/nm
    10.2112.91×10–33.05×10–53.80×10–4
    20.20.211.16×10–46.11×10–61.52×10–5
    30.2157.28×10–27.62×10–49.48×10–3
    40.5112.91×10–21.22×10–43.80×10–3
    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}}}$
    Al14.67024.6724.7401.16622.191
    Al20249.68015.7993.0440.67413.892
    Al60619.69514.7532.8600.73314.444
    Al707513.72322.2714.2931.02421.227
    下载: 导出CSV
  • [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 273Google Scholar

    [4]

    陈海霞, 林书玉 2021 物理学报 70 114302Google Scholar

    Chen H X, Lin S Y 2021 Acta Phys. Sin. 70 114302Google Scholar

    [5]

    Zhang S, Li X, Chen C, Jeong H, Xu G 2019 J. Nondestr. Eval. 38 88Google Scholar

    [6]

    Liu Y, He A, Liu J, Mao Y, Liu X 2020 Chin. Phys. B 29 054301Google Scholar

    [7]

    Li W, Lan Z, Hu N, Deng M 2021 Ultrasonics 113 106356Google Scholar

    [8]

    Zhang J, Xuan F Z, Yang F 2013 J. NonCryst. Solids 378 101Google 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 41Google Scholar

    [10]

    Kim G, Park S, Kim J Y, Kurtis K E, Hayes N W, Jacobs L J 2018 Constr. Build. Mater. 186 1114Google Scholar

    [11]

    Torello D, Selby N, Kim J Y, Qu J, Jacobs L 2017 Ultrasonics 81 107Google Scholar

    [12]

    张世功, 吴先梅, 张碧星, 安志武 2016 物理学报 65 104301Google Scholar

    Zhang S G, Wu X M, Zhang B X, An Z W 2016 Acta Phys. Sin. 65 104301Google Scholar

    [13]

    Kube C M, Argulles A P 2017 J. Acoust. Soc. Am. 142 EL224Google Scholar

    [14]

    Wang X, Gong X, Qin C, Zhang D, Wu H, Zhang D 2019 Mech. Syst. Sig. Process. 130 790Google 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 2569Google Scholar

    [17]

    Herrmann J, Kim J, Jacobs L J, Qu J, Littles J W, Savage M F 2006 J. Appl. Phys. 99 124913Google Scholar

    [18]

    Masurkar F, Tse P 2020 Ultrasonics 108 106036Google 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 EL289Google Scholar

    [23]

    Norris A 1991 J. Elasticity 25 247Google Scholar

    [24]

    Nagy P B, Qu J, Jacobs L J 2013 J. Acoust. Soc. Am. 134 1760Google Scholar

    [25]

    Morlock M B, Kim J, Jacobs L J, Qu J 2015 J. Acoust. Soc. Am. 137 281Google Scholar

    [26]

    Muir D D. 2009 Ph. D. Dissertation (Atlanta: Georgia Institute of Technology)

    [27]

    Zhong B, Zhu J 2021 Appl. Phys. Lett. 118 261903Google 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 4431Google Scholar

    [30]

    Shui Y, Solodov I Y 1988 J. Appl. Phys. 64 6155Google Scholar

    [31]

    Shull D J, Kim E E, Hamilton M F, Zabolotskaya E A 1995 J. Acoust. Soc. Am. 97 2126Google Scholar

  • [1] 申潇卓, 吴鹏飞, 林伟军. 脉动气泡在黏性介质中的声发射. 物理学报, 2024, 73(17): 174701. doi: 10.7498/aps.73.20240826
    [2] 丁怡洋, 邹帅, 孙华, 苏晓东. 金属网格-透明导电氧化物复合型透明电极的瑞利分析和仿真. 物理学报, 2024, 73(14): 146801. doi: 10.7498/aps.73.20240230
    [3] 许龙, 汪尧. 双泡耦合声空化动力学过程模拟. 物理学报, 2023, 72(2): 024303. doi: 10.7498/aps.72.20221571
    [4] 张羚翔, 魏薇, 张志明, 廖文英, 杨振国, 范万德, 李乙钢. 环形光子晶体光纤中涡旋光的传输特性研究. 物理学报, 2017, 66(1): 014205. doi: 10.7498/aps.66.014205
    [5] 刘珍黎, 宋亮华, 白亮, 许凯亮, 他得安. 长骨中振动声激发超声导波的方法. 物理学报, 2017, 66(15): 154303. doi: 10.7498/aps.66.154303
    [6] 李迎兵, 梁果, 洪伟毅, 任占梅, 郭旗. 负性向列相液晶中1+1维空间光孤子:微扰法. 物理学报, 2016, 65(9): 094204. doi: 10.7498/aps.65.094204
    [7] 陈卫军, 卢克清, 惠娟利, 王春香, 于会敏, 胡凯. LiNbO3晶体界面非线性表面波的研究. 物理学报, 2015, 64(1): 014204. doi: 10.7498/aps.64.014204
    [8] 刘梅, 王松岭, 吴正人. 非平整基底上受热液膜流动稳定性研究. 物理学报, 2014, 63(15): 154702. doi: 10.7498/aps.63.154702
    [9] 王伟, 杨博, 宋鸿儒, 范岳. 八边形高双折射双零色散点光子晶体光纤特性分析. 物理学报, 2012, 61(14): 144601. doi: 10.7498/aps.61.144601
    [10] 尹桂来, 李建英, 尧广, 成鹏飞, 李盛涛. ZnO压敏陶瓷冲击老化的电子陷阱过程研究. 物理学报, 2010, 59(9): 6345-6350. doi: 10.7498/aps.59.6345
    [11] 姜凌红, 侯蓝田. 双零色散光子晶体光纤结构参数的变化对其性能的影响. 物理学报, 2010, 59(2): 1095-1100. doi: 10.7498/aps.59.1095
    [12] 姜凌红, 侯蓝田, 杨倩倩. 三种典型结构光子晶体光纤基本特性的比较和分析. 物理学报, 2010, 59(7): 4726-4731. doi: 10.7498/aps.59.4726
    [13] 闫海峰, 俞重远, 田宏达, 刘玉敏, 韩利红. 八角光子晶体光纤传输特性与非线性特性研究. 物理学报, 2010, 59(5): 3273-3277. doi: 10.7498/aps.59.3273
    [14] 任新成, 郭立新. 具有二维fBm特征的分层介质粗糙面电磁散射的特性研究. 物理学报, 2009, 58(3): 1627-1634. doi: 10.7498/aps.58.1627
    [15] 朱叶青, 龙学文, 胡 巍, 曹龙贵, 杨平保, 郭 旗. 非局域程度对向列相液晶中空间光孤子的影响. 物理学报, 2008, 57(4): 2260-2265. doi: 10.7498/aps.57.2260
    [16] 刘炳灿, 潘学琴, 任志明. 非线性系数对超晶格透射的影响. 物理学报, 2006, 55(12): 6595-6599. doi: 10.7498/aps.55.6595
    [17] 吴中, 王奇, 周炯昴, 李春芳, 施解龙. 微波激发下非线性磁表面波传播特性. 物理学报, 2002, 51(7): 1612-1620. doi: 10.7498/aps.51.1612
    [18] 吴中, 王奇. 两种非线性反铁磁晶体交界面上磁表面波的传播特性. 物理学报, 2001, 50(6): 1178-1184. doi: 10.7498/aps.50.1178
    [19] 马大猷. 微扰法求解非线性驻波问题. 物理学报, 1996, 45(5): 796-800. doi: 10.7498/aps.45.796
    [20] 贺凯芬, 胡岗. 微扰法解由正弦波驱动的非线性漂移波的分岔. 物理学报, 1991, 40(12): 1948-1954. doi: 10.7498/aps.40.1948
计量
  • 文章访问数:  4675
  • PDF下载量:  81
  • 被引次数: 0
出版历程
  • 收稿日期:  2021-12-31
  • 修回日期:  2022-04-19
  • 上网日期:  2022-07-29
  • 刊出日期:  2022-08-20

/

返回文章
返回