复杂介质中任意阶频率依赖耗散声波的分数阶导数模型

蔡伟; 陈文

doi:10.6052/0459-1879-16-186

复杂介质中任意阶频率依赖耗散声波的分数阶导数模型

doi: 10.6052/0459-1879-16-186

蔡伟,
陈文^,

河海大学力学与材料学院软物质力学研究所, 南京 211100

基金项目: 国家自然科学基金（11372097）和111引智计划（B12032）资助项目.

详细信息

通讯作者:
陈文,教授,主要研究方向:固体力学、计算力学.E-mail:chenwen@hhu.edu.cn

中图分类号: O242;O302;O422.4;O426.2
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出版历程
- 收稿日期: 2016-07-08
- 修回日期: 2016-08-24
- 刊出日期: 2016-11-18

FRACTIONAL DERIVATIVE MODELING OF FREQUENCY-DEPENDENT DISSIPATIVE MECHANISM FORWAVE PROPAGATION IN COMPLEX MEDIA

Cai Wei,
Chen Wen^,

Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China

摘要

摘要: 实验现象表明，声波在复杂介质中传播时，其衰减往往呈现频率的任意次幂律依赖现象.鉴于复杂介质的力学和物理性质的记忆性和长程相关性，频率幂律依赖的声波衰减现象难以用经典的声波方程描述，因为经典的阻尼波方程和近似热黏性波方程只能分别描述与频率无关和频率二次方依赖的声衰减.近年来，带有分数阶导数项的声波方程已被成功用于描述这一声衰减现象.基于课题组对声波衰减分数阶导数建模的研究，对已有的分数阶导数声波方程的研究进展及获得的成果做一个系统的综述，重点讨论这些模型的力学本构、统计力学解释等.简述了软物质中声波传播的时间分数阶导数唯象模型和本构模型，空间分数阶导数唯象模型和本构模型，并深入讨论了各种模型之间的联系与区别；介绍了分数阶导数声波模型在多孔介质中的成功应用，该部分内容涉及了均匀和非均匀多孔介质，刚性固体骨架和可变形固体骨架多孔介质等；通过空间分数阶扩散方程与Lévy稳定分布之间的联系，给出了频率幂律依赖指数的变化区间为[0，2]的统计力学解释.最后，讨论了声波传播耗散行为的分数阶导数建模领域仍然存在的问题，并对今后的研究方向进行了探讨和展望.
- 复杂介质 /
- 分数阶导数 /
- 声波 /
- 衰减 /
- 幂律依赖
Abstract: The existing experimental data indicate that the attenuations of acoustic waves propagating in complex media always exhibit a non-integer power-law dependence on frequency. Such phenomenon is di cult to be characterized by traditional damping wave equation or approximate thermo-viscous wave equation, which can only describe the frequency independent or frequency-squared dependent attenuation, respectively. With the dynamic development and wide applications of fractional calculus, wave equations with fractional derivative terms have been successfully applied to depicting the frequency dependent attenuation. Based on the research achievements of our group, this paper aims at presenting a review of the various fractional derivative wave equations, discussing the corresponding mechanical constitutive relationships and statistical interpretation, and laying the foundation for the in-depth study in the future. The time-and space-fractional derivative wave equations for soft matters are introduced, which can be classified into two groups:the constitutive models and the phenomenological models. The connections and di erences between such models are also discussed. Then, the successful applications of fractional derivative in modeling wave propagation in porous media are also summarized. The statistical interpretation for the power-law dependent exponent covering[0, 2] is presented via linking the space-fractional diffusion equation with Lévy stable distribution. Finally, the key problems in such area for future explorations are highlighted.
- complex media /
- fractional derivative /
- acoustic wave /
- attenuation /
- power-law dependent

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参考文献(100)

1 陈文,孙洪广,李西成. 力学与工程问题的分数阶导数建模. 北京:科学出版社,2010(Chen Wen, Sun Hongguang, Li Xicheng. Fractional Derivative Modeling in Mechanical and Engineering Problems. Beijing:Science Press, 2010(in Chinese))

2 庞国飞,陈文,张晓棣等. 复杂介质中扩散和耗散行为的分数阶导数唯象建模. 应用数学和力学,2015,36(11):1117-1134(Pang Guofei, Chen Wen, Zhang Xiaodi, et al. Fractional differential phenomenological modeling for diffusion and dissipation behaviors of complex media. Applied Mathematics and Mechanics, 2015, 36(11):1117-1134(in Chinese))

3 Foster FS, Hunt JW. Transmission of ultrasound beams through human tissue-focussing and attenuation studies. Ultrasound in Medicine & Biology, 1979, 5(3):257-268

4 D'Astous FT, Foster FS. Frequency dependence of ultrasound attenuation and backscatter in breast tissue. Ultrasound in Medicine & Biology, 1986, 12(10):795-808

5 Lin T, Ophir J, Potter G. Frequency-dependent ultrasonic differentiation of normal and diffusely diseased liver. The Journal of the Acoustical Society of America, 1987, 82(4):1131-1138

6 He P. Simulation of ultrasound pulse propagation in lossy media obeying a frequency power law. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1998, 45(1):114-125

7 Duck FA. Physical Properties of Tissues:A Comprehensive Reference Book. New York:Academic Press, 2013

8 Szabo TL, Wu J. A model for longitudinal and shear wave propagation in viscoelastic media. The Journal of the Acoustical Society of America, 2000, 107(5):2437-2446

9 Sams MS, Neep JP, Worthington MH, et al. The measurement of velocity dispersion and frequency-dependent intrinsic attenuation in sedimentary rocks. Geophysics, 1997, 62(5):1456-1464

10 Blackstock DT. Transient solution for sound radiated into a viscous fluid. The Journal of the Acoustical Society of America, 1967, 41(5):1312-1319

11 Szabo TL. Time domain wave equations for lossy media obeying a frequency power law. The Journal of the Acoustical Society of America, 1994, 96(1):491-500

12 Szabo TL. Causal theories and data for acoustic attenuation obeying a frequency power law. The Journal of the Acoustical Society of America, 1995, 97(1):14-24

13 Pritz T. Frequency power law of material damping. Applied Acoustics, 2004, 65(11):1027-1036

14 Waters KR, Hughes MS, Mobley J, et al. On the applicability of Kramers-Krönig relations for ultrasonic attenuation obeying a frequency power law. The Journal of the Acoustical Society of America, 2000, 108(2):556-563

15 Waters KR, Hughes MS, Mobley J, et al. Di erential forms of the Kramers-Kronig dispersion relations. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2003, 50(1):68-76

16 Waters KR, Mobley J, Miller JG. Causality-imposed (Kramers-Kronig) relationships between attenuation and dispersion. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 2005, 52(5):822-823

17 Horton Sr CW. Dispersion relationships in sediments and sea water. The Journal of the Acoustical Society of America, 1974, 55(3):547-549

18 Horton Sr CW. Comment on Kramers-Kronig relationship between ultrasonic attenuation and phase velocity. The Journal of the Acoustical Society of America, 1981, 70(4):1182

19 Nachman AI, Smith Ⅲ JF,Waag RC. An equation for acoustic propagation in inhomogeneous media with relaxation losses. The Journal of the Acoustical Society of America, 1990, 88(3):1584-1595

20 Lu JF, Hanyga A. Numerical modelling method for wave propagation in a linear viscoelastic medium with singular memory. Geophysical Journal International, 2004,159(2):688-702

21 Ginter S. Numerical simulation of ultrasound-thermotherapy combining nonlinear wave propagation with broadband soft-tissue absorption. Ultrasonics, 2000, 37(10):693-696

22 Rossikhin YA, Shitikova MV. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Applied Mechanics Reviews, 1997, 50(1):15-67

23 Yuan X, Borup D,Wiskin J, et al. Simulation of acoustic wave propagation in dispersive media with relaxation losses by using FDTD method with PML absorbing boundary condition. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 1999, 46(1):14-23

24 Podlubny I. Fractional Di erential Equations:An Introduction to Fractional Derivatives, Fractional Di erential Equations, to Methods of Their Solution and Some of Their Applications. New York:Academic Press, 1998

25 Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity:An Introduction to Mathematical Models. Singapore:World Scientific, 2010

26 Metzler R, Klafter J. The random walk's guide to anomalous diffusion:a fractional dynamics approach. Physics Reports, 2000, 339(1):1-77

27 Sokolov IM, Chechkin AV, Klafter J. Distributed-order fractional kinetics. arXiv preprint cond-mat/0401146, 2004

28 Hanyga A.An anisotropic Cole-Cole model of seismic attenuation. Journal of Computational Acoustics, 2003, 11(1):75-90

29 Chen W, Holm S. Modified Szabo's wave equation models for lossy media obeying frequency power law. The Journal of the Acoustical Society of America, 2003, 114(5):2570-2574

30 Liebler M, GinterS, Dreyer T, et al. Full wave modeling of therapeutic ultrasound:Efficient time-domain implementation of the frequency power-law attenuation. The Journal of the Acoustical Society of America, 2004, 116(5):2742-2750

31 Chen W, Zhang X, Cai X. A study on modified Szabo's wave equation modeling of frequency-dependent dissipation in ultrasonic medical imaging. Physica Scripta, 2009(T136):014014

32 Zhang X, Chen W, Zhang C. Modified szabo's wave equation for arbitrarily frequency-dependent viscous dissipation in soft matter with applications to 3D ultrasonic imaging. Acta Mechanica Solida Sinica, 2012, 25(5):510-519

33 Kelly JF, McGough RJ, Meerschaert MM. Analytical time-domain Green's functions for power-law media. The Journal of the Acoustical Society of America, 2008, 124(5):2861-2872

34 Meerschaert MM, Sikorskii A. Stochastic Models for Fractional Calculus. Berlin:Walter de Gruyter, 2012

35 Wismer MG. Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation. The Journal of the Acoustical Society of America, 2006, 120(6):3493-3502

36 Ochmann M, Makarov S. Representation of the absorption of nonlinear waves by fractional derivatives. The Journal of the Acoustical Society of America, 1993, 94(6):3392-3399

37 Coppens AB, Sanders JV. Finite-amplitude standing waves in rigidwalled tubes. The Journal of the Acoustical Society of America, 1968, 43(3):516-529

38 Caputo M, Mainardi F.A new dissipation model based on memory mechanism. Pure and Applied Geophysics, 1971, 91(1):134-147.

39 Rossikhin YA.Reflections on two parallel ways in the progress of fractional calculus in mechanics of solids. Applied Mechanics Reviews, 2010, 63(1):010701

40 Rossikhin YA, Shitikova MV. Application of fractional calculus for dynamic problems of solid mechanics:novel trends and recent results. Applied Mechanics Reviews, 2010, 63(1):010801

41 殷德顺,任俊娟,和成亮等. 一种新的岩土流变模型元件. 岩石力学与工程学报, 2007, 26(9):1899-1903(Yin Deshun, Ren Junjuan, He Chengliang, et al. A new rheological model element for geomaterials. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(9):1899-1903(in Chinese))

42 Nutting PG. A new general law of deformation. Journal of the Franklin Institute, 1921, 191(5):679-685

43 Blair GWS, Caffyn JE. An application of the theory of quasiproperties to the treatment of anomalous strain-stress relations. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1949, 40(300):80-94

44 Caputo M. Linear models of dissipation whose Q is almost frequency independent-Ⅱ. Geophysical Journal International, 1967, 13(5):529-539

45 Holm S, Sinkus R. A unifying fractional wave equation for compressional and shear waves. The Journal of the Acoustical Society of America, 2010, 127(1):542-548

46 Holm S, Näsholm SP, Prieur F, et al. Deriving fractional acoustic wave equations from mechanical and thermal constitutive equations. Computers & Mathematics with Applications, 2013, 66(5):621-629

47 Holm S, Näsholm SP.A causal and fractional all-frequency wave equation for lossy media. The Journal of the Acoustical Society of America, 2011, 130(4):2195-2202

48 Holm S, Näsholm SP. Comparison of fractional wave equations for power law attenuation in ultrasound and elastography. Ultrasound in Medicine & Biology, 2014, 40(4):695-703

49 Caputo M, Carcione JM, Cavallini F. Wave simulation in biologic media based on the Kelvin-Voigt fractional-derivative stress-strain relation. Ultrasound in Medicine & Biology, 2011, 37(6):996-1004

50 Zhao X, McGough RJ. Time-domain comparisons of power law attenuation in causal and noncausal time-fractional wave equations. The Journal of the Acoustical Society of America, 2016, 139(5):3021-3031

51 Konjik S, Oparnica L, Zorica D. Waves in fractional Zener type viscoelastic media. Journal of Mathematical Analysis and Applications, 2010, 365(1):259-268

52 Näsholm SP, Holm S. On a fractional Zener elastic wave equation. Fractional Calculus and Applied Analysis, 2013, 16(1):26-50

53 Näsholm SP, Holm S. Linking multiple relaxation, power-law attenuation, and fractional wave equations. The Journal of the Acoustical Society of America, 2011, 130(5):3038-3045

54 Buckingham MJ. Theory of acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments. The Journal of the Acoustical Society of America, 1997, 102(5):2579-2596

55 Buckingham MJ. Wave propagation, stress relaxation, and grain-tograin shearing in saturated, unconsolidated marine sediments. The Journal of the Acoustical Society of America, 2000, 108(6):2796-2815

56 Pandey V, Holm S. Connecting the grain-shearing mechanism of wave propagation in marine sediments to fractional calculus. arXiv preprint arXiv:1512.05336, 2015

57 Prieur F, Holm S. Nonlinear acoustic wave equations with fractional loss operators.The Journal of the Acoustical Society of America, 2011, 130(3):1125-1132

58 Chen W, Holm S. Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency.The Journal of the Acoustical Society of America, 2004, 115(4):1424-1430

59 张晓棣. 医学超声和振动中频率依赖耗散的分数阶导数模型研究.[博士论文]. 南京:河海大学,2012(Zhang Xiaodi. A study on fractional derivative models of frequency dependent dissipative behaviors in medical ultrasound and vibration.[PhD Thesis]. Nanjing, Hohai University, 2012(in Chinese))

60 TreebyBE, Cox BT. Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian. The Journal of the Acoustical Society of America, 2010, 127(5):2741-2748

61 庞国飞. 非局部声波耗散和阻尼振动的空间分数阶微积分黏弹性本构建模研究.[博士论文]. 南京:河海大学,2015(Pang Guofei. Space-fractional calculus viscoelastic constitutive models for describing non-local acoustic wave dissipation and vibration damping.[PhD Thesis]. Nanjing:Hohai University, 2015(in Chinese))

62 TreebyBE, Cox BT. Modeling power law absorption and dispersion in viscoelastic solids using a split-field and the fractional Laplaciana. The Journal of the Acoustical Society of America, 2014, 136(4):1499-1510

63 Treeby BE, Cox BT. k-Wave:MATLAB toolbox for the simulation and reconstruction of photoacoustic wave fields. Journal of Biomedical Optics, 2010, 15(2):021314-021314-12

64 Carcione JM, Cavallini F, Mainardi F, et al. Time-domain modeling of constant-Q seismic waves using fractional derivatives. Pure and Applied Geophysics, 2002, 159(7-8):1719-1736

65 Carcione JM. Theory and modeling of constant-Q P-and S-waves using fractional time derivatives. Geophysics, 2008, 74(4):T1-T11

66 Carcione JM. A generalization of the fourier pseudospectral method. Geophysics, 2010, 75(6):A53-A56

67 Zhu T. Time-reverse modelling of acoustic wave propagation in attenuating media. Geophysical Journal International, 2014, 197(1):483-494

68 Zhu T, Harris JM.Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians. Geophysics, 2014, 79(3):T105-T116

69 汪越胜,于桂兰,章梓茂等. 复杂界面(界面层)条件下的弹性波传播问题研究综述. 力学进展,2000, 30(3):378-390(Wang Yuesheng, Yu Guilan, Zhang Zimao, et al. Review on elastic wave propagation under complex interface (interface layer) conditions. Advances in Mechanics, 2000, 30(3):378-390(in Chinese))

70 Meerschaert MM, McGough RJ. Attenuated fractional wave equations with anisotropy. Journal of Vibration and Acoustics, 2014, 136(5):050902

71 Meerschaert MM, Straka P, Zhou Y, et al. Stochastic solution to a time-fractional attenuated wave equation. Nonlinear Dynamics, 2012, 70(2):1273-1281

72 Kilbas AA, Marichev OI, Samko SG. Fractional Integral and Derivatives (Theory and Applications). Gordon and Breach, Switzerland, 1993, 1(1993):1

73 Delany ME, Bazley EN. Acoustical properties of fibrous absorbent materials. Applied Acoustics, 1970, 3(2):105-116

74 Chen W, Hu S, Cai W. A causal fractional derivative model for acoustic wave propagation in lossy media. Archive of Applied Mechanics, 2016, 86(3):529-539

75 胡帅. 多孔材料的振动和黏性波耗散的分数阶导数建模.[博士论文]. 南京:河海大学,2013(Hu Shuai. Fractional derivative modeling for vibration and viscous wave dissipation of porous material.[PhD Thesis]. Nanjing:Hohai University, 2013(in Chinese))

76 Johnson DL, Koplik J, Dashen R. Theory of dynamic permeability and tortuosity in fluid-saturated porous media. Journal of Fluid Mechanics, 1987, 176:379-402

77 Allard JF, Champoux Y. New empirical equations for sound propagation in rigid frame fibrous materials. The Journal of the Acoustical Society of America, 1992, 91(6):3346-3353

78 Fellah ZEA, Depollier C. Transient acoustic wave propagation in rigid porous media:a time-domain approach. The Journal of the Acoustical Society of America, 2000, 107(2):683-688

79 Fellah ZE A, Fellah M, Lauriks W, et al. Direct and inverse scattering of transient acoustic waves by a slab of rigid porous material. The Journal of the Acoustical Society of America, 2003, 113(1):61-72

80 Fellah ZEA, Berger S, Lauriks W, et al. Measuring the porosity and the tortuosity of porous materials via reflected waves at oblique incidence. The Journal of the Acoustical Society of America, 2003, 113(5):2424-2433

81 Fellah ZEA, Depollier C, Berger S, et al. Determination of transport parameters in air-saturated porous materials via reflected ultrasonic waves. The Journal of the Acoustical Society of America, 2003, 114(5):2561-2569

82 Fellah ZEA, Chapelon JY, Berger S, et al. Ultrasonic wave propagation in human cancellous bone:Application of Biot theory. The Journal of the Acoustical Society of America, 2004, 116(1):61-73

83 Fellah ZEA, Sebaa N, Fellah M, et al. Ultrasonic characterization of air-saturated double-layered porous media in time domain. Journal of Applied Physics, 2010, 108(1):014909

84 Hanyga A, Lu JF. Wave field simulation for heterogeneous transversely isotropic porous media with the JKD dynamic permeability. Computational Mechanics, 2005,36(3):196-208

85 Lu JF, Hanyga A. Wave field simulation for heterogeneous porous media with singular memory drag force. Journal of Computational Physics, 2005, 208(2):651-674

86 Blanc E. Time-domain numerical modeling of poroelastic waves:the Biot-JKD model with fractional derivatives.[PhD thesis]. Marseille:Aix-Marseille Université. 2014

87 Blanc E, Chiavassa G, Lombard B. Wave simulation in 2D heterogeneous transversely isotropic porous media with fractional attenuation:a Cartesian grid approach. Journal of Computational Physics, 2014, 275:118-142

88 Blanc E, Chiavassa G, Lombard B. Biot-JKD model:simulation of 1D transient poroelastic waves with fractional derivatives. Journal of Computational Physics, 2013, 237:1-20

89 Chen W. Lévy stable distribution and[0, 2] power law dependence of acoustic absorption on frequency in various lossy media. Chinese Physics Letters, 2005, 22(10):2601

90 Saichev AI, Zaslavsky GM. Fractional kinetic equations:solutions and applications. Chaos:An Interdisciplinary Journal of Nonlinear Science, 1997, 7(4):753-764

91 Mandelbrot BB.The fractal geometry of nature. Macmillan. 1982

92 Sato KI.Lévy Processes and Infinitely Divisible Distributions. Cambridge university Press, 1999

93 Mandelbrot BB, Van Ness JW. Fractional Brownian motions, fractional noises and applications. SIAM Review, 1968, 10(4):422-437

94 Berry MV. Di ractals. Journal of Physics A:Mathematical and General, 1979, 12(6):781

95 Wyss W. Fractional noise. Foundations of Physics Letters, 1991, 4(3):235-246

96 Tatom FB.The relationship between fractional calculus and fractals. Fractals, 1995, 3(1):217-229

97 Yao K, Su WY, Zhou SP. On the connection between the order of fractional calculus and the dimensions of a fractal function. Chaos, Solitons & Fractals, 2005, 23(2):621-629

98 Liang YS, Su WY. The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus. Chaos, Solitons & Fractals, 2007, 34(3):682-692

99 Liang YS, SuWY. Connection between the order of fractional calculus and fractional dimensions of a type of fractal functions. Analysis in Theory and Applications, 2007, 23(4):354-362

100 Henry BI, Wearne SL. Fractional reaction-diffusion. Physica A:Statistical Mechanics and its Applications, 2000, 276(3):448-455

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