基于DQEM的分层流体饱和热弹性多孔介质轴对称问题的动力响应分析
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摘要
研究了分层不可压流体饱和热弹性多孔介质轴对称问题的动力响应问题,基于多孔介质理论(PMT),给出了该问题的数学模型.在空间域内采用微分求积单元法(DQEM)设置离散的控制微分方程、边界条件和连接条件,在时间域内采用二阶向后差分格式处理时间导数.在离散化的初始条件下,运用Newton-Raphson法进行迭代求解,得到各离散点处未知物理量的数值结果.研究表明:该方法有效、可靠,且具备精度较高、计算量较小、数值稳定等优点.
In this paper,the dynamic response of axisymmetric problem for layered incompressible fluid-saturated thermo-elastic porous media is studied.Based on the porous media theory(PMT),the mathematical model of the problem is given.The differential quadrature element method(DQEM) is used to set the discrete differential equations,boundary conditions and connected conditions on the space domain.The second-order backward difference scheme is used to process the time derivative on the time domain.Newton-Raphson method is used to solve the discrete problem iteratively and get the numerical results of unknown physical quantities at discrete point.The results show that the proposed method is effective and reliable,and it has the advantages of high precision,small amount of calculation,stable value and so on.
In this paper,the dynamic response of axisymmetric problem for layered incompressible fluid-saturated thermo-elastic porous media is studied.Based on the porous media theory(PMT),the mathematical model of the problem is given.The differential quadrature element method(DQEM) is used to set the discrete differential equations,boundary conditions and connected conditions on the space domain.The second-order backward difference scheme is used to process the time derivative on the time domain.Newton-Raphson method is used to solve the discrete problem iteratively and get the numerical results of unknown physical quantities at discrete point.The results show that the proposed method is effective and reliable,and it has the advantages of high precision,small amount of calculation,stable value and so on.
引文
[1] BIOT M A.Theory of elasticity and consolidation for a porous anisotropic solid [J].Journal of Applied Physics,1955,26:182-185.
[2] CARTER J R,SAVVIDOU C.Consolidation around a spherical heat source [J].International Journal of Solids and Structures,1984,20:1079-1090.
[3] CUI Y J,SULTAN N,DELAGE P.A thermomechanical model for saturated clays [J].Canadian Geotechnical Journal,2000,37:607-620.
[4] 刘干斌,姚海林,杨洋,等.考虑热-水-力耦合效应多孔弹性地基的动力响应 [J].岩土力学,2007,28(9):1784-1795.LIU G B,YAO H L,YANG Y,et al.Coupling thermo-hydro-mechanical dynamic response of a porous elastic medium [J].Rock and Soil Mechanics,2007,28(9):1784-1795.
[5] 白冰.循环温度荷载作用下饱和多孔介质热-水-力耦合响应 [J].工程力学,2007,24(5):87-92.BAI B.Thermo-hydro-mechanical response of saturated porous media under cycle thermal loading [J].Engineering Mechanics,2007,24(5):87-92.
[6] 戴清晨,何录武.热局部非平衡条件下含柱形空洞横观各向同性饱和多孔介质的热应力分析 [J].力学季刊,2014,35(1):1-9.DAI Q C,HE L W.Thermal stresses around a cylindrical hole in a transversely isotropic poroelastic medium considering local thermal non-equilibrium [J].Chinese Quarterly of Mechanics,2014,35(1):1-9.
[7] de BOER R.Theoretical poroelasticity:a new approach [J].Chaos Solitons & Fractals,2005,25(4):861-878.
[8] de BOER R,KOWALSKI S J.Thermodynamics of fluid-saturated porous media with a phase change [J].Acta Mechanica,1995,109(1/2/3/4):167-189.
[9] de BOER R,EHLERS W,LIU Z.One-dimensional transient wave propagation in fluid-saturated incompressible porous media [J].Archive of Applied Mechanics,1993,63(1):59-72.
[10] 刘占芳,姜乃斌,李思平.饱和多孔介质一维瞬态波动问题的解析分析 [J].工程力学,2006,23(7):19-24.LIU Z F,JIANG N B,LI S P.An analysis on one-dimensional transient wave motion in saturated porous media [J].Engineering Mechanics,2006,23(7):19-24.
[11] 李向约,李向维.饱和多孔介质的热固结理论 [J].固体力学学报,1990,11(4):330-338.LI X Y,LI X W.Theory of thermo-consolidation for saturated porous elastic media [J].Acta Mechanica Solida Sinica,1990,11(4):330-338.
[12] HE L W,JIN Z H.A local thermal non-equilibrium poroelastic theory for fluid saturated [J].Journal of Thermal Stresses,2010,33:799-813.
[13] YANG X.Gurtin-type variational principles for dynamics of a non-local thermal equilibrium saturated porous medium [J].Acta Mechanica Solida Sinica,2005,18(1):37-45.
[14] 朱媛媛,胡育佳,程昌钧,等.基于DQM的空间轴对称流体饱和多孔热弹性柱体动力学特性研究 [J].振动与冲击,2017,36(23):83-91.ZHU Y Y,HU Y J,CHENG C J,et al.The study on dynamic characteristics for a space-axisymmetrical fluid-saturated porous thermo-elastic cylinder based on DQM [J].Journal of Vibration and Shock,2017,36(23):83-91.
[15] 严俊,魏迎奇,蔡红,等.非饱和多孔介质水-热-力耦合数学模型研究 [J].水利学报,2014(增刊2):152-160.YAN J,WEI Y Q,CAI H,et al.A mathematical thermal hydraulic-mechanical coupling model for unsaturated porous media [J].Journal of Hydraulic Engineering,2014(Suppl.2):152-160.
[16] BELLMAN R E,CASTI J.Differential quadrature and long term integration [J].Journal of Mathematical Analysis & Applications,1970,34(2):235-238.
[17] BELLMAM R E,KASHEF B G,CASTI J.Differential quadrature:a technique for the rapid solution of nonlinear partial differential equations [J].Journal of Computational Physics,1972,10(1):40-52.
[18] ZHU Y Y,HU Y J,CHENG C J.DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media [J].Acta Mechanica,2013,224(9):1977-1998.
[19] 聂国隽,仲政.用微分求积法求解梁的弹塑性问题 [J].工程力学,2005,22(1):59-62.NIE G J,ZHONG Z.Elasto-plastic analysis of beams by differential quadrature method [J].Engineering Mechanics,2005,22(1):59-62.
[2] CARTER J R,SAVVIDOU C.Consolidation around a spherical heat source [J].International Journal of Solids and Structures,1984,20:1079-1090.
[3] CUI Y J,SULTAN N,DELAGE P.A thermomechanical model for saturated clays [J].Canadian Geotechnical Journal,2000,37:607-620.
[4] 刘干斌,姚海林,杨洋,等.考虑热-水-力耦合效应多孔弹性地基的动力响应 [J].岩土力学,2007,28(9):1784-1795.LIU G B,YAO H L,YANG Y,et al.Coupling thermo-hydro-mechanical dynamic response of a porous elastic medium [J].Rock and Soil Mechanics,2007,28(9):1784-1795.
[5] 白冰.循环温度荷载作用下饱和多孔介质热-水-力耦合响应 [J].工程力学,2007,24(5):87-92.BAI B.Thermo-hydro-mechanical response of saturated porous media under cycle thermal loading [J].Engineering Mechanics,2007,24(5):87-92.
[6] 戴清晨,何录武.热局部非平衡条件下含柱形空洞横观各向同性饱和多孔介质的热应力分析 [J].力学季刊,2014,35(1):1-9.DAI Q C,HE L W.Thermal stresses around a cylindrical hole in a transversely isotropic poroelastic medium considering local thermal non-equilibrium [J].Chinese Quarterly of Mechanics,2014,35(1):1-9.
[7] de BOER R.Theoretical poroelasticity:a new approach [J].Chaos Solitons & Fractals,2005,25(4):861-878.
[8] de BOER R,KOWALSKI S J.Thermodynamics of fluid-saturated porous media with a phase change [J].Acta Mechanica,1995,109(1/2/3/4):167-189.
[9] de BOER R,EHLERS W,LIU Z.One-dimensional transient wave propagation in fluid-saturated incompressible porous media [J].Archive of Applied Mechanics,1993,63(1):59-72.
[10] 刘占芳,姜乃斌,李思平.饱和多孔介质一维瞬态波动问题的解析分析 [J].工程力学,2006,23(7):19-24.LIU Z F,JIANG N B,LI S P.An analysis on one-dimensional transient wave motion in saturated porous media [J].Engineering Mechanics,2006,23(7):19-24.
[11] 李向约,李向维.饱和多孔介质的热固结理论 [J].固体力学学报,1990,11(4):330-338.LI X Y,LI X W.Theory of thermo-consolidation for saturated porous elastic media [J].Acta Mechanica Solida Sinica,1990,11(4):330-338.
[12] HE L W,JIN Z H.A local thermal non-equilibrium poroelastic theory for fluid saturated [J].Journal of Thermal Stresses,2010,33:799-813.
[13] YANG X.Gurtin-type variational principles for dynamics of a non-local thermal equilibrium saturated porous medium [J].Acta Mechanica Solida Sinica,2005,18(1):37-45.
[14] 朱媛媛,胡育佳,程昌钧,等.基于DQM的空间轴对称流体饱和多孔热弹性柱体动力学特性研究 [J].振动与冲击,2017,36(23):83-91.ZHU Y Y,HU Y J,CHENG C J,et al.The study on dynamic characteristics for a space-axisymmetrical fluid-saturated porous thermo-elastic cylinder based on DQM [J].Journal of Vibration and Shock,2017,36(23):83-91.
[15] 严俊,魏迎奇,蔡红,等.非饱和多孔介质水-热-力耦合数学模型研究 [J].水利学报,2014(增刊2):152-160.YAN J,WEI Y Q,CAI H,et al.A mathematical thermal hydraulic-mechanical coupling model for unsaturated porous media [J].Journal of Hydraulic Engineering,2014(Suppl.2):152-160.
[16] BELLMAN R E,CASTI J.Differential quadrature and long term integration [J].Journal of Mathematical Analysis & Applications,1970,34(2):235-238.
[17] BELLMAM R E,KASHEF B G,CASTI J.Differential quadrature:a technique for the rapid solution of nonlinear partial differential equations [J].Journal of Computational Physics,1972,10(1):40-52.
[18] ZHU Y Y,HU Y J,CHENG C J.DQEM for analyzing dynamic characteristics of layered fluid-saturated porous elastic media [J].Acta Mechanica,2013,224(9):1977-1998.
[19] 聂国隽,仲政.用微分求积法求解梁的弹塑性问题 [J].工程力学,2005,22(1):59-62.NIE G J,ZHONG Z.Elasto-plastic analysis of beams by differential quadrature method [J].Engineering Mechanics,2005,22(1):59-62.