衰变热源作用下饱和多孔介质热固结问题的扩展精细积分法
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摘要
热源作用下饱和多孔介质热固结效应是土木及能源工程领域的一个重要课题.由于问题的复杂性,已有的研究大多将介质假定为均匀各向同性,且将热源假定为恒定强度.实际工程中,天然饱和多孔介质常表现出明显的分层特性,热源强度也存在衰变性,为此本工作采用扩展精细积分法对衰变热源作用下层状饱和多孔介质的热固结问题进行研究.借助于积分变换,将饱和多孔介质热固结问题的偏微分方程转化为变换域内的常微分方程;然后对饱和多孔介质微层元进行合并消元,并结合边界条件,推导出衰变热源作用下层状饱和多孔介质热固结问题在积分变换域内的扩展精细积分解;对所得解答进行相应的数值积分逆变换,可获得所求温度、超静孔压及竖向位移在物理域内的解答.基于上述求解过程,编制相应的计算程序进行数值计算,通过与已有文献对比,验证本文扩展精细积分法在求解层状饱和多孔介质热固结问题中的适应性和正确性;最后通过几组算例,分析热源衰变周期、热源埋深及介质的成层性对热固结效应的影响.结果表明:热源衰变周期对温度和超静孔压的峰值、以及达到峰值的时间均有明显影响,衰变周期越长,二者峰值均越大,且达到峰值所需时间越长;热源埋深对超静孔压及竖向位移变化影响显著,深埋热源作用时热源两侧竖向位移呈对称分布,而浅埋热源两侧则无此现象;饱和多孔介质的分层特性对热固结效应影响明显.
The thermal consolidation of saturated porous media subjected to a heat source is an important subject in civil engineering and energy engineering. For the complexity of the problem, the porous media are usually treated as homogeneous isotropic media and the heat source is assumed to be a heat source with constant strength in the existing studies.In engineering practice,natural saturated porous media usually show obvious layered characteristics and the heat source is decaying with time.In this case,the extended precise integration method(EPIM)is presented in this study to investigate the thermal consolidation problems of layered saturated porous media subjected to a decaying heat source.The partial differential equations are reduced to ordinary ones by means of the integral transform techniques.Combining the adjacent layer elements and considering the boundary conditions,the EPIM solutions in the transformed domain of the problems are deduced.With the aid of corresponding numerical integral inversion,the temperatures,excess pore pressures and vertical displacements in the physical domain are obtained.A numerical example with the corresponding calculation program is performed to compare with the existing results,which confirm the applicability and validity of the presented method in dealing with the thermal consolidation problems of layered saturated porous media.Finally,numerical examples are carried out to analyse the influence of the heat source’s half-life and buried depth,as well as the stratification of medium on the thermal consolidation behaviour.Numerical results show that:the decay period of heat sources has significant influence on the peak values and peak time of temperature and excess pore pressure,the longer the decay period,the greater the peak values and the longer the peak time of temperature and excess pore pressure;burial depths have obvious influence on the variations of excess pore pressure and vertical displacement,the evolutions of vertical displacements against time on both side of the deeply buried heat source are symmetrical,while there is no such phenomenon for the shallow heat source;stratification characteristics of the saturated porous media shows prominent effects on the thermal consolidation.
The thermal consolidation of saturated porous media subjected to a heat source is an important subject in civil engineering and energy engineering. For the complexity of the problem, the porous media are usually treated as homogeneous isotropic media and the heat source is assumed to be a heat source with constant strength in the existing studies.In engineering practice,natural saturated porous media usually show obvious layered characteristics and the heat source is decaying with time.In this case,the extended precise integration method(EPIM)is presented in this study to investigate the thermal consolidation problems of layered saturated porous media subjected to a decaying heat source.The partial differential equations are reduced to ordinary ones by means of the integral transform techniques.Combining the adjacent layer elements and considering the boundary conditions,the EPIM solutions in the transformed domain of the problems are deduced.With the aid of corresponding numerical integral inversion,the temperatures,excess pore pressures and vertical displacements in the physical domain are obtained.A numerical example with the corresponding calculation program is performed to compare with the existing results,which confirm the applicability and validity of the presented method in dealing with the thermal consolidation problems of layered saturated porous media.Finally,numerical examples are carried out to analyse the influence of the heat source’s half-life and buried depth,as well as the stratification of medium on the thermal consolidation behaviour.Numerical results show that:the decay period of heat sources has significant influence on the peak values and peak time of temperature and excess pore pressure,the longer the decay period,the greater the peak values and the longer the peak time of temperature and excess pore pressure;burial depths have obvious influence on the variations of excess pore pressure and vertical displacement,the evolutions of vertical displacements against time on both side of the deeply buried heat source are symmetrical,while there is no such phenomenon for the shallow heat source;stratification characteristics of the saturated porous media shows prominent effects on the thermal consolidation.
引文
1 Delage P,Sultan N,Cui YJ.On the thermal consolidation of Boom clay.Canadian Geotechnical Journal,2000,37(2):343-354
2王铁行,李宁,谢定义.土体水热力耦合问题研究意义、现状及建议.岩土力学,2005,26(3):488-493(Wang Tiehang,Li Ning,Xie Dingyi.Necessity and means in research on soil coupled heatmoisture-stress issues.Rock and Soil Mechanics,2005,26(3):488-493(in Chinese))
3蒋中明,Dashnor H.核废料贮存库围岩体热响应耦合场研究.岩土工程学报,2006,28(8):953-956(Jiang Zhongming,Dashnor H.Studies on coupled field of thermal response in rock mass of nuclear waste repository.Chinese Journal of Geotechnical Engineering,2006,28(8):953-956(in Chinese))
4 Bai B,Guo LJ,Han S.Pore pressure and consolidation of saturated silty clay induced by progressively heating/cooling.Mechanics of Materials,2014,75:84-94
5 Ai ZY,Wang LJ.Axisymmetric thermal consolidation of multilayered porous thermoelastic media due to a heat source.International Journal for Numerical and Analytical Methods in Geomechanics,2015,39(17):1912-1931
6 Biot MA.General theory of three-dimensional consolidation.Journal of Applied Physics,1941,12(2):155-164
7 Biot MA.Thermoelasticity and irreversible thermodynamics.Journal of Applied Physics,1956,27(3):240-253
8 Booker JR,Savvidou C.Consolidation around a spherical heat source.International Journal of Solids and Structures,1984,20(11-12):1079-1090
9 Savvidou C,Booker JR.Consolidation around a heat source buried deep in a porous thermoelastic medium with anisotropic flow properties.International Journal for Numerical and Analytical Methods in Geomechanics,1989,13(1):75-90
10 Mctigue DF.Thermoelastic response of fluid-saturated porous rock.Journal of Geophysical Research Atmospheres,1986,91(B9):9533-9542
11 Bai M,Abousleiman Y.Thermoporoelastic coupling with application to consolidation.International Journal for Numerical and Analytical Methods in Geomechanics,1997,21(2):121-132
12白冰.岩土介质非稳态热固结耦合问题的热源函数法.力学学报,2004,36(4):427-434(Bai Bing.Heat source function method for coupling analyses of thermal consolidation in saturated soil.Acta Mechanica Sinica,2004,36(4):427-434(in Chinese))
13郑荣跃,刘干斌,梧松.半空间饱和土内置点载荷作用下的热弹性波动.力学学报,2008,40(3):413-420(Zhen Rongyue,Liu Ganbin,Wu Song.Coupling thermo-hydro-mechanical dynamic response of saturated soil subjected to internal excitation.Chinese Journal of Theoretical and Applied Mechanics,2008,40(3):413-420(in Chinese))
14吴瑞潜,谢康和,程永锋.变载荷下饱和土一维热固结解析理论.浙江大学学报(工学版),2009,43(8):1532-1537(Wu Ruiqian,Xie Kanghe,Cheng Yongfeng.Analytical theory for one-dimensional thermal consolidation of saturated soil under time-dependent loading.Journal of Zhejiang University(Engineering Science),2009,43(8):1532-1537(in Chinese))
15 Lu JCC,Lin F.Thermal consolidation of a poroelastic full space subjected to a decaying point heat source//Proceedings of the 2nd International ISCM Symposium and the 12nd International EPMESCConference,2010:407-412
16 Selvadurai APS,Suvorov AP.Thermo-poromechanics of a fluidfilled cavity in a fluid-saturated geomaterial//Proceedings of the royal society a mathematical physical and engineering sciences,2014,470(2163):20130634
17 Giraud A,Homand F,Rousset G.Thermoelastic and thermoplastic response of a double-layer porous space containing a decaying heat source.International Journal for Numerical and Analytical Methods in Geomechanics,1998,22(2):133-149
18白冰.变温度载荷作用下半无限成层饱和介质的热固结分析.应用数学和力学,2006,27(11):1341-1348(Bai Bing.Thermal consolidation of layered porous half-space to variable thermal loading.Applied Mathematics and Mechanics,2006,27(11):1341-1348(in Chinese))
19 Yue ZQ,Yin JH.Backward transfer-matrix method for elastic analysis of layered solids with imperfect bonding.Journal of Elasticity,1998,50(2):109-128
20艾智勇,吴超.渗透各向异性可压缩地基固结的平面应变分析.力学学报,2009,41(5):801-807(Ai Zhiyong,Wu Chao.Analysis on plane strain consolidation of a multi-layered soil with anisotropic permeability and compressbility constituents.Chinese Journal of Theoretical and Applied Mechanics,2009,41(5):801-807(in Chinese))
21赵宇昕,陈少林.关于传递矩阵法分析饱和成层介质响应问题的讨论.力学学报,2016,48(5):1145-1158(Zhao Yunxin,Chen Shaolin.Discussion on the matrix propagator method to analyze the response of saturated layered media.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1145-1158(in Chinese))
22 Booker JR,Small JC.Finite layer analysis of consolidation.I.International Journal for Numerical and Analytical Methods in Geomechanics,1982,6(2):151-171
23宰金珉,梅国雄.有限层法求解三维比奥固结问题.岩土工程学报,2002,24(1):31-33(Zai Jinmin,Mei Guoxiong.Finite layer analysis of three dimensional Biot consolidation.Chinese Journal of Geotechnical Engineering,2002,24(1):31-33(in Chinese))
24钟阳,耿立涛.多层弹性平面问题解的精确刚度矩阵法.岩土力学,2008,29(10):2829-2832(Zhong Yang,Geng Litao.Explicit solution of multiplayer elastic plane by exact stiffness matrix method.Rock and Soil Mechanics,2008,29(10):2829-2832(in Chinese))
25艾智勇,曹国军,成怡冲.平面应变Biot固结的解析层元.力学学报,2012,44(2):401-407(Ai Zhiyong,Cao Guojun,Cheng Yichong.Analytical layer-element of plane strain Biot’s consolidation.Chinese Journal of Theoretical and Applied Mechanics,2012,44(2):401-407(in Chinese))
26艾智勇,王路君,曾凯.稳定温度场下层状路面体系的解析层元解.同济大学学报(自然科学版),2014,42(11):1665-1669(Ai Zhiyong,Wang Lujun,Zeng Kai.Analytical layer-element solution for layered pavement in stable temperature field.Journal of Tongji University(Natural Science),2014,42(11):1665-1669(in Chinese))
27钟万勰.结构动力方程的精细时程积分法.大连理工大学学报,1994,34(2):131-136(Zhong,Wanxie.On precise time-integration method for structural dynamics.Journal of Dalian University of Technology,1994,34(2):131-136(in Chinese))
28钟万勰.弹性力学求解新体系.大连:大连理工大学出版社,1995(Zhong Wanxie.A New Systematic Methodology for Theory of Elasticity.Dalian:Dalian University of Technology Press,1995(in Chinese))
29韩泽军,林皋,李建波.二维层状地基格林函数的求解.土木工程学报,2015,48(10):99-107(Han Zejun,Lin Gao,Li Jianbo.The solution of Green’s functions for two-dimensional layered ground.China Civil Engineering Journal,2015,48(10):99-107(in Chinese))
30 Ai ZY,Cheng YC.Extended precise integration method for consolidation of transversely isotropic poroelastic layered media.Computers&Mathematics with Applications,2014,68(12):1806-1818
31 Wang LJ,Ai ZY.Plane strain and three-dimensional analyses for thermo-mechanical behavior of multilayered transversely isotropic materials.International Journal of Mechanical Sciences,2015,103:199-211
32 Talbot A.The accurate numerical inversion of Laplace transforms.Journal of Institute of Mathematics and Its Application,1979,23(1):97-120
33 Sneddon IN.The Use of Integral Transform.New York:Mc GrawHill,1972
34 Zhong WX,Lin JH,Gao Q.The precise computation for wave propagation in stratified materials.International Journal for Numerical Methods in Engineering,2004,60(1):11-25
35 Bailey DH,Swarztrauber PN.A fast method for the numerical evaluation of continuous Fourier and Laplace transforms.SIAM Journal on Scientific Computing,1994,15(5):1105-1110
36 Abate J,Valko PP.Multi-precision Laplace transform inversion.International Journal for Numerical Methods in Engineering,2004,60(5):979-993
37 Ai ZY,Yue ZQ,Tham LG,et al.Extended Sneddon and Muki solutions for multilayered elastic materials.International Journal of Engineering Science,2002,40(13):1453-1483
2王铁行,李宁,谢定义.土体水热力耦合问题研究意义、现状及建议.岩土力学,2005,26(3):488-493(Wang Tiehang,Li Ning,Xie Dingyi.Necessity and means in research on soil coupled heatmoisture-stress issues.Rock and Soil Mechanics,2005,26(3):488-493(in Chinese))
3蒋中明,Dashnor H.核废料贮存库围岩体热响应耦合场研究.岩土工程学报,2006,28(8):953-956(Jiang Zhongming,Dashnor H.Studies on coupled field of thermal response in rock mass of nuclear waste repository.Chinese Journal of Geotechnical Engineering,2006,28(8):953-956(in Chinese))
4 Bai B,Guo LJ,Han S.Pore pressure and consolidation of saturated silty clay induced by progressively heating/cooling.Mechanics of Materials,2014,75:84-94
5 Ai ZY,Wang LJ.Axisymmetric thermal consolidation of multilayered porous thermoelastic media due to a heat source.International Journal for Numerical and Analytical Methods in Geomechanics,2015,39(17):1912-1931
6 Biot MA.General theory of three-dimensional consolidation.Journal of Applied Physics,1941,12(2):155-164
7 Biot MA.Thermoelasticity and irreversible thermodynamics.Journal of Applied Physics,1956,27(3):240-253
8 Booker JR,Savvidou C.Consolidation around a spherical heat source.International Journal of Solids and Structures,1984,20(11-12):1079-1090
9 Savvidou C,Booker JR.Consolidation around a heat source buried deep in a porous thermoelastic medium with anisotropic flow properties.International Journal for Numerical and Analytical Methods in Geomechanics,1989,13(1):75-90
10 Mctigue DF.Thermoelastic response of fluid-saturated porous rock.Journal of Geophysical Research Atmospheres,1986,91(B9):9533-9542
11 Bai M,Abousleiman Y.Thermoporoelastic coupling with application to consolidation.International Journal for Numerical and Analytical Methods in Geomechanics,1997,21(2):121-132
12白冰.岩土介质非稳态热固结耦合问题的热源函数法.力学学报,2004,36(4):427-434(Bai Bing.Heat source function method for coupling analyses of thermal consolidation in saturated soil.Acta Mechanica Sinica,2004,36(4):427-434(in Chinese))
13郑荣跃,刘干斌,梧松.半空间饱和土内置点载荷作用下的热弹性波动.力学学报,2008,40(3):413-420(Zhen Rongyue,Liu Ganbin,Wu Song.Coupling thermo-hydro-mechanical dynamic response of saturated soil subjected to internal excitation.Chinese Journal of Theoretical and Applied Mechanics,2008,40(3):413-420(in Chinese))
14吴瑞潜,谢康和,程永锋.变载荷下饱和土一维热固结解析理论.浙江大学学报(工学版),2009,43(8):1532-1537(Wu Ruiqian,Xie Kanghe,Cheng Yongfeng.Analytical theory for one-dimensional thermal consolidation of saturated soil under time-dependent loading.Journal of Zhejiang University(Engineering Science),2009,43(8):1532-1537(in Chinese))
15 Lu JCC,Lin F.Thermal consolidation of a poroelastic full space subjected to a decaying point heat source//Proceedings of the 2nd International ISCM Symposium and the 12nd International EPMESCConference,2010:407-412
16 Selvadurai APS,Suvorov AP.Thermo-poromechanics of a fluidfilled cavity in a fluid-saturated geomaterial//Proceedings of the royal society a mathematical physical and engineering sciences,2014,470(2163):20130634
17 Giraud A,Homand F,Rousset G.Thermoelastic and thermoplastic response of a double-layer porous space containing a decaying heat source.International Journal for Numerical and Analytical Methods in Geomechanics,1998,22(2):133-149
18白冰.变温度载荷作用下半无限成层饱和介质的热固结分析.应用数学和力学,2006,27(11):1341-1348(Bai Bing.Thermal consolidation of layered porous half-space to variable thermal loading.Applied Mathematics and Mechanics,2006,27(11):1341-1348(in Chinese))
19 Yue ZQ,Yin JH.Backward transfer-matrix method for elastic analysis of layered solids with imperfect bonding.Journal of Elasticity,1998,50(2):109-128
20艾智勇,吴超.渗透各向异性可压缩地基固结的平面应变分析.力学学报,2009,41(5):801-807(Ai Zhiyong,Wu Chao.Analysis on plane strain consolidation of a multi-layered soil with anisotropic permeability and compressbility constituents.Chinese Journal of Theoretical and Applied Mechanics,2009,41(5):801-807(in Chinese))
21赵宇昕,陈少林.关于传递矩阵法分析饱和成层介质响应问题的讨论.力学学报,2016,48(5):1145-1158(Zhao Yunxin,Chen Shaolin.Discussion on the matrix propagator method to analyze the response of saturated layered media.Chinese Journal of Theoretical and Applied Mechanics,2016,48(5):1145-1158(in Chinese))
22 Booker JR,Small JC.Finite layer analysis of consolidation.I.International Journal for Numerical and Analytical Methods in Geomechanics,1982,6(2):151-171
23宰金珉,梅国雄.有限层法求解三维比奥固结问题.岩土工程学报,2002,24(1):31-33(Zai Jinmin,Mei Guoxiong.Finite layer analysis of three dimensional Biot consolidation.Chinese Journal of Geotechnical Engineering,2002,24(1):31-33(in Chinese))
24钟阳,耿立涛.多层弹性平面问题解的精确刚度矩阵法.岩土力学,2008,29(10):2829-2832(Zhong Yang,Geng Litao.Explicit solution of multiplayer elastic plane by exact stiffness matrix method.Rock and Soil Mechanics,2008,29(10):2829-2832(in Chinese))
25艾智勇,曹国军,成怡冲.平面应变Biot固结的解析层元.力学学报,2012,44(2):401-407(Ai Zhiyong,Cao Guojun,Cheng Yichong.Analytical layer-element of plane strain Biot’s consolidation.Chinese Journal of Theoretical and Applied Mechanics,2012,44(2):401-407(in Chinese))
26艾智勇,王路君,曾凯.稳定温度场下层状路面体系的解析层元解.同济大学学报(自然科学版),2014,42(11):1665-1669(Ai Zhiyong,Wang Lujun,Zeng Kai.Analytical layer-element solution for layered pavement in stable temperature field.Journal of Tongji University(Natural Science),2014,42(11):1665-1669(in Chinese))
27钟万勰.结构动力方程的精细时程积分法.大连理工大学学报,1994,34(2):131-136(Zhong,Wanxie.On precise time-integration method for structural dynamics.Journal of Dalian University of Technology,1994,34(2):131-136(in Chinese))
28钟万勰.弹性力学求解新体系.大连:大连理工大学出版社,1995(Zhong Wanxie.A New Systematic Methodology for Theory of Elasticity.Dalian:Dalian University of Technology Press,1995(in Chinese))
29韩泽军,林皋,李建波.二维层状地基格林函数的求解.土木工程学报,2015,48(10):99-107(Han Zejun,Lin Gao,Li Jianbo.The solution of Green’s functions for two-dimensional layered ground.China Civil Engineering Journal,2015,48(10):99-107(in Chinese))
30 Ai ZY,Cheng YC.Extended precise integration method for consolidation of transversely isotropic poroelastic layered media.Computers&Mathematics with Applications,2014,68(12):1806-1818
31 Wang LJ,Ai ZY.Plane strain and three-dimensional analyses for thermo-mechanical behavior of multilayered transversely isotropic materials.International Journal of Mechanical Sciences,2015,103:199-211
32 Talbot A.The accurate numerical inversion of Laplace transforms.Journal of Institute of Mathematics and Its Application,1979,23(1):97-120
33 Sneddon IN.The Use of Integral Transform.New York:Mc GrawHill,1972
34 Zhong WX,Lin JH,Gao Q.The precise computation for wave propagation in stratified materials.International Journal for Numerical Methods in Engineering,2004,60(1):11-25
35 Bailey DH,Swarztrauber PN.A fast method for the numerical evaluation of continuous Fourier and Laplace transforms.SIAM Journal on Scientific Computing,1994,15(5):1105-1110
36 Abate J,Valko PP.Multi-precision Laplace transform inversion.International Journal for Numerical Methods in Engineering,2004,60(5):979-993
37 Ai ZY,Yue ZQ,Tham LG,et al.Extended Sneddon and Muki solutions for multilayered elastic materials.International Journal of Engineering Science,2002,40(13):1453-1483