Fractal ladder models and power law wave equations
Fractal ladder models and power law wave equations
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DOI：
10.1121/1.3204304
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发表时间：
2009-10-01
影响因子：
2.4
通讯作者：
McGough, Robert J.
中科院分区：
文献类型：
--
作者：
Kelly, James F.;McGough, Robert J.
哺乳动物组织中的超声衰减系数近似为频率依赖的幂律，频率小于100 MHz。为了描述软组织中的这种幂律行为，提出了一种分层分形网络模型。组织的粘弹性和自相似特性被捕获的本构方程的基础上的集总参数无限梯形拓扑结构，包括交替弹簧和阻尼器。在低频极限下，该梯形网络产生具有时间分数阶导数的应力-应变本构方程。通过结合线性化守恒原理和绝热状态方程的本构方程，分数阶偏微分方程，描述幂律衰减推导。结果表明，衰减系数呈幂律分布，其指数范围在1 ~ 2之间，而相速度符合Kramers-Kronig关系。分形梯形模型相比，公布的衰减系数数据，从而提供等效的集总参数。(C)2009年，美国声学学会。[DOI：10.1121/1.3204304]
The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers-Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters. (C) 2009 Acoustical Society of America. [DOI: 10.1121/1.3204304]