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Coupled and Generalized Thermoelasticity

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

Abstract

A structure under thermal shock load, when the period of shock is of the same order of magnitude as the lowest natural frequency of the structure, should be analyzed through the coupled form of the energy and thermoelasticity equations. Analytical solutions of this class of problems are mathematically complex and are limited to those of an infinite body or a half-space, where the boundary conditions are simple. This chapter begins with the analytical solutions of a number of classical problems of coupled thermoelasticity for an infinite body, a half-space, and a layer. Coupled thermoelasticity problem for a thick cylinder is discussed when the inertia terms are ignored. The generalized thermoelasticity problems for a layer, based on the Green–Naghdi, Green–Lindsay, and the Lord–Shulman models are discussed, when the analytical solution in the space domain is found. The solution in the time domain is obtained by numerical inversion of Laplace transforms. The generalized thermoelasticity of thick cylinders and spheres, in a unified form, is discussed, and problems are solved analytically in the space domain, while the inversion of Laplace transforms are carried out by numerical methods.

References

  1. 1.
    Ignaczak J (1989) Generalized thermoelasticity and its applications. In: Hetnarski RB (ed) Thermal stresses III. Elsevier, AmsterdamGoogle Scholar
  2. 2.
    Nowacki W (1986) Thermoelasticity, 2nd edn. PWN-Polish Scientific Publishers, Pergamon Press, Warsaw, OxfordGoogle Scholar
  3. 3.
    Kovalenko AD (1969) Thermoelasticity: basic theory and application. Wolters-Noordhoff Groningen, The NetherlandszbMATHGoogle Scholar
  4. 4.
    Nowacki W (1961) On some dynamic problems of thermoelasticity, contributed to the book Problems of continuum mechanics. SIAM, PhiladelphiaGoogle Scholar
  5. 5.
    Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New YorkzbMATHGoogle Scholar
  6. 6.
    Bahar LY, Hetnarski RB (1978) State-space approach to thermoelasticity. J Therm Stress 1:135–145CrossRefGoogle Scholar
  7. 7.
    Bahar LY, Hetnarski RB (1979) Direct approach to thermoelasticity. J Therm Stress 2:135–147CrossRefGoogle Scholar
  8. 8.
    Bahar LY, Hetnarski RB (1979) Connection between the thermoelastic potential and the state-space approach of thermoelasticity. J Therm Stress 2:283–290CrossRefGoogle Scholar
  9. 9.
    Bahar LY, Hetnarski RB (1980) Coupled thermoelasticity of layered medium. J Therm Stress 3:141–152CrossRefGoogle Scholar
  10. 10.
    Sherief HH (2014) State-space approach to generalized thermoelasticity. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 8. Springer, Dordrecht, pp 4537–4545CrossRefGoogle Scholar
  11. 11.
    Sherief HH (1993) State-space formulation for generalized thermoelasticity with one relaxation time including heat sources. J Therm Stress 16:163–180MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ezzat MA, Othman MI, El-Karamany AS (2002) State-space approach to generalized thermo-viscoplasticity with two relaxation times. Int J Eng Sci 40:283–302CrossRefGoogle Scholar
  13. 13.
    Ezzat MA, Othman MI, El-Karamany AS (2002) State-space approach to two-dimensional generalized thermoelasticity with one relaxation time. J Therm Stress 25:295–316CrossRefGoogle Scholar
  14. 14.
    Samanta SC, Maishal RK (2009) A study on magneto-thermo-viscoplastic interactions in an elastic half-space subjectd to a temperature pulse, using state-space approach. J Therm Stress 32(3)Google Scholar
  15. 15.
    Uflyand YS (1965) Survey of articles on the applications of integral theorems in the theory of elasticity, Applied Mathematical Research Group, North Carolina State University, Raleigh, pp 20–23Google Scholar
  16. 16.
    Lebedev NN, Skalskaya IP, Uflyand YS (1968) Problems of mathematical physics. Prentice Hall, New Jersey, pp 337–338Google Scholar
  17. 17.
    Sneddon IN (1951) Fourier transforms. McGraw-Hill, New YorkzbMATHGoogle Scholar
  18. 18.
    Jabbari M, Moradi A (2014) Exact solution for classic coupled thermoelasticity in cylindrical coordinates. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 3. Springer, Dordrecht, pp 1337–1353CrossRefGoogle Scholar
  19. 19.
    Jabbari M, Dehbani H (2014) Exact solution for classic coupled thermoelasticity in spherical coordinates. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 3. Springer, Dordrecht, pp 1353–1365CrossRefGoogle Scholar
  20. 20.
    Dillon OW Jr (1965) Thermoelasticity when the mechanical coupling parameter is unity. J Appl Mech, ASME 32:378–382CrossRefGoogle Scholar
  21. 21.
    Boley BA, Hetnarski RB (1968) Propagation of discontinuities in coupled thermoelastic problems. J Appl Mech, ASME 35:489–494CrossRefGoogle Scholar
  22. 22.
    Myshkina VV (1968) A coupled dynamic problem of thermoelasticity for a layer in the case of short intervals of time. Mechanics of solids. Allerton, New York, pp 103–106Google Scholar
  23. 23.
    Sherief HH, Anwar MN (1994) State-space approach to two-dimensional generalized thermoelasticity problems. J Therm Stress 17(4):567–590MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hetnarski RB (1969) The generalized D’Alembert solution to the coupled equations of thermoelasticity. In: Nowacki WK (ed) Progress in thermoelasticity, VIII European mechanics colloquium, Warsaw, 1967. PWN – Polish Scientific Publishers, Warsaw, pp 121–131Google Scholar
  25. 25.
    Agaryev VA (1963) The method of initial functions in two-dimensional problems of the theory of elasticity (in Russian). Isdatelstvo Akademii Nauk Ukrainskoi SSR, KievGoogle Scholar
  26. 26.
    Chandrasekharaiah DS (1998) Hyperbolic thermoelasticity, a review of recent literature. Appl Mech Rev 51:705–729CrossRefGoogle Scholar
  27. 27.
    Rossikhin YA, Shitikova MV (2009) D’Alembert’s solution in thermo-elasticity – impact of a rod against a heated barrier, Part I, a case of uncoupled strain and temperature fields. J Therm Stress 32(1–2)CrossRefGoogle Scholar
  28. 28.
    Rossikhin YA, Shitikova MV (2009) D’Alembert’s solution in thermo-elasticity – impact of a rod against a heated barrier, Part II, a case of coupled strain and temperature fields. J Therm Stress 32(3)CrossRefGoogle Scholar
  29. 29.
    Boley BA (1962) Discontinuities in integral-transform solution. Q Appl Math 19:273–284MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wagner P (1994) Fundamental matrix of the system of dynamic linear thermoelasticity. J Therm Stress 17(4):549–565MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ortner N, Wagner P (1992) On the fundamental solution of the operator of dynamic linear thermoelasticity. J Math Anal Appl 170:524–550MathSciNetCrossRefGoogle Scholar
  32. 32.
    Hetnarski RB (1964) Solution of the coupled problem of thermoelasticity in the form of series of functions. Arch Mech Stosow 16:919–941MathSciNetzbMATHGoogle Scholar
  33. 33.
    Jakubowska M (1982) Kirchhoff’s formula for thermoelastic solid. J Therm Stress 5:127–144MathSciNetCrossRefGoogle Scholar
  34. 34.
    Hetnarski RB (1964) Coupled thermoelastic problem for the half-space. Bull Acad Pol Sci Ser Sci Tech 12:49–57zbMATHGoogle Scholar
  35. 35.
    Hetnarski RB (1975) An algorithm for generating some inverse Laplace transforms of exponential form. J Appl Math Phys ZAMP 26(2):249–253MathSciNetzbMATHGoogle Scholar
  36. 36.
    Hetnarski RB (1961) Coupled one-dimensional thermal shock problem for small times. Arch Mech Stosow 13:295–306MathSciNetzbMATHGoogle Scholar
  37. 37.
    Danilovskaya VI (1950) Thermal stresses in an elastic half-space arising after a sudden heating at its boundary [in Russian]. Prikl Math Mekh 14(3)Google Scholar
  38. 38.
    Mura T (1952) Thermal strains and stresses in transient state. Proc Sec Jpn Congr Appl MechGoogle Scholar
  39. 39.
    Sternberg E, Chakravorty JG (1958) On inertia effects in a transient thermoelastic problem, Technical report No 2, Contract Nonr-562 (25), Brown UniversityGoogle Scholar
  40. 40.
    Gosn AH, Sabbaghian M (1982) Quasi-static coupled problems of thermoelasticity for cylindrical regions. J Therm Stress 5(3–4):299–313CrossRefGoogle Scholar
  41. 41.
    Green AE, Naghdi PM (1991) A re-examination of the basic postulates of thermomechanics. Proc Roy Soc Lond Ser A 432:171–194MathSciNetCrossRefGoogle Scholar
  42. 42.
    Green AE, Naghdi PM (1993) Thermoelasticity without energy dissipation. J Elast 31:189–208MathSciNetCrossRefGoogle Scholar
  43. 43.
    Green AE, Naghdi PM (1992) On undamped heat waves in an elastic solid. J Therm Stress 15:253–264MathSciNetCrossRefGoogle Scholar
  44. 44.
    Chandrasekharaiah DS (1996) A uniqueness theorem in the theory of thermoelasticity without energy dissipation. J Therm Stress 19:267–272MathSciNetCrossRefGoogle Scholar
  45. 45.
    Chandrasekharaiah DS (1996) One-dimensional wave propagation in the linear theory of thermoelasticity without energy dissipation. J Therm Stress 19:695–710CrossRefGoogle Scholar
  46. 46.
    Chandrasekharaiah DS, Srinath KS (1997) Axisymmetric thermoelastic interactions without energy dissipation in an unbounded body with cylindrical cavity. J Elast 46:19–31CrossRefGoogle Scholar
  47. 47.
    Chandrasekharaiah DS, Srinath KS (1998) Thermoelastic interactions without energy dissipation due to a point heat source. J Elast 50:97–108CrossRefGoogle Scholar
  48. 48.
    Chandrasekharaiah DS (1997) Complete solutions in the theory of thermo-elasticity without energy dissipation. Mech Res Commun 24:625–630MathSciNetCrossRefGoogle Scholar
  49. 49.
    Sharma JN, Chauhan RS (2001) Mechanical and thermal sources in a generalized thermoelastic half-space. J Therm Stress 24:651–675CrossRefGoogle Scholar
  50. 50.
    Li H, Dhaliwal RS (1996) Thermal shock problem in thermoelasticity without energy dissipation. Indian J Pure App Math 27:85–101Google Scholar
  51. 51.
    Taheri H, Fariborz S, Eslami MR (2004) Thermoelasticity solution of a layer using the Green-Naghdi model. J Therm Stress 27(8):691–704CrossRefGoogle Scholar
  52. 52.
    Durbin F (1974) Numerical inversion of Laplace transforms: an efficient improvement to Dubner and abate’s method. Comput J 17:371–376MathSciNetCrossRefGoogle Scholar
  53. 53.
    Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 15:299–309CrossRefGoogle Scholar
  54. 54.
    Ignaczak J, Hetnarski RB (2014) In: Hetnarski RB (ed) Generalized thermoasticity, mathematical formulation, encyclopedia of thermal stresses. Springer, Dordrecht, pp 1974–1986Google Scholar
  55. 55.
    Green AE, Lindsay KE (1972) Thermoelasticity. J Elast 2:1–7CrossRefGoogle Scholar
  56. 56.
    Chen J, Dargush GF (1995) Boundary element method for dynamic poroelastic and thermoelastic analysis. Int J Solids Struct 32(15):2257–2278CrossRefGoogle Scholar
  57. 57.
    Chen H, Lin H (1995) Study of transient coupled thermoelastic problems with relaxation times. Trans ASME, J Appl Mech 62:208–215CrossRefGoogle Scholar
  58. 58.
    Hosseini Tehrani P, Eslami MR (2000) Boundary element analysis of coupled thermoelasticity with relaxation time in finite domain. J AIAA 38(3):534–541CrossRefGoogle Scholar
  59. 59.
    Bagri A, Eslami MR (2004) Generalized coupled thermoelasticity of disks based on the Lord-Shulman model. J Therm Stress 27(8):691–704CrossRefGoogle Scholar
  60. 60.
    Ocłoń P, Łopata S (2014) Hyperbolic heat conduction equation. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 5. Springer, Dordrecht, pp 2332–2342CrossRefGoogle Scholar
  61. 61.
    Bagri A, Taheri H, Eslami MR, Fariborz S (2006) Generalized coupled thermoelasticity of a layer. J Therm Stress 29(4):359–370CrossRefGoogle Scholar
  62. 62.
    Honig G, Hirdes U (1984) A method for the numerical inversion of Laplace transforms. J Comput Appl Math 10:113–132MathSciNetCrossRefGoogle Scholar
  63. 63.
    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):53–64CrossRefGoogle Scholar
  64. 64.
    Bagri A, Eslami MR (2007) A unified generalized thermoelasticity formulation: application to thick functionally graded cylinders. J Therm Stress, special issue devoted to the 70th Birthday of Józef Ignaczak 30(9 and 10):911–930Google Scholar
  65. 65.
    Bagri A, Eslami MR (2007) A unified generalized thermoelasticity: solution for cylinders and spheres. Int J Mech Sci 49:1325–1335CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

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

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