软黏土弹性模量衰减的分数阶模型
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摘要
通过研究表明,在循环荷载的作用下,软黏土的动弹性模量衰减规律可以分为急剧衰减阶段和缓慢衰减阶段,且缓慢衰减阶段的时间要明显长于急剧衰减阶段,而其缓慢衰减过程具有幂律衰减的特征,可以用分数阶导数方程模型来描述,其解为Mittag-Leffler分布的统计函数。观察分析可以发现,分数阶导数方程的Mittag-Leffler分布与软黏土的弹性模量缓慢衰减阶段能够较好的吻合,与目前常用对数衰减模型相比,分数阶导数模型所用参数都是两个但拟合的精度更高且适用范围更广。
Research has shown that under cyclic loading, the attenuation of dynamic modulus of soft clay can be divided into sharp attenuation and slow attenuation phases, and the time of slow attenuation phase is significantly longer than sharp attenuation phase.The slow attenuation phase which appears a power-law decay can be described by the fractional order derivative equation model whose solution is Mittag-Leffler distribution statistics function. Observation and analysis can be found that the fractional order derivative equation Mittag-Leffler distribution can fit the elastic modulus of soft clay attenuation better. Compared with the usual logarithmic model, the fractional derivative model uses two parameters, but the fitting precision is higher and the scope of application is wider.
Research has shown that under cyclic loading, the attenuation of dynamic modulus of soft clay can be divided into sharp attenuation and slow attenuation phases, and the time of slow attenuation phase is significantly longer than sharp attenuation phase.The slow attenuation phase which appears a power-law decay can be described by the fractional order derivative equation model whose solution is Mittag-Leffler distribution statistics function. Observation and analysis can be found that the fractional order derivative equation Mittag-Leffler distribution can fit the elastic modulus of soft clay attenuation better. Compared with the usual logarithmic model, the fractional derivative model uses two parameters, but the fitting precision is higher and the scope of application is wider.
引文
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[6]徐毅青,唐益群,任兴伟,王元东,叶桂明.地铁振动荷载作用隧道周围加固软黏土动弹性模量试验[J].工程力学, 2012,29(7):250-255.(XU Yiqing, TANG Yiqun, REN Xingwei,WANG Yuandong, YE Guiming. Dynamic elastic modulus test of soft clay around tunnel strengthened by subway vibration load[J].Engineering Mechanics, 2012, 29(7):250-255(in Chinese)).
[7] Liang Ying-jie, CHEN Wen, Magin R L. Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation[J]. Physica A:Statistical Mechanics and Its Applications, 2016, 453:327-335.
[8] Chen Wen,Zhang Xiao-di,Koroak D. Investigation on fractional and fractal derivative relaxation-oscillation models[J]. International J ournal of Nonlinear Sciences&Numerical Simulation,2010,11(1):3-10.
[9] Chen Wen, Liang Ying-jie, Hu Shuai, et al. Fractional derivative anomalous diffusion equation modeling prime number distribution[J].Fractional Calculus&Applied Analysis, 2015, 18(3):789-798.
[10]胡海波,王林.幂律分布研究简史[J].物理,2005,34(12):889-896.(HU Haibo, WANGLin. Brief history of power law distribution[J]. physics, 2005,34(12):889-896(in Chinese)).
[11] D.A. Murio, On the stable numerical evaluation of Caputo fractional derivatives. Comput. Math. Appl. 51(2006), 1539-1550.
[12] PodlubnyI.FractionalDifferentialEquations[M].San Diego:Academic Press,1999.
[2]叶耀东,朱合华,王如路.软土地铁运营隧道病害现状及成因分析[J].地下空间与工程学报,2007, 3(1):157-160+166.(YE Yaodong, ZHU Hehua, WANG Rulu. Analysis on the present situation and cause of tunnel diseases in soft land railway operation[J].Chinese Journal of Underground Space and Engineering,2007, 3(1):157-160+166(in Chinese)).
[3] TARUMIN, SUNAGAM.Behaviors of low embankments on soft grounds during train passage[C]//ProcJSCF.[S.l.]:[s.n.],1988,(400/Ⅲ-10):1-11.
[4] ASSIMAKI D, Kausel E, Wittle A. Model for dynamic shear modulus and damping for granular soils[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2000, 126(10):859-869.
[5] Liu Feiyu, Cai Yuanqiang, Xu Changjie, Wang Jun.Degradation of dynamic elastic modulus of soft clayunder cyclic loading[J]. Journal of Zhejiang University(Engineering Science), 2008, 42(9):1479-1483.
[6]徐毅青,唐益群,任兴伟,王元东,叶桂明.地铁振动荷载作用隧道周围加固软黏土动弹性模量试验[J].工程力学, 2012,29(7):250-255.(XU Yiqing, TANG Yiqun, REN Xingwei,WANG Yuandong, YE Guiming. Dynamic elastic modulus test of soft clay around tunnel strengthened by subway vibration load[J].Engineering Mechanics, 2012, 29(7):250-255(in Chinese)).
[7] Liang Ying-jie, CHEN Wen, Magin R L. Connecting complexity with spectral entropy using the Laplace transformed solution to the fractional diffusion equation[J]. Physica A:Statistical Mechanics and Its Applications, 2016, 453:327-335.
[8] Chen Wen,Zhang Xiao-di,Koroak D. Investigation on fractional and fractal derivative relaxation-oscillation models[J]. International J ournal of Nonlinear Sciences&Numerical Simulation,2010,11(1):3-10.
[9] Chen Wen, Liang Ying-jie, Hu Shuai, et al. Fractional derivative anomalous diffusion equation modeling prime number distribution[J].Fractional Calculus&Applied Analysis, 2015, 18(3):789-798.
[10]胡海波,王林.幂律分布研究简史[J].物理,2005,34(12):889-896.(HU Haibo, WANGLin. Brief history of power law distribution[J]. physics, 2005,34(12):889-896(in Chinese)).
[11] D.A. Murio, On the stable numerical evaluation of Caputo fractional derivatives. Comput. Math. Appl. 51(2006), 1539-1550.
[12] PodlubnyI.FractionalDifferentialEquations[M].San Diego:Academic Press,1999.