Structural Engineering and Mechanics

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Exact and complete fundamental solutions for penny-shaped crack in an infinite transversely isotropic thermoporoelastic medium: mode I problem

  • LI, Xiang-Yu (School of Mechanics and Engineering, Southwest Jiaotong University) ;
  • Wu, J. (Laboratoire d'Etudes Aerodynamiques, Universite de Poitiers) ;
  • Chen, W.Q. (Department of Engineering Mechanics, Zhejiang University) ;
  • Wang, Hui-Ying (E.N.S.M.A., University of Poitiers) ;
  • Zhou, Z.Q. (Department of Mechanical and Aerospace Engineering, University of Miami)
  • Received : 2011.01.05
  • Accepted : 2012.03.24
  • Published : 2012.05.10
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Abstract

This paper examines the problem of a penny-shaped crack in a thermoporoelastic body. On the basis of the recently developed general solutions for thermoporoelasticity, appropriate potentials are suggested and the governing equations are solved in view of the similarity to those for pure elasticity. Exact and closed form fundamental solutions are expressed in terms of elementary functions. The singularity behavior is then discussed. The present solutions are compared with those in literature and an excellent agreement is achieved. Numerical calculations are performed to show the influence of the material parameters upon the distribution of the thermoporoelastic field. Due to its ideal property, the present solution is a natural benchmark to various numerical codes and simplified analyses.

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References

  1. Abousleiman, Y. and Ekbote, S. (2000), "Porothermoelasticity in transversely isotropic porous materials", IUTAM Symposium on Theoretical and Numerical Method in Continuum Mechanics of Porous Materials, 145-152.
  2. Abousleiman, Y., Cheng, A.H.D., Cui, L., Detournay, E. and Roegiers, J.C. (1996), "Mandel's problem revisited", Geotechnique, 46, 187-195. https://doi.org/10.1680/geot.1996.46.2.187
  3. Adelaide, F.F., Bui, H.D. and Ehrlacher, A. (2003), "Hydrostatic interaction of a wetting fluid and a circular crack in an elastic material", Mech. Mater., 35, 581-586. https://doi.org/10.1016/S0167-6636(02)00281-8
  4. Atkinson, C. and Craster, R.V. (1991), "Plane strain fracture in poroelastic media", Proc. Roy. Soc. London. A434, 605-633.
  5. Biot, M.A. (1941), "General solutions of three-dimensional consolidation", J. Appl. Phys., 12, 155-164. https://doi.org/10.1063/1.1712886
  6. Biot, M.A. (1955), "Theory of elasticity and consolidation for a porous anisotropic solid", J. Appl. Phys., 26, 182-185. https://doi.org/10.1063/1.1721956
  7. Boone, T.J. and Detournay, E. (1990), "Response of a vertical hydraulic fracture intersecting a poroelastic formation bounded by a semi-infinite impermeable elastic layer", Int. J. Rock. Mech. Min. Sci. Geomech. Abstr., 27, 189-197. https://doi.org/10.1016/0148-9062(90)94327-P
  8. Chen, J. (1994a), "Time domain fundamental solution to Biot's complete equations of poroelasticity: Part I twodimensional solution", Int. J. Solids Struct., 31, 1447-1490. https://doi.org/10.1016/0020-7683(94)90186-4
  9. Chen, J. (1994b), "Time domain fundamental solution to Biot's complete equations of poroelasticity: Part II three-dimensional solution", Int. J. Solids Struct., 3, 169-202.
  10. Chen, W.Q. and Ding, H.J. (2004), "Potential theory method for 3D crack and contact problems of multi-field coupled media: A survey", J. Zhejiang Univ. SCI., 5(9), 1009-1021. https://doi.org/10.1631/jzus.2004.1009
  11. Chen, W.Q., Ding, H.J. and Ling, D.S. (2004), "Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution", Int. J. Solids Struct., 41, 69-83. https://doi.org/10.1016/j.ijsolstr.2003.08.020
  12. Cheng, A.H.D. and Detournay, E. (1988), "A direct boundary element for plane strain poroelasticity", Int. J. Numer. Anal. Meth. Geomech., 12, 551-572. https://doi.org/10.1002/nag.1610120508
  13. Cheng, A.H.D. and Liggett, J.A. (1984a), "Boundary integral equation method for linear porous-elasticity with applications to soil consolidation", Int. J. Numer. Meth. Eng., 20, 255-278. https://doi.org/10.1002/nme.1620200206
  14. Cheng, A.H.D. and Liggett, J.A. (1984b), "Boundary integral equation method for linear porous-elasticity with applications to fracture propagation", Int. J. Numer. Meth. Eng., 20, 279-296. https://doi.org/10.1002/nme.1620200207
  15. Cleary, M.P. (1977), "Fundamental solutions for a fluid-saturated porous solid", Int. J. Solids Struct., 13, 785-806. https://doi.org/10.1016/0020-7683(77)90065-8
  16. Coussy, O. (1995), Poromechanics, John Wiley & Sons, Ltd.
  17. Ding, H.J., Chen, W.Q. and Zhang, L.C. (2006), Elasticity of Transversely Isotropic Materials, Springer, Dordrecht.
  18. Fabrikant, V.I. (1989), "Applications of potential theory in mechanics: A selection of new results", Kluwer Academic Publishers, The Netherlands.
  19. Gatmiri, B. and Jabbari, E. (2005a), "Time-domain Green's functions for unsaturated soils. Part I: Twodimensional solution", Int. J. Solids Struct., 42, 5971-5990. https://doi.org/10.1016/j.ijsolstr.2005.03.039
  20. Gatmiri, B. and Jabbari, E. (2005b), "Time-domain Green's functions for unsaturated soils. Part II: Threedimensional solution", Int. J. Solids Struct., 42, 5991-6002. https://doi.org/10.1016/j.ijsolstr.2005.03.040
  21. Gatmiri, B. and Nguyen, K.V. (2005), "Time 2D fundamental solution for saturated porous media with incompressible fluid", Commun. Numer. Meth. Eng., 21(3), 119-132.
  22. Gibson, R.E. (1967), "Some results concerning displacements and stresses in a non-homogeneous elastic halfspace", Geotechnique, 17, 58-67. https://doi.org/10.1680/geot.1967.17.1.58
  23. Gibson, R.E. and McNamee, J. (1963), "A three-dimensional problem of consolidation of a semi-infinite clay stratum", Quart. J. Mech. Appl. Math., 16, 115-127. https://doi.org/10.1093/qjmam/16.1.115
  24. Gordeyev, Y.N. (1995), "A hydraulic fracture in a transversely isotropic poroelastic medium", J. Appl. Math. Mech., 59, 645-656. https://doi.org/10.1016/0021-8928(95)00075-5
  25. Gradsbteyn, I.S. and Ryzbik, L.M. (2004), Table of Integrals, Series, and Products, 6th edition, Elsevier, Pte. Ltd., Singapore.
  26. Kanj, M. and Abousleiman, Y. (2005), "Porothermoelastic analyses of anisotropic hollow cylinders with applications", Int. J. Numer. Anal. Meth. Geomech., 29, 103-126. https://doi.org/10.1002/nag.406
  27. Li, X.Y., Chen, W.Q. and Wang, H.Y. (2010), "General steady state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application", Eur. J. Mech. A/Solids, 29, 317-326. https://doi.org/10.1016/j.euromechsol.2009.11.007
  28. Liu, L. and Kardomateas, G.A. (2005), "Thermal stress intensity factors for a crack in an anisotropic half plane", Int. J. Solids Struct., 42, 5208-5223. https://doi.org/10.1016/j.ijsolstr.2005.02.025
  29. Nowacki, W. (1966), "Green's functions for a thermoelastic medium (quasi-static problem)", Buletinul Institutului Politehnic din Iasi, Serie Noua12, 83-92.
  30. Norris, A.N. (1985), "Radiation from a point source and scattering theory in a fluid-saturated porous solid", J. Acoustic. Soc. Am., 77, 2012-2023. https://doi.org/10.1121/1.391773
  31. Singh, B.M., Danyluk, H.T. and Selvachurai, A.P.S. (1987), "Thermal stresses in a transversely isotropic elastic solid weakened by an external circular crack", Int. J. Solids Struct., 23, 403-412. https://doi.org/10.1016/0020-7683(87)90044-8

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