汶川地震断层应力强度因子分析超奇异积分法
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摘要
地震作为复杂瞬态断裂破坏物理过程,其机制研究是一个多学科交叉问题;同震断层处应力/能量积累释放规律,为从断裂角度研究地震机制提供了一个切入点;应力强度因子(应力角度)和应变能强度因子(能量角度)是断裂力学/物理学的两个最基本的参数。本文应用非线性边界元和主部分析法,将汶川地震断层破坏过程转化为解以断层位移间断为未知函数超奇异积分方程组问题,定义了断层处应力强度因子;利用有限部积分概念及体积力法,为方程组建立了数值解法,编制了Fortran程序,得到断层处应力强度因子数值结果;通过研究应力强度因子随断层位置变化规律,分析了汶川地震断层破坏过程;结合应变能强度因子理论,通过对宏微观电磁破坏过程进一步深入模拟研究可得到电磁辐射破坏变化规律,为地震短临预测提供理论支持和帮助。
Based on the nonlinear boundary element and the main-part analysis method, the problem of earthquake fault in Wenchuan is reduced to a set of hypersingular integral equations coupled with nonlinear boundary integral equations, in which the unknown functions are the general displacement discontinuities. The behavior of the general singular stress indexes around the crack front terminating at the fault surface is analyzed and the stress intensity factors are defined. A numerical method for the problem is put forward with the extended displacement discontinuities being approximated by the product of basic density functions and polynomials. Finally, the radiation distributions simulated by stress intensity factors against fault length, width and height at the crack surface are obtained, which can reveal the mechanism on co-seismal slip. The future work will focus on using the extended volume energy density function and Richter magnitude scale geophysics theory to analyze the extended macro/micro electromagnetic earthquake fault mechanism.
Based on the nonlinear boundary element and the main-part analysis method, the problem of earthquake fault in Wenchuan is reduced to a set of hypersingular integral equations coupled with nonlinear boundary integral equations, in which the unknown functions are the general displacement discontinuities. The behavior of the general singular stress indexes around the crack front terminating at the fault surface is analyzed and the stress intensity factors are defined. A numerical method for the problem is put forward with the extended displacement discontinuities being approximated by the product of basic density functions and polynomials. Finally, the radiation distributions simulated by stress intensity factors against fault length, width and height at the crack surface are obtained, which can reveal the mechanism on co-seismal slip. The future work will focus on using the extended volume energy density function and Richter magnitude scale geophysics theory to analyze the extended macro/micro electromagnetic earthquake fault mechanism.
引文
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[3]Pan E.Three-dimensional Green's functions in anisotropic magneto-electro-elastic bimaterials[J].Zeitschrift für Angewandte Mathematik und Physik,2002,53(5):815-838.
[4]Ding H J,Jiang A M,Hou P F,et al.Green's functions for two-phase transversely isotropic magneto-electro-elastic media[J].Engineering Analysis with Boundary Elements,2005,29(6):551-561.
[5]Chen W Q,Lee K Y,Ding H J.General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method[J].International Journal of Engineering Science,2004,42(13-14):1361-1379.
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[7]Bigoni D,Capuani D.Green's function for incremental nonlinear elasticity:Shear bands and boundary integral formulation[J].Journal of the Mechanics and Physics of Solids,2002,50(3):471-500.
[8]Bigoni D,Capuani D.Time-harmonic Green's function and boundary integral formulation for incremental nonlinear elasticity:Dynamics of wave patterns and shear bands[J].Journal of the Mechanics and Physics of Solids,2005,53(5):1163-1187.
[9]陈运泰.地震参数——数字地震学在地震预测中的应用[M].北京:地震出版社,2003.Chen Yuntai.Parameters of earthquake—Application of digital seismology in earthquake prediction[M].Beijing:Earthquake Press,2003.
[10]Okada Y.Surface deformation due to shear and tensile faults in half-space[J].Bulletin of the Seismological Society of America,1985,75(4):1135-1154.
[11]Bojing Z,Taiyan Q.Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites[J].International Journal of Solids and Structures,2007,44(18-19):5994-6012.
[12]张海明.半无限空间中平面断层的三维自发破裂传播的理论研究[D].北京:北京大学地球物理系,2004.Zhang Hiaoming.Theoretical research of three-dimensional spontaneous rupture spread of the fault plane in a semi-infinite space[D].Beijing:Department of Geophysics,Peking University,2004.
[13]Zhang H M,Chen X F.Dynamic rupture on a planar fault in three-dimensional half space-I.Theory[J].Geophysical Journal International,2006,164(3):633-652.
[14]Zhang H M,Chen X F.Dynamic rupture on a planar fault in three-dimensional half space II.Validations and numerical experiemnts[J].Geophysical Journal International,2006,167(2):917-932.
[15]Zhu B,Qin T.3D modeling of crack growth in electro-magneto-thermo-elastic coupled viscoplastic multiphase composites[J].Applied Mathematical Modelling,2010,in press.
[16]Qin T Y,Tang R J.Finite-part integral and boundary-element method to solve embedded planar crack problems[J].International Journal of Fracture,1993,60(4):373-381.
[17]Zhu B J,Shi Y L,Sukop M C,et al.Analysis of3D fluid driven crack propagation problem in co-seismic slip under P and S-waves by hybrid hypersingular integral method[J].Computer Methods in AppliedMechanics and Engineering,2009,198(30-32):2446-2469.
[18]Zhu B J,Shi Y L.Three-dimensional flow driven pore-crack networks in porous composites:Boltzmann Lattice method and hybrid hypersingular integrals[J].Theoretical and Applied Fracture Mechanics,2010,corrected proof.
[19]Qin T Y,Noda N A.Stress intensity factors of a rectangular crack meeting a bimaterial interface[J].International Journal of Solids and Structures,2003,40(10):2473-2486.
[20]Qin T Y,Noda N A.Analysis of a three-dimensional crack terminating at an interface using a hypersingular integral equation method[J].Journal of Applied Mechanics,2002,69(5):626-631.
[21]Coulomb C A.Sur une application des règles de maximis et minimis a quelques problèmes de stratique relatifisàl’architecture[J].Acad Roy des Sciences Memoires de Math et de Physique par Divers Savans,1773,7:343-382.
[22]Anderson E M.The dynamics of faultiong and dyke formation with applications to britain[M].2nd ed,Edinburgh:Oliver and Boyd,1951.
[23]Jagger J C.Elasticity,fracture and flow[M].2nd ed,London:Methuen,1962.
[24]Griffith A A.The phenomena of rupture and flow in solids[J].Philosophical Transactions of the Royal Society of London,Series A,Mathematical and Physical Sciences,1921,221:163-198.
[25]Griffith A A.The theory of rupture[C]//Proceedings of the First International Congress for Applied Mechanics.Delft:Delft J Waltman Jr Publishers,1924.
[26]Irwin G R.Fracture,hanbuch der physik[M].Flügge S,6ed.Berlin:Springer-Verlag,1958.
[27]Kassir M K,Sih G C.Griffith's theory of brittle fracture in three dimensions[J].International Journal of Engineering Science,1967,5:899-918.
[28]Sih G C,Liebowitz H.On the Griffith energy criterion for brittle fracture[J].International Journal of Solids and Structures,1967,3(1):1-22.
[2]陈运泰.地震预测——进展、困难与前景[J].地震地磁观测与研究,2007,28(2):1-24.Chen Yuntai.Observation and Research of Earthquake Geomagnetism,2007,28(2):1-24.
[3]Pan E.Three-dimensional Green's functions in anisotropic magneto-electro-elastic bimaterials[J].Zeitschrift für Angewandte Mathematik und Physik,2002,53(5):815-838.
[4]Ding H J,Jiang A M,Hou P F,et al.Green's functions for two-phase transversely isotropic magneto-electro-elastic media[J].Engineering Analysis with Boundary Elements,2005,29(6):551-561.
[5]Chen W Q,Lee K Y,Ding H J.General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method[J].International Journal of Engineering Science,2004,42(13-14):1361-1379.
[6]朱合华,陈清军,杨林德.边界元法及其在岩土工程中的应用[M].上海:同济大学出版社,1996:162.Zhu Hehua,Chen Qingjun,Yang Linde.Boundary element method and its application in rock soil engineering[M].Shanghai:Tongji University Press,1996:162.
[7]Bigoni D,Capuani D.Green's function for incremental nonlinear elasticity:Shear bands and boundary integral formulation[J].Journal of the Mechanics and Physics of Solids,2002,50(3):471-500.
[8]Bigoni D,Capuani D.Time-harmonic Green's function and boundary integral formulation for incremental nonlinear elasticity:Dynamics of wave patterns and shear bands[J].Journal of the Mechanics and Physics of Solids,2005,53(5):1163-1187.
[9]陈运泰.地震参数——数字地震学在地震预测中的应用[M].北京:地震出版社,2003.Chen Yuntai.Parameters of earthquake—Application of digital seismology in earthquake prediction[M].Beijing:Earthquake Press,2003.
[10]Okada Y.Surface deformation due to shear and tensile faults in half-space[J].Bulletin of the Seismological Society of America,1985,75(4):1135-1154.
[11]Bojing Z,Taiyan Q.Application of hypersingular integral equation method to three-dimensional crack in electromagnetothermoelastic multiphase composites[J].International Journal of Solids and Structures,2007,44(18-19):5994-6012.
[12]张海明.半无限空间中平面断层的三维自发破裂传播的理论研究[D].北京:北京大学地球物理系,2004.Zhang Hiaoming.Theoretical research of three-dimensional spontaneous rupture spread of the fault plane in a semi-infinite space[D].Beijing:Department of Geophysics,Peking University,2004.
[13]Zhang H M,Chen X F.Dynamic rupture on a planar fault in three-dimensional half space-I.Theory[J].Geophysical Journal International,2006,164(3):633-652.
[14]Zhang H M,Chen X F.Dynamic rupture on a planar fault in three-dimensional half space II.Validations and numerical experiemnts[J].Geophysical Journal International,2006,167(2):917-932.
[15]Zhu B,Qin T.3D modeling of crack growth in electro-magneto-thermo-elastic coupled viscoplastic multiphase composites[J].Applied Mathematical Modelling,2010,in press.
[16]Qin T Y,Tang R J.Finite-part integral and boundary-element method to solve embedded planar crack problems[J].International Journal of Fracture,1993,60(4):373-381.
[17]Zhu B J,Shi Y L,Sukop M C,et al.Analysis of3D fluid driven crack propagation problem in co-seismic slip under P and S-waves by hybrid hypersingular integral method[J].Computer Methods in AppliedMechanics and Engineering,2009,198(30-32):2446-2469.
[18]Zhu B J,Shi Y L.Three-dimensional flow driven pore-crack networks in porous composites:Boltzmann Lattice method and hybrid hypersingular integrals[J].Theoretical and Applied Fracture Mechanics,2010,corrected proof.
[19]Qin T Y,Noda N A.Stress intensity factors of a rectangular crack meeting a bimaterial interface[J].International Journal of Solids and Structures,2003,40(10):2473-2486.
[20]Qin T Y,Noda N A.Analysis of a three-dimensional crack terminating at an interface using a hypersingular integral equation method[J].Journal of Applied Mechanics,2002,69(5):626-631.
[21]Coulomb C A.Sur une application des règles de maximis et minimis a quelques problèmes de stratique relatifisàl’architecture[J].Acad Roy des Sciences Memoires de Math et de Physique par Divers Savans,1773,7:343-382.
[22]Anderson E M.The dynamics of faultiong and dyke formation with applications to britain[M].2nd ed,Edinburgh:Oliver and Boyd,1951.
[23]Jagger J C.Elasticity,fracture and flow[M].2nd ed,London:Methuen,1962.
[24]Griffith A A.The phenomena of rupture and flow in solids[J].Philosophical Transactions of the Royal Society of London,Series A,Mathematical and Physical Sciences,1921,221:163-198.
[25]Griffith A A.The theory of rupture[C]//Proceedings of the First International Congress for Applied Mechanics.Delft:Delft J Waltman Jr Publishers,1924.
[26]Irwin G R.Fracture,hanbuch der physik[M].Flügge S,6ed.Berlin:Springer-Verlag,1958.
[27]Kassir M K,Sih G C.Griffith's theory of brittle fracture in three dimensions[J].International Journal of Engineering Science,1967,5:899-918.
[28]Sih G C,Liebowitz H.On the Griffith energy criterion for brittle fracture[J].International Journal of Solids and Structures,1967,3(1):1-22.
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