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Second-order perturbation solution and analysis of nonlinear surface waves

Zeng Sheng-Yang, Jia Lu, Zhang Shu-Zeng, Li Xiong-Bing, Wang Meng
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• 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.

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.

作者及机构信息

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

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).

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

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.