基于积分型分数导数本构方程的黏弹性土层中单桩的竖向复刚度与导纳研究
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摘要
建立了基于三维积分型本构方程的单桩在黏弹性土体的竖向振动微分方程。利用三维分数导数Kelvin黏弹性本构关系描述土体的应力应变关系,考虑分数导数和卷积的Fourier变换性质,利用势函数和分离变量法在频域内求解了分数导数黏弹性土层的竖向振动,并在此基础上求解了黏弹性土层中单桩的竖直振动。研究了土体本构模型参量对桩顶复刚度和桩顶导纳的影响。研究表明,分数导数的阶数对桩顶复刚度和桩顶导纳有较大的影响,土体黏性系数对桩顶复刚度的实部和桩顶导纳影响较大,而对桩顶复刚度虚部影响相对较小。
Differential equations for the vertical vibration of single piles in viscoelastic soil were established based on the three dimensional and integral constitutive equations. The stress-strain relationship of soil was described by the three dimensional fractional derivatives for Kelvin viscoelastic constitutive relationship. The vertical vibration of fractional derivative viscoelastic soil was solved by potential functions and method of separation variables in frequency range by considering the properties of Fourier transform of fractional derivative and convolution integral. The influence of constitutive model parameters on complex stiffness and admittance at pile head was investigated, the research indicated that the order of fractional derivative has great effect on the complex stiffness and admittance at pile head, and the influence of viscosity coefficient of soil on the real part of complex stiffness and admittance at pile head was greater, but the effect of viscosity coefficient of soil on the image part of complex stiffness was relatively minor.
Differential equations for the vertical vibration of single piles in viscoelastic soil were established based on the three dimensional and integral constitutive equations. The stress-strain relationship of soil was described by the three dimensional fractional derivatives for Kelvin viscoelastic constitutive relationship. The vertical vibration of fractional derivative viscoelastic soil was solved by potential functions and method of separation variables in frequency range by considering the properties of Fourier transform of fractional derivative and convolution integral. The influence of constitutive model parameters on complex stiffness and admittance at pile head was investigated, the research indicated that the order of fractional derivative has great effect on the complex stiffness and admittance at pile head, and the influence of viscosity coefficient of soil on the real part of complex stiffness and admittance at pile head was greater, but the effect of viscosity coefficient of soil on the image part of complex stiffness was relatively minor.
引文
[1]李军强,刘宏昭,王忠民.线性黏弹性本构方程及其动力学应用研究综述[J].振动与冲击,2005,24(2):116-121.
[2]Novak M.Dynamic stiffness and damping of piles[J].Canadian Geotechnical Journal,1974,11(4):574-598.
[3]胡昌斌,黄晓明.成层黏弹性土中桩土耦合纵向振动时域响应研究[J].地震工程与工程振动,2006,26(4):205-211.
[4]周绪红,蒋建国,皱银生.黏弹性介质中考虑轴力作用时桩的动力分析[J].土木工程学报,2005,38(2):87-91.
[5]杨骁,潘元.饱和黏弹性土层中端承桩纵向振动的轴对称解析解[J].应用数学和力学,2010,31(2):180-190.
[6]杨骁,王琛.饱和黏弹性土层中悬浮桩的纵向振动[J].应用力学学报,2009,26(3):490-494.
[7]李强.饱和土中端水桩非完全黏结下的竖向振动特性[J].水利学报,2007,38(3):349-354.
[8]Timoshenko S P.Theory of elasticity[M].McGraw-Hill Book,New York,1934.
[9]杨挺青,罗文波,徐平,等.黏弹性理论与应用[M].北京:科学出版社,2004.
[10]刘林超,杨骁.竖向集中力作用下分数导数型半无限体黏弹性地基变形分析[J].工程力学,2009,26(1):13-17.
[11]朱鸿鹄,刘林超,叶肖伟.分数导数型黏弹性地基上巨型板的受荷响应[J].应用基础与工程科学学报,2011,19(2):271-277.
[12]Miller K S,Ross B.An introduction to the fractional calculus and fractional differential equations[M].New York,Wiley,1993.
[13]刘林超,杨骁.分数导数模型描述的饱和土桩纵向振动分析[J].岩土力学,2011,32(2):526-532.
[2]Novak M.Dynamic stiffness and damping of piles[J].Canadian Geotechnical Journal,1974,11(4):574-598.
[3]胡昌斌,黄晓明.成层黏弹性土中桩土耦合纵向振动时域响应研究[J].地震工程与工程振动,2006,26(4):205-211.
[4]周绪红,蒋建国,皱银生.黏弹性介质中考虑轴力作用时桩的动力分析[J].土木工程学报,2005,38(2):87-91.
[5]杨骁,潘元.饱和黏弹性土层中端承桩纵向振动的轴对称解析解[J].应用数学和力学,2010,31(2):180-190.
[6]杨骁,王琛.饱和黏弹性土层中悬浮桩的纵向振动[J].应用力学学报,2009,26(3):490-494.
[7]李强.饱和土中端水桩非完全黏结下的竖向振动特性[J].水利学报,2007,38(3):349-354.
[8]Timoshenko S P.Theory of elasticity[M].McGraw-Hill Book,New York,1934.
[9]杨挺青,罗文波,徐平,等.黏弹性理论与应用[M].北京:科学出版社,2004.
[10]刘林超,杨骁.竖向集中力作用下分数导数型半无限体黏弹性地基变形分析[J].工程力学,2009,26(1):13-17.
[11]朱鸿鹄,刘林超,叶肖伟.分数导数型黏弹性地基上巨型板的受荷响应[J].应用基础与工程科学学报,2011,19(2):271-277.
[12]Miller K S,Ross B.An introduction to the fractional calculus and fractional differential equations[M].New York,Wiley,1993.
[13]刘林超,杨骁.分数导数模型描述的饱和土桩纵向振动分析[J].岩土力学,2011,32(2):526-532.
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