An incremental-iterative BEM methodology to solve 3D thermoelastic contact problem including variable thermal resistance in the contact zone


This work presents a new incremental-iterative method based on the boundary element method to solve 3D thermoelastic contact problems including a variable thermal contact resistance. The consideration of a variable thermal contact resistance implies that both thermal and thermoelastic problems are coupled since the variable thermal contact resistance is a function of the contact normal traction. Thus the proposed method determines incrementally the equilibrium configuration of the system by solving the coupled thermal and thermoelastic problems. The correct value of the thermal resistance is obtained iteratively. The robustness and accuracy of the proposed method is shown by solving a numerical problem and comparing the results with other methods presented in the literature.

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  1. 1.

    Martynyak, R.M., Chumak, K.A.: Thermoelastic contact of half-spaces with equal thermal distortivities in the presence of a heat-permeable intersurface gap. J. Math. Sci. 165(3), 355–370 (2010)

    Article  Google Scholar 

  2. 2.

    Comninou, M., Dundurs, J.: On the Barber boundary conditions for thermoelastic contact. J. Appl. Mech. 46, 849–853 (1979)

    ADS  Article  MATH  Google Scholar 

  3. 3.

    Comninou, M., Dundurs, J., Barber, J.R.: Planar Hertz contact with heat conduction. J. Appl. Mech. 48, 549–554 (1981)

    ADS  Article  MATH  Google Scholar 

  4. 4.

    Dundurs, J.: Distortion of a body caused by free thermal expansion. Mech. Res. Commun. 1(3), 121–124 (1974)

    Article  Google Scholar 

  5. 5.

    Alonso, P., Garrido García, J.: BEM applied to 2D thermoelastic con- tact problems including conduction and forced convection in interstitial zones. Eng. Anal. Bound. Elem. 15(3), 249–259 (1995)

    Article  Google Scholar 

  6. 6.

    Vallepuga Espinosa, J., Foces Mediavilla, A.: Boundary element method applied to three dimensional thermoelastic contact. Eng. Anal. Bound. Elem. 36(6), 928–933 (2012)

    Article  MATH  Google Scholar 

  7. 7.

    Giannopoulos, G.I., Anifantis, N.K.: A BEM analysis for thermomechanical closure of interfacial cracks incorporating friction and thermal resistance. Comput. Methods Appl. Mech. Eng. 196(4–6), 1018–1029 (2007)

    ADS  Article  MATH  Google Scholar 

  8. 8.

    Garrido, J., Foces, A., París, F.: An incremental procedure for three- dimensional contact problems with friction. Comput. Struct. 50, 201–215 (1994)

    Article  MATH  Google Scholar 

  9. 9.

    Man, K.W., Aliabadi, M.H.: BEM frictional contact analysis: modelling considerations. Eng. Anal. Bound. Elem. 11, 77–85 (1993)

    Article  Google Scholar 

  10. 10.

    Man, K.W., Aliabadi, M.H., Rooke, D.P.: BEM frictional con- tact analysis: load Incremental technique. Comput. Struct. 47(6), 893–905 (1993)

    Article  MATH  Google Scholar 

  11. 11.

    Aliabadi, F.M.H.: The Boundary Element Method: Applications in Solids and Structures, vol. 2. Wiley, Chichester (2002)

    Google Scholar 

  12. 12.

    Sfantos, G.K., Aliabadi, M.H.: A boundary element formulation for three-dimensional sliding wear simulation. Wear 262, 672–683 (2007)

    Article  Google Scholar 

  13. 13.

    Sfantos, G.K., Aliabadi, M.H.: Wear simulation using an incremental sliding boundary element method. Wear 260, 1119–1128 (2006)

    Article  Google Scholar 

  14. 14.

    Rodríguez-Tembleque, L., Abascal, R., Aliabadi, M.H.: Anisotropic wear framework for 3D contact and rolling problems. Comput. Methods Appl. Mech. Eng. 241–244, 1–19 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Rodríguez-Tembleque, L., Sáez, A., Aliabadi, M.: Indentation re- sponse of piezoelectric films under frictional contact. Int. J. Eng. Sci. 107, 36–53 (2016)

    Article  MATH  Google Scholar 

  16. 16.

    Rodríguez-Tembleque, L., García-Macías, E., Sáez, A.: Cnt-polymer nanocomposites under frictional contact conditions. Compos. Part B Eng. 154, 114–127 (2018)

    Article  Google Scholar 

  17. 17.

    Zhang, S., Li, X.: A self-adaptive projection method for contact problems with the BEM. Appl. Math. Model. 55, 145–159 (2018)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Zhang, S., Li, X., Ran, R.: Self-adaptive projection and boundary element methods for contact problems with tresca friction. Commun. Nonlinear Sci. Numer. Simul. 68, 72–85 (2019)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Suárez, A., González, R., Sánchez, L., Vallepuga, J.: Iterative proce- dure to solve thermoelastic contact problems between 3D solids using BEM and OOP. Actas XXII Jornadas de Paralelismo La Laguna, (2011)

  20. 20.

    Vallepuga, J., Sánchez-González, L., Ubero, I.: Java application to solve thermoelastic contact problems using the boundary element method. Adv. Bound. Elem. Meshless Tech. XIX, 147–156 (2018)

    Google Scholar 

  21. 21.

    Vallepuga, J., Sánchez-González, L., Ubero, I., Castillo, V. R.-D.: The boundary element method applied to the resolution of problems in strength of materials and elasticity. In: International joint conference SOCO’18-CISIS’18-ICEUTE’18, 544–552 (2018)

  22. 22.

    Brebbia, C.A., Telles, J.C.F., Wrobel, L.C.: Boundary Element Techniques, pp. 177–236. Springer, Berlin (1984)

    Google Scholar 

  23. 23.

    Hartmann, F.: The somigliana identity on piecewise smooth surfaces. J Elast. 11(4), 403–423 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Paris, F., Foces, A., Garrido, J.A.: Application of boundary element method to solve three-dimensional elastic contact problems without friction. Comput. Struct. 43, 19–30 (1992)

    ADS  Article  MATH  Google Scholar 

  25. 25.

    Cooper, M.G., Mikic, B.B., Yovanovich, M.M.: Thermal contact conductance. J. Heat Mass Transf. 12, 279–300 (1969)

    Article  Google Scholar 

  26. 26.

    Paris, F., Garrido, J.A.: Aspectos numericos de la aplicacion del metodo de los elementos de contorno al problema de contacto. Revista Internacional de Metodos Numericos para Calculo y Diseño en Ingenieria 2(1), 43 (1986)

    Google Scholar 

  27. 27.

    MATLAB and Statistics Toolbox Release: The MathWorks Inc. Natick, Massachusetts, United States (2016b)

  28. 28.

    Youssef, H.M., El-Bary, A.A.: Thermal shock problem of generalized thermoelastic layered composite material with variable thermal conductivity. Math. Probl. Eng. 2006, 1–14 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Aouadi, Moncef: Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion. A. Angew. Math. Phys. 57, 351–367 (2006)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Wang, Yingze, Liu, Dong, Wang, Qian, Shu, Chang: Thermoelastic response of thin plate with variable material properties under transient thermal shock. Int. J. Mech. Sci. 104, 200–206 (2015)

    Article  Google Scholar 

  31. 31.

    Sherief, Hany H., Hamza, Farid A.: Modeling of variable thermal condutivity in a generalized thermoelastic infinitely long hollow cylinder. Meccanica 51, 551–558 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Li, Chenlin, Guo, Huili, Tian, Xin, Tian, Xiaogeng: Transient reponse for a half-space with variable thermal conductivity and diffusivity under thermal and chemical shock. J Therm. Stress. 40(3), 389–401 (2017)

    Article  Google Scholar 

  33. 33.

    Li, D., He, T.: Investigation of generalized piezoelectri-thermoelastic problem with nonlocal effect and temperature-dependent properties. Heliyon 4(10), e00860 (2018)

    Article  Google Scholar 

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Correspondence to Iván Ubero-Martínez.

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Vallepuga-Espinosa, J., Ubero-Martínez, I., Sánchez-González, L. et al. An incremental-iterative BEM methodology to solve 3D thermoelastic contact problem including variable thermal resistance in the contact zone. Continuum Mech. Thermodyn. 31, 1543–1558 (2019).

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  • Boundary element method
  • Incremental procedure
  • Elastic contact problem
  • Thermoelastic contact problem
  • Contact thermal resistance