Advertisement

Finite and Boundary Element Methods

  • Richard B. HetnarskiEmail author
  • M. Reza Eslami
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 158)

Abstract

Because the analytical solutions to coupled and generalized thermoelasticity problems are mathematically complicated, the numerical methods, such as the finite and the boundary element methods, have become powerful means of analysis. This chapter presents a new treatment of the finite and the boundary element methods for this class of problems. The finite element method based on the Galerkin technique is employed in order to model the general form of the coupled equations, and the application is then expanded to the two- and one-dimensional cases. The generalized thermoelasticity problems for a functionally graded layer, a thick sphere, a disk, and a beam are discussed using the Galerkin finite element technique. To show the strong rate of convergence of the Galerkin-based finite element, a problem for a radially symmetric loaded disk with three types of shape functions, linear, quadratic, and cubic, is solved. It is shown that the linear solution rapidly converges to that of the cubic solution. When the temperature change compared to the reference temperature may not be ignored, the heat conduction becomes nonlinear. The problem of thermally nonlinear generalized thermoelasticity of a layer based on the Lord–Shulman model is presented in this section, and it is indicated that how and when this assumption is essential to be used. The chapter concludes with the boundary element formulation for the generalized thermoelasticity. A unique principal solution satisfying both the thermoelasticity and the coupled energy equations is employed to obtain the boundary element formulation.

References

  1. 1.
    Eslami MR (1992) A note on finite element of coupled thermoelasticity. In: Proceedings of the ICEAM, Sharif University, Tehran, 9–12 June 1992Google Scholar
  2. 2.
    Eslami MR, Shakeri M, Sedaghati R (1994) Coupled thermoelasticity of axially symmetric cylindrical shells. J Therm Stress 17(1):115–135Google Scholar
  3. 3.
    Eslami MR, Shakeri M, Ohadi AR (1995) Coupled thermoelasticity of shells. In: Proceedings of the Thermal Stresses’95, Shizuoka University, Hamamatsu, JapanGoogle Scholar
  4. 4.
    Eslami MR, Shakeri M, Ohadi AR, Shiari B (1999) Coupled thermoelasticity of shells of revolution: the effect of normal stress. AIAA J 37(4):496–504Google Scholar
  5. 5.
    Eslami MR (2014) Finite elements methods in mechanics. Springer International Publishing, SwitzerlandzbMATHGoogle Scholar
  6. 6.
    Eslami MR, Salehzadeh A (1987) Application of Galerkin method to coupled thermoelasticity problems. In: Proceedings of the 5th international modal analysis conference, New York, 6–9 April 1987Google Scholar
  7. 7.
    Eslami MR, Vahedi H (1989) Coupled thermoelasticity beam problems. AIAA J 27(5):662–665Google Scholar
  8. 8.
    Eslami MR, Vahedi H (1991) A general finite element stress formulation of dynamic thermoelastic problems using Galerkin method. J Therm Stress 14(2):143–159Google Scholar
  9. 9.
    Maxwell JC (1867) On the dynamical theory of gases. Philos Trans R Soc N Y 157:49–88Google Scholar
  10. 10.
    Landau EM (1941) The theory of superfluidity of helium II. J Phys USSR 5:71–90Google Scholar
  11. 11.
    Peshkov V (1944) Second sound in helium II. J Phys USSR 8:131–138Google Scholar
  12. 12.
    Cattaneo MC (1948) Sulla Conduzione de Calor. Atti Sem Mat Fis Del Univ Modena 3:3Google Scholar
  13. 13.
    Vernotte P (1958) Les Paradoxes de la Theorie Continue de l’equation de la Chaleur. C R Acad Sci 246:3154–3155Google Scholar
  14. 14.
    Chester M (1963) Second sound in solids. Phys Rev 131:2013–2015Google Scholar
  15. 15.
    Ignaczak J (1981) Linear dynamic thermoelasticity, a survey. Shock Vib Dig 13:3–8Google Scholar
  16. 16.
    Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 15:299–309zbMATHGoogle Scholar
  17. 17.
    Green AE, Lindsay KA (1972) Thermoelasticity. J Elast 2(1):1–7Google Scholar
  18. 18.
    Chandrasekharaiah DS (1986) Thermoelasticity with second sound: a review. Appl Mech Rev 39(3):355–376MathSciNetzbMATHGoogle Scholar
  19. 19.
    Joseph DD, Preziosi L (1989) Heat waves. Rev Mod Phys 61:41–73MathSciNetzbMATHGoogle Scholar
  20. 20.
    Joseph DD, Preziosi L (1990) Addendum to the paper heat waves. Rev Mod Phys 62:375–391Google Scholar
  21. 21.
    Lee WY, Stinton DP, Berndt CC, Erdogan F, Lee Y, Mutasim Z (1996) Concept of functionally graded materials for advanced thermal barrier coating applications. J Am Ceram Soc 79(12)Google Scholar
  22. 22.
    Bagri A, Taheri H, Eslami MR, Fariborz S (2006) Generalized coupled thermoelasticity of a layer. J Therm Stress 29(4):359–370Google Scholar
  23. 23.
    Green AE, Naghdi PM (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208MathSciNetzbMATHGoogle Scholar
  24. 24.
    Green AE, Naghdi PM (1991) A re-examination of the basic postulates of thermomechanics. Proc R Soc Lond Ser A 432:171–194MathSciNetzbMATHGoogle Scholar
  25. 25.
    Takeuti Y, Furukawa T (1981) Some consideration on thermal shock problems in plate. ASME J Appl Mech 48:113–118zbMATHGoogle Scholar
  26. 26.
    Amin AM, Sierakowski RL (1990) Effect of thermomechanical coupling of the response of elastic solids. AIAA J 28:1319–1322Google Scholar
  27. 27.
    Tamma K, Namburu R (1997) Computational approaches with application to non-classical and classical thermomechanical problems. Appl Mech Rev 50:514–551Google Scholar
  28. 28.
    Chen H, Lin H (1995) Study of transient coupled thermoelastic problems with relaxation times. ASME J Appl Mech 62:208–215zbMATHGoogle Scholar
  29. 29.
    Taler J, Ocłoń P (2014) Finite element method in steady-state and transient heat conduction. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 4. Springer, Dordrecht, pp 1604–1633Google Scholar
  30. 30.
    Sherief HH, El-Latief AM (2014) Boundary element method in generalized thermoelasticity. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 1. Springer, Dordrecht, pp 407–415Google Scholar
  31. 31.
    Nowak AJ (2014) Boundary element method in heat conduction. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 1. Springer, Dordrecht, pp 415–424Google Scholar
  32. 32.
    Hosseini Tehrani P, Eslami MR (2000) Boundary element analysis of coupled thermoelasticity with relaxation times in finite domain. AIAA J 38(3):534–541Google Scholar
  33. 33.
    Hosseini Tehrani P, Eslami MR (2003) Boundary element analysis of finite domains under thermal and mechanical shock with the Lord–Shulman theory. J Strain Anal 38(1):53–64Google Scholar
  34. 34.
    Zhang QJ, Zhang LM, Yuan RZ (1993) A coupled thermoelasticity model of functionally gradient materials under sudden high surface heating. Ceram Trans Funct Gradient Mater 34:99–106Google Scholar
  35. 35.
    Praveen GN, Reddy JN (1998) Nonlinear transient thermoelastic analysis of functionally graded ceramic-metal plates. Int J Solids Struct 35:4457–4476zbMATHGoogle Scholar
  36. 36.
    Bagri A, Eslami MR, Samsam-Shariat B (2005) Coupled thermoelasticity of functionally graded layer. In: Proceedings of the 5th international congress thermal stresses, Vienna University of Technology, 26–29 May 2005, pp 721–724Google Scholar
  37. 37.
    Bagri A, Eslami MR, Samsam-Shariat B (2006) Generalized coupled thermoelasticity of functionally graded layers. In: ASME conference, ESDA2006, Torino, Italy, 4–7 July 2006Google Scholar
  38. 38.
    Honig G, Hirdes U (1984) A method for the numerical inversion of Laplace transforms. J Comput Appl Math 10:113–132MathSciNetzbMATHGoogle Scholar
  39. 39.
    Li YY, Ghoneim H, Chen Y (1983) A numerical method in solving a coupled thermoelasticity equation and some results. J Therm Stress 6:253–280Google Scholar
  40. 40.
    Ghoneim H (1986) Thermoviscoplasticity by finite element: dynamic loading of a thick walled cylinders. J Therm Stress 9:345–358Google Scholar
  41. 41.
    Eslami MR, Vahedi H (1989) A Galerkin finite element formulation of dynamic thermoelasticity for spherical problems. In: Proceedings of the 1989 ASME-PVP conference, Hawaii, 23–27 July 1989Google Scholar
  42. 42.
    Eslami MR, Vahedi H (1992) Galerkin finite element displacement formulation of coupled thermoelasticity spherical problems. Trans ASME J Press Vessel Technol 14(3):380–384Google Scholar
  43. 43.
    Obata Y, Noda N (1994) Steady thermal stresses in a hollow circular cylinder and a hollow sphere of functionally gradient material. J Therm Stress 17:471–488Google Scholar
  44. 44.
    Lutz MP, Zimmerman RW (1996) Thermal stresses and effective thermal expansion coefficient of a functionally graded sphere. J Therm Stress 19:39–54Google Scholar
  45. 45.
    Eslami MR, Babai MH, Poultangari R (2005) Thermal and mechanical stresses in a functionally graded thick sphere. Int J Press Vessel Pip 82:522–527Google Scholar
  46. 46.
    Bagri A, Eslami MR (2007) Analysis of thermoelastic waves in functionally graded hollow spheres based on the Green–Lindsay theory. J Therm Stress 30(12):1175–1193Google Scholar
  47. 47.
    Bagri A, Eslami MR (2004) Generalized coupled thermoelasticity of disks based on the Lord-Shulman model. J Therm Stress 27(8):691–704Google Scholar
  48. 48.
    Reddy JN, Chin CD (1998) Thermomechanical analysis of functionally graded cylinders and plates. J Therm Stress 21:593–626Google Scholar
  49. 49.
    Bahtui A, Eslami MR (2007) Coupled thermoelasticity of functionally graded cylindrical shells. Mech Res Commun 34(1):1–18zbMATHGoogle Scholar
  50. 50.
    Bakhshi M, Bagri A, Eslami MR (2006) Coupled thermoelasticity of functionally graded disk. Mech Adv Mater Struct 13(3):219–225Google Scholar
  51. 51.
    Eslami MR, Bagri A (2004) Higher order elements for the analysis of the generalized thermoelasticity of disk based on the Lord Shulman model. In: Proceedings of the international conference on computational methods in science and engineering, Athena, Greece, pp 19–23Google Scholar
  52. 52.
    Jones PJ (1966) Thermoelastic vibration of beams. J Acoust Soc Am 39:542–548zbMATHGoogle Scholar
  53. 53.
    Seibert AG, Rice JS (1973) Coupled thermally induced vibrations of beams. AIAA J 11:1033–1035Google Scholar
  54. 54.
    Massalas CV, Kalpakidis VK (1983) Coupled thermoelastic vibration of a simply supported beam. J Sound Vib 88:425–429zbMATHGoogle Scholar
  55. 55.
    Massalas CV, Kalpakidis VK (1984) Coupled thermoelastic vibration of a Timoshenko beam. Lett Appl Eng Sci 22:459–465zbMATHGoogle Scholar
  56. 56.
    Maruthi Rao D, Sinha PK (1997) Finite element coupled thermostructural analysis of composite beams. Comput Struct 63:539–549zbMATHGoogle Scholar
  57. 57.
    Manoach E, Ribeiro P (2004) Coupled thermoelastic large amplitude vibrations of Timoshenko beams. J Mech Sci 46:1589–1606zbMATHGoogle Scholar
  58. 58.
    Sankar BV (2001) An elasticity for functionally graded beams. J Compos Sci Technol 61:686–696Google Scholar
  59. 59.
    Bhavani V, Sankar BV, Tzeng JT (2002) Thermal stresses in functionally graded beams. AIAA J 40:1228–1232Google Scholar
  60. 60.
    Babai MH, Abbasi M, Eslami MR (2008) Coupled thermoelasticity of functionally graded beams. J Therm Stress 31:680–697Google Scholar
  61. 61.
    McQuillen EJ, Brull MA (1970) Dynamic thermoelastic response of cylindrical shell. J Appl Mech 37:661–670zbMATHGoogle Scholar
  62. 62.
    Bateni M, Eslami MR (2017) Thermally nonlinear generalized thermoelasticity of a layer. J Therm Stress 40(10).  https://doi.org/10.1080/01495739.2017.1320776Google Scholar
  63. 63.
    Dedicated to the memory of Professor Franz Ziegler. Bateni M, Eslami MR (2017) Thermally nonlinear generalized thermoelasticity; a note on the thermal boundary conditions. Acta Mech.  https://doi.org/10.1007/s00707-017-2001-6MathSciNetGoogle Scholar
  64. 64.
    Kiani Y, Eslami MR (2017) Thermally nonlinear generalized thermoelasticity of a layer based on the Lord-Shulman theory. Eur J Mech A/Solid 61:245–253.  https://doi.org/10.1016/j.euromechsol.2016.10.004MathSciNetzbMATHGoogle Scholar
  65. 65.
    Kiani Y, Eslami MR, A GDQ approach to thermally nonlinear generalized thermoelasticity of disks. J Therm Stress.  https://doi.org/10.1080/01495739.2016.1217179Google Scholar
  66. 66.
    Kiani Y, Eslami MR (2016) The GDQ approach to thermally nonlinear generalized thermoelasticity of spheres. Int J Mech Sci 118:195–204.  https://doi.org/10.1016/j.ijmecsci.2016.09.019Google Scholar
  67. 67.
    Nayfeh AH (1977) Propagation of thermoelastic distribution in non-Fourier solids. AIAA J 15:957–960zbMATHGoogle Scholar
  68. 68.
    Nayfeh AH, Nemat-Nasser S (1971) Thermoelastic waves in solids with thermal relaxation. Acta Mech 12:53–69zbMATHGoogle Scholar
  69. 69.
    Puri P (1973) Plane waves in generalized thermoelasticity. Int J Eng Sci 11:735–744zbMATHGoogle Scholar
  70. 70.
    Agarwal VK (1979) On plane waves in generalized thermoelasticity. Acta Mech 31:185–198MathSciNetzbMATHGoogle Scholar
  71. 71.
    Tamma KK, Zhou X (1998) Macroscale thermal transport and thermo-mechanical interactions: some noteworthy perspectives. J Therm Stress 21:405–450Google Scholar
  72. 72.
    Zhou X, Tamma KK, Anderson VDR (2001) On a new C- and F-processes heat conduction constitutive mode and associated generalized theory of thermoelasticity. J Therm Stress 24:531–564Google Scholar
  73. 73.
    Hosseini Tehrani P, Eslami MR (1998) Two-dimensional time harmonic dynamic coupled thermoelasticity analysis by BEM formulation. Eng Anal Bound Elem 22:245–250Google Scholar
  74. 74.
    Prevost JH, Tao D (1983) Finite element analysis of dynamic coupled thermoelasticity problems with relaxation times. ASME J Appl Mech 50:817–822zbMATHGoogle Scholar
  75. 75.
    Tamma KK, Railkar SB (1990) Evaluation of thermally induced non-Fourier stress wave disturbances via specially tailored hybrid transfinite formulations. Comput Struct 34:5Google Scholar
  76. 76.
    Tamma KK (1989) Numerical simulations for hyperbolic heat conduction/dynamic problems influenced by non-Fourier/Fourier effects. In: Symposium of heat waves, University of Minnesota, MinneapolisGoogle Scholar
  77. 77.
    Tamma KK (1996) An overview of non-classical/classical thermal structural models and computational methods for analysis of engineering structures. In: Hetnarski RB (ed) Thermal stresses IV. Elsevier Science, AmsterdamGoogle Scholar
  78. 78.
    Chen J, Dargush GF (1995) BEM for dynamic poroelastic and thermoelastic analysis. Int J Solids Struct 32(15):2257–2278zbMATHGoogle Scholar
  79. 79.
    Eslami MR, Hosseini Tehrani P (1999) Propagation of thermoelastic waves in a two-dimensional finite domain by BEM. In: Proceedings of the third international congress on thermal stresses, 13–17 June, Cracow, PolandGoogle Scholar
  80. 80.
    Hector LG Jr, Hetnarski RB (1996) Thermal stresses in materials due to laser heating. In: Hetnarski RB (ed) Thermal stresses IV. Elsevier Science, AmsterdamGoogle Scholar
  81. 81.
    Hector LG Jr, Hetnarski RB (1996) Thermal stresses due to a laser pulse: the elastic solution. ASME J Appl Mech 63:38–46zbMATHGoogle Scholar
  82. 82.
    Kim WS, Hector LG Jr, Hetnarski RB (1997) Thermoelastic stresses in a bonded layer due to repetitively pulsed laser radiation. Acta Mech 125:107–128zbMATHGoogle Scholar
  83. 83.
    Tahrani PH, Hector LG Jr, Hetnarski RB, Eslami MR (2001) Boundary element formulation for thermal stresses during pulsed laser heating. ASME J Appl Mech 68:480–489zbMATHGoogle Scholar
  84. 84.
    Hetnarski RB, Ignaczak J (1993) Generalized thermoelasticity: closed form solutions. J Therm Stress 16(4):473–498MathSciNetGoogle Scholar
  85. 85.
    Hetnarski RB, Ignaczak J (1994) Generalized thermoelasticity: response of semi-space to a short laser pulse. J Therm Stress 17(3):377–396MathSciNetGoogle Scholar
  86. 86.
    Fichera G (1997) A boundary value problem connected with response of semi-space to a short laser pulse. Rend Mat Acc Lincei Serie IX 8:197–227Google Scholar
  87. 87.
    Hetnarski RB, Ignaczak J (2000) Nonlinear dynamical thermoelasticity. Int J Solids Struct 37:215–224zbMATHGoogle Scholar
  88. 88.
    Tosaka N (1986) Boundary integral equation formulations for linear coupled thermoelasticity. In: Proceedings of the 3rd Japan symposium on BEM, Tokyo, pp 207–212Google Scholar
  89. 89.
    Durbin F (1974) Numerical inversion of Laplace transforms: an efficient improvement to Dubner’s and Abate’s method. Comput J 17:371–376MathSciNetzbMATHGoogle Scholar
  90. 90.
    Sternberg E, Chakravorty JG (1959) On inertia effects in a transient thermoelastic problem. ASME J Appl Mech 26:503–509MathSciNetGoogle Scholar
  91. 91.
    Hosseini Tehrani P, Eslami MR (2003) Boundary element analysis of finite domain under thermal and mechanical shock with the Lord–Shulman theory. Proc I Mech E J Strain Anal 38(1)Google Scholar
  92. 92.
    Hosseini Tehrani P, Eslami MR (2000) Boundary element analysis of thermal and mechanical shock in a two-dimensional finite domain considering coupled thermoelasticity. J Eng Anal Bound Elem 24:249–257Google Scholar
  93. 93.
    Hosseini Tehrani P, Eslami MR (2000) Boundary element analysis of Green and Lindsay theory under thermal and mechanical shock in a finite domain. J Therm Stress 23(8):773–792Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.NaplesUSA
  2. 2.Department of Mechanical EngineeringAmirkabir University of TechnologyTehranIran

Personalised recommendations