Abstract
The transient response of an unlimited cylindrical cavity buried in the infinite elastic soil subjected to an anti-plane impact load along the cavern axis direction was studied. Using Laplace transform combining with contour integral of the Laplace inverse transform specifically, the general analytical expressions of the soil displacement and stress are obtained in the time domain, respectively. And the numerical solutions of the problem computed by analytical expressions are presented. In the time domain, the dynamic responses of the infinite elastic soil are analyzed, and the calculation results are compared with those from numerical inversion proposed by Durbin and the static results. One observes good agreement between analytical and numerical inversion results, lending the further support to the method presented. Finally, some valuable shear wave propagation laws are gained: the displacement of the soil remains zero before the wave arrival, and after the shear wave arrival, the stress and the displacement at this point increase abruptly, then reduce and tend to the static value gradually at last. The wave attenuates along the radial, therefore the farther the wave is from the source, the smaller the stress and the displacement are, and the stress and the displacement are just functions of the radial distance from the axis.
Similar content being viewed by others
References
HAYWOOD J H. Response of an elastic cylindrical shell to a pressure pulse [J]. Journal of Mechanics and Applied Mathematics, 1958, 11(2): 130–140.
PAO Yih-hsing, MOW C C. Diffraction of elastic waves and dynamic stress concentrations [M]. US: Adam Hillier Ltd, 1973: 120–131.
LEE J K, MAL A K. A volume integral equation technique for multiple scattering problems in elastodynamics [J]. Applied Mathematics and Computation, 1995, 67(1/2/3): 135–159.
DAVIS C A, LEE V W, BARDET J P. Transverse response of underground cavities and pipes to incident SV waves [J]. Earthquake Engineering and Structural Dynamics, 2001, 30(3): 395–410.
IAKOVLEV S. Interaction of a spherical shock wave and a submerged fluid-filled circular cylindrical shell [J]. Journal of Sound and Vibration, 2002, 255(4): 615–633.
MANOLIS G D. Elastic wave scattering around cavities in inhomogeneous continua by the BEM [J]. Journal of Sound and Vibration, 2003, 266(2): 281–305.
FANG Xue-qian, HU Chao, HUANG Wen-hu. Dynamic stress of a circular cavity buried in a semi-infinite functionally graded piezoelectric material subjected to shear waves [J]. European Journal of Mechanics A/Solids, 2007, 26(6): 1016–1028.
OHTAKI H, KOTOSAKA S, NAGASAKA Y. Deriving the dynamic stress function using complex function and its application to the analysis of the stress distribution around an elliptical hole [J]. International Journal of Engineering Science, 2008, 46(1): 66–85.
JIANG Ling-fa, ZHOU Xiang-lian, WANG Jian-hua. Scattering of a plane wave by a lined cylindrical cavity in a poroelastic half-plane [J]. Computers and Geotechnics, 2009, 36(5): 773–786.
SMERZINI C, AVILES J, PAOLUCC R, SANCHEZ-SESMA F J. Effect of underground cavities on surface earthquake ground motion under SH wave propagation [J]. Earthquake Engineering and Structural Dynamics, 2009, 38(12): 1441–1460.
LIN Chi-hsin, LEE V W, TODOROVSKA M I, TRIFUNAC M D. Zero-stress, cylindrical wave functions around a circular underground tunnel in a flat, elastic half-space: Incident P-waves [J]. Soil Dynamics and Earthquake Engineering, 2010, 30(10): 879–894.
MIKLOWITZ J. Plane-stress unloading waves emanating from a suddenly punched hole in a stretched elastic plate [J]. Journal of Applied Mechanics, 1960, 27(1): 165–171.
EASON G. Propagation of waves from spherical and cylindrical cavities [J]. ZAMP, 1963, 14(1): 12–22.
HERMAN H, KLOSNER J M. Transient response of a periodically supported cylindrical shell immersed in a fluid medium [J]. Journal of Applied Mechanics, 1965, 32(3): 562–568.
GEERS T L. Excitation of an elastic cylindrical shell by a transient acoustic wave [J]. Journal of Applied Mechanics, 1969, 36(3): 459–469.
FORRESTAL M J, SAGARTZ M J. Radiated pressure in an acoustic medium produced by pulsed cylindrical and spherical shells [J]. Journal of Applied Mechanics, 1971, 38(4): 1057–1060.
EASON G. The propagation of waves from a cylindrical cavity [J]. Journal of Composite Materials, 1973, 7: 90–99.
DUFFEY T A. Transient Response of viscoplastic and viscoelastic shells submerged in fluid media [J]. Journal of Applied Mechanics, 1976, 43(1): 137–143.
MOODIE T B, BARCLAY D W. Wave propagation from a cylindrical cavity [J]. Acta Mechanica, 1977, 27(1/2/3/4): 103–120.
MOODIE T B, HADDOW J B, MIODUCHOWSKI A, TAIT R J. Plane elastic waves generated by dynamical loading applied to edge of circular hole [J]. Journal of Applied Mechanics, 1981, 48(3): 577–581.
WATANABE K. Transient response of an inhomogeneous elastic solid to an impulsive SH-wave [J]. The Japan Society of Mechanical Engineers, 1982, 25(201): 315–319.
GLENN L A, KIDDER R E. Blast loading of a spherical container surrounded by an infinite elastic medium [J]. Journal of Applied Mechanics, 1983, 50(4): 723–726.
AKKAS N, ERDOGOGAN F. The residual variable method applied to the diffusion equation in cylindrical coordinates [J]. ACTA Mechanic, 1989, 79(3/4): 207–219.
ZHANG Qing-yuan, ZHAN Ren-rui. Dynamic response of a spherical cavity subjected to blast loads [J]. Explosion and Shock Wave, 1994, 14(2): 182–185. (in Chinese)
LIU Guo-li, ZHAO Hui-bin, XU Yi-yan. Transient response of semi-circular canyon under step SH wave-long term solution [J]. Earthquake Engineering and Engineering Vibration, 1995, 15(1): 92–99. (in Chinese)
YANG Jun, GONG Qun-mei, WU Shin-ming. Dynamic analysis of cylindrical holes in saturated soil [J]. Shanghai Mechanics, 1996, 17(1): 37–45. (in Chinese)
ZAKOUT U, AKKAS N. Transient response of a cylindrical cavity with and without a bonded shell in an infinite elastic medium [J]. International Journal of Engineering Science, 1997, 35(12/13): 1203–1220.
LI X. Stress and displacement fields around a deep circular tunnel with partial sealing [J]. Computers and Geotechnics, 1999, 24(2): 125–140.
LI X, FLORES-BERRONES R. Time-dependent behavior of partially sealed circular tunnels [J]. Computers and Geotechnics, 2002, 29(6): 433–449.
HADDOW J B, JIANG Lei. Finite amplitude elastic waves due to non-uniform traction at the surface of a cylindrical cavity [J]. International Journal of Non-linear Mechanics, 2006, 41(2): 231–241.
FELDGUN V R, KOCHETKOV A V, KARINSKI Y S, YANKELEVSKY D Z. Internal blast loading in a buried lined tunnel [J]. International Journal of Impact Engineering, 2008, 35(3): 172–183.
FELDGUN V R, KOCHETKOV A V, KARINSKI Y S, YANKELEVSKY D Z. Blast response of a lined cavity in a porous saturated soil [J]. International Journal of Impact Engineering, 2008, 35(9): 953–966.
IAKOVLEV S. Interaction between a submerged evacuated cylindrical shell and a shock wave-Part I: Diffraction-radiation analysis [J]. Journal of Fluids and Structures, 2008, 24(7): 1077–1097.
IAKOVLEV S. Interaction between a submerged evacuated cylindrical shell and a shock wave-Part II: Numerical aspects of the solution [J]. Journal of Fluids and Structures, 2008, 24(7): 1098–1119.
GAO Meng, GAO Guang-yun, WANG Ying, YANG Cheng-bin, XIONG Hao. Dynamic solutions of cylindrical cavities with the lining under internal load in the saturated soil [J]. Chinese Journal of Solid Mechanics, 2009, 30(5): 481–487. (in Chinese)
GAO Meng, GAO Guang-yun, WANG Ying, XIONG Hao. Solution on dynamic response of a cylindrical cavity under internal load [J]. Chinese Quarterly of Mechanics, 2009, 30(2): 266–272. (in Chinese)
GIANNOPOULOS G, LARCHER M, CASADEI F, SOLOMOS G. Risk assessment of the fatality due to explosion in land mass transport infrastructure by fast transient dynamic analysis [J]. Journal of Hazardous Materials, 2010, 173(1/2/3): 401–408.
CAI Yuan-qiang, CHEN Cheng-zhen, SUN Hong-lei. Dynamic response of tunnel in viscoelastic saturated soil subjected to blast loads [J]. Journal of Zhejiang University (Engineering Science), 2011, 45(9): 1657–1663. (in Chinese)
HE Wei, CHEN Jian-yun, GUO Jing. Dynamic analysis of subway station subjected to internal blast loading [J]. Journal Central South University of Technology, 2011, 18(3): 917–924.
HADDOW J B, LORIMER S A, TAIT R J. Nonlinear axial shear wave propagation in a hyperelastic incompressible solid [J]. Acta Mechanica, 1987, 66(1/2/3/4): 205–216.
WATANABE K, PAYTON R G.. SH wave in a cylindrically anisotropic elastic solid a general solution for a point source [J]. Wave Motion, 1996, 25(2): 197–212.
BARCLAY D W. Wavefront expansion for nonlinear axial shear wave propagation [J]. International Journal of Non-linear Mechanics, 1998, 33(2): 259–274.
BARCLAY D W. Shock front analysis for axial shear wave propagation in a hyperelastic incompressible solid [J]. International Journal of Non-linear Mechanics, 1999, 133(1/2/3/4): 105–129.
BARCLAY D W. Shock calculations for axially symmetric shear wave propagation in a hyperelastic incompressible solid [J]. International Journal of Non-linear Mechanics, 2004, 39(1): 101–121.
HADDOW J B, JIANG L. Finite amplitude azimuthal shear waves in a compressible hyperelastic solid [J]. Journal of Applied Mechanics, 2001, 68(2): 145–152.
BARCLAY D W. The propagation and reflection of finite amplitude cylindrically symmetric shear waves in a hyperelastic incompressible solid [J]. Acta Mechanica, 2007, 193(1/2): 17–42.
PAYTON R G. Diffraction in a cylindrically orthotropic elastic solid containing a stress free crack [J]. Z Angew Math Phys, 2007, 58(5): 876–888.
DAROS C H. A fundamental solution for SH-waves in a class of inhomogeneous anisotropic media [J]. International Journal of Engineering Science, 2008, 46(8): 809–817.
DURBIN F. Numerical inversion of Laplace transformation: An efficient improvement to durbin and abatep’s method [J]. The Computer Journal, 1974, 17(4): 371–376.
LIANG Kun-miao. Methods of mathematics and physics [M]. Beijing: Higher Education Press, 2010: 355–361. (in Chinese)
WATSON G N. Theory of bessel function [M]. England: Cambridge University Press, 1995: 215–221.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhai, Cj., Xia, Td., Du, Gq. et al. Dynamic response of cylindrical cavity to anti-plane impact load by using analytical approach. J. Cent. South Univ. 21, 405–415 (2014). https://doi.org/10.1007/s11771-014-1954-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11771-014-1954-z