Acta Geophysica

, Volume 63, Issue 3, pp 605–633 | Cite as

Quasi-static Planar Deformation in a Medium Composed of Elastic and Thermoelastic Solid Half Spaces Due to Seismic Sources in an Elastic Solid

Open Access


A two-dimensional problem of quasi static deformation of a medium consisting of an elastic half space in welded contact with thermoelastic half space, caused due to seismic sources, is studied. Source is considered to be in the elastic half space. The basic equations, governed by the coupled theory of thermoelasticity, are used to model for thermoelastic half space. The analytical expressions for displacements, strain and stresses in the two half spaces are obtained first for line source and then for dip slip fault. The results for two particular cases, adiabatic conditions and isothermal conditions, are also obtained. Numerical results for displacements, stresses and temperature distribution have also been computed and are shown.

Key words

seismic sources thermoelastic quasi-static deformation 


  1. Abd-Alla, A.M. (1995), Thermal stress in a transversely isotropic circular cylinder due to an instantaneous heat source, Appl. Math. Comput. 68, 2-3, 113–124, DOI: 10.1016/0096-3003(94)00085-I.Google Scholar
  2. Ahrens, T.J. (ed.) (1995), Mineral Physics and Crystallography: A Handbook of Physical Constants, American Geophysical Union, Washington, D.C.Google Scholar
  3. Aki, K., and P.G. Richards (1980), Quantitative Seismology: Theory and Methods, Vol. I and II, W.H. Freeman & Co., San Francisco.Google Scholar
  4. Attetkov, A.V., I.K. Volkov, and S.S. Pilyavskii (2009), Temperature field of a solid body containing a spherical heating source with a uniformly moving boundary, J. Eng. Phys. Thermophys. 82, 2, 368–375, DOI: 10.1007/s10891-009-0185-x.CrossRefGoogle Scholar
  5. Ben-Menahem, A., and S.J. Singh (1981), Seismic Waves and Sources, 2nd ed., Springer Verlag, New York.CrossRefGoogle Scholar
  6. Burridge, R., and L. Knopoff (1964), Body force equivalents for seismic dislocations, Bull. Seismol. Soc. Am. 54, 6a, 1875–1888.Google Scholar
  7. Dziewonski, A.M., and D.L. Anderson (1981), Preliminary reference Earth model, Phys. Earth Planet. In. 25, 4, 297–356, DOI: 10.1016/0031-9201(81)90046-7.CrossRefGoogle Scholar
  8. Freund, L.B., and D.M. Barnett (1976), A two-dimensional analysis of surface deformation due to dip-slip faulting, Bull. Seismol. Soc. Am. 66, 3, 667–675.Google Scholar
  9. Garg, N.R., and S.J. Singh (1987), 2-D static response of a transversely isotropic multilayered half-space to surface loads, Indian J. Pure Appl. Math. 18, 8, 763–777.Google Scholar
  10. Garg, N.R., D.K. Madan, and R.K. Sharma (1996), Two-dimensional deformation of an orthotropic elastic medium due to seismic sources, Phys. Earth Planet. In. 94, 1-2, 43–62, DOI: 10.1016/0031-9201(95)03095-6.CrossRefGoogle Scholar
  11. Garg, N.R., R. Kumar, A. Goel, and A. Miglani (2003), Plane strain deformation of an orthotropic elastic medium using an eigenvalue approach, Earth Planets Space 55, 1, 3–9, DOI: 10.1186/BF03352457.CrossRefGoogle Scholar
  12. Ghosh, M.K., and M. Kanoria (2007), Displacements and stresses in composite multi-layered media due to varying temperature and concentrated load, Appl. Math. Mech. 28, 6, 811–822, DOI: 10.1007/s10483-007-0611-5.CrossRefGoogle Scholar
  13. Heaton, T.H., and R.E. Heaton (1989), Static deformations from point forces and force couples located in welded elastic Poissonian half-spaces: Implications for seismic moment tensors, Bull. Seismol. Soc. Am. 79, 3, 813–841.Google Scholar
  14. Hou, P.-F., A.Y.T. Leung, and C.-P. Chen (2008a), Fundamental solution for transversely isotropic thermoelastic materials, Int. J. Solids Struct. 45, 2, 392–408, DOI: 10.1016/j.ijsolstr.2007.08.024.CrossRefGoogle Scholar
  15. Hou, P.-F., A.Y.T. Leung, and C.-P. Chen (2008b), Green’s functions for semiinfinite transversely isotropic thermoelastic materials, ZAMM J. Appl. Math. Mech. 88, 1, 33–41, DOI: 10.1002/zamm.200710355.CrossRefGoogle Scholar
  16. Maruyama, T. (1966), On two-dimensional elastic dislocations in an infinite and semi-infinite medium, Bull. Earthq. Res. Inst. Univ. Tokyo 44, 811–871.Google Scholar
  17. Nowacki, W. (1975), Dynamical Problems of Thermoelasticity, PWN Polish Sci. Publ., Warszawa, Noordhoff Int. Publ., Leyden.Google Scholar
  18. Okada, Y. (1985), Surface deformation due to shear and tensile faults in a halfspace, Bull. Seismol. Soc. Am. 75, 4, 1135–1154.Google Scholar
  19. Okada, Y. (1992), Internal deformation due to shear and tensile faults in a halfspace, Bull. Seismol. Soc. Am. 82, 2, 1018–1040.Google Scholar
  20. Pan, E. (1989a), Static response of a transversely isotropic and layered half-space to general surface loads, Phys. Earth Planet. In. 54, 3–4, 353–363, DOI: 10.1016/0031-9201(89)90252-5.Google Scholar
  21. Pan, E. (1989b), Static response of a transversely isotropic and layered half-space to general dislocation sources, Phys. Earth Planet. In. 58, 2–3, 103–117, DOI: 10.1016/0031-9201(89)90046-0.CrossRefGoogle Scholar
  22. Pan, E. (1990), Thermoelastic deformation of a transversely isotropic and layered half-space by surface loads and internal sources, Phys. Earth Planet. Int. 60, 1–4, 254–264, DOI: 10.1016/0031-9201(90)90266-Z.CrossRefGoogle Scholar
  23. Rani, S., S.J. Singh, and N.R. Garg (1991), Displacements and stresses at any point of a uniform half-space due to two-dimensional buried sources, Phys. Earth Planet. Int. 65, 3–5, 276–282, DOI: 10.1016/0031-9201(91)90134-4.CrossRefGoogle Scholar
  24. Rongved, L., and J.T. Frasier (1958), Displacement discontinuity in the elastic halfspace, J. Appl. Mech. 25, 125–128.Google Scholar
  25. Rundle, J.B. (1982), Some solutions for static and pseudo-static deformation in layered, nonisothermal, porous media, J. Phys. Earth 30, 5, 421–440, DOI: 10.4294/jpe1952.30.421.CrossRefGoogle Scholar
  26. Sato, R. (1971), Crustal deformation due to dislocation in a multi-layered medium, J. Phys. Earth 19, 1, 31–46, DOI: 10.4294/jpe1952.19.31.CrossRefGoogle Scholar
  27. Sato, R., and M. Matsu’ura (1973), Static deformations due to the fault spreading over several layers in a multi-layered medium. Part I: Displacement, J. Phys. Earth 21, 3, 227–249, DOI: 10.4294/jpe1952.21.227.CrossRefGoogle Scholar
  28. Schapery, R.A. (1962), Approximate methods of transform inversion for viscoelastic stress analysis. In: Proc. 4th U.S. National Congress of Applied Mechanics, 18–21 June 1962, Berkeley USA, Vol. 2, 1075–1085, American Society of Mechanical Engineers, New York.Google Scholar
  29. Shevchenko, V.P., and A.S. Gol’tsev (2001), The thermoelastic state of orthotropic shells heated by concentrated heat sources, Int. Appl. Mech. 37, 5, 654–661, DOI: 10.1023/A:1012364530719.CrossRefGoogle Scholar
  30. Singh, K., D.K. Madan, A. Goel, and N.R. Garg (2005), Two-dimensional static deformation of an anisotropic medium, Sadhana 30, 4, 565–583, DOI: 10.1007/BF02703280.CrossRefGoogle Scholar
  31. Singh, S.J. (1970), Static deformation of a multilayered half-space by internal sources, J. Geophys. Res. 75, 17, 3257–3263, DOI: 10.1029/JB075i017p03257.CrossRefGoogle Scholar
  32. Singh, S.J., and A. Ben-Menahem (1969), Displacement and strain fields due to faulting in a sphere, Phys. Earth Planet. In. 2, 2, 77–87, DOI: 10.1016/0031-9201(69)90003-X.CrossRefGoogle Scholar
  33. Singh, S.J., and N.R Garg (1985), On two-dimensional elastic dislocations in a multilayered half-space, Phys. Earth Planet. In. 40, 2, 135–145, DOI: 10.1016/0031-9201(85)90067-6.CrossRefGoogle Scholar
  34. Singh, S.J., and N.R. Garg (1986), On the representation of two-dimensional seismic sources, Acta Geophys. Pol. 34, 1, 1–12.Google Scholar
  35. Singh, S.J., A. Ben-Menahem, and M. Vered (1973), A unified approach to the representation of seismic sources, Proc. Roy. Soc. London A 331, 1587, 525–551, DOI: 10.1098/rspa.1973.0006.CrossRefGoogle Scholar
  36. Singh, S.J., S. Rani, and N.R. Garg (1992), Displacements and stresses in two welded half-spaces caused by two-dimensional sources, Phys. Earth Planet. In. 70, 1, 90–101, DOI: 10.1016/0031-9201(92)90164-Q.CrossRefGoogle Scholar
  37. Singh, S.J., G. Kumari, and K. Singh (1993), Static deformation of two welded elastic half-spaces caused by a finite rectangular fault, Phys. Earth Planet. In. 79, 3, 313–333, DOI: 10.1016/0031-9201(93)90112-M.CrossRefGoogle Scholar
  38. Singh, S.J., A. Kumar, and J. Singh (2003), Deformation of a monoclinic elastic half-space by a long inclined strike-slip fault, ISET J. Earthq. Technol. 40, 1, 51–59.Google Scholar
  39. Small, J.C., and J.R. Booker (1986), The behaviour of layered soil or rock containing a decaying heat source, Int. J. Numer. Anal. Meth. Geomech. 10, 5, 501–519, DOI: 10.1002/nag.1610100504.CrossRefGoogle Scholar
  40. Stein, S., and M. Wysession (2003), An Introduction to Seismology, Earthquakes, and Earth Structure, Blackwell Publishing, Oxford.Google Scholar
  41. Steketee, J.A. (1958), On Volterra’s dislocations in a semi-infinite elastic medium, Can. J. Phys. 36, 2, 192–205, DOI: 10.1139/p58-024.CrossRefGoogle Scholar
  42. Tomar, S.K., and N.K. Dhiman (2003), 2-D deformation analysis of a half-space due to a long dip-slip fault at finite depth, J. Earth Syst. Sci. 112, 4, 587–596, DOI: 10.1007/BF02709782.CrossRefGoogle Scholar
  43. Youssef, H.M. (2006), Problem of generalized thermoelastic infinite medium with cylindrical cavity subjected to a ramp-type heating and loading, Arch. Appl. Mech. 75, 8–9, 553–565, DOI: 10.1007/s00419-005-0440-3.CrossRefGoogle Scholar
  44. Youssef, H.M. (2009), Generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source, Mech. Res. Commun. 36, 4, 487–496, DOI: 10.1016/j.mechrescom.2008.12.004.CrossRefGoogle Scholar
  45. Youssef, H.M. (2010), Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source, Arch. Appl. Mech. 80, 11, 1213–1224, DOI: 10.1007/s00419-009-0359-1.CrossRefGoogle Scholar

Copyright information

© Vashisth et al. 2015

Authors and Affiliations

  1. 1.Department of MathematicsKurukshetra UniversityKurukshetraIndia
  2. 2.Department of MathematicsGovernment Post Graduate CollegeHisarIndia
  3. 3.Department of MathematicsGuru Jambheshwar University of Science and TechnologyHisarIndia

Personalised recommendations