Quasi-Static Deformation of a Uniform Thermoelastic Half –Space Due to Seismic Sources and Heat Source
محورهای موضوعی : Engineering
A.K Vashishth
1
(Department of Mathematics, Kurukshetra University, Kurukshetra 136119, India)
K Rani
2
(Department of Mathematics, CMG Govt. College for Women, Bhodia Khera, Fatehabad 125050, India)
کلید واژه: Thermoelastic, Heat source, Seismic source, Dip-slip dislocation, Plane deformation,
چکیده مقاله :
This paper investigates the quasi-static plane deformation of an isotropic thermoelastic half-space due to buried seismic sources and heat source. Governing equations of thermo-elasticity are solved to obtain solutions for seismic sources in a thermoelastic half-space. The general solutions are acquired with the aid of Laplace and Fourier transforms and with the use of boundary conditions. The case of dip-slip line dislocation is studied in detail along with line heat source. Analytical solutions for two limiting cases: adiabatic and isothermal, are obtained. The solutions for displacement, stresses and temperature in space-time domain are obtained by using a numerical inversion procedure. The accuracy of the proposed method is verified through a comparison of the results obtained with the existing solutions for elastic medium. In addition, numerical results for displacements, stresses and temperature function, induced by a vertical dip-slip dislocation and line heat source, are presented graphically to illustrate the effect of inclusion of thermal effect in simulation of the problem.
[1] Steketee J. A., 1958, On Volterra's dislocations in a semi-infinite elastic medium, Canadian Journal of Physics 36: 192-205.
[2] Maruyama T., 1964, Statical elastic dislocations in an infinite and semi-infinite medium, Bulletin of the Earthquake Research Institute 42: 289-368.
[3] Maruyama T., 1966, On two-dimensional elastic dislocations in an infinite and semi-infinite medium, Bulletin of the Earthquake Research Institute 44: 811-871.
[4] Savage J.C., 1974, Dislocations in Seismology, Dislocation Theory: A Treatise, New York, Marcel Dekker.
[5] Savage J.C., 1980, Dislocations in Seismology, Dislocations in Solids, Amsterdam, North Holland.
[6] Freund L.B., Barnett D.M., 1976, A two dimensional analysis of surface deformation due to dip-slip faulting, Bulletin of the Seismological Society of America 66: 667-675.
[7] 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.
[8] Okada Y., 1992, Internal deformation due to shear and tensile faults in a half-space, Bulletin of the Seismological Society of America 82(2): 1018-1040.
[9] Rani S., Singh S.J., Garg N.R., 1991, Displacements and stresses at any point of a uniform half space due to two-dimensional buried sources, Physics of the Earth and Planetary Interiors 65: 276-282.
[10] Cohen S.C., 1992, Post seismic deformation and stresses diffusion due to visco-elasticity and comments on the modified Elsasser model, Journal of Geophysical Research 97: 15395-15403.
[11] Singh S.J., Rani S., 1996, 2-D modeling of the crustal deformation associated with strike-slip and dip-slip faulting in the Earth, Proceedings of the Indian Academy of Science 66: 187-215.
[12] Singh M., Singh S.J., 2000, Static deformation of a uniform half-space due to a very long tensile fault, Journal of Earthquake Technology 37: 27-38.
[13] Singh S.J., Kumar A., Rani S., Singh M., 2002, Deformation of a uniform half-space due to a long inclined tensile fault, Geophysical Journal International 148: 687-691.
[14] Tomar S.K., Dhiman N.K., 2003, 2-D Deformation analysis of a half-space due to a long dip-slip fault at finite depth, Proceedings of the Indian Academy of Science 112(4): 587-596.
[15] Rani S., Verma R.C., 2013, Two-dimensional deformation of a uniform half-space due to non-uniform movement accompanying a long vertical tensile fracture, Journal of Earth System Science 122(4): 1055-1063.
[16] Gade M., Raghukanth S.T.G., 2015, Seismic ground motion in micro polar elastic half-space, Applied Mathematical Modelling 39: 7244-7265.
[17] Sahrawat R.K., Godara Y., Singh M., 2014, Static deformation of a uniform half- space with rigid boundary due to a long dip-slip fault of finite width, International Journal of Engineering and Technical Research 2(5): 189-194.
[18] Volkov D., 2009, An inverse problem for faults in elastic half space, ESAIM: Proceedings 26: 1-23.
[19] Volkov D., Vousin C., Ionescu I.R., 2017, Determining fault geometries from surface displacements, Pure and Applied Geophysics 174(4): 1659-1678.
[20] Singh S.J., Garg N. R., 1986, On the representation of two-dimensional seismic sources, Acta Geophysica 34: 1-12.
[21] Singh S.J., Rani S., 1991, Static deformation due to two dimensional seismic sources embedded in an isotropic half-space in welded contact with an orthotropic half-space, Journal of Physics of the Earth 39: 599-618.
[22] Rani S., Singh S. J., 1992, Static deformation of a uniform half-space due to a long dip-slip fault, Geophysical Journal International 109: 469-476.
[23] Rani S., Singh S.J., 1992, Static deformation of two welded half-spaces due to dip-slip faulting, Proceedings of the Indian Academy of Science 101: 269-282.
[24] Singh S.J., Rani S., Garg N. R., 1992, Displacement and stresses in two welded half spaces caused by two-dimensional buried sources, Physics of the Earth and Planetary Interiors 70: 90-101.
[25] Garg N.R., Madan D.K., Sharma R.K., 1996, Two-dimensional deformation of an orthotropic elastic medium due to seismic sources, Physics of the Earth and Planetary Interiors 94(1): 43-62.
[26] Singh S. J., Punia M., Kumari G., 1997, Deformation of a layered half-space due to a very long dip-slip fault, Proceedings of the Indian National Science Academy 63(3): 225-240.
[27] Rani S., Bala N., 2006, 2-D deformation of two welded half-spaces due to a blind dip-slip fault, Journal of Earth System Science 115: 277-287.
[28] Rani S., Bala N., Verma R.C., 2012, Displacement field due to non-uniform slip along a long dip-slip fault in two welded half-spaces, Journal of Earth Science 23(6): 864-872.
[29] Malik M., Singh M., Singh J., 2013, Static deformation of a uniform half-space with rigid boundary due to a vertical dip-slip line source, IOSR Journal of Mathematics 4(6): 26-37.
[30] Debnath S. K., Sen S., 2013, Pattern of stress-strain accumulation due to a long dip slip fault movement in a viscoelastic layered model of the lithosphere –asthenosphere system, International Journal of Applied Mechanics and Engineering 18(3): 653-670.
[31] Godara Y., Sahrawat R. K., Singh M., 2014, Static deformation due to two-dimensional seismic sources embedded in an isotropic half-space in smooth contact with an orthotropic half-space, Global Journal of Mathematical Analysis 2(3): 169-183.
[32] Verma R.C., Rani S., Singh S. J., 2016, Deformation of a poroelastic layer overlying an elastic half-space due to dip-slip faulting, International Journal for Numerical and Analytical Methods in Geomechanics 40: 391-405.
[33] Pan E., 1990, Thermoelastic deformation of a transversely isotropic and layered half-space by surface loads and internal sources, Physics of the Earth and Planetary Interiors 60: 254-264.
[34] Ghosh M.K., Kanoria M., 2007, Displacements and stresses in composite multi-layered media due to varying temperature and concentrated load, Applied Mathematics and Mechanics 28(6): 811-822.
[35] Hou P.F., Tong J., Xiong S.M., Hu J.F., 2012, Two-dimensional Green’s functions for semi-infinite isotropic thermoelastic plane, Zeitschrift für Angewandte Mathematik und Physik 64: 1587-1598.
[36] Jacquey A.B., Cacace M., Blocher G., Wenderoth M. S., 2015, Numerical investigation of thermoelastic effects on fault slip tendency during injection and production of geothermal fluids, Energy Procedia 76: 311-320.
[37] Marin M., Florea O., Mahmoud S.R., 2015, A result regarding the seismic dislocations in micro stretch thermoelastic bodies, Mathematical Problems in Engineering 2015: 1-8.
[38] Vashisth A.K., Rani K., Singh K., 2015, Quasi-static planar deformation in a medium composed of elastic and thermoelastic solid half spaces due to seismic sources in an elastic solid, Acta Geophysica 63(3): 605-633.
[39] Naeeni M.R., Eskandari-Ghadi M., Ardalan A.A., Rahimian M., Hayati Y., 2013, Analytical solution of coupled thermoelastic axisymmetric transient waves in a transversely isotropic half-space, Journal of Applied Mechanics 80(2): 024502 (1-7).
[40] Hayati Y., Eskandari-Ghadi M., Raoofian M., Rahimian M., Ardalan A.A., 2013, Dynamic Green's functions of an axisymmetric thermoelastic half-space by a method of potentials, Journal of Engineering Mechanics 139(9): 1166-1177.
[41] Naeeni M.R., Eskandari-Ghadi M., Ardalan A.A., Pak R.Y.S., 2014, Asymmetric motion of a transversely isotropic thermoelastic half-space under time-harmonic buried source, Zeitschrift für Angewandte Mathematik und Physik 65(5): 1031-1051.
[42] Naeeni M.R., Ghadi M.E., Ardalan A.A., Sture S., Rahimian M., 2015, Transient response of a thermoelastic half-space to mechanical and thermal buried sources, Journal of Applied Mathematics and Mechanics 95(4): 354-376.
[43] Eskandari‐Ghadi M., Raoofian‐Naeeni M., Pak R.Y.S., Ardalan A.A., Morshedifard A., 2017, Three dimensional transient Green's functions in a thermoelastic transversely isotropic half‐space, Journal of Applied Mathematics and Mechanics 97: 1611-1624.
[44] Kordkheili H.M., Amiri G.G., Hosseini M., 2016, Axisymmetric analysis of a thermoelastic isotropic half-space under buried sources in displacement and temperature potentials, Journal of Thermal Stresses 40: 237-254.
[45] Nowacki W., 1966, Green’s functions for a thermoelastic medium (quasi-static problem), Bulletin of Institute Political Jasi 12(3-4): 83-92.
[46] Cohen S.C., 1996, Convenient formulas for determining dip-slip fault parameters from geophysical observables, Bulletin of the Seismological Society of America 86(5): 1642-1644.
[47] Kato N., 2001, Simulation of seismic cycles of buried intersecting reverse faults, Journal of Geophysical Research: Solid Earth 106(B3): 4221-4232.
[48] Cattin R., Loevenbruck A., Pichon X.L., 2004, Why does the co-seismic slip of the 1999 Chi-Chi (Taiwan) earthquake increase progressively northwestward on the plane of rupture? Tectonophysics 386: 67-80.
[49] Mitsui Y., Hirahara K., 2007, Two‐dimensional model calculations of earthquake cycle on a fluid‐infiltrated plate interface at a subduction zone: Focal depth dependence on pore pressure conditions, Geophysical Research Letters 34(9): L09310 (1-6).
[50] Mitsui Y., Hirahara K., 2008, Long-term slow slip events are not necessarily caused by high pore fluid pressure at the plate interface: An implication from two-dimensional model calculations, Geophysical Journal International 174: 331-335.
[51] Kanda R.V., Simons M., 2012, Practical implications of the geometrical sensitivity of elastic dislocation models for field geologic surveys, Tectonophysics 560-561: 94-104.
[52] Zakian P., Khaji N., Soltani M., 2017, A Monte Carlo adapted finite element method for dislocation simulation of faults with uncertain geometry, Journal of Earth System Science 126(7): 105(1-22).
[53] Lay T., Wallace T. C., 1995, Modern Global Seismology, Academic Press, New York.
[54] Banerjee P.K., 1994, The Boundary Element Methods in Engineering, McGraw-Hill book company, New York.
[55] Ben-Menahem A., Singh S. J., 1981, Seismic Waves and Sources, Springer-Verlag, New York.
[56] Barber J.R., 2004, Elasticity, Kluwer academic publishers, New York.
[57] Erdelyi A., 1954, Bateman Manuscript Project-Tables of Integral Transforms, McGraw Hill book company, New York.
[58] Ahrens T.J., 1995, Mineral Physics and Crystallography: A Handbook of Physical Constants, American Geophysical Union, Washington.
[59] Aki K., Richards P.G., 1980, Quantitative Seismology: Theory and Methods, Freeman and Company, San Francisco.
[60] Schapery R.A., 1962, Approximate methods of transform inversion for viscoelastic stress analysis, Proceedings of the Fourth US National Congress of Applied Mechanics 2: 1075-1085.
[61] Naeeni M.R., Campagna R., Eskandari-Ghadi M., Ardalan A.A.,2015, Performance comparison of numerical inversion methods for Laplace and Hankel integral transforms in engineering problems, Applied Mathematics and Computations 250: 759-775.
