功能梯度圆柱板中的周向衰逝导波
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摘要
与传播波模态不同,衰逝波模态波数为纯虚数或复数,它们随传播距离呈指数或阻尼式衰减.具有复波数的衰逝波对结构缺陷形状和尺寸的导波检测具有重要作用,但其求解却是非常困难的,通常要借助于各种迭代技术.论文提出一种计算求解衰逝波问题的改进的Legendre正交多项式方法,该方法可将复杂的变系数微分方程组计算转换为特征值求解问题,无需迭代便能计算得到包含实波数域、虚波数域和复波数域的完整的频散曲线.通过具体算例验证了提出方法的正确性.应用提出方法计算了不同梯度圆柱板中的衰逝导波,绘制了三维频散曲线,研究了不同径厚比和梯度场对衰逝导波频散的影响,分析了衰逝导波的位移和应力分布,讨论了衰逝导波的传播特性.
Evanescent wave modes are substantially different from the propagating modes,which are associated with non-real wavenumbers and decay exponentially or damp with propagation distance.Such modes with complex wavenumbers play an important role in the detection of the shape and size of defects.However,it is very difficult to obtain the complex solutions of the dispersion equation corresponding to the evanescent waves,especially for demanding cases,such as those involving composite materials and curved structures.In this study,an improved orthogonal polynomial method is proposed to deal with the problems of evanescent guided waves in a functionally graded cylindrical curved plate.The proposed method can convert the complex differential equations with variable coefficients into an eigenvalue problem and obtain all the purely real,purely imaginary and complex solutions without iterative process.The validity of the proposed method is illustrated by specific numerical examples.Three dimensional full-dispersion curves of the guided waves in various graded cylindrical curved plates are obtained,and the effects of different radiusthickness ratios and graded fields on the dispersion characteristics of guided evanescent waves are illustrated.The displacement amplitude and stress distributions are also discussed to analyze the specificities of the evanescent guided waves.Numerical results show that there are only purely real and purely imaginary branches for circumferential shear horizontal(SH)waves;but for circumferential Lamb-like waves,there are purely real,purely imaginary and complex branches.Most complex branches collapse onto the minima of purely imaginary branches and local inflection points,and occasionally appear on higher-order real branches.The complex branches exhibit both local vibration and local propagation,which turn into real branches with increasing frequency.Both the radius-thickness ratio and the graded field have significant effects on dispersion curves.With the increase of radius-thickness ratio,the decay of evanescent waves is faster.All the results presented in this work can provide theoretical guidance for ultrasonic nondestructive evaluation.
Evanescent wave modes are substantially different from the propagating modes,which are associated with non-real wavenumbers and decay exponentially or damp with propagation distance.Such modes with complex wavenumbers play an important role in the detection of the shape and size of defects.However,it is very difficult to obtain the complex solutions of the dispersion equation corresponding to the evanescent waves,especially for demanding cases,such as those involving composite materials and curved structures.In this study,an improved orthogonal polynomial method is proposed to deal with the problems of evanescent guided waves in a functionally graded cylindrical curved plate.The proposed method can convert the complex differential equations with variable coefficients into an eigenvalue problem and obtain all the purely real,purely imaginary and complex solutions without iterative process.The validity of the proposed method is illustrated by specific numerical examples.Three dimensional full-dispersion curves of the guided waves in various graded cylindrical curved plates are obtained,and the effects of different radiusthickness ratios and graded fields on the dispersion characteristics of guided evanescent waves are illustrated.The displacement amplitude and stress distributions are also discussed to analyze the specificities of the evanescent guided waves.Numerical results show that there are only purely real and purely imaginary branches for circumferential shear horizontal(SH)waves;but for circumferential Lamb-like waves,there are purely real,purely imaginary and complex branches.Most complex branches collapse onto the minima of purely imaginary branches and local inflection points,and occasionally appear on higher-order real branches.The complex branches exhibit both local vibration and local propagation,which turn into real branches with increasing frequency.Both the radius-thickness ratio and the graded field have significant effects on dispersion curves.With the increase of radius-thickness ratio,the decay of evanescent waves is faster.All the results presented in this work can provide theoretical guidance for ultrasonic nondestructive evaluation.
引文
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[9]Cao X S,Shi J P,Jin F.Lamb wave propagation in the functionally graded piezoelectric-piezomagnetic material plate[J].Acta Mechanica,2012,233(5):1081-1091.
[10]刘睫,王子昆,张陵.梯度材料层状结构中的Love波[J].固体力学学报,2004,25(2):165-170.(Liu J,Wang Z K,Zhang L.Love waves in layered structures with graded materials[J].Chinese Journal of Solid Mechanics,2004,25(2):165-170.(in Chinese))
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[17] Pagneux V,Maurel A.Determination of Lamb mode eigenvalues[J].Journal of the Acoustical Society of America,2001,110(3):1307-1314.
[18] Damljanovic V,Weaver R L.Propagating and evanescent elastic waves in cylindrical waveguides of arbitrary cross section[J].Journal of the Acoustical Society of America,2004,115(4):1572-1581.
[19] An Y K.Measurement of crack-induced non-propagating Lamb wave modes under varying crack widths[J].International Journal of Solids and Structures,2015,62:134-143.
[20] Krushynska A A,Meleshko V V.Normal waves in elastic bars of rectangular cross section[J].Journal of the Acoustical Society of America,2011,129(3):1324-1335.
[21] Quintanilla F H,Lowe M J S,Craster R V.Full 3D dispersion curve solutions for guided waves in generally anisotropic media[J].Journal of Sound and Vibration,2016,363:545-559.
[22]李念,钱征华,王彬.复波数域弥散曲线的求解方法研究[J].应用力学学报,2016,33(3):365-370.(Li N,Qian Z H,Wang B.Study on computational methods of dispersion curves in complex wavenumber range[J].Chinese Journal of Applied Mechanics,2016,33(3):365-370.(in Chinese))
[23] Lefebvre J E,Yu J G,Ratolojanahary F E,Elmaimouni L,Xu W J,Gryba T.Mapped orthogonal functions method applied to acoustic waves-based devices[J].AIP Advances,2016,6(6):1-18.
[24] Onoe M,Mcniven H D,Mindlin R D.Dispersion of axially symmetric waves in elastic rods[J].Journal of Applied Mechanics,1962,29(4):729-734.
[2]Han X,Liu G R,Xi Z C,Lam K Y.Characteristics of waves in a functionally graded cylinder[J].International Journal for Numerical Methods in Engineering,2002,53(3):653-676.
[3]Santare M H,Thamburaj P,Gazonas G A.The use of graded finite elements in the study of elastic wave propagation in continuously nonhomogeneous materials[J].International Journal of Solids and Structures,2003,40:5621-5634.
[4]Chakraborty A,Gopalakrishnan S.A higher-order spectral element for wave propagation analysis in functionally graded materials[J].Acta Mechanica,2004,172(1-2):17-43.
[5]洪轲,袁玲,沈中华,倪晓武.利用Taylor展开法研究Lamb波在功能梯度材料中的传播特性[J].物理学报,2011,60(10):426-432.(Hong K,Yuan L,Shen Z H,Ni X W.Analysis of Lamb waves propagation in functional gradient materials using Taylor expansion method[J].Acta Physica Sinica,2011,60(10):426-432.(in Chinese))
[6]Chen W Q,Wang H M,Bao R H.On calculating dispersion curves of waves in a functionally graded elastic plate[J].Composite Structures,2007,8l(2):233-242.
[7]郭永强.回传射线矩阵法的理论及其应用[D].杭州:浙江大学,2008.(Guo Y Q.The Method of Reverberation-ray Matrix and Its Applications[D].Hang Zhou:Zhejiang University,2008.(in Chinese))
[8]曹小杉,金峰,王子昆.功能梯度材料板中Lamb波传播特性研究[J].固体力学学报,2009,30(1):35-41.(Cao X S,Jin F,Wang Z K.Propagation properties of lamb waves in a functionally graded material plate[J].Chinese Journal of Solid Mechanics,2009,30(1):35-41.(in Chinese))
[9]Cao X S,Shi J P,Jin F.Lamb wave propagation in the functionally graded piezoelectric-piezomagnetic material plate[J].Acta Mechanica,2012,233(5):1081-1091.
[10]刘睫,王子昆,张陵.梯度材料层状结构中的Love波[J].固体力学学报,2004,25(2):165-170.(Liu J,Wang Z K,Zhang L.Love waves in layered structures with graded materials[J].Chinese Journal of Solid Mechanics,2004,25(2):165-170.(in Chinese))
[11] Elmaimouni L,Lefebvre J E,Zhang V,Gryba T.Guided waves in radially graded cylinders:apolynomial approach[J].NDT&E International,2005,38(5):344-353.
[12] Yu J G,Wu B,He C F.Characteristics of guided waves in graded spherical curved plates[J].International Journal of Solids and Structures,2007,44(11-12):3627-3637.
[13] Yu J G,Zhang V.Influences of initial stresses on guided waves in functionally graded hollow cylinders[J].Acta Mechanica,2013,224(4):745-757.
[14]吴斌,禹建功,何存富.功能梯度空心圆柱体中的周向超声导波[J].爆炸与冲击,2007,27(3):266-271.(Wu B,Yu J G,He C F.Guided circumferential waves in functionally graded cylinders[J].Explosion and Shock Waves,2007,27(3):266-271.(in Chinese))
[15] Auld B A.Acoustic Fields and Waves in Solids[M].Florida:2nd Ed.Krieger Publishing Company,1990.
[16] Huang K H,Dong S B.Propagating waves and edge vibrations in anisotropic composite cylinders[J].Journal of Sound and Vibration,1984,96(3):363-379.
[17] Pagneux V,Maurel A.Determination of Lamb mode eigenvalues[J].Journal of the Acoustical Society of America,2001,110(3):1307-1314.
[18] Damljanovic V,Weaver R L.Propagating and evanescent elastic waves in cylindrical waveguides of arbitrary cross section[J].Journal of the Acoustical Society of America,2004,115(4):1572-1581.
[19] An Y K.Measurement of crack-induced non-propagating Lamb wave modes under varying crack widths[J].International Journal of Solids and Structures,2015,62:134-143.
[20] Krushynska A A,Meleshko V V.Normal waves in elastic bars of rectangular cross section[J].Journal of the Acoustical Society of America,2011,129(3):1324-1335.
[21] Quintanilla F H,Lowe M J S,Craster R V.Full 3D dispersion curve solutions for guided waves in generally anisotropic media[J].Journal of Sound and Vibration,2016,363:545-559.
[22]李念,钱征华,王彬.复波数域弥散曲线的求解方法研究[J].应用力学学报,2016,33(3):365-370.(Li N,Qian Z H,Wang B.Study on computational methods of dispersion curves in complex wavenumber range[J].Chinese Journal of Applied Mechanics,2016,33(3):365-370.(in Chinese))
[23] Lefebvre J E,Yu J G,Ratolojanahary F E,Elmaimouni L,Xu W J,Gryba T.Mapped orthogonal functions method applied to acoustic waves-based devices[J].AIP Advances,2016,6(6):1-18.
[24] Onoe M,Mcniven H D,Mindlin R D.Dispersion of axially symmetric waves in elastic rods[J].Journal of Applied Mechanics,1962,29(4):729-734.