三维层状黏弹性半空间中球面SH、P和SV波源自由场
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摘要
采用刚度矩阵方法结合Hankel积分变换,求解了层状黏弹性半空间中球面SH、P和SV波的自由波场.首先,在柱坐标系下建立层状黏弹性半空间的反轴对称(柱面SH波)和轴对称(柱面P-SV波)情况精确动力刚度矩阵.进而由Hankel变换将空间域内的球面波展开为波数域内柱面波的叠加,然后将球面波源所在层的上下端面固定,求得固定层内的动力响应和固定端面反力,将固端反力反向施加到层状黏弹性半空间上,采用直接刚度法求得固端反力的动力响应,叠加固定层内和固端反力动力响应,求得波数域内球面波源动力响应.最后由Hankel积分逆变换求得频率-空间域内球面波源自由场,时域结果由傅里叶逆变换求得.文中验证了方法的正确性,并以均匀半空间和基岩上单一土层中球面SH、P和SV波为例分别在频域和时域内进行了数值计算分析.研究表明基岩上单一土层中球面波自由场与均匀半空间情况有着本质差异;基岩上单一土层中球面波位移频谱峰值频率与场地固有频率相对应,基岩面的存在使得基岩上单一土层地表点的位移时程非常复杂,振动持续时间明显增长;阻尼的增大显著降低了动力响应的峰值,同时也显著减少了波在土层的往复次数.
Free-field responses of spherical sources embedded in a half-space,especially in a layered half-space is of fundamental importance in studying various wave scattering problem and soil-structures interaction problem.However,few studies have been reported to investigate the free-field responses of spherical sources.In this paper,dynamic responses of spherical SH-,Pand SV-wave sources embedded in a layered visco-elastic half-space are studied.The method of direct stiffness combining with the technique of Hankel transform is used to calculate the wave propagation of spherical sources.Firstly,the exact dynamic stiffness matrices of the layered visco-elastic half-space corresponding to the anti-symmetric cylindrical SH-waves and to the symmetric cylindrical P-and SV-waves are established,respectively.Then,the spherical sources expressed in the space domain are expanded as the summation of cylindrical waves in the wave-number domain.The layer in which the spherical sources locate is fixed at itstop and bottom interfaces and the dynamic responses restricted in the fixed layer and the corresponding reaction forces are obtained.The directions of these forces are then reversed,and they are applied as loads on the whole layered visco-elastic half-space.The dynamic responses induced by the reactions forces can be determined by using the direct stiffness method.And the dynamic responses restricted in the fixed layer are added to the dynamic responses of the reaction forces to determine the global responses in the wave-number domain.Finally,the free-field responses are obtained by using the inverse Hankel transform.And results in time domain can be easily obtained by using the inverse Fourier transform.The accuracy of the new method is verified by comparing results with those obtained by Lamb′s method.And by taking spherical SH-,P-and SV-wave sources embedded in a uniform half-space and in a single layered overlying on bedrock as examples,the following numerical calculations are performed.(1)Dynamic responses for different stiffness ratio between the bedrock and the layer are illustrated,and numerical results show that both the real and imaginary parts of the displacement and shear stress have kinks at the interface between the layer and the underlying half-space.(2)Spectrums of surface displacement amplitudes for different layer′s thickness are given,and numerical results show that the spectrums for the single layered halfspace have definite peak frequencies,which vary with thickness of the layer.In addition,the dynamic responses of the spherical SH-and SV-wave sources are less sensitive to the layer′s thickness.(3)Effects of material damping ratio on the free field responses are studied,and numerical results show that both the real and imaginary parts of the dynamic responses are decreased significantly with the increase of the material damping ratio,especially for peak displacement and stress.(4)Time domain results are illustrated by using the inverse Fourier transform,and numerical results show that only reflected SH-waves are observed for spherical SH sources,and both the reflected P-and SV-waves can be observed for P-or SV-sources due to Wave Mode Conversion.Additionally,in cases of the single layer half-space,reflected waves from the surface of the bedrock can be observed in the time histories.The free-field responses for the single layer half-space can be significantly different from those of the uniform half-space case;The peak frequencies of the surface displacement amplitude are strongly related to the fundamental frequencies of the single layer site;The existence of the bedrock makes the time histories of the surface displacement amplitudes very complicated and the duration of vibration very long;And the peak values of the dynamic responses decreased greatly and the times of wave propagating up and down in the layer reduced greatly with the increase of the material damping.
Free-field responses of spherical sources embedded in a half-space,especially in a layered half-space is of fundamental importance in studying various wave scattering problem and soil-structures interaction problem.However,few studies have been reported to investigate the free-field responses of spherical sources.In this paper,dynamic responses of spherical SH-,Pand SV-wave sources embedded in a layered visco-elastic half-space are studied.The method of direct stiffness combining with the technique of Hankel transform is used to calculate the wave propagation of spherical sources.Firstly,the exact dynamic stiffness matrices of the layered visco-elastic half-space corresponding to the anti-symmetric cylindrical SH-waves and to the symmetric cylindrical P-and SV-waves are established,respectively.Then,the spherical sources expressed in the space domain are expanded as the summation of cylindrical waves in the wave-number domain.The layer in which the spherical sources locate is fixed at itstop and bottom interfaces and the dynamic responses restricted in the fixed layer and the corresponding reaction forces are obtained.The directions of these forces are then reversed,and they are applied as loads on the whole layered visco-elastic half-space.The dynamic responses induced by the reactions forces can be determined by using the direct stiffness method.And the dynamic responses restricted in the fixed layer are added to the dynamic responses of the reaction forces to determine the global responses in the wave-number domain.Finally,the free-field responses are obtained by using the inverse Hankel transform.And results in time domain can be easily obtained by using the inverse Fourier transform.The accuracy of the new method is verified by comparing results with those obtained by Lamb′s method.And by taking spherical SH-,P-and SV-wave sources embedded in a uniform half-space and in a single layered overlying on bedrock as examples,the following numerical calculations are performed.(1)Dynamic responses for different stiffness ratio between the bedrock and the layer are illustrated,and numerical results show that both the real and imaginary parts of the displacement and shear stress have kinks at the interface between the layer and the underlying half-space.(2)Spectrums of surface displacement amplitudes for different layer′s thickness are given,and numerical results show that the spectrums for the single layered halfspace have definite peak frequencies,which vary with thickness of the layer.In addition,the dynamic responses of the spherical SH-and SV-wave sources are less sensitive to the layer′s thickness.(3)Effects of material damping ratio on the free field responses are studied,and numerical results show that both the real and imaginary parts of the dynamic responses are decreased significantly with the increase of the material damping ratio,especially for peak displacement and stress.(4)Time domain results are illustrated by using the inverse Fourier transform,and numerical results show that only reflected SH-waves are observed for spherical SH sources,and both the reflected P-and SV-waves can be observed for P-or SV-sources due to Wave Mode Conversion.Additionally,in cases of the single layer half-space,reflected waves from the surface of the bedrock can be observed in the time histories.The free-field responses for the single layer half-space can be significantly different from those of the uniform half-space case;The peak frequencies of the surface displacement amplitude are strongly related to the fundamental frequencies of the single layer site;The existence of the bedrock makes the time histories of the surface displacement amplitudes very complicated and the duration of vibration very long;And the peak values of the dynamic responses decreased greatly and the times of wave propagating up and down in the layer reduced greatly with the increase of the material damping.
引文
Gao Y F,Jin J X,Xie K H,et al.1999.General solution to the soil seismic response on stratified foundations.Chinese Journal of Geotechnical Engineering(in Chinese),21(4):498-500.
Haskell N A.1953.The dispersion of surface waves on multilayered media.Bull.Seismol.Soc.Am.,43(1):17-34.
Idriss I M,Seed H B.1968.Seismic response of horizontal soil layers.Journal of the Soil Mechanics and Foundations Division,94(4):1003-1031.
Jin X,Kong G,Ding H P.2004.Nonlinear seismic response analysis of horizontal layered site.Earthquake Engineering and Engineering Vibration(in Chinese),24(3):38-43.
Kausel E,Roёsset J M.1981.Stiffness matrices for layered soils.Bull.Seismol.Soc.Am.,71(6):1743-1761.
Lamb H.1904.On the propagation of tremors over the surface of an elastic solid.Phil.Trans.Roy.Soc.London,Ser.A,203(359-371):1-42.
Li S Y,Wang X L,Zhou Z H.2003.The time-step numerical simulation of free field motion of layered half-space for inclined seismic waves.Journal of Jilin University(Earth Science Edition)(in Chinese),33(3):372-376.
Li X J.1987.Seismic response analysis of soil layer and viscoelastoplastic model[Ph.D.](in Chinese).Harbin:Institute of Engineering Mechanics,China Seismological Bureau,34-51.
Liang J W,You H B.2004.Dynamic stiffness matrix of a poroelastic multi-layered site and its green′s functions.Earthq.Eng.Eng.Vib.,3(2):273-282.
Liang J W,Zhang A J,He Y.2014.2-D inversion of obliquely incident earthquake ground motion in layered elastic half-space.Journal of Vibration Engineering(in Chinese),27(3):441-450.
Lin C H,Lee V W,Trifunac M D.2005.The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid.Soil Dyn.Earthq.Eng.,25(3):205-223.
Liu J B,Wang Y.2006.A 1-D time-domain method for 2-D wave motion in elastic layered half-space by antiplane wave oblique incidence.Chinese Journal of Theoretical and Applied Mechanics(in Chinese),38(2):219-225.
Luan M T,Lin G.1992.Computational model for nonlinear analysis of soil site seimic response.Engineering Mechanics(in Chinese),9(1):94-103.
Thomson W T.1950.Transmission of elastic waves through a stratified solid medium.J.Appl.Phys.,21(2):89-93.
Wolf J P,Obernhuber P.1982a.Free-field response from inclined SV-and P-waves and Rayleigh-waves.Earthq.Eng.Struct.Dyn.,10(6):847-869.
Wolf J P,Obernhuber P.1982b.Free-field response from inclined SH-waves and Love-waves.Earthq.Eng.Struct.Dyn.,10(6):823-845.
Xue S T,Chen J,Chen R,et al.2000.The response analysis of damped TI layered soil for incident SH-waves.Journal of Vibration and Shock(in Chinese),19(4):54-56.
Xue S T,Xie L Y,Chen R,et al.2004.Dynamic analysis of response of transversely isotropic stratified media to incident P-SV waves.Chinese Journal of Rock Mechanics and Engineering(in Chinese),23(7):1163-1168.
高玉峰,金建新,谢康和等.1999.成层地基一维土层地震反应解析解.岩土工程学报,21(4):498-500.
金星,孔戈,丁海平.2004.水平成层场地地震反应非线性分析.地震工程与工程振动,24(3):38-43.
李山有,王学良,周正华.2003.地震波斜入射情形下水平成层半空间自由场的时域计算.吉林大学学报(地球科学版),33(3):372-376.
李小军.1987.粘弹塑性模型及土层地震反应分析[博士论文].哈尔滨:国家地震局工程力学研究所,34-51.
梁建文,张爱娟,何颖.2014.层状弹性场地基岩斜入射地震动二维反演.振动工程学报,27(3):441-450.
刘晶波,王艳.2006.成层半空间出平面自由波场的一维化时域算法.力学学报,38(2):219-225.
栾茂田,林皋.1992.场地地震反应一维非线性计算模型.工程力学,9(1):94-103.
薛松涛,陈军,陈镕等.2000.有限尼TI层状场地对平面入射SH波的响应分析.振动与冲击,19(4):54-56.
薛松涛,谢丽宇,陈镕等.2004.平面P-SV波入射时TI层状自由场地的响应.岩石力学与工程学报,23(7):1163-1168.
Haskell N A.1953.The dispersion of surface waves on multilayered media.Bull.Seismol.Soc.Am.,43(1):17-34.
Idriss I M,Seed H B.1968.Seismic response of horizontal soil layers.Journal of the Soil Mechanics and Foundations Division,94(4):1003-1031.
Jin X,Kong G,Ding H P.2004.Nonlinear seismic response analysis of horizontal layered site.Earthquake Engineering and Engineering Vibration(in Chinese),24(3):38-43.
Kausel E,Roёsset J M.1981.Stiffness matrices for layered soils.Bull.Seismol.Soc.Am.,71(6):1743-1761.
Lamb H.1904.On the propagation of tremors over the surface of an elastic solid.Phil.Trans.Roy.Soc.London,Ser.A,203(359-371):1-42.
Li S Y,Wang X L,Zhou Z H.2003.The time-step numerical simulation of free field motion of layered half-space for inclined seismic waves.Journal of Jilin University(Earth Science Edition)(in Chinese),33(3):372-376.
Li X J.1987.Seismic response analysis of soil layer and viscoelastoplastic model[Ph.D.](in Chinese).Harbin:Institute of Engineering Mechanics,China Seismological Bureau,34-51.
Liang J W,You H B.2004.Dynamic stiffness matrix of a poroelastic multi-layered site and its green′s functions.Earthq.Eng.Eng.Vib.,3(2):273-282.
Liang J W,Zhang A J,He Y.2014.2-D inversion of obliquely incident earthquake ground motion in layered elastic half-space.Journal of Vibration Engineering(in Chinese),27(3):441-450.
Lin C H,Lee V W,Trifunac M D.2005.The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid.Soil Dyn.Earthq.Eng.,25(3):205-223.
Liu J B,Wang Y.2006.A 1-D time-domain method for 2-D wave motion in elastic layered half-space by antiplane wave oblique incidence.Chinese Journal of Theoretical and Applied Mechanics(in Chinese),38(2):219-225.
Luan M T,Lin G.1992.Computational model for nonlinear analysis of soil site seimic response.Engineering Mechanics(in Chinese),9(1):94-103.
Thomson W T.1950.Transmission of elastic waves through a stratified solid medium.J.Appl.Phys.,21(2):89-93.
Wolf J P,Obernhuber P.1982a.Free-field response from inclined SV-and P-waves and Rayleigh-waves.Earthq.Eng.Struct.Dyn.,10(6):847-869.
Wolf J P,Obernhuber P.1982b.Free-field response from inclined SH-waves and Love-waves.Earthq.Eng.Struct.Dyn.,10(6):823-845.
Xue S T,Chen J,Chen R,et al.2000.The response analysis of damped TI layered soil for incident SH-waves.Journal of Vibration and Shock(in Chinese),19(4):54-56.
Xue S T,Xie L Y,Chen R,et al.2004.Dynamic analysis of response of transversely isotropic stratified media to incident P-SV waves.Chinese Journal of Rock Mechanics and Engineering(in Chinese),23(7):1163-1168.
高玉峰,金建新,谢康和等.1999.成层地基一维土层地震反应解析解.岩土工程学报,21(4):498-500.
金星,孔戈,丁海平.2004.水平成层场地地震反应非线性分析.地震工程与工程振动,24(3):38-43.
李山有,王学良,周正华.2003.地震波斜入射情形下水平成层半空间自由场的时域计算.吉林大学学报(地球科学版),33(3):372-376.
李小军.1987.粘弹塑性模型及土层地震反应分析[博士论文].哈尔滨:国家地震局工程力学研究所,34-51.
梁建文,张爱娟,何颖.2014.层状弹性场地基岩斜入射地震动二维反演.振动工程学报,27(3):441-450.
刘晶波,王艳.2006.成层半空间出平面自由波场的一维化时域算法.力学学报,38(2):219-225.
栾茂田,林皋.1992.场地地震反应一维非线性计算模型.工程力学,9(1):94-103.
薛松涛,陈军,陈镕等.2000.有限尼TI层状场地对平面入射SH波的响应分析.振动与冲击,19(4):54-56.
薛松涛,谢丽宇,陈镕等.2004.平面P-SV波入射时TI层状自由场地的响应.岩石力学与工程学报,23(7):1163-1168.