力学进展, 2020, 50(1): 202009-202009 DOI: 10.6052/1000-0992-18-019

有限单元法在超声导波检测技术中的应用

陈洪磊, 刘增华,†, 李子明, 吴斌, 何存富

北京工业大学机械工程与应用电子技术学院, 北京 100124

Application of finite element method in ultrasonic guided waves testing technique

CHEN Honglei, LIU Zenghua,†, LI Ziming, WU Bin, HE Cunfu

College of Mechanical Engineering and Applied Electronics Technology, Beijing University of Technology, Beijing 100124, China

通讯作者: †E-mail:liuzenghua@bjut.edu.cn

责任编辑: 戴兰宏

收稿日期: 2018-08-28   接受日期: 2019-03-24   网络出版日期: 2019-03-29

Corresponding authors: †E-mail:liuzenghua@bjut.edu.cn

Received: 2018-08-28   Accepted: 2019-03-24   Online: 2019-03-29

作者简介 About authors

刘增华,1973年生,2006年于北京工业大学获得机械设计及理论专业博士学位.现任北京工业大学机电学院教授,博士生导师.兼任中国无损检测学会超声检测专业委员会副主任委员;中国仪器仪表学会设备结构健康监测与预警分会常务理事;《无损检测》《北京工业大学学报》(自然科学版)编委.主要研究领域有:实验固体力学、超声无损检测与结构健康监测、机械测试理论与技术、现代传感和信号处理技术等.主持国家和省部级科研项目10余项.合作出版译著1部,在国内外期刊及学术会议上发表和录用学术论文140余篇,获浙江省科学技术奖二等奖一项.

摘要

超声导波检测技术具有对波导结构中的缺陷进行远距离无损检测的能力,多年来一直是无损检测领域关注的热点之一.有限单元法具有对各种复杂动力学问题进行计算的能力,已成为超声导波检测技术研究的重要工具.本文结合超声导波检测技术研究领域中的热点问题,对相关的有限单元法进行了简要综述.介绍了有限单元法的发展及其在多物理场耦合机制下导波的激励与接收、线弹性和黏弹性结构中导波的传播特性、非线性超声导波等多个方面的应用研究情况. 最后,基于超声导波检测技术研究趋势展望了相关有限单元法的未来研究重点和发展方向.

关键词： 有限单元法 ; 数值计算 ; 超声导波 ; 波导 ; 传播特性 ; 无损检测

Abstract

Ultrasonic guided waves have the ability of long-distance nondestructive testing for defects in waveguide structures, and have been one of the hotspots in the field of nondestructive testing for many years. Finite element method (FEM) has the ability to calculate various complex dynamics problems and has become an important tool in the research of ultrasonic guided wave testing technique. Considering the hot issues in the research, a brief review of the relevant FEM is proposed. The development of FEM and its application in the excitation and reception of guided waves under multi-physical coupled field mechanism, the propagation characteristics of guided waves in linear elasticity and viscoelastic structures and nonlinear ultrasonic guided waves are introduced. Finally, the research emphasis and development direction of the relevant FEM in the future is prospected based on the research trend of ultrasonic guided wave testing technique.

Keywords： finite element method ; numerical calculation ; ultrasonic guided waves ; waveguide ; propagation characteristics ; nondestructive testing

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本文引用格式

陈洪磊, 刘增华, 李子明, 吴斌, 何存富. 有限单元法在超声导波检测技术中的应用. 力学进展[J], 2020, 50(1): 202009-202009 DOI:10.6052/1000-0992-18-019

CHEN Honglei, LIU Zenghua, LI Ziming, WU Bin, HE Cunfu. Application of finite element method in ultrasonic guided waves testing technique. Advances in Mechanics[J], 2020, 50(1): 202009-202009 DOI:10.6052/1000-0992-18-019

1 引言

以板壳、管道和杆件等波导结构为基本组成单元的工程设施在航空航天、石油化工、交通运输等领域有着大量的应用. 受原材料特性、加工工艺和工况环境等的影响, 波导结构中会出现夹杂、裂纹、腐蚀等缺陷, 降低结构的力学性能并危及工程安全. 超声导波是一种由纵波和横波等体波在波导结构中经谐振叠加而形成的弹性波, 具有模态多、衰减小、结构截面全分布等特点. 超声导波检测技术采用信号处理算法对检测到的导波信号进行分析, 研究结构缺陷对超声导波传播特性的影响, 建立超声信号声学参数同缺陷类型、大小、位置等之间的关系, 实现波导结构的无损检测(Rose 2011, Zhang et al. 2017, Guan 2017, 何存富等 2001, Mitra & Gopalakrishnan 2016, Willberg et al. 2015). 为了适应工程设施多样化的检测需求, 现代超声导波检测技术发展出了面向结构宏观缺陷的常规超声导波检测和结构早期微观损伤的非线性超声导波检测两大研究方向(Shen 2014, 何存富等 2016, Cawley 2018,刘瑶璐等 2017). 图1 为超声导波结构检测技术原理及相关动力学计算问题示意图. 超声导波结构检测中, 采用分布在结构表面的接触或非接触式传感器进行超声导波信号的激励与接收. 当结构中出现缺陷时, 超声导波会与缺陷发生相互作用, 改变导波传播路径和能量分布, 并形成缺陷散射波. 进而影响检测信号特征, 具体体现为信号幅值、抵达时间、频谱等的变化. 为了应对具体工况环境下结构的超声导波检测需要, 研究中需要考虑温度、结构应力等对超声导波传播特性的影响, 以便在检测中对信号进行修正, 准确提取结构缺陷信息. 可见, 不论是常规超声导波检测技术还是非线性超声导波检测技术的研究, 均面临以下3个重要问题: (1)明确超声导波在具体波导结构中的传播特性(Koshiba et al. 1981, 1984; Ng et al. 2012; Wilcox et al. 2007; Chen et al. 2017; Harding et al. 2006; Thompson et al. 2009); (2) 分析结构缺陷或工况环境对超声导波传播特性的影响; (3) 实现结构中特定模态超声导波的激励与接收(Sébastien et al. 2002). 考虑到超声导波激励和接收机制、波导结构材料属性, 超声导波检测技术研究中的动力学计算问题可大体归结为, 涉及压电-结构、电磁-结构和热-结构等耦合场动力学计算问题, 线弹性和黏弹性结构中的动力学计算问题, 涵盖健康的均质和非均质结构中导波传播特性分析及超声导波与缺陷相互作用特性分析及非线性超声导波动力学计算问题. 其中,许多动力学计算问题难以通过解析技术进行计算,诸如弯曲结构、浇筑结构中的动力学计算、复杂缺陷与导波的相互作用关系等问题的计算(Saravanos et al. 1997, Han et al. 2001). 有限单元法(finite element method, FEM)将求解域离散成众多的小单元进行分析, 可用于复杂边界条件、非均质结构及存在腐蚀、裂纹、材料性能退化等缺陷的结构中动力学问题的计算, 且具有易于编程、成熟的大型商业软件多等优点, 已成为超声导波检测技术研究的重要计算方法(Willberg et al. 2015, Augustyniak & Usarek 2016, Rose 2018, Ong 2018). 但是受制于单元形函数类型、数值积分算法、特殊结构缺陷模型等方面的不足, 常规的FEM在应用于具体的超声导波动力学问题的计算时会存在计算效率低,甚至是无法建模的问题, 特别是针对高频高波数超声导波计算问题、耦合场动力学显式计算问题、非均质结构中的动力学计算问题. 为此, 近几十年来研究人员提出并发展了各种FEM应用于超声导波检测技术的研究.

图1

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图1   超声导波结构检测技术研究中的动力学计算问题示意图


三维弹性波导结构求解域$\Omega $ 中的动力学问题描述方程有波动偏微分方程

$ \sigma _{ij,j} + f_i - \rho \ddot {u}_i - \mu \dot {u}_i = 0$

应变-位移几何方程

$ \varepsilon _{ij} = 0.5\left( {u_{i,j} + u_{j,i} } \right)$

和应力-应变物理本构方程

$ \sigma _{ij} = D_{ijkl} \varepsilon _{kl}$

求解域边界$\Gamma $上的位移和力边界方程

$ \left. \begin{array}{l} u_i = \bar {u}_i \\ \sigma _{ij} n_j = \bar {T}_i \\ \end{array} \right\}$

式中, $\{\sigma_{ij}\} $为应力张量, $\{f_i \}$为体力张量, $\rho $为密度, $\mu $为阻尼系数, $\{u_{i}\}$, $\{\dot {u}_i \}$和$\{\ddot {u}_i \}$分别为位移、速度和加速度矢量, $\{\varepsilon _{ij}\}$为应变张量, $\{D_{ijkl}\}$为材料弹性系数矩阵, $\{n_{i}\}$ 为边界外法方向单位矢量, $\{\bar {T}_{i}\}$ 为边界力. FEM依据求解域空间离散和基于加权余量法或变分原理的波动偏微分方程的弱积分形式建立动力学问题分析数值模型. 此时, 位移$\pmb u$是关于求解域物理空间和时间的四维参量, 离散模型中采用位移插值函数建立求解域局部节点自由度到位移的映射

$ \pmb u=\pmb N a$

式中, $\pmb u$为位移矩阵, $\pmb N$为空间离散插值形函数矩阵, $a$为节点自由度. 加权余量法波动偏微分方程弱积分形式为动力学偏微分方程和边界力方程同具有正交性质的权函数乘积的差. 此时, 波动偏微分方程的求解转化为寻找使方程最小化的位移函数, 即试探函数. 采用统一形式进行权函数和试探函数描述的伽辽金(Galerkin) FEM是目前超声导波检测技术研究中应用较多的一种方法, 其中权函数为位移的变分, 波动偏微分方程的弱积分形式为

$ \int_{\Omega} \delta u_i \left( {\sigma _{ij,j} + f_i - \rho \ddot {u}_i - \mu \dot {u}_i } \right){\rm d} \Omega - \int_{\Gamma} \delta u_i \left( {\sigma _{ij} n_j - \bar {T}_i } \right){\rm d} \Gamma = 0$

式中, $\delta u$为位移变分. 对应力分量$\int_{\Omega} {\delta u_i \sigma _{ij,j} {\rm d} \Omega } $进行分部积分, 并带入应力-应变物理本构方程和位移插值方程(5), 得到FEM动力学问题通用常微分矩阵方程式

$ \pmb M\ddot {\pmb u}(t) +\pmb C\dot {\pmb u}(t) +\pmb K\pmb u(t) = \pmb F(t),\qquad t \in I = \left( {0,t_{\rm e} } \right)$

式中, $\pmb M$为质量矩阵, $\pmb C$为阻尼矩阵, $\pmb K$为刚度矩阵, $\pmb F(t)$为载荷矢量, $t_{\rm e}$为求解时间长度, $I$为持续时长为$t_{\rm e}$的时间求解域. $\pmb M\ddot {\pmb u}$, $\pmb C\dot {\pmb u}(t)$, $\pmb K\pmb u(t)$分别对应惯性力、阻尼力和弹性力. 对应单元的质量矩阵、刚度矩阵、阻尼矩阵及单元载荷矢量为

$ \left. \begin{array}{l} \pmb M^{\rm e} = \int_{\Omega _{\rm e} } \rho\pmb N^{\rm T}\pmb N{\rm d} \Omega \\ \pmb K^{\rm e} = \int_{\Omega _{\rm e} } \pmb B^{\rm T}\pmb D\pmb B{\rm d} \Omega \\ \pmb C^{\rm e} = \int_{\Omega _{\rm e} } \mu\pmb N^{\rm T}\pmb N{\rm d} \Omega \\ \pmb F^{\rm e} = \int_{\Omega _{\rm e} } \pmb N^{\rm T}f{\rm d} \Omega + \int_{ \Gamma_\sigma ^{\rm e} } \pmb N^{\rm T}\bar {\pmb T}{\rm d} \Gamma \\ \end{array} \right\}$

式中, $\pmb M^{\rm e}$, $\pmb c^{\rm e}$, $\pmb K ^{e}$, $\pmb F ^{\rm e}$分别为单元质量矩阵、单元阻尼矩阵、单元刚度矩阵和单元载荷矢量, $\pmb B$为形函数导数矩阵, T为矩阵转置运算符.当所求解的模型为线弹性问题时, 阻尼矩阵$\pmb C=0$.

FEM中用于动力学问题的计算方法主要有直接积分法和振型叠加法两种. 直接积分法利用泰勒(Taylor)展开和差分的思想假定位移、速度和加速度等参量的近似形式, 建立离散时间点上的递推公式, 直接在离散单元层面进行模型积分计算. 依据计算参量与当下时刻的关系, 直接积分法又分为以中心差分法和Newmark法为代表的显式积分法和隐式积分法两类,分别表示为式(9)和式(10). 式(9)中模型$t+\Delta t$时刻的位移可由$t$及$t-\Delta t$时刻的位移和$t$时刻的载荷共同确定. 当$\lambda =1$时, 方程为经典中心差分公式. 式(10)中模型$t+\Delta t$时刻的位移是通过求解$t+\Delta t$时刻的波动方程得到. 其中, $ \gamma $为振幅衰减参数, 一般取值很小, ANSYS软件中$ \gamma $默认设置为0.005. 较大的$ \gamma $取值会使隐式积分计算结果与理论值发生较大偏差. 隐式积分法本身不受时间积分步长的制约, 是一种无条件稳定数值计算方法. 但为了得到与显式算法同样的精度, 隐式积分法中的时间积分步长通常不能大于显式积分法的时间积分步长(Dauksher & Emery 1999). 与此同时, 隐式积分法在求解位移时需要进行刚度矩阵求逆运算, 增加了计算量; 在涉及非线性问题时, 结构刚度矩阵、质量矩阵和阻尼矩阵均可能随时间发生变化, 此时只能采用显式积分法进行计算. 振型叠加法属于频域计算方法, 首先利用系统固有振型将方程组转换为互不耦合的方程组进行解析或数值积分得到每个振型的响应, 然后将各振型的响应进行叠加得到系统的响应. 研究表明直接积分和振型叠加法在低阶超声波场的计算上具有相似的效果, 当求解域中关注波的频率成分较集中时, 推荐采用直接积分方法(Bathe & Wilson 1973, Sun et al. 2000). 因此, 在超声导波检测技术研究中, 特别是在进行超声导波与缺陷相互作用关系的动力学问题计算中多采用显式时间积分算法进行FEM计算. 但该积分法在运用中对时间积分步长和单元矩阵形式均有严格的要求. 显式积分法运用中的质量矩阵为便于求逆运算的对角矩阵, 时间积分步长需要满足库朗-弗里德里希斯-列维(Courant-Friedrich-Levy, CFL)条件, 即保证单位时间步长内波不会传播经过一个最小单元宽度

$ \left( {\dfrac{\lambda }{\Delta t^2}\pmb M + \dfrac{\lambda }{2\Delta t}\pmb C} \right)u_{t + \Delta t} =\pmb F_t - \left( {\pmb K - \dfrac{2\lambda }{\Delta t^2}\pmb M} \right)u_t - \left( {\dfrac{\lambda }{\Delta t^2}\pmb M - \dfrac{1}{2\Delta t}\pmb C} \right)u_{t - \Delta t}$

其中, $\lambda $为加速度调节常数.

$ \left( {\pmb K + \dfrac{1}{\alpha _1 \Delta t^2}\pmb M + \dfrac{\delta }{\alpha \Delta t}\pmb C} \right)u_{t + \Delta t} = \pmb F_{t + \Delta t} + \left[\dfrac{1}{\alpha _1 \Delta t^2}u_t + \dfrac{1}{\alpha _1 \Delta t}\dot { u}_t + \left( {\dfrac{1}{2\alpha _1 } - 1} \right)\ddot {u}_t \right]\pmb M + \\ \left[\dfrac{\delta }{\alpha _1 \Delta t}u_t + \dfrac{\delta }{\alpha _1 }\dot { u}_t + \delta \left( {\dfrac{1}{2\alpha _1 } - 1} \right)\Delta t \cdot \ddot {u}_t \right]\pmb C$

其中, $\alpha _{1}$和$\delta $分别为积分精度和稳定性控制参数, 不同取值代表不同的积分方案. $\delta =0.5+ \gamma $, $\alpha \geq 0.25(1+ \gamma )^{2}$.

最初关于FEM的研究和学术著作中用于构建空间离散单元的形函数多为具有一阶连续偏导性质 的等间距$C_{0}$型拉格朗日(Lagrange)多项式. 该类FEM被称为$h$-FEM (Hayashi & Endoh 2000),亦是目前商用有限单元软件所采用的主流FEM类型. 在动力学问题计算中, FEM为了获得理想的计算结果, 单元网格的边长比例最好接近为1、单位波长下单元的个数不低于10 (Bayliss 1985, Alleyne & Cawley 1991). 否则, 计算中可能会出现网格频散(Lee & Andreas 1992, Dauksher & Emery 1999, Guddati & Yue 2004)、幅值频散、各向异性、吉伯斯现象、伪振荡等错误(Düster et al. 2001). 在有限单元积分算法方面, 由于高斯积分法在对$n$点多项式进行积分计算时可以得到$2n-1$阶的精度, 所以$h$-FEM多采用高斯积分法进行单元形函数积分. 但该方法的计算效率低、内存消耗较大; 与此同时生成的单元矩阵为协调矩阵, 无法采用显式积分法对动力学问题进行计算. 为此, 人们从模型空间离散精度、单元积分算法等方面出发, 提出了不同的方法进行FEM优化. 常规FEM模型空间离散精度的优化主要有$h$型优化或$p$型优化, 即减小单元网格尺寸或提升单元形函数的阶数. 但该方法会增加计算模型的自由度, 降低了计算效率. $h$-FEM在单元积分算法优化方面主要有算法本身优化和单元矩阵对角化方法改进两个方面. Jensen (1996)通过对拉格朗日单元数值积分点进行调整的方法实现质量矩阵对角化. Guddati和Yue (2004)将二维模型中双线性四边形单元的积分点由传统高斯或高斯-洛巴托积分点偏移到$\left( {\pm \sqrt {2 /3} ,\pm \sqrt {2/3} } \right)$点, 优化有限元和无限元的计算效率. Ainsworth和Wajid (2010)采用新型的非标准求积算法组合质量矩阵和刚度矩阵代替高斯积分准则, 提高数值积分的执行效率. Idesman等(2011)在分析线性波动理论时通过将积分点调整到正交位置, 并结合质量矩阵平均法实现数值频散抑制. 由于方程单元形函数积分算法的优化是在单元级上进行的, 具有对单元扭曲不敏感的特点, 在规则均匀单元和不规则扭曲单元中均具有较好的应用效果, 而且修改数值积分方式不会改变FEM计算的收敛率. 集中质量矩阵将单元质量集中至节点自由度, 并独立于单元形函数, 运用中不改变计算的收敛, 是时域显式积分算法运用的基础. Ainsworth和Wajid (2010)采用平均质量法构建的集中质量矩阵代替协调质量矩阵, 以优化一维结构中谐波传播问题的计算. 但该方法不适用于二维和三维模型动力学问题的求解. Yue和Guddati (2005)通过修改质量矩阵和刚度矩阵积分方法的方式改善二维波场计算中的数值频散问题; 相对于平均质量矩阵法, 该方法将相速度的计算精度提升了2个数量级. Marfurt (1984)采用协调质量矩阵和集中质量矩阵加权的方式组建混合质量矩阵, 以平衡质量集中矩阵计算中节点误差和协调质量矩阵计算效率低的不足. Seriani和Oliveira (2007)对混合质量矩阵构建方法进行了改进. 但混合质量矩阵构建需要对方程组进行求解. 虽然合理的选择网格类型、大小及积分类型和时间积分步长能够抑制高波数仿真中的频散, 但无法消除频散(Babuška et al. 1997a, 1997b; Babuška & Sauter 1997; Ihlenburg & Babuška 1995, 1997). 然而, 在动力学问题计算中对单元形函数积分算法进行修改可能会在计算中引入伪高频振荡, 而调整质量矩阵表达形式则可能引入污染错误. 综上可知, 就计算算法本身而言, $h$-FEM在动力学问题的计算中存在诸多问题, 需要发展新型的FEM以提升具体问题的计算效率.

波导结构及导波的波结构相对于传统超声波具有一定的特殊性和复杂性, 因此在采用FEM进行相关动力学问题的计算时不仅需要遵循传统超声动力学计算的稳定性条件, 还面临着自身存在的问题. 依据质点振动形式的差异, 导波分为不同的模态. 板壳中的导波主要为兰姆波(Lamb waves), 有反对称模态(anti-symmetric modes, A)和对称模态(symmetric modes, S)两种; 管道中的超声导波主要有纵向模态(longitudinal modes, L)、弯曲模态(flexural modes, F)和扭转模态(torsional modes, T). 以板中兰姆波为例, 图2 为3 mm厚铝板中兰姆波频散曲线. 其中, 图2(a) 为群速度频散曲线, 图2(b) 为波长频散曲线. 如图所示, 各模态导波均具有不同阶数的子模态类型, 且均表现出频散特性, 即速度随频率发生变化; 而各模态导波的波长是频率的函数, 且随着频率的增加而快速降低.图3 为兰姆波质点振形和波结构示意图. 其中, 图3(a)和图3(b)分别为A模态和S模态的振动形式图. 如图所示板壳中A模态和S模态导波在结构上下面的位移形式分别以结构中轴线呈反对称和对称关系; 图3(c)和图3(d)分别为频率140 kHz的 A$_{0}$模态和S$_{0}$模态在3 mm厚铝板中的波结构图, 兰姆波的面内位移和离面位移在整个板壳横截面的分布形式随厚度发生变化. 不同模态中面内位移和离面位移的相对大小不同, A模态中位移以离面位移为主, 而S模态的位移以面内位移为主. 受超声导波的波结构的影响, 采用FEM进行相关动力学问题计算时, 相同网格离散形式对不同模态导波的计算效果影响的存在差异, 不同模态导波对单元扭曲的敏感程度不同. 例如,相对于兰姆波S模态, A模态对单元扭曲和单元长短边的比值更为敏感. 采用$h$-FEM在薄壁板结构厚度方向进行粗糙离散会使数值计算中出现单元锁定, 即刚度效应, 降低兰姆波计算精度. 采用缩减单元积分可减少单元锁定对计算精度的影响, 而低积分法又可能使刚度矩阵出现秩亏, 引起沙漏问题, 即零能模式, 此时需要采用沙漏控制技术保障计算的有效性(Flanagan & Belytschko 1981, Belytschko et al. 1984, Willberg et al. 2015). 缩减积分仅提升了兰姆波A模态的计算精度, 对S模态的改善效果并不理想; 而沙漏控制的不足会降低A模态和S模态的计算精度.

图2

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图2   3 mm厚铝板中兰姆波频散曲线. (a) 群速度频散曲线, (b) 波长频散曲线


图3

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图3   兰姆波质点振形和波结构示意图. (a) A模态质点振形, (b) S模态质点振形, (c) 3 mm 厚铝板中频率为140 kHz的 A$_{0}$模态波结构示意图, (d) 3 mm 厚铝板中频率为140 kHz 的S$_{0}$模态波结构示意图


此外, 在具体的超声导波检测技术研究中涉及的动力学问题会更为复杂. 在耦合场动力学问题的计算方面, $h$-FEM多采用隐式积分算法进行分析, 计算效率低. 由于$h$-FEM采用低阶拉格朗日函数构建的有限单元对单元网格扭曲具有较高的敏感性,在包含裂纹、夹杂或异质材料的波导结构中的动力学计算中, $h$-FEM需要足够细的网格对裂纹类强间断区域进行处理, 以保证计算精度; 然而在模型网格处理中, 不同网格离散区域交界处的网格粗细的差异不易过大, 否则在计算中会形成虚假波(Casadei & Ruzzene 2012, Wagner & Liu 2003). 在夹杂问题计算中, 有限单元的边需要位于夹杂与基体的界面处, 大大增加了模型网格离散难度. 超声导波检测技术研究中为验证导波的远场、大范围和多类型缺陷的检测特性,模型在导波传播方向上的尺寸常常处于几十到上千波长的等级, 模型较大, 增加了内存消耗和计算时间成本. 为满足波长与单元尺寸比和时间积分稳定条件, 基于低阶插值形函数的$h$-FEM在应用中很容易达到极限, 出现模型大、耗时多的问题, 特别是在三维波动问题及复杂非线性问题研究中, 该问题会更加突出(Satyarnarayan et al. 2008). 因此, 为实现耦合场和不同结构、材料属性下超声导波检测技术研究中动力学问题的高效计算, 发展和探索新的FEM已成为超声导波检测研究中特别需要关注的问题.

鉴于常规$h$-FEM算法本身在动力学问题计算中存在的局限性、波导结构和超声导波的特殊性和复杂性,以及现有超声导波检测技术研究中对数值计算高效性和复杂问题的适用性的要求, 本文结合超声导波检测技术研究中关注的动力学问题, 介绍了有限单元法的发展及其在导波检测中的典型应用. 第2节基于单元形函数、时间域谱处理方法和数值模型构建方式, 介绍了导波检测研究中现有的典型有限单元计算方法; 第3节介绍了有限单元法对超声导波检测技术研究中的具体问题的处理方式及研究结果; 最后, 展望了超声导波检测技术研究相关有限单元法的未来发展趋势.

2 有限单元法

本节从超声导波波场的高效计算和非均匀复杂波导中导波波场的计算两个方面出发, 介绍近年来应用于超声导波检测技术研究的几类有限单元法. 依据数值模型中形函数构建形式、时间域处理方式、特殊单元建模方式等, 将有限单元法分为高阶有限单元法、时间域谱有限单元法、波有限单元法、有限胞元法和多尺度有限单元法等. 最后, 简要介绍了FEM和其他计算方法组合进行超声导波与结构缺陷相互作用的研究.

2.1 高阶有限单元法

相对于$h$-FEM, 高阶有限单元法采用高阶正交多项式构建单元形函数. 相同自由度数目下的高阶有限元比低阶有限元有更高的计算精确度(Mulder 1996), 而且高阶单元较低阶单元可以在较粗网格情况下获得相同等级的精度. 目前高阶有限元形函数主要包括: (1) 基于勒让德(Legendre)多项式归一化积分构建的阶谱积分函数(Duczek & Gabbert 2013), 称为$p$-FEM. 勒让德多项式归一化积分函数和勒让德多项式形函数见式(11)和式(12). (2) 定义在高斯-洛巴托-勒让德(Gauss-Lobatto-Legendre, GLL) (Schmicker 2011, Kudela & Ostachowicz 2009)或切比雪夫-高斯-洛巴托(Chebyshev-Gauss-Lobatto, CGL)网格节点(Komatitsch et al. 2009, Moll et al. 2010, Rucka 2010, Sridhar et al. 2006)上的高阶拉格朗日多项式或高斯-勒让德节点上的切比雪夫多项式(Dauksher & Emery 1999), 即谱元法(spectral element method, SEM). 拉格朗日形函数和GLL网格点见式(13)和式(14). (3) 非均匀有理B样条(non-uniform rational B-splines, NURBS)形函数(Kumar & Pandit 2012, Li & Chen 2014, Mitra & Gopalakrishnan 2005, Ostachowicz 2008, Lucena 2016), 即N-FEM. 对应插值形函数和基函数见式(16)和式(17). 相较于$C_{0}$型低阶拉格朗日插值形函数, 高阶谱单元中的节点参数不一定都代表节点函数值. $p$-FEM中低阶形函数是高阶形函数的子集, 前面形成的单元刚度矩阵是阶数增加后单元刚度矩阵的子块, 原有低阶单元构造的质量矩阵和刚度矩阵可以继续使用,可以有效降低网格划分对计算结果的影响, 大大节省编程和运算时间. SEM的基本理论由Patera (1984)在进行流体动力学分析时提出, 所采用的形函数是近似解展开成的正交多项式, 具有质量矩阵对角化的优势, 又称为空间域谱元法, 其理论的发展依赖于函数插值和正交多项式理论的研究. NURBS是为了弥补有限元和计算机辅助设计工具之间的缝隙而提出的一种形函数, 之后被用于动力学问题的计算(Hughes 2005). 表1 为高阶形函数特性对比表. $p$-FEM和SEM中单元标准定义域为$[-1, 1]$, 单元插值节点为定义域上一类适当选取的正交多项式的零点, 坐标满足强制条件以保证单元展开式具有$C_{0}$连续性; 而N-FEM标准单元域定义在[0, 0.5]或[0.5, 1]上, 单元间连续性为$C_{p - 1}$, 即单元间的连续性随着形函数阶数的提高而增加; $p$-FEM和SEM中相邻单元自由度均为1, 而N-FEM中相邻单元自由度与形函数阶数相关

$ \Phi _n \left( \xi \right) = \sqrt {\dfrac{2n - 1}{2}} \int_{x = - 1}^\xi {L_{n - 1} \left( x \right){\rm d}x} = \dfrac{1}{\sqrt {4n - 2} }\left[ {L_n \left( \xi \right) - L_{n - 2} \left( \xi \right)} \right],\qquad n = 2,3, \cdots$
$ N_n^{\rm Legendre} \left( \xi \right) = \Phi _{n - 1} \left( \xi \right),\qquad n = 3,4, \cdots ,p - 1$

式中, $L_{n}$为勒让德多项式, $\xi $为积分节点; $\Phi $为修正勒让德多项式, 当$\xi = \mp 1$时$\Phi =0$; $N_n^{\rm Legendre} $为勒让德多项式形函数; $n$为多项式阶数, 对应的最大取值为 $p-1$.

$ N_n^{{\rm Legendre},p} \left( \xi \right) = \prod\limits_{j = 1,i \ne n}^{p + 1} {\dfrac{\xi - \xi _j }{\xi _n - \xi _j }} ,\qquad n = 1,2, \cdots ,p + 1$
$ \xi _i = \left\{ \begin{array}{ll} - 1 ,&\quad i = 1\\ \xi _{0,i - 1}^{Lo,p - 1},&\quad 2 \leq i < p + 1\\ + 1,&\quad i = p + 1 \\ \end{array} \right.$

式中, $N_n^{{\rm Legendre},p} $为$p$阶拉格朗日形函数, $\xi _{0,i - 1}^{Lo,p - 1} $为$p-1$阶洛巴托多项式(15)的根

$ Lo_{p - 1} \left( \xi \right) = \dfrac{1}{2^pp!}\dfrac{{\rm d}^{p + 1}}{{\rm d}\xi ^{p + 1}}\left[ {\left( {\xi ^2 - 1} \right)^p} \right]$
$ N_i^{{\rm NURBS},p} = \dfrac{R_{i,p} \left( \xi \right)w_i }{\sum\limits_{j = 1}^{n_{\rm cont} } {R_{j,p} \left( \xi \right)w_j } },\qquad i = 1,2, \cdots ,p + 1$
$R_{i,p} \left( \xi \right) = \dfrac{\xi - \xi _i }{\xi _{i + p} - \xi _i }R_{i,p - 1} \left( \xi \right) + \dfrac{\xi _{i + p + 1} - \xi }{\xi _{i + p + 1} - \xi _{i + 1} }R_{i + 1,p - 1} \left( \xi \right)$

式中, $R_{i, p}$为基函数, $w_{i}$为基函数权重, $i$为基函数编号.

$p$-FEM和SEM具有和$h$-FEM相似的收敛特性, 而N-FEM具有更高的收敛率. 在动力学问题的计算中, 通常采用$5\sim 10$ 阶 多 项 式 构 造 SEM 中 形 函 数 (Komatitsch & Vilot 1998). 与此同时SEM的单元密度可以在积分点之间改变, 能够用于全异质结构体中弹性波问题的分析(Komatitsch & Tromp 1999). 相对于拉格朗日形函数, NURBS形函数的阶数可以达到100以上, 但在超声波场计算中NURBS形函数的阶数不要高于6. 在中低频超声导波的波场计算中, 六阶NURBS形函数具有和$p$型优化拉格朗日形函数具有相同的收敛率; 在高频导波分析中, 相同阶数下的NURBS形函数较拉格朗日形函数拥有更快的收敛率(Gravenkamp et al. 2017). $p$-FEM中阶谱形函数可以通过提升网格密度或单元形函数阶数的方式增加模型空间离散精度, 然而这些形函数会形成完全填充的质量矩阵, 制约了显式积分算法在数值计算中的应用. 然而, 相对于$p$-FEM和N-FEM, 基于GLL网格的节点通过GLL积分法连接的SEM中的质量矩阵具有自对角化的特点, 能够改善数值计算中的舍入误差, 使得显式积分算法能够直接运用于动力学问题的计算. Dauksher和Emery (1999)研究发现基于切比雪夫形函数的SEM较基于拉格朗日多项式的$p$-FEM在波动问题计算中具有更高的计算效率. Willberg等(2012)采用单位波长8个节点的SEM实现高阶兰姆波A$_{1}$模态的精确计算. Ostachowicz 等(2012)提出时域谱有限单元(time-domain spectral finite element method, TDSFEM)用于波动问题分析, 采用显式积分算法实现导波传播问题的快速计算. Willberg等(2015)对比介绍了高阶有限单元、时域和频域有限单元在导波结构检测研究中的应用.

表1   高阶形函数特性对比表(Willberg et al. 2012)

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高阶有限单元法同时具有通过$h$型优化和$p$型优化改善数值模型计算效率的能力,又被称为hp-FEM. hp型优化是同时提升模型网格密度和形函数的阶数实现数值计算优化(Cottrell et al. 2007), 数值计算中具有比$p$-FEM和$h$-FEM更快的收敛率(Evans et al. 2009). 美国内华达大学(University of Nevada)、捷克西波希米亚大学(University of West Bohemia)和捷克共和国科学院(Academy of Sciences of the Czech Republic)联合开发了hp-FEM开源软件包, 用于多物理场偏微分方程求解(Chleboun & Solin 2013, Solin & Korous 2012, Solín et al. 2008). Cantwell等(2015)介绍了一种开源hp有限元框架软件 Nektar++. 但采用网格细化或高阶有限单元均是通过增加模型空间离散度的方式进行数值计算优化, 增加了计算时间成本和内存消耗. 在hp-FEM数值计算稳定性研究方面, Ainsworth (2004)给出了单元阶数对离散频散的显式表达关系式.

2.2 时间域谱有限单元法

时间域谱有限单元法(time-domain spectral finite element method, TDSFEM)是一种将波动偏微分方程在时间域上进行展开, 进而进行动力学问题数值求解的一种方法. 该方法的理论基础是波动偏微分方程在时间和空间上的导数所具有的线性特征. Narayanan 和Beskos (1978)最早进行偏微分方程时间域谱分析技术研究, Doyle (1997)将该技术命名为谱元法. 与基于形函数的SEM不同, 该方法在时间域进行, 又称为时间域谱元法. 基于谱展开采用的基函数的差别, 时间域谱单元法有傅里叶谱有限单元法(Fourier spectral finite element method, FSFEM)和小波谱有限单元法(wavelet spectral finite elements method, WSFEM)两类. FSFEM采用快速傅里叶变换将偏微分方程转化为以频率为参数的常微分方程; 通过对常微分方程进行求解, 得到用于谱单元空间离散表达的插值形函数. 之后由计算得到的动态刚度矩阵确定系统传递函数, 即频域响应函数. 最后采用快速傅里叶逆变换对传递函数与载荷的卷积结果进行处理, 得到结构中载荷的历史时间响应(Doyle 1988). WSFEM在时域内采用小波函数将偏微分方程转换为空间域内的常微分方程. 与FSFEM不同, 所构建的常微分方程是关于不同小波时间尺度的耦合常微分方程. 需要通过特征值分解技术将耦合方程解耦, 然后求解非耦合常微分方程的通解, 得到模型空间离散形函数. 随后采用和FSFEM类似的处理方式进行模型计算. WSFEM中耦合常微分方程的解耦分析会增加数值计算的时间成本, 但因为它与具体问题无关, 所以可以事先进行计算和存储. 在计算中根据需要再添加进模型中, 从而改善WSFEM的计算效率. 由于很难通过连续小波变换采用的广义积分运算进行数值求解, 所以WSFEM均基于离散小波进行建模, 其中应用较多的是拥有不同紧支撑特性的多贝西小波(Daubechies, db)和正交紧支撑特性的哈儿(Haar)小波. Hong和Kennett (2002), Kumar和Pandit (2012), Li和Chen (2014)等对小波数值分析方法进行较详细的介绍. 时域谱有限单元法中的系统传递矩阵是反问题研究中的主要成分之一, 在反问题的研究中具有天然的优势. 相对于$h$-FEM, 基于FSFEM中形函数构造的质量矩阵和动态刚度矩阵更为精确. 常微分通解构建的插值形函数能够克服常规FEM无法通过网格细化消除的吉伯斯现象(Ham & Bathe 2012).

FSFEM受制于有限时间窗特性, 在进行短波导结构中超声导波波场的计算时, 不能保证在短时间尺度内形成的短程力与多次反射波的乘积在选定的时间窗内消失, 致使响应中出现环绕问题(Mitra & Gopalakrishnan 2005), 此时需要采用小时间积分步长进行波动问题计算, 以避免环绕错误造成的时域响应扭曲. 相对于WSFEM, 基于紧支撑小波的WSFEM能够对强梯度或振动区域的偏微分方程进行准确、稳定的表征. Mitra和Gopalakrishnan (2005)利用基于db小波的WSFEM分析一维波导结构中冲击力激励的超声导波的传播特性. 在有限域内采用外推技术对自适应小波进行处理, 以消除FSFEM中周期边界假设而引起的环绕问题. 但在针对A$_{0}$模态与各向同性和各项异性板结构中分层缺陷的相互作用的研究显示, 时间域谱有限元无法直接应用于复杂边界条件和带有裂纹、夹渣等不规则结构中超声导波的计算(Ng et al. 2012), 此时需要建立对应的缺陷单元进行相应波动力学问题的计算. Ostachowicz (2008)基于FSFEM分析铁木辛柯梁结构中兰姆波同裂纹的相互作用. Lucena和Santos (2016)结合时间反转算法和FSFEM对杆件缺陷进行检测研究. Yang等(2016)采用B样条小波有限单元法分析膜性结构中波的传播特性. Joglekar和Mitra (2016)利用db小波有限元分析梁中弯曲波同呼吸裂纹的非线性作用.

2.3 波有限单元法

波有限单元法(wave finite element method, WFEM)用于对均匀或周期性结构及缓慢变化的非均匀结构进行动力学问题分析. 建模的基本理论是基于Mead (1975a, 1975b)对周期性结构中波动问题的研究, 周期性结构耦合界面处的波数目是界面最小自由数目的两倍, 可以分解为前向和后向传播的两个波成分; 波传播经过周期性结构中单个区段时的位移和作用力满足布洛赫理论(Bloch theory), 式(18). 即区段前后端的位移和力大小为传播常数$\lambda $的函数. 图4 为波有限单元法原理示意图. 分析中首先将整体波导结构分成长度为$d$的不同区段进行建模, 区段相连界面的节点分布相同, 并假设波导壁结构具有无限大的阻抗值. 各区段对应的有限元计算方程和角频率下的谐波方程见式(19)和式(20). 导波求解中通过对各区段动态刚度矩阵进行排列组合得到求解域的传递矩阵, 通过求解传递矩阵的特征值问题得到代表自由波传递的波数、模态和波形等(Duhamel et al. 2006). 相对于常规FEM, 对于具有周期性特点的结构, WFEM能够通过最小重复单元的建模实现整个模型的分析, 极大降低了分析模型的维度. 在WFEM研究中可以利用结构截止频率对应的波形进行坐标转换矩阵设置, 进行WFEM优化. Droz等(2014)通过将截面传递矩阵映射到一组简化的和波传播有关的形函数以提升WFEM分析效率, 实现夹层复合材料梁和流体弹性管道中导波频散曲线的快速计算. 但该方法对所选择的波有很强的依赖性. Zhou等(2015)基于相邻区段模态振型设置传递函数, 进行WFEM优化. 然而该方法不适用于区段中存在局部阻尼器或压电元件的情况. Yu等(2017)通过缩减模型和改进特征值计算方法优化WFEM, 实现周期性压电结构中波场的计算. WFEM可单独用于周期性结构的分析也可和常规FEM结合应用于一维周期结构(Zhou et al. 2015)、二维各向同性结构(Mace & Manconi 2008)、截面弯曲波导结构(Ichchou et al. 2007)中导波波场的运算. Kessentini等(2016)采用WFEM分析充水管中泄漏导波模态的传播和衰减. Chronopoulos (2018)运用WFEM分析铝和分层复合材料梁中导波与局部非均匀区域和局部非弹性区域相互作用形成的非线性特征.

图4

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图4   波有限单元法原理示意图


$ \left. \begin{array}{l} \pmb u_{\rm r} = \lambda \pmb u_{\rm l} \\ \pmb f_{\rm r} = - \lambda \pmb f_{\rm l} \\ \end{array} \right\}$

式中, $\lambda ={\rm e}^{ -{\rm j}kd}$为传播常量, 表征波由左向右传播过一个区段时幅值和相位的变化, $k$为波数, $d$为传播距离, 此分析中等于区段长度.

$ \pmb M\left[ \begin{array}{l} \ddot {\pmb u}_{\rm l} \\ \ddot {\pmb u}_{\rm r} \\ \end{array} \right] +\pmb C\left[ \begin{array}{l} \dot {\pmb u}_{\rm l} \\ \dot {\pmb u}_{\rm r} \\ \end{array} \right] +\pmb K\left[ {{\begin{array}{l} \pmb u_{\rm l} \\ \pmb u_{\rm r} \\ \end{array} }} \right] = \left[ {{\begin{array}{l} \pmb f_{\rm l} \\ \pmb f_{\rm r} \\ \end{array} }} \right]$
$ \left[ \begin{array}{*{20}c} {\pmb D_{\rm ll} } & \pmb D_{\rm lr} \\ {\pmb D_{\rm rr} } & \pmb D_{\rm rl} \\ \end{array} \right]\left[ \begin{array}{*{20}c} \pmb u_{\rm l} \\ \pmb u_{\rm r} \\ \end{array} \right] = \left[ {{\begin{array}{*{20}c} \pmb f_{\rm l} \\ \pmb f_{\rm r} \\ \end{array} }} \right]$

式中, $\pmb f$为应力矢量, l和 r分别为左右边界标识符, $\pmb D$为动态刚度矩阵, 等于 $-\omega^{2}\pmb M+{\rm j}\omega \pmb C+\pmb K$.

WFEM中的数值计算问题除常规FEM的模型离散误差、周期结构伪缺陷问题外, 还存在动态刚度矩阵惯性项截断及数值条件和特征值问题. WFEM模型是集成的离散弹簧-质量结构, 在波传播方向存在空间周期性, 当区段长度和波长相当时, 会出现周期结构现象(Mead 1996), 即波模态和频率是传播常数的周期函数, 此时会形成频率混叠效应(Ichchou et al. 2007). 而较大的区段长度将限制波数有效域, 形成频带滤波效应, 引起折叠失真. 当波传播入新的区段时, 新区段的传播模态和前一区段模态在进行能量转换时产生的耦合效应可能激起波的振动, 引发奇异变形. 同时受周期性结构通带和阻带特性的影响, 只有特定范围内的频率能够自由传播. 数值计算中, 如果刚度矩阵相对于惯性项过大, 则会在计算中产生惯性项截断, 降低计算精度. 此时可以通过对过长区段进行网格重新划分或采用高精度的计算方法减少截断错误. 对于区段长度不是很大的情况, 可以将原有区段进行串联形成一个超级区段, 以降低刚度项和惯性项间的差距. 该方法的优点是不需要对区段进行重新建模, 新的全局刚度和质量矩阵可以由原始区段组合形成. 数值计算中对于由给定的特征值求解特征频率的情况, 通常不会发生特殊的数值状况. 而对于给定特征频率求解特征值的情况, 特别是当特征值很小时则可能发生病态数值条件, 此时需要在矩阵计算中引入奇异值分解方法克服该问题. Waki等(2009)对WFEM数值计算问题及处理方式进行了较详细的介绍.

2.4 有限胞元法

有限胞元法(finite cell method, FCM)结合了虚拟域方法和FEM进行复杂几何结构处理, 可以自适应性地对模型进行网格划分, 有效避免了常规FEM中网络离散困难的问题, 能够方便地对泡沫结构、复合材料等具有复杂微结构的对象进行建模. FCM最初由Parvizian et al. (2007)提出, 之后被用于线性热弹问题(Zander et al. 2012), 骨骼结构(Ruess et al. 2012), 弹塑性问题(Abedian et al. 2014), 损伤机制(Ranjbar et al. 2014)等方面的研究. 图5 为有限胞元法原理示意图. 计算中将包含复杂微结构的求解域视为物理域; 将物理域外的规则区域视为虚拟域; 通过虚拟域方法的运用, 将包含复杂微结构的物理域嵌入虚拟域中, 形成扩展域, 随后基于FEM对扩展域进行离散. 模型处理上, 采用非常软的软材料属性进行虚拟域材料属性的设置, 以保证原始物理域和扩展域中解的应变能保持不变. 通常虚拟域方法中结构的质量和刚度矩阵为零(Allaire et al. 2003). FCM不需要确定物理边界, 对应胞元矩阵的积分不具有连续性, 因此需要采用不连续数值积分技术对胞元矩阵进行计算. 但是不连续函数的积分计算较复杂模型网格生成任务更为便捷, 可以执行自动运算, 能够减少用户操作对数值模型计算的干扰. Duczek等(2014)融合FCM和SEM提出谱胞元法(spectral cell method, SCM), 用于分析非均匀材质和复杂几何结构中兰姆波的传播问题. SCM在弹性动力学迦辽金形式变分方程中引入了指示函数(indicator function) $\varsigma $对扩展域中的求解域进行标记, 见式(21)和式(22). 式(22)中指示函数大小控制参数$q$的取值控制在$5\sim 10$之间, 以保证虚拟域中指示函数的值足够小,以避免构建方程出现病态问题; Parvizian et al. (2007)在基于$h$-和$p$-FEM扩展的FCM研究中令$q=10$以避免全局刚度矩阵的病态. SEM中的质量矩阵为集中质量矩阵, 可以保证数值计算的精度和效率. 然而, 虚拟域中胞元同几何边界相交, 致使SCM模型中的质量矩阵不具有对角性. 针对SCM中GLL积分无法实现截断胞元的质量矩阵对角化的问题, Joulaian等(2014)采用截断胞元质量对标准质量矩阵进行对角线缩放尺度处理, 构造截断胞元集中质量矩阵, 进行非均质材料中波场显式动力学计算.

图5

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图5   有限胞元法原理示意图. (a) 物理域$\Omega $, (b) 虚拟域$\Omega _{\rm e}/ \Omega $, (c) 扩展域, (d) FCM网格划分和指示函数


$ \int_{\Omega} {\varsigma \delta u_i \left( {\sigma _{ij,j} + f_i - \rho \ddot {\pmb u}_i - \mu \dot {\pmb u}_i } \right){\rm d} \Omega } - \int_{\Gamma} {\varsigma \delta u_i \left( {\sigma _{ij} n_j - \bar {T}_i } \right){\rm d} \Gamma } = 0$
$ \varsigma (x) = \left\{ \begin{array}{ll} 1, \quad & \forall {x} \in \Omega \\ \varsigma _0 = 10^{ - q} , \quad & \forall {x} \in {\Omega _{\rm e} } /\Omega \\ \end{array} \right.$

式中, $\varsigma $为指示函数, $q$为指示函数控制参数.

在具体应用中,FCM可以和$h$-FEM, $p$-FEM组合进行数值模型设计. 在对常规平滑问题进行计算时, FCM和$h$-FEM的计算收敛率和$h$-FEM一致, FCM和$p$-FEM组合模型计算的收敛率则呈指数形式, 而且能对奇异问题实现较高精度的计算. 基于虚拟域方法的FCM在边界条件实施方面和FEM一样自然满足均匀纽曼(Neumann)边界条件; 但在不均匀纽曼边界条件下, 需要对牵引力作用表面进行参数化描述. 受有限单元内部不连续性的影响, 胞元中截断单元的质量矩阵和刚度矩阵只能采用不连续数值积分技术进行计算. 当胞元的边界同物理域边界不重合时, 需要采用分布式拉格朗日法(distributed Lagrangian)或罚函数法使界面能量平衡(Parvizian et al. 2007). 为了利用显式积分算法进行FCM模型计算, 需要探索和发展各种胞元质量矩阵对角化技术, 实现波动问题的显式计算.

2.5 多尺度有限单元法

多尺度有限单元法(multiscale finite element method, MsFEM)将多尺度特征并入形函数的定义, 而非采用高阶多项式进行形函数优化. MsFEM是建立在Babuška和Osborn (1983)提出而后经Strouboulis等(2000)进一步发展的广义有限单元法(generalized finite element method, GFEM)的基础上. 该方法的提出最初是为了进行存在快速振荡系数的椭圆问题的分析. MsFEM通过对一定边界条件下, 均匀微尺度问题的求解, 实现单元基函数的调整. 计算中微尺度信息通过全局刚度矩阵耦合, 被引入宏观尺度. 多尺度单元(multiscale-elements, MSEs)将精细尺度融合到单元形函数中, 处理胞元内几何和材料的不连续性. 缺陷周围区域节点的位移通过插值进行确定, 即单元内精细尺度网格的自由度和粗尺度网格的自由度通过多尺度插值插值函数进行映射(Casadei et al. 2016), 见式(23). 图6 为多尺度方法单元处理原理示意图, 其中全局节点为矩形, 多尺度单元1和2中的精细尺度离散节点为圆点. 通过一系列经过辅助细网格离散得到的有限数目节点进行MSE宏观自由度描述, 其中辅助网格离散对应的局部自由度是宏观尺度自由度的子集. MSEs中的精细尺度网格相关的局部自由度不参与粗网格的宏观计算, 精细尺度对应的多尺度形函数仅用于计算MSE的质量和刚度矩阵. 模型计算中, 首先在分析预处理阶段计算得到单元质量和刚度矩阵, 将MSEs组装到有限元模型中进行求解. MsFEM可用于构建嵌入式缺陷多尺度单元数据库, 所构建模型不需要进行局部网格细划, 亦不会对不规则缺陷周围单元产生网格扭曲. Casadei 等(2013)将MSEs的思想同MsFEM方法的数值形函数的概念相结合, 提出了几何多尺度有限单元法(geometric multiscale finite element method, GMsFEM), 用于三维非均匀材料中弹性波动问题的计算, 其中详细介绍了线弹性固体中多尺度形函数及相关单元矩阵的构建过程. 为使局部自由度沿MSE边沿呈线性化, Vernerey和Kabiri (2012)运用拉格朗日乘法器处理局部自由度和MSE形函数的兼容性问题. Capuano等(2018)运用单元法向振动模态形函数对GMsFEM单元进行优化, 计算一维杆和二维板中超声导波的波场. 但该方法的计算精度对选择的优化模态具有依赖性. Tchalla等(2013)采用Python脚本和FORTRAN子程序开发了一套用于ABAQUS软件的MsFEM工具包, 用于计算各向同性介质中的线性和非线性力学问题.

$\pmb u =\pmb N\pmb c \to \left[ {{\begin{array}{*{20}c} \pmb c \\ \pmb w \\ \end{array} }} \right] = \left[ {{\begin{array}{*{20}c} {\pmb I_{n_c } } \\ {\pmb E} \\ \end{array} }} \right]\pmb c = \\ \left[ {{\begin{array}{*{20}c} {\pmb I_{n_c } } \\ { - \pmb K_{ww}^{ - 1} \left[ {\pmb K_{cw}^{\rm T} + \pmb B_w^{\rm T} \left( {\pmb B_w\pmb K_{ww}^{ - 1} \pmb B_w^{\rm T} } \right)^{ - 1}\left( {\pmb B_c - \pmb B_w \pmb K_{ww}^{ - 1} \pmb B_{cw}^{\rm T} } \right)} \right]} \\ \end{array} }} \right]\pmb c$

式中, $\pmb u$为精细尺度网格自由度, $\pmb c$为粗尺度网格自由度, $w$为未在粗尺度网格中显示的局部精细自由度, $\pmb I$ 为$n_{c}\times n_{c}$单位矩阵, $n_{c}$ 为粗自由度数目, $\pmb E$为粗尺度网格自由度$\pmb c$到局部精细自由度$w$的映射矩阵.

图6

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图6   多尺度方法单元处理原理示意图(Casadei et al. 2013)


2.6 有限单元组合计算方法

为充分利用FEM对复杂求解域的适用性和其他计算方法的高效性, 发展出了多种有限单元组合计算方法用于超声导波与结构缺陷相互作用关系的计算. 有限单元组合计算方法主要有不同FEM间或FEM和其他数值计算方法组合进行导波动力学问题的计算和FEM和解析计算方法组合两大类.

不同FEM间或FEM同其他数值计算方法组合进行导波动力学问题的计算, 在该类方法中主要利用FEM对复杂求解域进行数值计算, 而后将复杂求解域计算结果作为边界条件引入其他FEM或数值计算方法, 最终实现整个模型中导波波场的计算. Liu GR等(1991a, 1991b)提出混合数值法(hybrid numerical method, HNM)用于计算各向异性复合材料板中瞬态波的传播问题. 在WFEM和FEM组合数值计算方面. Loveday (2007, 2008)采用WFEM和传统FEM分别对无限等横截面波导和压电传感器进行建模, 研究压电激励的超声波导传播特性. Renno和Mace (2013)融合FEM和WFEM对三维结构中联合点处兰姆波的散射和透射特性进行研究. Fan等(2016)分别采用FEM和WFEM分析组合结构中导波的近场和远场特性. Benmeddour等(2011)采用半解析有限单元法(semi-analytical finite element mothod, SAFEM)与FEM混合的方法研究实心柱体结构中纵向模态、扭转模态和弯曲模态与非轴对称裂纹的相互作用关系, 采用三维FEM模型对缺陷近场区域进行分析, 将计算结果得到的导波模态分解到进口横截面和出口横截面, 随后使用SAFEM将FEM计算得到的导波模态映射到二维模型进行计算. Liu GR (2002)采用FEM各向异性层合板中含裂纹、夹渣区域进行建模, 采用有限条元法对非缺陷半无限区域进行建模, 研究缺陷的弹性波散射特性. 有限条元区域与FEM区域均采用二阶形函数进行模型离散, 以保证两区域交接处的兼容性. Duan和Kirby (2015)采用迭代加权残差法构建SAFE-FE公式, 研究导波与非对称缺陷的相互作用. 通过伽辽金法进行波动方程求解, 避免了基于虚功和变分原理推导SAFE-FE公式中对全局矩阵的乘法操作. 此外, Shen和Cesnik (2016)将FEM和局部相互作用仿真法 (local interaction simulation approach, LISA)结合, 实现FEM对局部压电-结构耦合计算的精确性和LISA显式算法高效性的有效融合, 对复杂复合材料板结构中兰姆波的形成、传播及其与缺陷的相互作用进行研究. 该混合方法中采用FEM模型计算传感器的压电效应, 由LISA对非压电耦合作用区域中波的传播及波与缺陷间的相互作用进行求解.

FEM与解析技术组合进行超声导波动力学问题计算的典型应用有全局-局部法(global-local methods, GLM). 相较于FEM同其他数值计算方法组合的计算技术, GLM中同样采用FEM进行模型中复杂区域的计算, 随后将FEM计算得到的局部缺陷散射场带入解析技术进行计算, 得到整个求解域中波场. Koshiba 和Morita (1981, 1984)分析二维不连续介质中SH模态传播问题和S模态与楔形内部裂纹的相互作用, 由二阶三角形单元对结构缺陷区域进行建模. 发现表面裂纹的楔尖角对S模态反射系数影响较小, 随着楔尖角的增加, 反射系数减少. Gresil和Giurgiutiu (2013)将全局解析计算和局部FEM分析结合计算导波与缺陷的相互作用, FEM部分基于ABAQUS软件, 采用显式动力学算法进行分析. Bhuiyan等(2016)分析兰姆波与含裂纹缺陷过孔的相互作用关系. 采用频域有限元技术计算兰姆波与局部缺陷的相互作用, 得到兰姆波与缺陷的相互作用系数, 再由解析技术对有限元计算得到的系数矩阵进行处理, 得到在板中缺陷散射场的远场分布. Bhuiyan等(2017)、Shen和Giurgiutiu (2016)在局部区域FEM计算兰姆波与板中凹槽及铆钉孔裂纹的相互作用, 解析技术计算其余区域波动传播问题. 解析求解的模态节点应力通过FEM边沿处的边界积分作用到FEM区域. Bhuiyan等(2017)分析不同入射角度下钻孔边沿的裂纹缺陷的兰姆波散射场, 观察到兰姆波S模态、A模态同缺陷相互作用形成的SH模态.

3 典型应用实例

本节分别从超声导波激励与接收技术、导波传播特性分析及非线性超声导波检测技术三个方面出发, 介绍了耦合场超声导波检测技术研究、复杂线弹性和黏弹性波导结构中的导波传播特性分析和非线性超声导波检测技术研究中的有限单元法及相关研究成果.

3.1 耦合场中超声导波

3.1.1 压电-结构耦合

压电元件具有较高的机电转换效率, 在超声导波检测中可贴覆于结构表面或埋置于结构内部用于超声导波信号的激励与接收. FEM中的压电单元融合了机械自由度和电自由度, 进行压电材料的机械-电耦合特性表征(Benjeddou 2000, Gaudenzi & Bathe 1995, Devonshire 1954, Willberg & Gabbert 2012, Ghandi & Hagood 1997), Wang J等(1999, 2000, 2008)和Chen YY等(2018)基于FEM和高阶Mindlin板理论分析了电极效应、温度效应和高频非线性效应等对压电材料板振动的影响. 压电-结构耦合分析中模型应力-应变物理本构方程和静力平衡方程受压电材料特性的影响. 应力-应变物理本构方程和压电本构方程为

$ \left. \begin{array}{*{20}c} \pmb \sigma = \pmb D \pmb \varepsilon - \pmb {e}^{\rm T} \pmb E \\ \pmb u_{\rm e} = \pmb e \pmb \varepsilon + \pmb D_{\rm e} \pmb E \\ \end{array} \right\}$

式中, $\pmb e$为压电应力系数矩阵, $\pmb D_{\rm e} $为零应变条件下的介电常数矩阵, $\pmb u_{\rm e} $为电位移矢量, $\pmb E$为电场矢量,是电势$\varphi $的函数, $\pmb E = - \nabla \varphi $. 基于Hamilton变分原理和位移插值方程,得到压电-结构耦合动力学常微分矩阵方程(Avdiaj et al. 2009)

$ \left[ {{\begin{array}{*{20}c} \pmb M &\pmb 0 \\ \pmb 0 & \pmb 0 \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} \ddot {\pmb u} \\ \ddot {\varphi } \\ \end{array} }} \right] + \left[ {{\begin{array}{*{20}c} \pmb C &\pmb 0 \\ \pmb 0 &\pmb 0 \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} \dot {\pmb u} \\ \dot {\varphi } \\ \end{array} }} \right] + \left[ {{\begin{array}{*{20}c} \pmb K & {\pmb K^{\rm z}} \\ {\left( {\pmb K^{\rm z}} \right)^{\rm T}} & \pmb K \\ \end{array} }} \right]\left[ {{\begin{array}{*{20}c} \pmb u \\ \varphi \\ \end{array} }} \right] = \left[ {{\begin{array}{*{20}c} \pmb F \\ \pmb V \\ \end{array} }} \right]$

式中, $\varphi $为电势, $\pmb V$为压电元件两端电荷矢量, $\pmb K ^{\rm z}$为压电耦合矩阵. 现有超声导波检测技术研究中, 基于压电元件的耦合场计算问题主要集中于以下两个方面: 压电元件激励和接收超声导波动力学问题的高效计算方法, 压电元件与波导结构的粘接效果对超声导波检测的影响.

压电元件激励和接收超声导波动力学问题的高效计算方法研究方面, $h$-FEM采用隐式算法或将隐式积分算法与显式积分算法融合的方式进行压电-结构耦合场动力学问题的计算. Liu 和Giurgiutiu (2007)采用软件ANSYS中的耦合场方法分析压电元件直接贴覆在梁结构上时的机电阻抗效应和兰姆波激励特性, 研究梁中A$_{0}$模态和S$_{0}$模态与裂纹的相互作用关系. Soorgee等(2013)基于ABAQUS/co-simulation仿真, 分别采用隐式瞬态分析和显式分析方法对压电纤维复合材料传感器部分和结构中导波激励、传播和接收特性进行分析. 研究显示压电纤维复合材料传感器适用于做激励器, 不适用于接收器. 相对于压电位移参量, 面内应变和离面应变的线性组形式, 见式(26), 能够更好的表征压电元件检测的电压信号特征. 压电-结构耦合场动力学问题的高效FEM计算方法研究方面. Ha和Chang (2010)采用二节点高斯正交(two-point-Gauss quadrature)构造厚度方向单元形函数, 在平面方向采用传统谱单元构建混合谱单元进行压电传感器高频导波激励仿真建模, 实现压电-结构耦合场的显式分析. Jang等(2014)和Lee等(2013)采用FSFEM进行复合材料梁中压电元件激励高频导波特性研究. 鉴于薄板中S模态和A模态振动受剪切变形影响较弱的特点, 在厚度方向采用两节点线性插值函数进行插值. 依据明德林-赫尔曼(Mindlin-Herrmann)杆理论设置横向收缩效应以增强复合材料梁轴向变形. 研究观察到横向收缩效应对S$_{0}$模态的影响大于对A$_{0}$模态的影响, 有效约束力的增加随着压电层厚度的增加而减低. Kim等(2008)采用SEM研究了压电元件嵌入结构时的兰姆波激励、接收效果. Kalkowski等(2016)构建压电半解析单元用于模拟压电超声激励, 研究中考虑了传感器的动态特性, 但忽略剪切压电系数. Rekatsinas和Saravanos (2017)基于多物理泛义分层理论提出一种新的时域谱方法, 实现复合材料板中压电传感器兰姆波激励特性的显式积分运算. 由三阶埃尔米特(Hermite)样条函数近似表征厚度方向位移和电势. 观察到正交碳纤维/树脂多层板中波指向性及粘结层对压电元件高频导波激励的影响.

$ s_v =e_{31} \varepsilon _x + e_{33} \varepsilon _z$

式中, $s_{v}$为检测信号, $e_{31}$和$e_{33}$为压电应力系数.

压电元件与波导结构的粘接效果对超声导波检测的影响方面. Lanzara等(2009)采用SEM分析粘接层材料退化及不完全黏贴对压电元件激励导波的影响. 压电片脱粘截面行为和悬臂梁类似, 在粘接结构边缘处应力奇异点上的应力集中会引起能量损失, 严重降低激励信号幅值. 随着应力波长接近于或小于传感器尺寸, 粘结层的截面特征将会严重影响压电元件的响应. 部分脱粘会降低接收信号幅值, 而脱粘面积的增加会增加相位的延迟; 粘结界面外围的脱粘会使信号能量的降低最大. Ha和Chang (2010)采用混合谱元法(hybrid spectral element method, HSEM) 研究粘结层对压电元件兰姆波激励特性的影响. 研究显示检测信号幅值的改变主要受压电元件共振效应和剪切滞后效应的影响. 如果粘结层厚度大于一定的值, 剪力滞后效应将会大于共振效应对信号检测产生的影响. 随着粘结剪切模量的降低, 共振频率向低频偏移; 如果粘结剪切模量足够低、粘结层足够厚, 即使处于压电元件共振频率下, 粘结层的剪切滞回效应仍会对检测信号起主要影响. 随粘结层厚度的变化变化,压电片直径-厚度比对能量转换效应的影响形式也会发生改变. 对于40 $\mu $m粘结层厚度, 受剪力滞后效应的影响, 100 kHz信号幅值随粘结层厚度的增加而降低; 对于500 kHz信号, 受共振效应的影响, 在60 $\mu $m粘结层厚度时, 检测信号的幅值达到最大, 随后开始下降. Agrahari和Kapuria (2016)采用ABAQUS标准求解器分析粘接层、结构、压电元件厚度对时间反转信号相似度的影响. 不同粘结层厚度下获得最佳时反重构信号所需要的信号中心频率不同. 在较厚粘结层下, 特定频率检测信号重构结果同激励信号的相似性指数表现出多样性, 不一定随激励粘结层厚度的增加而降低. 同时观察到较厚压电元件激励的高频信号存在扭曲, 所以推荐在基于时间反转法的超声导波结构检测中使用薄的压电元件进行信号激励与接收. Ren和Lissenden (2018)构建由电力驱动电路、压电单元、线弹性波导三个部分组成的全耦合有限元模型. 研究长方形压电元件高阶兰姆波激励调制特性, 分析压电元件大小、板厚、激励信号周期对导波激励的影响.研究指出通过合理选择压电片的几何参数可以实现高阶兰姆波模态的激励, 较宽的压电单元可实现高纯度模态的激励. 在压电-结构耦合场FEM模型简化研究方面, 研究人员分析了应力或位移加载模型对数值计算效果的影响. Chaudhry和Rogers (1994)、 Nieuwenhuis等(2005)发现压电传感器理想粘结条件下结构厚度与压电片厚度比大于6时, 基于集中力载荷加载的超声导波波场的计算结果与实验结果具有较好的一致性. 翻边圆形压电元件激励导波周向分布的均匀性受激励频率及翻边影响较大, 随着激励频率的增加, 均匀性降低. FEM分析中需要采用集中力方式模拟翻边处压电应变引起的振动(Moll et al. 2012). Khalili和Cawley (2015)在FEM模型中, 采用集中力阵列模拟斜入射压电传感器的兰姆波激励, 研究高频厚积情况下单模态兰姆波的激励方法.

3.1.2 电-磁-结构耦合

电磁声传感器(electromagnetic acoustic transducer, EMAT)基于电磁结构耦合机制实现超声导波的激励和接收. 当交变电流通过置于铁磁性材料近表面的线圈时, 线圈会在铁磁性材料表面趋肤层产生涡流和动磁场, 而涡流与动、静磁场相互作用产生洛伦兹力, 铁磁性材料中磁畴结构受动、静磁场作用形成磁致伸缩力、磁化力. 洛伦兹力、磁致伸缩力和磁化力在结构中耦合形成应力波源, 实现导波激励; 结构中的导波带动磁畴结构运动切割动静磁场时, 会在导线中形成感应电流, 实现导波检测(Remo et al. 2010, 刘增华 等 2017). Thompson (1973)和Kawashima (1976)于20世纪70年代最先开始EMAT的数值计算研究, 之后FEM在EMAT优化设计中得到大量研究与应用(Jia et al. 2017; Xie et al. 2016; Seung et al. 2016; Liu ZH et al. 2010, 2015, 2016, 2018a, 2018b). 现有电-磁-结构耦合导波检测研究多采用基于$h$-FEM的商用软件进行, 主要研究包括电-磁-结构耦合场超声导波高效计算问题及材料电磁性能对EMAT换能机制的影响两个方面.

电-磁-结构耦合场超声导波的有限单元法高效计算问题研究方面, Oh等(2013)采用线性化时域-谐波分析法仿真强偏置静磁场作用下由薄磁致伸缩片和励磁线圈组成传感器的导波激励特性. 首先分别基于线性化的磁场-应变曲线方程和电磁转换理算计算得到结构中的磁致伸缩应变和动静磁场, 然后将计算得到的磁致伸缩应变和动静磁场带入时域-谐波弹性动力学方程进行计算. 由于磁线圈提供的磁场在磁致伸缩片中不是均匀分布的, 所以仿真中对标准线性磁致伸缩方程进行修改以计算线性化磁致伸缩耦合矩阵. 研究显示, 受磁致伸缩片中动磁场椭圆分布特性的影响, 激励的超声波场呈辐射模式. Su等(2014)采用分步仿真模型进行EMAT超声计算研究. 首先采用时域谐波分析法进行模型求解, 得到水平偏置磁场下EMAT的动、静磁场、洛伦兹力和磁致伸缩力, 将计算的两种力以体力的形式加载到结构动力学计算模型中, 而磁致伸缩应变以初始应变的形式作用于波场中. 研究显示水平磁场作用下的低碳钢中磁致伸缩力是兰姆波形成的主要机制. 针对COMSOL隐式求解器在大体积、小网格模型中高频导波计算模型计算存在的收敛速度慢的问题, Dhayalan和Balasubramaniam (2010)结合COMSOL电磁结构耦合模型和ABAQUS显式弹性动力学模型分析EMAT在铝板中激励超声波的传播特性. 首先由COMSOL仿真得到EMAT在弹性体中产生的洛伦兹力, 之后将求解结果以边界条件的形式加载到ABAQUS显式动力学分析模型, 进行兰姆波传播特性分析. 研究显示超声导波与缺陷相互作用形成的模态转换信号的幅值随着缺陷深度的增加而增加. Wang S等(2014)构建多物理场洛伦兹力机制折线圈电磁超声换能器激励-接收系统研究电磁噪声对EMAT性能的影响. 采用高斯随机函数设置随机空间和时间分布的环境电磁噪声. 环境电磁场$\pmb B _{\rm E}$在试件中引起的电流噪声密度$\pmb J _{\rm N}$和自由导电电流密度$\pmb J _{\rm L}$共同影响磁性矢量, 进而影响接收线圈接收电压, 见式(27). 分析了PCB技术和漆包线包裹在胶木板骨架两种情况下回折线圈的导波接收性能. 在固定导线厚度时PCB线圈中感生的电压强度随线宽的增加降低, 在固定线宽下感生电压强度随导线厚度的增加降低; 最佳的峰-峰值感应电压和信噪比感生电压信号对应的PCB线圈数存在差异; 骨架线圈中感应电压随线圈数目的增加而线性增加直至达到稳定; 单层骨架线圈比多层线圈具有更好的检测性能. Ribichini等(2012)采用COMSOL多物理场分析法向偏置磁场EMAT在不同铁磁性材料特性假设下洛伦兹力和磁致伸缩力成分的比重及其影响因素. 研究指出Ogi (1997)所提磁致伸缩力学边界条件将磁致伸缩力视为同洛伦兹力一样为不受表面牵引力影响的体力, 致使所提磁致伸缩力方程在应用时需要人为在结构表面引入牵引力, 而传统动态切向电磁场边界方程适用于均匀介质表面情况, 无法描述铁磁性材料与空气接触面存在感生涡流和磁场的情况. 作者基于惯性力方程和铁磁材料表面微变形假设条件得到磁致伸缩力修正式(28); 通过磁成像技术对空气与铁磁性材料表面电磁现象进行观测和EMAT线圈电流密度和动磁场感 生涡流一致的假设提出切向磁场边界修正式(29). 基于修正的力学和磁学边界条件进行FEM模型设置, 运用磁致伸缩本构方程代替等效力进行模型计算. 由于涡流集肤深度之外区域的动磁场十分微弱, 研究中仅对线圈下会引起涡流的区域采用全磁致伸缩本构方程进行建模, 其余区域采用弹性本构方程进行建模. Xiao等(2016)采用SEM计算得到模型标准特征矩阵, 依据式(28)和式(29)设置模型力学和磁学边界条件,计算无限功能梯度磁电弹性板中导波的频散曲线.

$ \left. \begin{array}{l} \pmb V = - l\sum\limits_{i = 1}^n \left( \dfrac{\partial \pmb A}{\partial t} \right)_i \\ - \dfrac{1}{\mu }\nabla ^2\pmb A + \sigma \dfrac{\partial \pmb A}{\partial t} = \pmb J_{\rm L} + \pmb J_{\rm N} \\ \pmb J_{\rm L} =\sigma\pmb V\times\pmb B_{\rm S}\\ \pmb \nabla \times\pmb J_{\rm N} = - \sigma \dfrac{\partial \pmb B_{\rm E} }{\partial t} \end{array} \right\}$

式中, $\pmb V$为接收线圈中的总电压, $\pmb J _{\rm L}$为静磁场$\pmb B _{\rm S}$作用下结构振动为$\pmb V$时线圈中的感应电流密度, $\pmb B _{\rm E}$为环境电磁场, $\pmb J _{N}$为环境电磁场引起的噪声电流密度, $\pmb A$ 为电流密度相关的磁矢, $\mu $为结构磁导率, $\pmb E_{i}$为第$i$根接收线圈对应导线感生的电场, $l$为导线长度.

$ f_1 = \dfrac{3\varepsilon _{\rm t} \Omega ^2\rho }{{ i}k}\dfrac{\tilde {H}_1 }{\bar {H}_3 }$
$ \tilde {H}_1 \left( {x_3 = 0} \right) = J$

式中, $ f_{\rm 1}$为磁致伸缩力, $\varepsilon _{\rm t}$为磁致伸缩应变, $\tilde {H}_1$ 为切向动磁场强度, $\bar {H}_3 $为法向静磁场强度, $k$为波数, $J$为单位长度电流密度. 低碳钢中$\bar {H}_3=5 $kA/m, $\varepsilon _{\rm t}=2.0\times 10^{-6}$.

超声导波检测中材料电磁性能对EMAT工作机制的影响方面, Remo等(2010)定量分析了EMAT在铁素体材料中的换能机制及磁致伸缩耦合常数、磁导率、偏置静磁场、驱动电流强度和频率对激励超声信号的影响, 证明材料磁特性及其与频率关系的不确定会使计算结果偏离实验值. 验证表明磁致伸缩EMAT灵敏性不仅是磁致伸缩耦合常数的函数, 也受磁导率的影响. 在偏置磁场大于动磁场时, 传感器灵敏性正比于动磁场和电流强度. Ribichini等(2011)研究发现磁致伸缩力激励超声信号幅值和涡流集肤深度与磁致伸缩应变乘积成正比, 与材料表面应力状态、磁场强度呈高度的非线性关系, 电导率和磁导率的降低会增加其工作效率;洛伦兹力大小与材料电导率、相对磁导率呈线性关系, 正比于涡流深度积分. 涡流集肤深度远小于波长时, 洛伦兹力工作效能可视为与肌肤深度和电导率无关. 此时, 洛伦兹力和磁致伸缩力分别正比于总灌电流和动磁场的积分. Ribichini等(2012)采用COMSOL多物理场分析法向偏置磁场EMAT在不同铁磁性材料特性假设下洛伦兹力和磁致伸缩力成分的比重及其影响因素. 研究显示应力自由表面建模不合理会引起磁致伸缩效应被高估, 而电磁边界设置错误会引起垂直偏置磁场下铁磁性材料中洛伦兹力和磁致伸缩力在换能中的作用强度被错误评估. Rouge等(2014)当检测线圈产生的动磁场是相对于永磁体产生的静磁场不可忽视时, 洛伦兹力的频率会是激励线圈加载电流频率的2倍; 推导出磁致伸缩机制EMAT和洛伦兹力EMAT中磁致伸缩两种响应表达式的统一形式; 给出磁致伸缩力同磁致伸缩应变的关系. 在高非线性磁致伸缩应变条件下, 磁致伸缩效应产生信号频谱包含有检测线圈加载电流的多个高阶谐波成分, 且谐波成分能量与动磁场相关. Ashigwuike等(2015)采用COMSOL软件建立EMAT有限元模型, 研究动态洛伦兹力、静态洛伦兹力和磁致伸缩力对超声导波激励位移强度的影响.

3.1.3 热-结构耦合

激光超声检测具有空间分辨率高、信号频带宽的特点, 被用于各项同性黏弹性结构(Sun et al. 2009)、各项异性结构(Song et al. 2013)中的疲劳裂纹(Yang & Ume 2008, Pei et al. 2012)、表面微缺陷(Guo et al. 2017)、残余应力 (Zhan et al. 2017)等缺陷的检测.超声导波检测技术研究中的热-结构耦合问题主要涉及两种情况, 一种为温度影响结构材料属性, 改变导波传播特性(Dhayalan et al. 2018); 另一种为激光超声检测技术涉及的热弹激光超声激励. 前一种情况在FEM分析可以通过设置对应温度下结构的材料属性, 研究温度对导波检测技术的影响. 分析过程简单, 后文将不做赘述. 激光束照射到材料表面时, 脉冲激光的热能被材料吸收后以热波的形式传入材料内部, 当热膨胀急剧变化时, 会在结构中形成弹性波源, 实现激光超声激励. 此时, 式(7)中的载荷矢量激光热弹形成的与温度相关的体力

$ \pmb F = \int_{\Omega _{\rm e} } {\pmb B^{\rm T}\pmb E\pmb \varepsilon _0 {\rm d} \Omega }$

式中, $\pmb \varepsilon _{0}$为热应变矢量, $\pmb E$为材料矩阵.由于激光脉冲的上升时间通常在纳秒级别, FEM在进行热-结构动力学问题的计算时,时间积分步长设置为$1/(180\times f_{\max})$以获得足够的时间分辨率, 但会极大增加计算成本. 为此, 研究人员探索了不同的有限单元计算方法以提升热-结构动力学耦合问题的计算效率.

FEM中常采用热弹区域和机械区域分别进行建模的方式来降低模型计算的复杂度, 提升激光超声动力学模型的计算效率. Lee和Burger (1995) 采用FEM激光激励兰姆波模型中的温度、位移、频谱和等效体力随时间的变化. 对热作用区域和力学区域采用不同的网格划分模式, 依据机械区域的网格尺寸设置时间步长. 通过狄拉克(Dirac)函数将热弹力施加到机械区域中进行后续FEM计算, 见式(31). 研究显示仅采用水平体力、忽略垂直体力成分的方式无法准确模拟激光超声信号激励过程; 激光热源影响区域虽然较小, 但热弹区域的网格粗化会使计算出现低能量的高阶模态. Liu和Hong (2015) 通过将模型分为机械区域和热力区域, 计算非聚焦和线聚焦激光脉冲激励下形成的激光兰姆波. 较全耦合模型, 该方法的计算效率提升了近6.3倍. 研究显示缺陷的出现会使A$_{0}$和S$_{0}$模态的抵达时间出现延迟. Liu P等 (2016) 基于COMSOL多物理场, 以热流的形式将空间呈高斯分布、时域呈脉冲分布的激光束作用于结构表面. 模型分为热热力区域和机械区域, 热区域呈规则矩形状态. 采用四面体多尺度网格划分技术进行模型离散; 通过双圆交叉确定裂纹短轴距离, 再通过两边延伸的方式构建疲劳裂纹, 满足导波传播过程中裂纹的呼吸变换; 依据机械区域动力平衡条件设置时间积分步长. 许伯强 (2014)采用应力-位移混合有限元分析技术进行激光超声计算. 采用傅里叶变换将时域线弹性材料中激光超声激励控制方程转换到频域, 由拉伸函数将微分关系变化到频率, 得到包含完美匹配层的频域控制方程, 再由傅里叶逆变换将包含完美匹配层的频域控制方程组变换到时域进行求解, 消除边界反射波, 降低模型尺寸并节省计算资源, 准确模拟了激光超声波在材料内的产生和传播特征. Xu等 (2004)考虑温度对材料热力参数影响, 研究瞬态温度场在铝板的超声波激励特性. 在薄板中激励的兰姆波包含典型的A$_{0}$和S$_{0}$模态, 随着板厚的增加, 激光激励信号中出现了高阶模态导波; 材料随温度变化见式 (32). 基于相同的材料属性随温度的变化情况, Soltani和Akbareian (2014)在激光超声检测研究中观察到, 激光脉冲的热作用使结构表面温度快速升高, 之后由于热传导的原因而缓慢冷却. 随着深度的增加, 温度下降的速度增快;激光照射参数, 表面积、脉冲持续时间对超声波有严重的影响. 仅增加激光脉冲持续时间或线宽参数时, 激励的信号均表现出波形逐渐变宽、能量不断降低的特点. 此外, 针对激光超声模拟存在的低精度、低计算效率、缺乏专业的激光源力学模型的问题, Zhan等(2016)在激光超声材料弹性常数测量研究中采用等效载荷法构建激光超声激励模型, 见式(33). 采用时间和空间呈高斯分布的脉冲形式的力学模型来模拟激光超声激励源, 引入校正系数对激光参数同载荷大小的等效关系进行修正.

$ \left. \begin{array}{l} F_{\rm t\ast } =F_{\rm t} \delta \left( {x - x_0 } \right)\delta \left( {y - y_0 } \right)\delta \left( {z - z_0 } \right) \\ F_{\rm m} = \int_{\Omega _{\rm e} } {\pmb N^{\rm T}F_{\rm t\ast } {\rm d} \Omega } \\ \end{array} \right\}$

式中, $x_{0}$, $y_{0}$, $z_{0}$分别为热域中网格节点坐标, $F_{\rm t}$为热域力, $F_{\rm m}$为机械力, $\delta $为狄拉克函数.

$ \left. \begin{array}{l} A(t) = 5.2\times 10^{ - 2} + 3.0\times 10^{ - 5}\left( {T - 300} \right)\\ \rho = - 0.22T + 2769,\qquad 300 \leq T \leq T_{\rm m}\\ C_{\rm p} (t) = \left\{ \begin{array}{ll} 3.971T, \quad & T < 200\\ 780.45 - 0.488T, \quad & 200 \leq T \leq T_{\rm m}\\ \end{array}\right.\\ k(t) = \left\{ \begin{array}{ll} 292.6, \quad & T < 200\\ 249.45 - 0.085T, \quad & 200 < T < 730\\ 198.47 - 0.014T, \quad & 730 < T_{\rm m} \\ \end{array} \right. \end{array} \right\}$

式中, $T$为开尔文温度, $T_{\rm m}$为熔点温度, $A$ 为光吸收率, $k$为热导率(单位: W·m$^{ - 1}$ $\cdot$ K$^{ - 1})$, $C_{\rm p}$ 为比热(单位: J·k·g$^{ - 1}$ $\cdot$ K$^{ - 1})$.

$ \left. \begin{array}{l} P = \alpha P_{\max} f(r)g(t)\\ P_{\max} = \varPhi \tau ^{ - \frac{1}{8}}\lambda ^{ - \frac{1}{4}}I_0^{\frac{3}{4}}\\ f(r) = \dfrac{1}{\sqrt {2\pi }}\dfrac{2}{r_0^2 }{\rm e}^{\frac{ - 2r^2}{r_0^2 }} \\ g(t) = \dfrac{8t^3}{\tau ^4}{\rm e}^{\frac{ - 2t^2}{\tau ^2}} \end{array} \right\}$

式中, $P_{\max}$, $f$, $g$分别为载荷峰值132 MPa、信号空间分布和时间分布, $\alpha $ 为校正系数, $\varphi $为材料相关系数 $2.3\times 10^{ - 5}$, $I_{0}$为能量密度峰值3.2 MW/cm2, $t$为脉冲作用时间, $\lambda $为激光波长, $\tau $为激光脉冲持续时间, $r_{0}$为高斯光束半径, $r$为高斯半径.

3.2 线弹性结构中的超声导波

线弹性材料为工程设施中波导结构的主要材料类型. 对于规则板壳、管道和杆件中的超声导波波场的计算问题可以采用解析技术获得高效求解, 但对于各向异性波导结构、弯曲波导结构、非均质结构、浇筑组合结构中的超声导波的动力学问题的计算, 则需要采用具有复杂求解域适用性的FEM进行计算. 针对导波结构本身和结构缺陷几何特征两个方面的复杂性对超声导波动力学计算问题的影响, 从板壳和管道两种基本导波结构类型出发, 介绍了FEM在线弹性复杂波导结构超声导波检测技术研究中的应用及研究情况.

复杂板壳结构中超声导波传播特性研究方面, Terrien等(2007)基于线性插值形函数构建FEM模型对航空结构中点腐蚀缺陷区域进行建模, 由模态分解法研究2.25 MHz兰姆波与缺陷相互作用形成的反射、透射特征. Peng等(2009)采用SEM分析铝板中兰姆波与结构穿透裂纹的相互作用. Moreau等(2011)基于FEM计算各向同性板中异形穿透腔体中的兰姆波散射场. Casadei等(2014, 2016)采用GMsFEM研究三维板中椭圆形穿透缺陷的SH$_{0}$, S$_{0}$模态散射场和非均匀周期固体结构中波的传播特性. Galán和Abascal (2002)将混合边界元和FEM组合, 用于计算各向同性夹芯板中障碍物与兰姆波相互作用形成的散射. Hosseini等 (2013)分别采用二维双线性三角形壳单元和恒定截面线性直桁架单元构建金属泡沫夹芯板中的闭合泡沫模型和开孔泡沫模型, 由隐式时间积分算法对数值模型进行计算, 得到结构中兰姆波波场. 对计算结果分析观察到闭合和开口单元中的导波传播特性具有相似形. Hosseini和Gabbert (2013)采用三维实体单元和二维壳单元对板皮和蜂窝单元进行建模, 研究蜂窝夹层板中夹板厚度、蜂窝厚度、蜂窝芯尺寸及激励信号频率对导波传播特性的影响. 通过ANSYS软件中自带的多点约束方程实现板皮和蜂窝单元连接区域应力和变形的准确建模(Surana 1980). Sikdar和Banerjee (2016)基于Soorgee等(2013)所提仿真技术研究纤维增强蜂窝复合夹层结构中高密度核结构对压电激励导波传播特性的影响. 观察到结构中出现的高密度核结构会降低兰姆波的群速度, 并增加了兰姆波的衰减率.在各向异性板壳结构中的超声导波传播特性的研究方面, Basri和Chiu (2004)研究了兰姆波与正交各向同性复合材料板中局部弹性刚度退化的相互作用, S$_{0}$模态较A$_{0}$模态对凹槽和局部弹性模量缩减模拟的非穿透刚度退化具有更高的灵敏性. 对于局部穿透弹性模量缩减模拟的缺陷, 当局部弹性模量为$0.2E$时, 即会发生S$_{0}$到A$_{0}$的模态转换现象, 反射信号出现延迟; 对于结构贯穿截面发生材料退化的情况, 入射波没有发生模态转换. Chakraborty和Gopalakrishnan (2003)基于一阶剪切变形理论设计了FSFEM谱单元, 用于功能梯度梁中导波的计算. Hedayatrasa等(2014)采用基于二维时域谱有限单元法分析功能梯度材料中兰姆波的传播特性, 通过单层六节点的五阶切比雪夫多项式构建的谱单元进行材料厚度方向的离散单元; 提取S模态和A模态的波结构, 通过零交叉技术和小波信号频谱分解技术计算导波相速度、群速度和波结构.

复杂管道和杆件中超声导波的传播特性研究方面, Zhu (2002)采用FEM分析时间延迟周期传感器环阵在各向同性钢道中激励低频L(0,1)和L(0,2)模态. 采用管壁切向力模拟以剪切振动为主的压电片环阵列进行L(0,2)模态激励. 观察到导波反射系数随腐蚀深度和周向展开分别呈非线性和呈线性变化, 而腐蚀缺陷的锐边会增强反射信号强度. 同时观察到不同频率下L(0,2)模态的主振动位移特性, 300 kHz以下时以非频散面内位移为主, 在$300\sim 700$ kHz区间以高频散离面位移为主. L(0,2)模态经过直-弯-直管道中的弯曲部分时会发生模态转换, 在后续直管道部分形成F(1,3)和L(0,2)模态. 导波通过弯曲部分时的传递系数和管道几何、弯曲处直径和弯曲长度有关(Demma et al. 2005). 在管道弯肘部分前端发生L模态发生模态转换形成高阶T模态(Furukawa et al. 2012). Heinlein 等(2018)研究管道弯曲处T(0,1)模态缺陷反射特性, 分析弯曲半径、周向和角度位置对缺陷反射的影响. Gresil等(2016)通过ABAQUS软件构建三维FEM模型分析碳/玻璃纤维混合复合材料管 中L模态和T模态与结构表面缺陷的相互作用; 采用集中力模式进行L模态和T模态激励. 研究显示频率为30 kHz的L(0,1)模态较频率为255 kHz的L(0,2)模态对粘结缺陷更为敏感. Chen J等(2017)分析得到L(0,2)模态能够穿过混凝土结构, 与混凝土覆盖部分相互作用中发生模态转换; 相同缺陷尺寸条件下, 混凝土边沿处缺陷的反射信号强度远大于覆盖区域缺陷的反射强度. Rong等(2017)采用ABAQUS显式动力学研究灌浆锚杆结构导波检测, 采用八节点线性单元、缩减积分、沙漏控制单元进行模型离散, 灌浆和钢筋之间采用刚性接触条件. 观察到L(0,3)模态仅在钢筋中传播, 而L(0,1)模态在钢筋和灌浆中均有传播, 对灌浆缺失更为敏感. Yu和Yan (2017)采用ANSYS软件研究由锤击产生的导波在嵌入土壤中的木电线杆中的传播特性. 采用摩擦接触和绑定接触两种方式处理木杆与周边土壤之间的相互作用关系, 和实验结果对比显示绑定接触能够更好地模拟导波在木杆中的传播, 而摩擦接触法能够精确地对首次抵达波进行模拟. 纵向的高弹性模量使得产生的导波速度更快, 而径向或切向的弹性模量对导波传播的影响较小; 在低密度区的导波速度更快.

3.3 黏弹性结构中的超声导波

黏弹性结构中应变响应滞后于应力作用, 此时应力-应变方程中的弹性模量是线弹性杨氏模量和黏弹性模量组合而成的复参量(Larcher et al. 2015), 对应的复刚度矩阵为

$ \tilde {\pmb C}=\pmb C'-{\rm i}\pmb C''$

式中, $\tilde {\pmb C}$为复刚度矩阵, $\pmb C'$为弹性模量矩阵, $\pmb C''$为黏性模量矩阵, i为虚数符. 黏弹性结构中的材料阻尼描述模型主要有Kelvin-Voigt模型、滞回模型, 相应的复刚度矩阵为(Bartoli et al. 2006)

$ \left. \begin{array}{ll} \tilde {\pmb C}=\pmb C'-{\rm i}\dfrac{\omega}{\bar{\omega}}\pmb \eta,&\quad \mbox{Kelvin-Voigt模型}\\ \tilde {\pmb C}=\pmb C'-{\rm i} \pmb \eta,&\quad \mbox{滞回模型}\\ \end{array}\right\}$

式中, $\pmb \eta $为黏度张量, $\bar {\omega }$为特征角频率. Kelvin-Voigt模型和滞回模型分别具有频率相关性和频率非相关性. 由于依赖频率的阻尼矩阵在应用中难以确定准确值, 且复刚度矩阵不适用于时域FEM分析. 因此, FEM常采用质量矩阵和刚度矩阵线性组合的瑞利(Rayleigh)阻尼矩阵进行相关问题研究. 瑞利阻尼矩阵为

$\pmb C=\alpha \pmb M+\beta \pmb K$

式中, $\alpha $为质量阻尼系数, $\beta $为刚度阻尼系数.瑞利阻尼力中将正比于质点运动速度量的介质阻尼矩阵简化成比例于单元质量矩阵, 将比例于应变率的阻尼矩阵简化为正比于单元刚度矩阵, 是非复刚度矩阵, 能够适用于频域、时域等不同的FEM. 超声导波检测技术研究中需要考虑材料黏弹性性的地方包括 黏弹性结构中超声导波传播特性分析及胶结界面的超声导波检测技术研究两个方面.

基于树脂基的碳纤维复合材料板壳和管道结构是一种典型的黏弹性结构, 受材料黏弹性特性的影响, 超声导波在复合材料导波结构中的传播特性及与缺陷的相互作用关系较线弹性结构具有明显的差异. 复合材料波导中的FEM研究需要准确建立单元的黏弹性模型. Hosten等(2006, 2007)基于COMSOL频域FEM研究各向异性黏弹性有机材料板中凹槽缺陷处S$_{0}$模态的反射及透射系数; 为准确获得导波与凹槽的相互作用, 对缺陷周围网格进行细化, 并通过分析凹槽上空区域表面的正应力和剪切应力是否为零, 评估凹槽周围网格划分质量. Gresil和Griugiutiu (2015)通过ABAQUS建立多物理场有限元模型, 隐式方法计算PWAS在复合材料板中激励兰姆波受瑞利阻尼模型影响下的衰减特性. 研究中首先推导出波阻尼系数$\eta $与瑞利阻尼系数的关系式(37), 再由实验测量数据对式(38)进行拟合确定波阻尼系数, 结合式(37)确定瑞利阻尼系数关系曲线, 评估兰姆波FEM分析模型中瑞利阻尼参数值. 研究显示黏弹性复合材料板中S$_{0}$模态能量的衰减同时受几何扩散和结构阻尼的影响, 而A$_{0}$模态的衰减主要与结构阻尼有关, 说明FEM中采用瑞利阻尼表征Lamb波衰减的合理性. 当$\alpha =0$, $\beta =6.1\times 10^{ - 8}$时, FEM计算所得的频率为150 kHz的S$_{0}$模态幅值较A$_{0}$模态更接近实验结果. Bartoli 等 (2006) 基于Kelvin-Voigt模型和滞回模型和SAFEM研究任意横截面阻尼波导结构中波的传播特性. 当材料黏弹性由Kelvin-Voigt模型表示时, 每次数值迭代计算中均需要根据关注频率和特征频率$\bar {\omega }$和式(35)中的Kelvin-Voigt复刚度矩阵方程进行刚度矩阵缩放; 采用滞回模型表示材料黏弹性时,计算中采用基于初始频率和式(35)中滞回模型复刚度矩阵缩放结果设置整个频域仿真期间的刚度矩阵参数. Marzani (2008)基于滞回模型与时频域谱有限单元法分析黏弹性轴对称管道中导波传播特性, 阻尼特征的增加使L模态和T模态出现明显的衰减. He和Ng (2015)基于明德林-赫尔曼(Mindlin-Herrmann)杆理论和铁摩辛柯(Timoshenko)梁理论和四自由度谱有限单元法构建各向同性梁模型, 计算不同表面裂纹下低频导波的模态转换和散射特性, 阻尼矩阵如式(35), $\eta= 550$ s$^{ - 1}$; 采用GLL近似多项式构建插值多项式和一维谱裂纹单元. Singh等(2011)建立参数化二维有限元技术进行弹性、黏弹性各向同性和各向异性材料板中条形缺陷厚度和宽度的尺寸检测. 采用缺陷反射和透射系数作为缺陷尺寸检测分析的输入参数, 当预测反射和透射系数同实验数据达到最佳匹配时停止反运算. 反演过程中, 缺陷区域的材料模量和密度被设置为零. 研究中认为冲击裂纹区域的材料发生了刚度退化, 而密度没有改变. Ren和Lissenden (2018)在全耦合压电超声FEM模型研究中采用阻抗分析仪确定1 mm厚、6.35 mm直径PZT5A片的两个共振频率点,计算出两共振频率对应的品质因数, 代入式(39)计算压电材料瑞利阻尼系数, $\alpha =7.3\times 10^{4}$, $\beta =5.48\times 10^{ - 9}$, 实现压电材料阻尼属性的准确建模.

$ c_{\rm p} \eta =\dfrac {\alpha + \beta \omega ^2}{2}$
$ \phi \left( {d,t} \right) = A\dfrac{1}{\sqrt d }{\rm e}^{ - \eta d}{\rm e}^{{\rm i}\left( {\omega t - kd} \right)}$

式中, $c_{\rm p}$为相速度, $\eta $为阻尼系数, $d$为传播距离, $k$为波数.

$ \dfrac{\alpha }{\omega _0 } + \beta \omega _0 =\dfrac{1}{Q}$

式中, $Q$为品质因数, $\omega _0$为共振角频率.

FEM在结构粘结层对超声导波传播特性影响研究中的典型应用. Seifried等(2002)基于ABAQUS显式动力学分析研究三层板中粘结层的低刚度、黏弹性特征对导波频率、位移场、频率与波数衰减曲线的影响. 采用CPE4R四节点二阶插值平面应变连续单元进行网格划分. 采用单位时间应变比例阻尼描述粘结层应变黏弹性, 该方法具有和Kelvin-Voigt模型相似的黏性应力-应变关系. 材料粘结属性的增加加快了高阶模态的衰减. 受模型频率效应的影响, 粘结层对350 kHz区域信号影响较弱, 所以在阻尼和非阻尼模型中该区域模态没有变化. Pahlavan (2016)采用高阶谱有限元模型和显式Newmark计算方法分析了温度条件下完美粘接和完全脱离两种情况时, 导波在铺沥青的多层钢结构桥梁中结构中的传播特性. 采用温度相关正比于质量矩阵的瑞丽阻尼矩阵设置材料黏弹性. 随着薄膜刚度的增加, 导波幅值减小; 通过对不同衰减系数下缺陷模型和健康模型检测信号进行对比分析指出, 用于缺陷检测的信号成分变化对衰减并不敏感;不同温度下沥青对加强筋结构S$_{0}$模态的反射率的影响可以忽略.

3.4 非线性超声导波

结构微裂纹的萌生宏观上起源于应力集中区、高应变区、弱强度基体与腐蚀坑等区域, 在裂纹附近的应变程度较大, 小角度晶界密度较高, 裂纹扩展时裂纹尖端产生塑性区域; 此外, 结构残余应力会增加裂纹尖端塑性尺寸, 拉伸残余应力会加剧裂纹尖端的塑形变形, 而材料的塑性会诱导裂纹闭合. 受微缺陷非线性材料特性和非线性行为的影响, 超声导波与结构中的微裂纹发生非线性相互作用, 形成高次谐波. 通过对高次谐波与基础波进行分析提取结构特征信息, 实现结构非线性导波检测. 常规FEM中通过降低单元弹性系数(Cawley et al. 1979)、杨氏模量(Yuen 1985)、裂纹处单元的截面积等方式无法模拟疲劳裂纹的非线性行为. 学者发展出单自由度双刚度振子理论、虚拟弹簧理论、迟滞闭合裂纹模型和界面引力模型(Ohara et al. 2008)等进行超声波与裂纹间的非线性相互作用的研究. Christides和Barr (1984)提出单元刚度与裂纹位置模型进行连续梁中裂纹描述. 应用中需要依据裂纹位置变化重新计算刚度矩阵. Lee和Chung (2000)基于断裂力学中的应力强度因子的经验表达式描述裂纹引起的结构柔度, 显式给出了包含裂纹的有限单元的刚度矩阵, 但该方法没有显式表达出刚度矩阵与裂纹位置的关系. Shen和Giurgiutiu (2014)利用单元生死法和罚函数法接触分析实现非线性超声的模拟. Broda等(2014)对裂纹与波的相互作用模型进行了综述介绍. FEM在非线性超声导波检测技术研究主要包括结构非线性超声导波和材料非线性超声导波两个方面.

结构非线性超声导波是由微缺陷与超声导波的非线性相互作用引起的, 即裂纹开闭引起的导波非线性. Friswell和Penny (2002)综合对比了梁结构中呼吸裂纹的建模方法, 指出裂纹开闭运动中单元的刚度矩阵和系统状态矢量均会发生瞬时改变, 见式(40), 此时采用显式动力学分析能够更好地进行非线性超声导波的计算. Joglekar和Mitra (2016)采用等效转动弹簧理论构建缺陷db小波谱单元, 分析各向同性欧拉-伯努利梁中呼吸裂纹与弯曲波的非线性相互作用. 研究中对裂纹闭合状态下的结构做无缺陷假设处理. He和Ng (2015)采用GLL近似多项式构建插值多项式和一维谱裂纹单元, 分析各向同性梁结构中导波与裂纹间相互作用形成的模态转换和散射; 裂纹单元每个节点的拥有轴向力、剪切向力和弯矩共3个自由度. 观察不同裂纹深度及宽度对导波反射和投射特性的影响, A$_{0}$和S$_{0}$模态法向幅值和裂纹尺寸有直接关系, 反射信号和模态转换幅度随裂纹尺寸的增加而增加, 而透射信号则相反; 裂纹深度与波长之比对应的法向幅度较裂纹宽度与波长之比对应的法向幅度复杂. Wan等(2014)采用硬接触无摩擦模型进行裂纹表面设置, 分析S$_{0}$模态同光滑椭圆形微裂纹相互作用形成的二次谐波和新波包的幅值比. S$_{0}$模态二次谐波的幅值随微裂纹长度的增加而线性增加, 新增波形随微裂纹长度增加到峰值后降低; S$_{0}$模态二次谐波幅值及幅值比均随裂纹宽度的增加而降低. Blanloeuil等(2014)结合预应力和单方向接触库伦摩擦理论构建闭合裂纹模型, 分析SH波、P波与裂纹的相互作用. 通过对裂纹局部进行$h$型优化, 以克服裂纹尖端应力奇异问题. Shen和Cesnik (2016, 2017)结合罚函数法和局部交互仿真方法进行疲劳裂纹非线性超声仿真. 依据库伦粘滑接触运动原理设置黏性接触和滑动接触力学模型;当达到近似接触刚度时, 渗透量接近于零, 使得数值解收敛到物理接触状态. 引入随机算法设置界面接触粗糙分布, 研究超声导波与初始存在开口和闭合状态的粗糙表面裂纹的相互作用. 观察到非线性高阶响应和直流成分只会发生在导波和疲劳裂纹非线性相互接触状态; 对于穿透裂纹, S模态入射波只会转换为S模态, A模态会转换成S和A两种模态; 而S模态和A模态与半穿透裂纹相互作用时, 均会发生向主模态类型的转换. 观察到导波与裂纹相互作用时的阈值行为. 因此, 为更好地利用超声非线性进行结构检测, 激励的波场需要足够强的能量, 以触发裂纹表面的非线性相互作用. Zhou等(2017)分别采用CONTA172和TARGE169两种单元构建接触裂纹的不同接触面, 并对裂纹区域进行网格加密; 采用时间反转分解算法对非线性兰姆波进行分析, 观察形成的二次谐波. Wang Y等(2011)设计了缺陷傅里叶谱单元对加强杆中裂纹、加强杆与混凝土脱粘缺陷建模, 采用传递矩阵法组装多个谱单元构建模型刚度矩阵, 并结合克隆选择算法对波传播数据进行分析, 实现一维杆中缺陷的反演识别.

$ \pmb K = \left\{ \begin{array}{ll} \pmb K_{\rm c} , &\quad u > u_0 \\ \pmb K_{\rm t}, &\quad u < u_0 \\ \end{array} \right.$

式中, $u_{0}$为裂纹开合位移阈值, $\pmb K_{\rm c}$为裂纹张开模型刚度, $\pmb K_{\rm t}$为裂纹闭合模型刚度.

材料非线性对导波传播特性的影响. Chillara和Lissenden (2014)采用COMSOL建立基于Murnaghan模型研究超声兰姆波非线性特征. Rauter和Lammering (2018)针对基于Murnaghan非线性弹性理论中需要14个材料参数进行横观各向同性材料属性描述的现象, 提出一种新的非线性弹性横观各向同性材料模型, 将独立材料参数个数减少至6个. 研究显示随着传播距离的增加, 频率500 kHz的S$_{0}$模态二阶幅值随着传播距离的增加而振荡. 而在单向复合材料板中的频率200 kHz的S$_{0}$模态谐波幅值随着传播距离的增加而呈线性增加. Zhu等(2016)通过ABAQUS用户交互子程序VUMAT将混合位错非线性本构方程式代入模型, 研究马氏体不锈钢位错演化形成的塑性损伤对超声纵波传播的影响. Radecki等(2018)采用ABAQUS模型验证LISA在金属疲劳对导波非线性的影响的计算. 基于Landau-Lifshitz方程的能量密度法构建超弹性材料属性, 由弹簧模型构建疲劳呼吸裂纹. 通过对裂纹交界面法向应力的评估结果设置裂纹开合状态. 裂纹深度为梁结构厚度的一半, 所提模型和ABAQUS模型计算得到的二次谐波具有很好的一致性, S$_{0}$模态及其二次谐波满足近似同步条件, 当材料存在非线性时会形成二次谐波累加现象. 随着裂纹尺寸的增加, 材料非线性对二次谐波形成的影响减弱.

4 总结与展望

$h$-FEM采用的低阶有限单元在数值计算中存在对单元扭曲敏感、耦合场动力学问题计算效率低、非均质结构建模困难等问题. 为适应现代超声导波结构检测中对耦合场、黏弹性复杂结构、非线性动力学问题的高效计算的需求, 近几十年人们提出和发展出了多种有限单元法. 其中, hp-FEM, TFFEM, WFEM及不同有限单元组合分析方法, 能够极大提升超声导波波场的计算效率; 近些年提出的MsFEM和 FCM能够有效解决非均质结构中波场的计算问题. 在超声导波结构检测研究中关注的耦合场导波检测技术、线弹性和黏弹结构中 超声导波传播特性、非均质结构中的导波传播特性及结构与材料非线性导波特性等方面, 均通过FEM的应用, 取得了众多突出的研究进展. 然而, 为进一步深化超声导波检测技术的研究, 推动该技术在工程的实际应用, 有限单元分析技术在以下问题上仍有诸多工作需要开展:

(1) 充分利用$hp$-FEM, TFFEM及WFEM的快速计算能力和MsFEM, FCM方法对非均质结构的处理能力, 将两类方法进行结合, 发展复杂波导结构中动力学问题的高效计算方法. 优化有限单元矩阵积分算法, 进行质量矩阵对角化处理, 特别是对MsFEM中的截断单元实现质量矩阵对角化, 推进显式算法在动力学问题数值计算中的应用.

(2) 在超声导波有限单元计算精度可靠性的研究方面, 目前主要集中在分析$h$-FEM和hp-FEM两类有限单元在兰姆波的计算精度的影响因素方面, 但在对目前超声导波检测研究中关注的高阶兰姆波和管道中超声导波的数值计算精度的分析较少. 尚未见TFFEM, WFEM, MsFEM等方法对不同模态导波计算效率评估的介绍. 由于不同模态超声导波的波结构和波长的多样性, 为保障超声导波检测技术数值计算的可靠性, 需要展开研究网格离散形式、界面网格差异、网格奇异、时间积分步长、单元积分算法等对板壳、管道中不同导波计算精度的影响形式和计算稳定条件.

(3) $h$-FEM多采用隐式积分算法进行耦合场动力学模型计算, 运算效率低. 现有研究中多采用不同物理场分区进行建模的方式, 然后将不同物理场进行融合进行计算, 在一定程度上改善了计算效率. 但该方法在不同物理域划分的选择上具有一定的任意性, 且操作过程繁琐. 因此需要发展耦合场动力学问题显式计算技术,通过高阶形函数建立压电耦合单元、电磁声耦合单元及热结构耦合单元, 发展耦合场单元矩阵对角化算法.

(4) 超声导波检测中使用的传感器及其与波导的接触状态均会随时间的增加发生改变. 研究人员针对压电与波导结构的耦合层老化及不完全粘结情况进行了一定的研究. 但在压电传感器本身老化或传感器焊接不合理等对超声导波激励影响的研究仍有待进一步加强. 电磁超声有限单元计算中的耦合单元模型中的磁致伸缩特性多为各向同性, 而对于各向异性磁致伸缩单元的模型仍有待研究.

(5) 工程中大量存在的异形结构、变截面结构、变厚度结构、非均质结构、浇筑结构、胶结结构等的检测问题已得到研究人员的关注, 但相关研究仍处于起步阶段. 对于各模态导波在以上结构中传播特性的分析、最佳检测频率及模态的选取、检测实验设置等需要进一步明确; 特别是浇筑结构及胶结结构, 需要确定合理的界面作用模型, 研究环境温度变化、结构受力载荷等对结构的影响, 实现导波传播特性的准确计算.

(6) 金属板壳、管道和钢索等结构在加工时形成结构空间非均匀分布的硬度层和残余应力, 在工程应用中会受到不同形式的载荷作用. 针对以上情况, 发展灵活的有限单元建模技术, 构建材料属性空间非均匀分布情况下的模型, 分析均匀结构和非均匀网格离散形式对数值计算的影响形式, 研究结构硬度及应力对超声导波传播特性的影响. 结合反演算法和FEM, 发展结构力学性能表征技术和成像技术.

(7) 复合材料、功能梯度材料中缺陷的尺寸定量检测技术, 建立功能梯度材料单元模型和结构缺陷模型, 研究导波在功能梯度材料中的传播特性, 分析各模态导波对材料中不同位置处各缺陷的相互作用关系及衰减特性. 优化选取缺陷检测导波模态、设计缺陷检测实验.

(8) 在结构缺陷和材料非线性两个方面引起的导波非线性问题均有了大量的研究, 但多将两种非线性问题分开进行研究. 实际结构早期损伤中材料非线性和结构非线性往往是同时存在的. 另外, 据调研显示目前的导波非线性研究仍集中于兰姆波非线性问题, 在管道中导波非线性的研究较少. 因此, 在材料非线性和结构非线性综合对导波非线性的影响形式、管中导波的非线性检测技术及相关材料非线性和结构非线性单元模型的建立等方面均有待深入研究.

(9) 超声导波检测技术在工程应用中会遇到各类干扰, 影响检测的可靠性. 因此, 综合概率统计分析算法、导波检测和有限单元计算方法, 开展基于模型辅助超声导波结构检测/监测技术可靠性性研究, 对推进超声导波检测技术的具体应用具有十分重要的意义. 基于数值计算技术, 结合具体检测环境及检测技术手段和检测目标特性, 发展超声导波结构健康检测/监测综合评价系统.

(责任编委: 戴兰宏)

致谢

国家重点研发计划(2018 YFC0809003), 国家自然科学基金(11772014,51475012)资助项目.

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The noncontact nature of electromagnetic acoustic transducers (EMATs) offers a series of advantages over traditional piezoelectric transducers, but these features are counter-balanced by their relatively low signal-to-noise ratio and their strong dependence on material properties such as electric conductivity, magnetic permeability, and magnetostriction. The implication is that full exploitation of EMATs needs detailed modeling of their operation. A finite element model, accounting for the main transduction mechanisms, has been developed to allow the optimization of the transducers. Magnetostriction is included and described through an analogy with piezoelectricity. The model is used to predict the performance of a simple EMAT: a single current-carrying wire, parallel to a bias magnetic field generating shear horizontal waves in a nickel plate close to it. The results are validated against experiments. The model is able to successfully predict the wave amplitude dependence on significant parameters: the static bias field, the driving current amplitude, and the excitation frequency. The comparison does not employ any arbitrary adjustable parameter; for the first time an absolute validation of a magnetostrictive EMAT model has been achieved. The results are satisfactory: the discrepancy between the numerical predictions and the measured values of wave amplitude per unit current is less than 20% over a 200 kHz frequency range. The study has also shown that magnetostrictive EMAT sensitivity is not only a function of the magnetostrictive properties, because the magnetic permeability also plays a significant role in the transduction mechanism, partly counterbalancing the magnetostrictive effects.

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This work concerned a technique for a health monitoring system based on the generation and sensing of Lamb waves in composite structures by thin surface-bonded piezoceramic transducers. The objective was to develop transducers that are adapted for the damage detection in orthotropic composites. The key problem with the investigated Lamb waves was to select a mode to be sensitive to the damage. A hybrid modeling technique was therefore used to conceive transducers that were adapted to achieve such a feature. This modeling technique enabled studying the influence of the transducer characteristics on the Lamb waves propagating in orthotropic plates. It was demonstrated that a Lamb mode could be generated dominantly to other modes by using a multi-element transducer. The effectiveness of this technique was successfully verified experimentally on composite plates. It was shown that the dominant Lamb mode, obtained by use of dual-element transducers, was an appropriate mode for successfully detecting a damage in composites.

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This article presents a parallel algorithm to model the nonlinear dynamic interactions between ultrasonic guided waves and fatigue cracks. The Local Interaction Simulation Approach (LISA) is further developed to capture the contact-impact clapping phenomena during the wave crack interactions based on the penalty method. Initial opening and closure distributions are considered to approximate the 3-D rough crack microscopic features. A Coulomb friction model is integrated to capture the stick-slip contact motions between the crack surfaces. The LISA procedure is parallelized via the Compute Unified Device Architecture (CUDA), which enables parallel computing on powerful graphic cards. The explicit contact formulation, the parallel algorithm, as well as the GPU-based implementation facilitate LISA's high computational efficiency over the conventional finite element method (FEM). This article starts with the theoretical formulation and numerical implementation of the proposed algorithm, followed by the solution behavior study and numerical verification against a commercial finite element code. Numerical case studies are conducted on Lamb wave interactions with fatigue cracks. Several nonlinear ultrasonic phenomena are addressed. The classical nonlinear higher harmonic and DC response are successfully captured. The nonlinear mode conversion at a through-thickness and a half-thickness fatigue crack is investigated. Threshold behaviors, induced by initial openings and closures of rough crack surfaces, are depicted by the proposed contact LISA model.

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A coordinated theoretical, numerical and experimental study is carried out in an effort to interpret the characteristics of propagating guided Lamb wave modes in presence of a high-density (HD) core region in a honeycomb composite sandwich structure (HCSS). Initially, a two-dimensional (2D) semi-analytical model based on the global matrix method is used to study the response and dispersion characteristics of the HCSS with a soft core. Due to the complex structural characteristics, the study of guided wave (GW) propagation in HCSS with HD-core region inherently poses many challenges. Therefore, a numerical simulation of GW propagation in the HCSS with and without the HD-core region is carried out, using surface-bonded piezoelectric wafer transducer (PWT) network. From the numerical results, it is observed that the presence of HD-core significantly decreases both the group velocity and the amplitude of the received GW signal. Laboratory experiments are then conducted in order to verify the theoretical and numerical results. A good agreement between the theoretical, numerical and experimental results is observed in all the cases studied. An extensive parametric study is also carried out for a range of HD-core sizes and densities in order to study the effect due to the change in size and density of the HD zone on the characteristics of propagating GW modes. It is found that the amplitudes and group velocities of the GW modes decrease with the increase in HD-core width and density.

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A spectral element model updating procedure is presented to identify damage in a structure using Guided wave propagation results. Two damage spectral elements (DSE1 and DSE2) are developed to model the local (cracks in reinforcement bar) and global (debonding between reinforcement bar and concrete) damage in one-dimensional homogeneous and composite waveguide, respectively. Transfer matrix method is adopted to assemble the stiffness matrix of multiple spectral elements. In order to solve the inverse problem, clonal selection algorithm is used for the optimization calculations. Two displacement-based functions and two frequency-based functions are used as objective functions in this study. Numerical simulations of wave propagation in a bare steel bar and in a reinforcement bar without and with various assumed damage scenarios are carried out. Numerically simulated data are then used to identify local and global damage of the steel rebar and the concrete-steel interface using the proposed method. Results show that local damage is easy to be identified by using any considered objective function with the proposed method while only using the wavelet energy-based objective function gives reliable identification of global damage. The method is then extended to identify multiple damages in a structure. To further verify the proposed method, experiments of wave propagation in a rectangular steel bar before and after damage are conducted. The proposed method is used to update the structural model for damage identification. The results demonstrate the capability of the proposed method in identifying cracks in steel bars based on measured wave propagation data.

Wilcox P D, Croxford A J, Konstantinidis G , et al. 2007.

Sensitivity limitations for guided wave structural health monitoring

// Conference on Health Monitoring of Structural and Biological Systems, March 19-22, 2007, San Diego, California, USA.

[本文引用: 1]

Willberg C, Duczek S, Perez J V, Schmicker D. 2012.

Comparison of different higher order finite element schemes for the simulation of Lamb waves

Comput. Meth. Appl. Mech. Eng., 241:246-261.

[本文引用: 2]

Willberg C, Gabbert U. 2012.

Development of a three dimensional piezoelectric isogeometric finite element for smart structure applications

Acta Mech., 223:1837-1850.

DOI      URL     [本文引用: 1]

In this paper, a 3D electromechanical-coupled isogeometric finite element is presented. An important advantage of such elements is the possibility to incorporate geometrical data of computer-aided design systems into the description of the finite element models. In this work, the isogeometric approach is extended to a coupled electromechanical finite element with mechanical and electrical degrees of freedom. The proposed piezoelectric finite element is based on nonuniform rational B-spline shape functions. In order to demonstrate the functionality and the advantages of the isogeometric piezoelectric element for smart structure and structural health monitoring applications, various examples are presented. The results obtained with isogeometric elements agree very well with analytical and numerical reference solutions. It is shown that the better geometrical approximation introduced by isogeometric elements results in an improvement of the solution quality.

Willberg C, Duczek S, Perez J M V, Ahmad Z A B. 2015.

Simulation methods for guided wave-based structural health monitoring: A review

Appl. Mech. Rev., 67:1-20.

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Xiao D, Han Q, Liu Y, Li C. 2016.

Guided wave propagation in an infinite functionally graded magneto-electro-elastic plate by the Chebyshev spectral element method

Compos. Struct., 153:704-711.

DOI      URL     [本文引用: 1]

Xie Y, Yin W, Liu Z, Peytona A. 2016.

Simulation of ultrasonic and EMAT arrays using FEM and FDTD

Ultrasonics, 66:154-165.

DOI      URL     PMID      [本文引用: 1]

This paper presents a method which combines electromagnetic simulation and ultrasonic simulation to build EMAT array models. For a specific sensor configuration, Lorentz forces are calculated using the finite element method (FEM), which then can feed through to ultrasonic simulations. The propagation of ultrasound waves is numerically simulated using finite-difference time-domain (FDTD) method to describe their propagation within homogenous medium and their scattering phenomenon by cracks. Radiation pattern obtained with Hilbert transform on time domain waveforms is proposed to characterise the sensor in terms of its beam directivity and field distribution along the steering angle.

Xu B, Shen Z, Ni X, Lu J. 2004.

Numerical simulation of laser-generated ultrasound by the finite element method

J. Appl. Phys., 95:2116-2122.

DOI      URL     [本文引用: 1]

Yang J, Ume I C. 2008.

Thermomechanical reliability study of flip-chip solder bumps: Using laser ultrasound technique and finite element method

// 58th Electronic Components and Technology Conference, May 27-30, 2008, Orlando, Florida, USA.

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Yang Z, Chen X, Xie Y, Zuo H. 2016.

Wave motion analysis and modeling of membrane structures using the wavelet finite element method

Appl. Math. Model., 40:2407-2420.

DOI      URL     [本文引用: 1]

Yu F, Collet M, Ichchou M, Lin L. 2017.

Enhanced wave and finite element method for wave propagation and forced response prediction in periodic piezoelectric structures

Chin. J. Aeronaut., 30:75-87.

DOI      URL     [本文引用: 1]

Yu X, Fan Z, Castaings M, Biateau C. 2017.

Feature guided wave inspection of bond line defects between a stiffener and a composite plate

NDT E Int., 89:44-55.

DOI      URL    

Yu Y, Yan N. 2017.

Numerical study on guided wave propagation in wood utility poles finite element modelling and parametric sensitivity analysis

Appl. Sci., 7:1063.

DOI      URL     [本文引用: 1]

Yue B, Guddati M N. 2005.

Dispersion-reducing finite elementsfor transient acoustics

J. Acoust. Soc. Am., 118:2132-2141.

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Yuen M M F. 1985.

A numerical study of the eigenparameters of a damaged cantilever

J. Sound Vib., 103:301-310.

DOI      URL     [本文引用: 1]

Zander N, Kollmannsberger S, Ruess M, Yosibash Z. 2012.

The Finite Cell Method for linear thermoelasticity

Comput. Math. Appl., 64:3527-3541.

DOI      URL     [本文引用: 1]

The recently introduced Finite Cell Method (FCM) combines the fictitious domain idea with the benefits of high-order Finite Elements. While previous publications concentrated on single-field applications, this paper demonstrates that the advantages of the method carry over to the multi-physical context of linear thermoelasticity. The ability of the method to converge with exponential rates is illustrated in detail with a benchmark problem. A second example shows that the Finite Cell Method correctly captures the thermoelastic state of a complex problem from engineering practice. Both examples additionally verify that, also for two-field problems, Dirichlet boundary conditions can be weakly imposed on non-conforming meshes by the proposed extension of Nitsche's Method. (C) 2012 Elsevier Ltd.

Zhan Y, Liu C, Kong X, Lin Z. 2017.

Experiment and numerical simulation for laser ultrasonic measurement of residual stress

Ultrasonics, 73:271-276.

DOI      URL     PMID      [本文引用: 1]

Laser ultrasonic is a most promising method for non-destructive evaluation of residual stress. The residual stress of thin steel plate is measured by laser ultrasonic technique. The pre-stress loading device is designed which can easily realize the condition of the specimen being laser ultrasonic tested at the same time in the known stress state. By the method of pre-stress loading, the acoustoelastic constants are obtained and the effect of different test directions on the results of surface wave velocity measurement is discussed. On the basis of known acoustoelastic constants, the longitudinal and transverse welding residual stresses are measured by the laser ultrasonic technique. The finite element method is used to simulate the process of surface wave detection of welding residual stress. The pulsed laser is equivalent to the surface load and the relationship between the physical parameters of the laser and the load is established by the correction coefficient. The welding residual stress of the specimen is realized by the ABAQUS function module of predefined field. The results of finite element analysis are in good agreement with the experimental method. The simple and effective numerical and experimental methods for laser ultrasonic measurement of residual stress are demonstrated.

Zhan Y, Liu C, Zhang F, Qiu Z. 2016.

Experimental study and finite element analysis based on equivalent load method for laser ultrasonic measurement of elastic constants

Ultrasonics, 69:243-247.

DOI      URL     PMID      [本文引用: 1]

The laser ultrasonic generation of Rayleigh surface wave and longitudinal wave in an elastic plate is studied by experiment and finite element method. In order to eliminate the measurement error and the time delay of the experimental system, the linear fitting method of experimental data is applied. The finite element analysis software ABAQUS is used to simulate the propagation of Rayleigh surface wave and longitudinal wave caused by laser excitation on a sheet metal sample surface. The equivalent load method is proposed and applied. The pulsed laser is equivalent to the surface load in time and space domain to meet the Gaussian profile. The relationship between the physical parameters of the laser and the load is established by the correction factor. The numerical solution is in good agreement with the experimental result. The simple and effective numerical and experimental methods for laser ultrasonic measurement of the elastic constants are demonstrated.

Zhang X, Tang Z, Lü F, Yang K. 2017.

Scattering of torsional flexural guided waves from circular holes and crack-like defects in hollow cylinders

NDT E Int., 89:56-66.

DOI      URL     [本文引用: 1]

Zhou C W, Lainé J P, Ichchou M N, Zine A M. 2015.

Wave finite element method based on reduced model for one-dimensional periodic structures

Int. J. Appl. Mech., 7:1550018.

DOI      URL     [本文引用: 2]

Zhou J, Xiao L, Qu W, Lu Y. 2017.

Nonlinear Lamb wave based DORT method for detection of fatigue cracks

NDT E Int., 92:22-29.

DOI      URL     [本文引用: 1]

Zhu W. 2002.

An FEM simulation for guided elastic wave generation and reflection in hollow cylinders with corrosion defects

J. Press. Vessel Technol. -Trans. ASME, 124:108-117.

DOI      URL     [本文引用: 1]

Zhu W, Deng M, Xiang Y, Xuan F. 2016.

Modeling of ultrasonic nonlinearities for dislocation evolution in plastically deformed materials: Simulation and experimental validation

Ultrasonics, 68:134-141.

DOI      URL     PMID      [本文引用: 1]

A nonlinear constitutive relationship was established to investigate nonlinear behaviors of ultrasonic wave propagation in plastically damaged media based on analyses of mixed dislocation evolution. Finite element simulations of longitudinal wave propagation in plastically deformed martensite stainless steel were performed based on the proposed nonlinear constitutive relationship, in which the contribution of mixed dislocation to acoustic nonlinearity was considered. The simulated results were validated by experimental measurements of plastically deformed 30Cr2Ni4MoV martensite stainless steels. Simulated and experimental results both reveal a monotonically increasing tendency of the normalized acoustic nonlinearity parameter as a function of plastic strain. Microscopic studies revealed that the changes of the acoustic nonlinearity are mainly attributed to dislocation evolutions, such as dislocation density, dislocation length, and the type and fraction of dislocations during plastic loading.

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