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Fundamental Solutions to Static and Dynamic Loads for Homogeneous Reduced Micropolar Half-Space

Pure and Applied Geophysics volume 176, pages 4881–4905 (2019)Cite this article

Abstract

Analytical expressions for displacements and rotations in a homogeneous reduced micropolar half-space subjected to a finite buried source are derived using the method of potentials. Explicit solutions for displacements and rotations are derived for a uniformly distributed force acting over a circular region in the horizontal or vertical direction. The obtained solutions for displacements are validated against those available in literature for a classical elastic half-space. Finally, Green’s functions for displacements and rotations are derived for a unit impulse applied in the horizontal or vertical direction. In addition to analytical solutions, the paper also compares the dispersion phenomenon of compression and shear waves propagating in a reduced micropolar half-space with different material properties.

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References

  • Abreu, R., Thomas, C., & Durand, S. (2018). Effect of observed micropolar motions on wave propagation in deep Earth minerals. Physics of the Earth and Planetary Interiors, 276, 215–225.

    Article  Google Scholar 

  • Bouchon, M. (1981). A simple method to calculate Green’s functions for elastic layered media. Bulletin of the Seismological Society of America, 71(4), 959–971.

    Google Scholar 

  • Cosserat, E., & Cosserat, F. (1909). Theory of deformable bodies. Hondo: Scientific Library.

    Google Scholar 

  • Eringen, A. C. (1999). Microcontinuum field theories: I. Foundations and solids. New York: Springer.

    Book  Google Scholar 

  • Gade, M., & Raghukanth, S. T. G. (2015). Seismic ground motion in micropolar elastic half-space. Applied Mathematical Modelling, 39(23–24), 7244–7265.

    Article  Google Scholar 

  • Grekova, E. F., Kulesh, M. A., & Herman, G. C. (2009). Waves in linear elastic media with microrotations, part 2: Isotropic reduced Cosserat model. Bulletin of the Seismological Society of America, 99(2B), 1423–1428.

    Article  Google Scholar 

  • Hart, G. C., Lew, M., & DiJulio, R. M. (1975). Torsional response of high-rise buildings. Journal of the Structural Division, 101(2), 397–416.

    Google Scholar 

  • Hassanpour, S., & Heppler, G. R. (2017). Micropolar elasticity theory: a survey of linear isotropic equations, representative notations, and experimental investigations. Mathematics and Mechanics of Solids, 22(2), 224–242.

    Article  Google Scholar 

  • Hisada, Y. (1994). An efficient method for computing Green’s functions for a layered half-space at large epicentral distances. Bulletin of the Seismological Society of America, 84(5), 1456–1472.

    Google Scholar 

  • Igel, H., Schreiber, U., Flaws, A., Schuberth, B., Velikoseltsev, A., & Cochard, A. (2005). Rotational motions induced by the M.81 Tokachi-oki earthquake. Geophysical Research Letters, 8, 32.

    Google Scholar 

  • Kanth, S. T. (2008). Source mechanism model for ground motion simulation. Applied Mathematical Modelling, 32(7), 1417–1435.

    Article  Google Scholar 

  • Kennett, B. (1983). Seismic wave propagation in stratified media. Canberra: ANU E Press.

    Google Scholar 

  • Kulesh, M. (2009). Waves in linear elastic media with microrotations, Part 1: Isotropic full Cosserat model. Bulletin of the Seismological Society of America, 99(2 B), 1416–1422.

    Article  Google Scholar 

  • Kurnosov, A., Marquardt, H., Frost, D. J., Boffa Ballaran, T., & Ziberna, L. (2017). Evidence for a \(\mathrm{Fe}^{3+}\)-rich pyrolitic lower mantle from (Al, Fe)-bearing bridgmanite elasticity data. Nature, 543(7646), 543–546.

    Article  Google Scholar 

  • Nigbor, R. L. (1994). Six-degree-of-freedom ground-motion measurement. Bulletin of the Seismological Society of America, 84(5), 1665–1669.

    Google Scholar 

  • Okada, Y. (1985). Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America, 75(4), 1135–1154.

    Google Scholar 

  • Pak, R. Y. S. (1987). Asymmetric wave propagation in an elastic half-space by a method of potentials. Journal of Applied Mechanics-Transactions of the ASME, 54(1), 121–126.

    Article  Google Scholar 

  • Pak, R. Y. S., & Guzina, B. (2002). Three-dimensional Green’s functions for a multilayered half-space in displacement potentials. Journal of Engineering Mechanics, 128(4), 449–461.

    Article  Google Scholar 

  • Pekeris, C. L., & Longman, I. M. (1958). The motion of the surface of a uniform elastic half-space produced by a buried torque-pulse. Geophysical Journal of the Royal Astronomical Society, 1(2), 146–153.

    Article  Google Scholar 

  • Piessens, R. (2000). The Hankel transform. The Transforms and Applications Handbook, 2, 9.

    Google Scholar 

  • Schwartz, L. M., Johnson, D. L., & Feng, S. (1984). Vibrational modes in granular materials. Physical Review Letters, 52(10), 831–834.

    Article  Google Scholar 

  • Takeo, M. (1998). Ground rotational motions recorded in near-source region of earthquakes. Geophysical Research Letters, 25(6), 789–792.

    Article  Google Scholar 

  • Teisseyre, R. (1973). Earthquake processes in a micromorphic continuum. Pure and Applied Geophysics PAGEOPH, 102(1), 15–28.

    Article  Google Scholar 

  • Teisseyre, R. (2011). Why rotation seismology: Confrontation between classic and asymmetric theories. Bulletin of the Seismological Society of America, 101(4), 1683–1691.

    Article  Google Scholar 

  • Tsai, N. C., & Housner, G. W. (1970). Calculation of surface motions of a layered half-space. Bulletin of the Seismological Society of America, 60(5), 1625–1651.

    Google Scholar 

  • Yin, J., Nigbor, R. L., Chen, Q., & Steidl, J. (2016). Engineering analysis of measured rotational ground motions at GVDA. Soil Dynamics and Earthquake Engineering, 87, 125–137.

    Article  Google Scholar 

  • Yong, Y., Zhang, R., & Yu, J. (1997a). Motion of foundation on a layered soil medium—I. Impedance characteristics. Soil Dynamics and Earthquake Engineering, 16(5), 295–306.

    Article  Google Scholar 

  • Yong, Y., Zhang, R., & Yu, J. (1997b). Motion of foundation on a layered soil medium—II. Response analysis. Soil Dynamics and Earthquake Engineering, 16(5), 307–316.

    Article  Google Scholar 

  • Zerva, A., & Zhang, O. (1997). Correlation patterns in characteristics of spatially variable seismic ground motions. Earthquake Engineering and Structural Dynamics, 26(1), 19–39.

    Article  Google Scholar 

  • Zhang, R. (2000). Some observations of modelling of wave motion in layer-based elastic media. Journal of Sound and Vibration, 229(5), 1193–1212.

    Article  Google Scholar 

  • Zhang, R., Yong, Y., & Lin, Y. K. (1991a). Earthquake ground motion modeling. I: Deterministic point source. Journal of Engineering Mechanics, 117(9), 2114–2132.

    Article  Google Scholar 

  • Zhang, R., Yong, Y., & Lin, Y. K. (1991b). Earthquake ground motion modeling. II: Stochastic line source. Journal of Engineering Mechanics, 117(9), 2133–2148.

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Civil Engineering, IIT Madras, Chennai, India

    Anjali C. Dhabu & S. T. G. Raghukanth

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  1. Anjali C. Dhabu
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  2. S. T. G. Raghukanth
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Correspondence to S. T. G. Raghukanth.

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Dhabu, A.C., Raghukanth, S.T.G. Fundamental Solutions to Static and Dynamic Loads for Homogeneous Reduced Micropolar Half-Space. Pure Appl. Geophys. 176, 4881–4905 (2019). https://doi.org/10.1007/s00024-019-02225-0

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  • DOI: https://doi.org/10.1007/s00024-019-02225-0

Keywords

  • Reduced micropolar medium
  • seismic wave propagation
  • Green’s functions
  • dispersion analysis