﻿ 复杂介质中任意阶频率依赖耗散声波的分数阶导数模型<sup>1)</sup>
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Cai Wei, Chen Wen. FRACTIONAL DERIVATIVE MODELING OF FREQUENCY-DEPENDENT DISSIPATIVE MECHANISM FORWAVE PROPAGATION IN COMPLEX MEDIA1)[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(6): 1265-1280. DOI: 10.6052/0459-1879-16-186.
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### 基金项目

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

### 通讯作者

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

### 文章历史

2016-07-08 收稿
2016-08-24 录用
2016-08-26网络版发表

 $\alpha =\alpha _0 \omega ^\eta$ (1)

Ochmann等[36]根据幂律衰减现象推导得到了一类广义的Burgers方程

 $\dfrac{1}{c_0^2 }\dfrac{\partial ^2P}{\partial t^2} + \left( {\eta + 1} \right)\nabla \left( {\nabla ^2P} \right) + \dfrac{2D}{c_0 }\dfrac{\partial ^{\eta + 1}P}{\partial t^{\eta + 1}} = \nabla ^2P$ (9)

 $\dfrac{\partial ^2P}{\partial t^2} + c_0^2 \left( {\gamma + 1} \right) \dfrac{\partial ^3P}{\partial x^3} + 2Dc_0 \dfrac{\partial ^{\gamma + 1}P}{\partial t^{\gamma + 1}} = c_0^2 \dfrac{\partial ^2P}{\partial x^2}$ (10)

1.1.2 本构模型

Caputo[44]早在1967年即从分数阶Kelvin黏弹性本构方程推导出分数阶Kelvin波动方程，其形式如下

 $\dfrac{1}{c_0^2 }\dfrac{\partial^2 P}{\partial t^2} - \tau _\sigma ^\eta \dfrac{\partial ^\eta }{\partial t^\eta }\left( {\nabla ^2P} \right) = \nabla ^2P$ (11)

 $\begin{gathered} {\nabla ^2}\psi = \frac{1}{{c_0^2}}\frac{{{\partial ^2}\psi }}{{\partial {t^2}}} - \\ \Gamma \left( {1 - \eta } \right)\left( {\frac{{{\lambda _{\text{p}}}}}{{{\rho _0}c_0^2}} + \frac{4}{3}\frac{{{\gamma _{\text{s}}}}}{{{\rho _0}c_0^2}}} \right){\tau ^{\eta - 1}}\frac{{{\partial ^\eta }}}{{\partial {t^\eta }}}{\nabla ^2}\psi \\ \end{gathered}$ (14)

 $\dfrac{\partial ^{2 - \eta }{\boldsymbol {A}}}{\partial t^{2 - \eta }} = \Gamma \left( {1 - \eta } \right)\dfrac{\gamma _{\rm s} }{\rho _0 }\tau ^{\eta - 1}\nabla ^2 {\boldsymbol {A}}$ (15)

 $\dfrac{1}{c_0^2 }\dfrac{\partial^2 P }{\partial t^2} - \tau _{\rm p}^{\eta _1 } \dfrac{\partial ^{\eta _1 }}{\partial t^{\eta _1 }}\left( {\nabla ^2P} \right) -\tau _s^{\eta _2 } \dfrac{\partial ^{\eta _2 }}{\partial t^{\eta _2 }}\left( {\nabla ^2P} \right) = \nabla ^2P$ (16)

Prieur等[57]采用分数阶Kelvin本构模型，与分数阶热传导模型相结合，得到的波动方程如下

 $\begin{gathered} {\nabla ^2}P - \frac{1}{{c_0^2}}\frac{{{\partial ^2}P}}{{\partial {t^2}}} + {L_{\text{v}}}\frac{{{\partial ^{\eta - 1}}}}{{\partial {t^{\eta - 1}}}}{\nabla ^2}P - \frac{{{L_{\text{t}}}}}{{c_0^2}}\frac{{{\partial ^{\eta + 1}}P}}{{\partial {t^{\eta + 1}}}} = \hfill \\ \;\;\;\;\;\;\; - \frac{\beta }{{{\rho _0}c_0^4}}\frac{{{\partial ^2}P}}{{\partial {t^2}}} \hfill \\ \end{gathered}$ (17)

1.2 空间分数阶声波方程

 $\dfrac{1}{c_0^2 }\dfrac{\partial^2 P}{\partial t^2} + \dfrac{2\alpha _0 }{c_0^{1 - \eta } }\dfrac{\partial }{\partial t}\left( { - \nabla^2 } \right)^{\eta / 2}P = \nabla ^2P$ (18)

 图 6 4种不同声波模型的比较[59] Figure 6 The comparison between different acoustic wave models[59]

 $\begin{gathered} \frac{1}{{c_0^2}}\frac{{{\partial ^2}P}}{{\partial {t^2}}} + \frac{{2{\alpha _0}}}{{c_0^{1 - \eta }}}\frac{\partial }{{\partial t}}{\left( { - {\nabla ^2}} \right)^{\eta /2}}P + \hfill \\ \;\;\;\;\;\;2{\alpha _0}c_0^\eta \tan \frac{{\pi \eta }}{2}{\left( { - {\nabla ^2}} \right)^{(\eta + 1)/2}}P = {\nabla ^2}P \end{gathered}$ (19)

 ${{\text{I}}^\gamma }u\left( x \right) = \left\{ \begin{gathered} \int {\frac{A}{{\Gamma \left( l \right)}}{{\left| {x - y} \right|}^{ - n + \gamma \left( l \right)}}u\left( y \right){\text{d}}y} \hfill \\ \;\;\;\;\;\;\;\left| {x - y} \right| \leqslant l{\mkern 1mu}, \gamma < n \hfill \\ 0, \;\;\;\;\;\left| {x - y} \right| > l \end{gathered} \right.$

Saichev等[90]指出，为了保证概率密度函数为正，Lévy分布的指数必须在(0, 2]之间.也就是说，在$\eta$大于2时，材料在统计上来说是不稳定的；当$\eta$为0时，此时的耗散是与频率无关的.如此，即从统计上解释了$\eta$的变化范围.

 $\eta = \dfrac{\ln \left( {\alpha / \alpha _0 } \right)}{\ln \omega }$ (33)

4 结论和展望

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FRACTIONAL DERIVATIVE MODELING OF FREQUENCY-DEPENDENT DISSIPATIVE MECHANISM FORWAVE PROPAGATION IN COMPLEX MEDIA1)
Cai Wei, Chen Wen2)
Institute of Soft Matter Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 211100, China
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.
Key words: complex media    fractional derivative    acoustic wave    attenuation    power-law dependent