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Transient dynamic analysis of generally anisotropic materials using the boundary element method

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Abstract

This work presents the use of an explicit-form Green’s function for 3D general anisotropy in conjunction with the dual reciprocity boundary element method and the radial integration method to analyse elastodynamic problems using BEM. The latter two schemes are to treat the inertial loads in the time-domain formulation of the dynamic problem. The direct analysis with the Houbolt’s algorithm is used to perform the transient analysis. The efficient and accurate computation of the fundamental solutions has been a subject of great interest for 3D general anisotropic materials due to their mathematical complexity. A recently proposed scheme based on double Fourier series representation of these solutions is used to this end. Numerical examples are presented to demonstrate the feasibility and successful implementation of applying these schemes in synthesis to treat 3D elastodynamic problems of anisotropic materials.

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References

  1. Lifshitz, I.M., Rozenzweig, L.N.: Construction of the green tensor fot the fundamental equation of elasticity theory in the case of unbounded elastic anisotropic medium. Z. Éksp. Teor. Fiz. 17, 783–791 (1947)

    Google Scholar 

  2. Wilson, R., Cruse, T.: Efficient implementation of anisotropic three dimensional boundary-integral equation stress analysis. Int. J. Numer. Methods Eng. 12, 1383–1397 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Sales, M.A., Gray, L.J.: Evaluation of the anisotropic Green’s function and its derivatives. Comput. Struct. 69, 247–254 (1998)

    Article  MATH  Google Scholar 

  4. Tonon, F., Pan, E., Amadei, B.: Green’s functions and boundary element method formulation for 3D anisotropic media. Comput. Struct. 79, 469–482 (2001)

    Article  Google Scholar 

  5. Phan, P.V., Gray, L.J., Kaplan, T.: On the residue calculus evaluation of the 3-D anisotropic elastic green’s function. Commun. Numer. Methods Eng. 20, 335–341 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Wang, C.Y., Denda, M.: 3D BEM for general anisotropic elasticity. Int. J. Solids Struct. 44, 7073–7091 (2007)

    Article  MATH  Google Scholar 

  7. Ting, T.C.T., Lee, V.G.: The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. Q. J. Mech. Appl. Math. 50, 407–426 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lee, V.: Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials. Mech. Res. Commun. 30, 241–249 (2003)

    Article  MATH  Google Scholar 

  9. Tavara, L., Ortiz, J.E., Mantic, V., Paris, R.: Unique real-variable expression of displacement and traction fundamental solutions covering all transversely isotropic materials for 3D BEM. Int. J. Numer. Methods Eng. 74, 776–798 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Shiah, Y.C., Tan, C.L., Lee, V.G.: Evaluation of explicit-form fundamental solutions for displacements and stresses in 3D anisotropic elastic solids. CMES Comput. Model. Eng. Sci. 34, 205–226 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Tan, C.L., Shiah, Y.C., Lin, C.W.: Stress analysis of 3D generally anisotropic elastic solids using the boundary element method. CMES Comput. Model. Eng. Sci. 41, 195–214 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Tan, C.L., Shiah, Y.C., Wang, C.Y.: Boundary element elastic stress analysis of 3D generally anisotropic solids using fundamental solutions based on fourier series. Int. J. Solids Struct. 50, 2701–2711 (2013)

    Article  Google Scholar 

  13. Shiah, Y.C., Tan, C.L.: The boundary integral equation for 3D general anisotropic thermoelasticity. Comput. Model. Eng. Sci. 102(6), 425–447 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Saez, A., Dominguez, J.: BEM analysis of wave scattering in transversely isotropic solids. Int. J. Numer. Methods Eng. 44, 1283–1300 (1999)

    Article  MATH  Google Scholar 

  15. Dominguez, J.: Boundary Elements in Dynamics. Computational Mechanics Publications, Southampton (1993)

    MATH  Google Scholar 

  16. Venturini, W.: A study of boundary element method and its application on engineering problems. Professorial Thesis, Sao Carlos, University of Sao Paulo (1988)

  17. Gao, X.W.: The radial integration method for evaluation of domain integrals with boundary-only discretization. Eng. Anal. Bound. Elem. 26, 905–916 (2002)

    Article  MATH  Google Scholar 

  18. Albuquerque, E., Sollero, P., Venturini, W., Aliabadi, M.: Boundary element method analysis of anisotropic Kirchhoff plates. Int. J. Solids Struct. 43, 4029–4046 (2006)

    Article  MATH  Google Scholar 

  19. Albuquerque, E.L., Sollero, P., de Paiva, W.P.: The BEM and the RIM in the dynamic analysis of symmetric laminate composite plates. In: Alves, M., da Costa Mattos, H.S. (eds.) Mechanics of Solids in Brazil 2007. Brazilian Society of Mechanical Sciences and Engineering, Sao Paulo (2007)

    Google Scholar 

  20. Albuquerque, E., Sollero, P., Aliabadi, M.: The boundary element method applied to time dependent problems in anisotropic materials. Int. J. Solids Struct. 39(5), 1405–1422 (2002)

    Article  MATH  Google Scholar 

  21. Albuquerque, E., Sollero, P., Fedelinski, P.: Dual reciprocity boundary element method in Laplace domain applied to anisotropic dynamic crack problems. Comput. Struct. 81, 1703–1713 (2003)

    Article  Google Scholar 

  22. Albuquerque, E., Sollero, P., Fedelinski, P.: Free vibration analysis of anisotropic material structures using the boundary element method. Eng. Anal. Bound. Elem. 27, 977–985 (2003)

    Article  MATH  Google Scholar 

  23. Albuquerque, E., Sollero, P., Aliabadi, M.: Dual boundary element method for anisotropic dynamic fracture mechanics. Int. J. Numer. Methods Eng. 59, 1187–1205 (2004)

    Article  MATH  Google Scholar 

  24. Galvis, A., Sollero, P.: 2D analysis of intergranular dynamic crack propagation in polycrystalline materials a multiscale cohesive zone model and dual reciprocity boundary elements. Comput. Struct. 164, 1–14 (2016)

    Article  Google Scholar 

  25. Kögl, M., Gaul, L.: A 3D boundary element method for dynamic analysis of anisotropic elastic solids. Comput. Model. Eng. Sci. 1(4), 27–43 (2000)

    MATH  Google Scholar 

  26. Kögl, M., Gaul, L.: A boundary element method for transient piezoelectric analysis. Eng. Anal. Bound. Elem. 24, 591–598 (2000)

    Article  MATH  Google Scholar 

  27. Kögl, M., Gaul, L.: Free vibration analysis of anisotropic solids with the boundary element method. Eng. Anal. Bound. Elem. 27, 107–114 (2003)

    Article  MATH  Google Scholar 

  28. Partridge, P.W., Brebbia, C.A., Wrobel, L.C.: The Dual Reciprocity Boundary Element Method. Elsevier, Amsterdam (1992)

    MATH  Google Scholar 

  29. Gaul, L., Kögl, M., Wagner, M.: Boundary Element Methods for Engineers and Scientists. Springer, Berlin (2003)

    Book  MATH  Google Scholar 

  30. Carrer, J., Fleischfresser, S., Garcia, L., Mansur, W.: Dynamic analysis of Timoshenko beams by the boundary element method. Eng. Anal. Bound. Elem. 37, 1602–1616 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Useche, J., Harnish, C.: A boundary element method formulation for modal analysis of doubly curved thick shallows shells. Appl. Math. Model. 40, 3591–3600 (2016)

    Article  MathSciNet  Google Scholar 

  32. Shiah, Y.C., Tan, C.L., Wang, C.Y.: An efficient numerical scheme for the evaluation of the fundamental solution and its derivatives in 3D generally anisotropic elasticity. In: Advances in Boundary Element and Meshless Techniques XIII, pp. 190–199 (2012)

  33. Lee, V.G.: Derivatives of the three-dimensional Green’s function for anisotropic materials. Int. J. Solids Struct. 46, 3471–3479 (2009)

    Article  MATH  Google Scholar 

  34. Shiah, Y.C., Tan, C.L., Lee, R.F.: Internal point solutions for displacements and stresses in 3D anisotropic elastic solids using the boundary element method. CMES Comput. Model. Eng. Sci. 69, 167–197 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Nardini, D., Brebbia, C.A.: A New Approach to Free Vibration Analysis Using Boundary Elements. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  36. Wrobel, L.C., Brebbia, C.A.: The dual reciprocity boundary element formulation for non-linear diffusion problems. Comput. Methods Appl. Mech. Eng. 65(2), 147–164 (1987)

    Article  MATH  Google Scholar 

  37. Grundemann, H.: A general procedure transferring domain integrals onto boundary integrals in BEM. Eng. Anal. Bound. Elem. 6(4), 214–222 (1989)

    Article  Google Scholar 

  38. Atkinson, K.E.: The numerical evaluation of particular solutions for Poisson’s equation. IMA J. Numer. Anal. 5, 319–338 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  39. Golberg, M.A.: The numerical evaluation of particular solutions in the BEM—a review. Bound. Elem. Commun. 6, 99–106 (1995)

    Google Scholar 

  40. Schclar, N.A.: Anisotropic Analysis Using Boundary Elements. Computational Mechanics Publications, Southampton (1994)

    MATH  Google Scholar 

  41. Gao, X.W.: Boundary only integral equations in boundary element analysis. In: Proceedings of the International Conference on Boundary Element Techniques (2001)

  42. Wood, W.L.: Practical Time Stepping Schemes. Clarendon Press, Oxford (1990)

    MATH  Google Scholar 

  43. Houbolt, J.C.: A recurrence matrix solution for the dynamic response of elastic aircraft. J. Aeronaut. Sci. 17, 540–550 (1950)

    Article  MathSciNet  Google Scholar 

  44. Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85, 67–94 (1959)

    Google Scholar 

  45. Park, K.C.: An improved stiffly stable method for direct integration of nonlinear structural dynamic equations. ASME J. Appl. Mech. 42, 464–470 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  46. Loeffler, C.F., Mansur, W.J.: Analysis of time integration schemes for boundary element applications to transient wave propagation problems. In: Boundary Element Techniques: Applications in Stress Analysis and Heat transfer. Computational Mechanics Publications, Southampton (1987)

  47. Agnantiaris, J.P., Polyzos, D., Beskos, D.E.: Some studies on dual reciprocity BEM for elastodynamic analysis. Comput. Mech. 17, 270–277 (1996)

    Article  MATH  Google Scholar 

  48. Timoshenko, S., Goodier, J.: Theory of Elasticity, 3rd edn. McGraw-Hill, New York (1970)

    MATH  Google Scholar 

  49. Lekhnitskii, S.G.: Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco (1963)

    MATH  Google Scholar 

  50. Clough, R.W., Penzien, J.: Dynamics of Structures, 3rd edn. Computer & Structures, Inc., Berkeley (2003)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the Center for Computational Engineering and Sciences (CCES-CEPID/UNICAMP) and the National Center for High Performance Computing in São Paulo (CENAPAD-SP) for the support of computational facilities. Also to the São Paulo Research Foundation (FAPESP) Grant No. 2015/22199-9 and the National Council for the Scientific and Technological Development (CNPq) Grant No. 54283/2014-2 for the financial support of this research.

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Author notes

  1. R. Q. Rodríguez

    Present address: School of Civil Engineering, State University of Mato Grosso Campus Tangará da Serra, Tangará da Serra, MT, 78300-000, Brazil

Authors and Affiliations

  1. Department of Computational Mechanics, School of Mechanical Engineering, University of Campinas, Campinas, SP, 13083-860, Brazil

    R. Q. Rodríguez, A. F. Galvis & P. Sollero

  2. Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, ON, K1S 5B6, Canada

    C. L. Tan

  3. Department of Mechanical Engineering, Faculty of Technology, University of Brasilia, Brasília, DF, 70910-900, Brazil

    E. L. Albuquerque

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  1. R. Q. Rodríguez
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  3. P. Sollero
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Correspondence to P. Sollero.

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Rodríguez, R.Q., Galvis, A.F., Sollero, P. et al. Transient dynamic analysis of generally anisotropic materials using the boundary element method. Acta Mech 229, 1893–1910 (2018). https://doi.org/10.1007/s00707-018-2108-4

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  • DOI: https://doi.org/10.1007/s00707-018-2108-4

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