摘要
研究地震时局部地形对地面运动的影响是地震工程学的重要课题之一。本文采用“分区”的思想，利用复变函数和移动坐标法研究了稳态SH波入射时，夹杂、裂纹附近的半圆形凸起地形，以及半圆形凸起与半圆形凹陷或半圆形沉积层相连地形的情况下，凸起地形部分的地震动问题。求解时，根据具体求解模型，将求解区域分为几个部分，其中之一为包括半圆形凸起在内的圆域，该圆域的下半部边界为“公共边界”。首先采用复变函数的形式，利用波函数展开法在各个区域内分别构造满足边界条件的位移解，然后通过移动坐标使之满足“公共边界”上位移、应力连续条件以及其他边界条件，从而建立起求解问题的无穷代数方程组．。本文具体工作如下：
1．研究了SH波作用下，浅埋弹性夹杂和可移动刚性夹杂附近的半圆形凸起地形的地震动问题。采用“分区”方法，将求解区域分为两个部分，在区域Ⅰ中构造一个上半部边界应力为零，其余部分位移、应力任意的驻波解，在区域Ⅱ中构造“公共”边界和夹杂产生的散射波，且要求其满足水平边界应力自由的边界条件。然后移动坐标，满足“公共边界”和夹杂的边界条件，从而建立起求解该问题的无穷代数方程组。通过具体算例，给出半圆形凸起地形及其附近两侧水平地表位移幅值的数值结果，并对其进行了讨论。
2．对SH波作用下，半圆形凸起与半圆形凹陷或半圆形沉积层相连地形的散射问题进行了研究。同样采用“分区”的方法进行求解。首先给出在半圆形凹陷的影响下，半圆形凸起地形对SH波散射的位移场；在此基础上，又求解了半圆形凸起地形在半圆形沉积层影响下对SH波的散射。分别给出具体算例，对凸起部分地表位移幅值随入射波波数，入射角，凹陷半径等各种参数的变化规律进行了讨论。
3．采用“分区”的方法，求解出适合本文问题的Green函数，即含有半圆形凸起的弹性半空间内任意一点承受时间谐和的出平面线源荷载作用时位移场的解答。
4．利用本文构造的Green函数，对SH波作用下浅埋裂纹附近的半圆形凸起地形的地震动问题进行了讨论。从半圆形凸起地形对SH波的散射问题出发，利用Green函数在基体内构造出裂纹，建立起浅埋裂纹附近的半圆形凸起地形的求解模型。给出SH波入射时，在裂纹影响下半圆形凸起地形的位移场。最后，通过具体算例给出了半圆形凸起地形地震动的数值结果。讨论了入射波波数、入射角等各种参数对半圆形凸起地形及其附近两侧水平地表位移幅值的影响。
Researching the influence of local irregular topography on ground motion is one of the most important problems in earthquake engineering. Based on the idea of domain division, the complex function and moving coordinate are used to study the scattering of steady-state SH wave by a semi-cylindrical hill above a subsurface inclusion or a crack, and a semi-cylindrical hill adjoins a semi-cylindrical canyon or a semi-cylindrical alluvial valley, as well as the ground motion of the hill. According to idiographic model, the whole solution domain is divided into several parts during the press of solving. One of them is a circle domain, which includes the boundary of the hill and the junction interface. First, the displacement solutions satisfying the boundary conditions are constructed in every domain by complex function and wave function expansion. Second, the method of moving coordinate is used to make the displacement solutions conjoin at the common boundary and satisfiy other boundary conditions. Then the solutions of the problem are reduced to solve several series of infinite linear algebraic equations. The work of this paper can be detailed as follows:
1. Scattering of SH wave by a semi-cylindrical hill above a subsurface elastic inclusion or a removable rigid inclusion is determined. According to the domain division idea, the solution domain is divided into two parts. A standing wave function is constructed in part I , which satisfies the condition that traction free at the hill edge and arbitrary at the other part. In domain II, including the junction interface and the subsurface inclusion in half-pace, the scattered wave function satisfying the condition of traction free at the horizontal surface is constructed. Then emploied to the method of moving coordinate, the two solutions are matched up on the junction interface, and satisfying the boundary conditions at the inclusion edge. Then the problem can be reduced to solve a series of infinite linear algebraic equations. The computational results of surface displacement are presented together with the discussion.
2. This paper presents the models and solutions for the scattering of SH wave by a semi-cylindrical hill adjoins a semi-cylindrical canyon or a semi-cylindrical alluvial valley. The idea of domain division is also the key step during the press of solving. At first, the displacement field of the semi-cylindrical hill including a semi-cylindrical canyon under SH wave is provided. Then the scattering of SH wave by a semi-cylindrical hill adjoins a semi-cylindrical alluvial valley is solved based on the idea of domain division. The calculating results are provided respectively, and the variation of surface displacement amplitudes with wave numbers, angle of incidence and the ratio of canyon are discussed.
3. A suitable Green function is constructed employed to the idea of domain division. That is the fundermental solution of displacement field for an elastic half space with a semi-cylindrical hill under an out-plane harmonic line source loading at an arbitrary point in matrix.
4. The Green function is employed to discuss the influence of subsurface crack on the motion of semi-cylindrical hill under SH wave. Based on the problem of scattering of SH wave by a semi-cylindrical hill, the Green function is used to make a subsurface crack, and then the model of a semi-cylindrical hill above a subsurface crack can be built. The displacement field of semi-cylindrical hill including a subsurface crack under SH wave is given. Finally, the calculating results of the hill motion are plotted to show the effect of wave numbers, angle of incidence and the other parameters on the surface displacement amplitudes.
引文
[1] 地震工程概论编写组．地震工程概论．北京：科学出版社，1977
[2] 黄茹蕙，张秋红．地震波散射问题研究进展．低温建筑技术．2006， 109(1)：68-70页
[3] 钟伟芳，聂国华著．弹性波的散射理论．武汉：华中理工大学出版社， 1997：137-189页
[4] Sezawa K. Scattering of Elastic Waves and Some Allied Problems. Bull Earthquake Res Inst Tokyo Imperial Univ. 1927,3:19P
[5] Wolf A. Motion of a Rigid Sphere in an Acoustic Wave Field. Geophysics. 1945,10:91P
[6] Nagase M. Diffraction of Elastic Wave by a Spherical Surface. J. Phys. Soc. Japan. 1957, 11(3): 279P
[7] Knopoff L. Scattering of Compressional Waves by .Spherical Obstacles. Geophysics. 1959, 24(1): 30P
[8] Nishimura G. Jimbo Y A. Dynamic Problem of Stress Concentration Stresses in the Vicinity of a Spherical Matter Included in an Elastic Solid Under Dynamical Force. J Faculty of Engineering, Univ of Tokyo. 1955,24:101P
[9] Ying C F, Truell R. scattering in an Isotropic Elastic Solid J Appl phys. 1956, 27(9): 1086-1097P
[10] Einspruch N G, Witterholt E J, Truell R. Scattering of a Plane Transverse Wave by a Spherical Obstacle in an Elastic Medium. J Appl Phys. 1960,31(5):806-818P
[11] Pao Y H, Mow C C. Dynamic Stress Concentration in an Elastic Plate with Rigid Circular Inclusion. Proc 4th National Cong of Appl Mech. 1962, 335P
[12] Mow C C, Mente L J. Dynamic Stresses and Displacements Around Cyindrical Discontinuities due to Plane Harmonic Shear Waves. J Appl Mech. 1963, 30(4):598P
[13] White R M. Elastic Wave Scattering at a Cylindrical Discontinuity in a Solid. J Acoust Soc Am. 1958, 30(8):771P
[14] Kato K. Reflections of Sound Wave due to a Hollow Cylinder in an Elastic Body. Mem Inst Sci Indus Res. Vol. 9 Osaka University, Japan, 1952
[15] Baron M L, Mattews A T. Diffraction of a Pressure Wave by a Cylindrical Cavity in an Elastic Medium. ASME J Appl Mech. 1961, 28: 347P
[16] Baron M L, Parnes R. Displacements and Velocities Produced by the Diffraction of a Pressure Wave by a Cylindrical Cavity in an Elastic Medium. ASME J Appl Mech. 1962, 29: 385P
[17] Mow C C. Transient Response of a Rigid Spherical Inclusion in an Elastic Medium. J Appl Mech. 1965, 32(3): 637P
[18] Mow C C. On the Transient Motion of a Rigid Spherical Inclusion in an Elastic Medium and its Inverse problem. J Appl Mech, 1966, 33(4): 807P
[19] Miklowitz J. Scattering of a plane Elastic Compressional Pulse by a Cylindrical Cavity. Proc llthInt Cong of Appl Mech, Munich, 1964, 469P
[20] Norwood F R, Miklowitz J. Diffraction of Transient Elastic Waves by a Spherical Cavity. J Appl Mech, 1967, 34(3): 735P
[21] Jain D L, Kanwal R P. Scattering of elastic Waves by Circular Cylindrical Flaws and Inclusions. J Appl Phys, 1979, 50(6): 4067-4108P
[22] Jain D L, Kanwal R P. Scattering of Elastic Waves by an Elastic Sphere. Int J Eng Sci, 1980, 18: 829-839P
[23] S. K. DATTA, A. H. SHAH. Scattering Of SH-waves By Embedded Cavities. Wave Motion 1982,4: 265-283P
[24] Datta S K, Akily N E. Diffraction of Elastic Waves in a Half Space I : integral Representations and Matched Asymptotic Expansions. in Modern problems in Elastic Wave Propagation ed. Miklowitz J. and ichenbach JD. iWiley-Interscience, 1978,197-218P
[25] Datta S K, Scattering of elastic Waves in Mechanics Today ed. Nemat-Nasser S. 1978,4:149-208P
[26] Datta S K, Wong K C, shah A H. Dynamic Stresses and Displacements Around Cylindrical Cavities of Arbitrary Shapes. ASME J Appl Mech, 1984, 51:798-803P
[27] Pao Y.H., Mow C.C. Diffraction of Elastic Waves and Dynamic Stress Concentrations. Crane and Russak, New York, 1973: 114-304P
[28] 鲍亦兴，毛昭宙著．刘殿魁，苏先樾译．弹性波的衍射与动应力集中．北京：科学出版社，1993：72-202页
[29] 黎在良，刘殿魁 著．固体中的波．北京：科学出版社，1995：286-319页，377-431页
[30] Liu D. K. Gal. B. Z. and Tao G.Y. ,Applications of the Method of Complex Function to Dynamic Stress Concentration , Wave motion ,1982(4): 293-304P
[31] Trifunac MD. Surface motion of a semi-cylindrical alluvial valley for incident plane SH waves. Bull Seism Soc Am. 1971. 61(2): 1755-1770P
[32] M. D. Trifunac. Scattering of Plane SH-waves by a Semi-Cylindrical Canyon. Earthquake Engineering and Structural Dynamics. 1973(1): 267-281P
[33] 韩峰，刘殿魁．SH波对半无限圆形凹陷地形造成的位移场的研究．哈尔滨建筑工程学院学报．1990，23(3)：24-31页
[34] 刘殿魁，韩峰．SH波对各向异性凹陷地形的散射．地震工程与工程振 动．1990，10(2)：11-25页
[35] Liu D.K. and Han F. Scattering of plane SH-wave by Cylindrical Canyon of Arbitrary Shapes. Int. H. Soil. Dyn. and Ear. Eng. 1991, 10(5): 249-255P
[36] 许贻燕，韩峰．平面SH波在相邻多个半圆形凹陷地形上的散射．地震工程与工程振动．1992，12(2)：12-18页
[37] Liu D.K. and Han F. Scattering of Plane SH-waves by Cylindrical Canyon of Arbitrary Shapes in Anisotropic Media. Acta Mechanica Sinica. 1990, 6(3): 256-266P
[38] 刘殿魁，许贻燕．各向异性介质中SH波与多个半圆形凹陷地形的相互作用．力学学报．1993，25(1)：93-102页
[39] 袁晓铭，廖振鹏．圆弧形凹陷地形对平面SH波散射问题的级数解答．地震工程与工程振动．1993，13(2)：1-11页
[40] 房营光．相邻多个浅圆弧凹陷地形对平面SH波散射的级数解．应用数学和力学．1995，16(7)：615-623页
[41] Han F, Wei Y, Liu D. K. The Interaction of Plane SH-wave and Circular Cavity Surfaced with Lining in Anisotropic Media. Applied Mathematics and Mechanics. 1995,16(2):1067-1078P
[42] Lee VW, Manoogian ME. Surface motion above an arbitrary shape underground cavity for incident SH wave. European Earthquake Engineering. 1995, 8(1):3-11P
[43] 刘殿魁，刘宏伟．曲线坐标在弹性波散射研究中的应用——SH波对不等深度凹陷地形的散射．地震工程与工程振动．1996，16(2)：14-24页
[44] V. W. Lee, S. Chen and I. R. Hsu. Antiplane Diffraction from Canyon Above Subsurface Unlined Tunnel. A. S. C. E. Eng. Mech. Div. 1999
[45] 袁晓铭，廖振鹏．圆弧形沉积盆地对平面SH波的散射．华南地震．1995， 15(2)：1-8页
[46] 梁建文，张郁山，顾晓鲁，Vincent W Lee．圆弧形层状凹陷地形对平面 SH波的散射．振动工程学报．2003，16(2)：158-165页
[47] 林皋．地下结构抗震分析综述(上)．世界地震工程．1999，15(2)：1-10页
[48] PAO Y H. Applied mechanics in science and engineering. Applied Mechanics Review. 1998,51(2): 141-153P
[49] 王国庆，刘殿魁．SH波对浅埋相邻多个圆孔作用的动力分析．哈尔滨工程大学学报．2003，24(1)：108-113页
[50] 王铎，马兴瑞，刘殿魁．弹性动力学最新进展．北京：科学出版社， 1995：13-15页
[51] 王铎，汪越胜．界面动力学研究近况．上海力学，1993(4)：1-15页
[52] 刘殿魁，刘宏伟．SH波散射与界面圆孔附近的动应力集中．力学学报．1998，30(5)：597-604页
[53] 刘宏伟，刘殿魁．界面圆孔对SH波的散射远场解．固体力学学报．1999， 20(4)：349-355页
[54] 刘殿魁，田家勇．SH波对界面圆柱形弹性夹杂散射及动应力集中．爆炸与冲击．1999，19(2)：115-123页
[55] Yang Zailin, Liu Dankui. Scattering far field solution of SH-wave by movable rigid cylindrical interface. Acta Mechanica Solida Sinica. 2002,15(3): 214-220P
[56] 刘殿魁，杨在林，刘建百．界面可移动圆柱形刚性夹杂对SH波散射及动应力集中．哈尔滨建筑大学学报．2001，34(6)：1-7页
[57] 刘殿魁，史守峡．界面上圆形衬砌结构对平面SH波散射．力学学报．2002，34(5)：796-803页
[58] 史守峡，刘殿魁．SH波与界面多圆孔的散射及动应力集中．力学学报．2001，33(1)：60-70页
[59] 王志伟，齐辉，刘殿魁．具有刚性覆盖层的界面圆环形衬砌对SH波的散射．哈尔滨工程大学学报．2002，23(4)：81-85页
[60] 陈志刚，刘殿魁．椭圆孔对SH波散射的远场解．哈尔滨工程大学学 报．2003，24(3)：334-338页
[61] 刘殿魁，林宏．SH波对双相介质界面附近圆形孔洞的散射．固体力学学报．2003，24(2)：197-204页
[62] Chen Zhigang, Liu Diankui and Yang Zailin. Dynamic stress concentration and scattering of SH-wave by interface elliptic cylindrical cavity. Earthquake engineering and engineering vibration. 2003, 2(2): 299-306P
[63] GREGORY R D. An expansion theorem applicable to problems of wave propagation in an elastic half space containing a cavity. Proc Camb, Philos Soc. 1967,63:1341-1367P
[64] GREGORY R D. Propagation of waves in an elastic half space containing a cylindrical cavity. Proc Camb, Philos Soc. 1970,67:689-710P
[65] 韩峰，韦扬，刘殿魁．各项异性介质中带衬砌的圆形结构与SH波相互作用．应用数学和力学．1995，16(12)：1067-1077页
[66] 林宏，史文谱，刘殿魁．SH波入射时浅埋结构的动力分析．哈尔滨工程大学学报．2001，22(6)：83-87页
[67] 王艳，刘殿魁．SH波入射时浅埋衬砌结构动力分析．哈尔滨工程大学学报．2002，23(6)：43-47页
[68] 林宏，刘殿魁．半无限空间中圆形孔洞周围SH波的散射．地震工程与工程振动．2002，22(2)：9-16页
[69] 齐辉，王艳，刘殿魁．半无限空间界面附近SH波对圆形衬砌的散射．地震工程与工程振动．2003，23(3)：41-46页
[70] 刘殿魁．林宏．浅埋的圆形孔洞对SH波散射与地震动．爆炸与冲击．2003，23(1)：6-12页
[71] 齐辉．界面圆环形衬砌对SH波散射及动应力集中．哈尔滨工程大学博士学位论文．2000．11：33-47页
[72] G.C. Shi. Stress Distribution Near Internal Crack Tips for Longitudinal Shear Problems.Journal of Applied Mechanics. 1965, 32(1), 51-58P
[73] 陆建飞，王建华，沈为平．曲线裂纹和反平面圆形夹杂相交问题．固体力学学报．2000，21(3)：205-210页
[74] 陆建飞，蔡兰，柳春图．圆夹杂内裂纹对SH波的动力响应．力学学报．2003，35(5)：623-627页
[75] 刘殿魁，刘宏伟．孔边裂纹对SH波散射及动应力强度因子，力学学报， 1999，31(3)：292-299页
[76] 史守峡，刘殿魁，杨庆山．SH波对内含裂纹衬砌结构的散射与动应力集中．爆炸与冲击．2000，20(3)：228-234页
[77] 田家勇，齐辉，刘殿魁．一类复合缺陷对SH波散射及动应力强度因子．哈尔滨工程大学学报．2000，21(3)：52-57页
[78] 刘殿魁，陈志刚．椭圆孔边裂纹对SH波的散射及其动应力强度因子．应用数学与力学．2004，25(9)：958-966页
[79] Liu Diankui, Lin Hong. Scattering of SH-waves by an Interface Linear Crack and a Circular Cavity near Bimaterial Interface. ACTA Mechanica Sinica. 2004,20(3):317-326P
[80] 李宏亮．SH波作用下孔洞、夹杂与直线形裂纹的相互作用．哈尔滨工程大学博士学位论文．2004．11：33-36页
[81] 扬在林．半空间夹杂与裂纹对SH波的散射．哈尔滨工程大学博士学位论文．2005．11：49-66页
[82] Xiaoming Yuan and Fulu Men. Scattering of Plane SH-waves by a Semi-cylindrical Hill" .Earthq. Eng. and Struct. Dynamics. 1992(21): 1091-1098P
[83] 崔志刚，曹新荣，刘殿魁．SH波对半圆形凸起地形的散射．地震工程与工程振动．1998，18(1)：140-146页
[84] 崔志刚，邹永超，刘殿魁．SH波对圆弧形凸起地形的散射．地震工程与工程振动．1998，18(4)：8-14页
[85] 刘殿魁，曹新荣，崔志刚．多个半圆形凸起地形对平面SH波散射．固体力学学报．1998(特刊)：178-185页
[86] 李彤，王国庆，刘殿魁．SH波在含圆形孔洞的半圆形凸起处的散射．地震工程与工程振动．2003，23(5)：26-31页
[87] 刘殿魁，王国庆．浅埋圆形孔洞附近的半圆形凸起对SH波的散射．力学学报．2006，38(2)：209-218页
[88] Abdul Hayir, Maria I. Todorovska, Mihailo D. Trifunac. Antiplane response of a dike with flexible soil-structure interface to incident SH waves. Soil Dynamics and Engineering. 2001,21:603-613P
[89] M.I. Totorovska, A. Hayir, M.D. Trifunac. Antiplane response of a dike on embedded foundation to incident SH waves. Soil Dynamics and Engineering. 2001,21: 593-601P
[90] Qiu F.Q and Liu D.K. Antiplane response of isosceles triangular hill to incident SH waves. Earthquake Engineering and Engineering Vibration. 2005, 4:1-10P
[91] H. CAO and V.W. LEE. Scattering and diffraction of plane P-wave by circular cylindrical canyons with variable depth-to-width ratio. Soil Dynamics and Earthquake Engineering. 1990,9(3): 141-150P
[92] 刘宏伟，刘殿魁．P波对具有不等深度凹陷地形的散射。固体力学学报．1997，18(4)：295-300页
[93] 梁建文，严林隽，Vincent W．Lee．圆弧形层状沉积谷地对入射P波的散射解析解．地震学报．2001，23(2)：167-184页
[94] 尤红兵，梁建文．层状半空间中洞室对入射平面SV波的散射．岩土力学．2006，27(3)：383-388页
[95] 杨桂通，张善元．弹性动力学．中国铁道出版社，1988：1-5页
[96] 阿肯巴赫．J．D著．徐植信．弹性固体中波的传播．上海：同济大学出版 社，1992：279-284页
[97] 王仁等．第十九届国际理论与应用力学大会(ICTAM)情况介绍．力学与践．1997，19(1)：57-64页
[98] 郑哲敏，周恒，张涵信，黄克智，白以龙．21世纪初的力学发展趋势．力学进展．1995，25(4)：433-449页
[99] 徐芝纶．弹性力学．北京：高等教育出版社，1978：1-161页，382-397页
[100] 林成森．数值计算方法(下册)．北京：科学出版社，1999：60-95页
[101] 张石生．积分方程．重庆：重庆出版社，1985：77-124页
[102] 梁昆淼．数学物理方法．第三版．北京：高等教育出版社，1998：374页
[103] 王竹溪，郭敦仁．特殊函数概论．北京：北京大学出版社，2000：337-417页，601-620页
