Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-17T02:12:02.420Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Efficiency of Analyzing 3D Anisotropic, Transversely Isotropic, and Isotropic Bodies in BEM

Published online by Cambridge University Press:  28 September 2011

Y. C. Shiah  and
W. X. Sun
Y. C. Shiah*
Affiliation:
Department of Aerospace and Systems Engineering, Feng Chia University, Taichung, Taiwan40724, R.O.C.
W. X. Sun*
Affiliation:
Department of Aerospace and Systems Engineering, Feng Chia University, Taichung, Taiwan40724, R.O.C.
*
* Professor, corresponding author
** Research associate
Get access

Abstract

Due to a lack of closed-form solutions for three dimensional anisotropic bodies, the computational burden of evaluating the fundamental solutions in the boundary element method (BEM) has been a research focus over the years. In engineering practice, transversely isotropic material has gained popularity in the use of composites. As a degenerate case of the generally anisotropic material, transverse isotropy still needs to be treated separately to ease the computations. This paper aims to investigate the computational efficiency of the BEM implementations for 3D anisotropic, transversely isotropic, and isotropic bodies. For evaluating the fundamental solutions of 3D anisotropy, the explicit formulations reported in [1,2] are implemented. For treating transversely isotropic materials, numerous closed form solutions have been reported in the literature. For the present study, the formulations presented by Pan and Chou [3] are particularly employed. At the end, a numerical example is presented to compare the computational efficiency of the three cases and to demonstrate how the CPU time varies with the number of meshes.

Keywords

Boundary element methodTransversely isotropic3D anisotropic
Type
Research Article
Information
Journal of Mechanics , Volume 26 , Issue 4 , December 2010 , pp. 483 - 491
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

1. Ting, T. C. T. and Lee, V. G., “The Three-Dimensional Elastostatic Green's Function for General Anisotropic Linear Elastic Solid.” Quarterly Journal of Mechanics Applied Mathematics, 50, pp. 407–426 (1997).CrossRefGoogle Scholar
2. Lee, V. G., “Explicit Expression of Derivatives of Elastic Green's Functions for General Anisotropic Materials,” Mechanics Research Communications, 30, pp. 241–249 (2003).CrossRefGoogle Scholar
3. Pan, Y. C. and Chou, T. W., “Point Force Solution for an Infinite Transversely Isotropic Solid,” Journal of Applied Mechanics, 29, pp. 225–236 (1976).Google Scholar
4. Lifshitz, I. M. and Rozenzweig, L.N., “Construction of the Green Tensor for the Fundamental Equation of Elasticity Theory in the Case of Unbounded Elastically Anisotropic Medium,” Journal of Experimental and Theoretical Physics, 17, pp. 783–791 (1947).Google Scholar
5. Synge, J. L., The Hypercircle in Mathematical Physics, Cambridge University Press, Cambridge (1957).Google Scholar
6. Barnett, D. M., “The Precise Evaluation of Derivatives of the Anisotropic Elastic Green's Functions,” Physica Status Solidi, (b) 49, pp. 741–748 (1972).CrossRefGoogle Scholar
7. Wilson, R. B. and Cruse, T. A., “Efficient Implementation of Anisotropic Three Dimensional Boundary Integral Equation Stress Analysis,” International Journal for Numerical Methods in Engineering, 12, pp. 1383–1397 (1978).Google Scholar
8. Chen, T. and Lin, F. Z., “Numerical Evaluation of Derivatives of the Anisotropic Elastic Green's Functions,” Mechanics Research Communications, 20, pp. 501–506 (2001).Google Scholar
9. Nakamura, G. and Tanuma, K., “A formula for the Fundamental Solution of Anisotropic Elasticity,” Quarterly Journal of Mechanics Applied Mathematics, 50, pp. 179–194 (1997).Google Scholar
10. Wang, C. Y., “Elastic Fields Produced by a Point Source in Solids of General Anisotropy,” Journal of Engineering Mathematics, 32, pp. 41–52 (1997).Google Scholar
11. Tonon, F., Pan, E. and Amadei, B., “Green's Functions and Boundary Element Method Formulation for 3D Anisotropic Media,” Composite and Structures, 79, pp. 469–482 (2001).CrossRefGoogle Scholar
12. Wang, C. Y. and Denda, M., “3D BEM for General Anisotropic Elasticity,” International Journal of Solids and Structures, 44, pp. 7073–7091 (2007).CrossRefGoogle Scholar
13. Tan, C. L., Shiah, Y. C. and Lin, C. W., “Stress Analysis of 3D Generally Anisotropic Elastic Solids Using the Boundary Element Method,” CMES-Computer Modeling in Engineering and Sciences, 41, pp. 159–214 (2009).Google Scholar
14. Eubanks, R. A. and Sternberg, E., “On the Axisymmetric Problem of Elasticity Theory for a Medium with Transverse Isotropy,” Journal of Rational Mechanics and Analysis, 3, pp. 89–101 (1954).Google Scholar
15. Gurtin, M. E., The linear theory of elasticity, in: Flugge, S., Truesdell, C. Eds., Handbuch der Physik, Mechanics of Solids II, Via/2, Springer, Berlin, pp. 1–295 (1972).Google Scholar
16. Michell, J. H., “The Stress in an Anisotropic Elastic Solid with an Infinite Plane Boundary,” Proceedings London Mathematics Society, 32, pp. 247–258 (1900).CrossRefGoogle Scholar
17. Hu, H. C. “On the Three Dimensional Problems of the Theory of Elasticity of a Transversely Isotropic Body,” Society Sinica, 2, pp. 145–151 (1953).Google Scholar
18. Nowacki, W., “The Stress Function in Three-Dimensional Problems Concerning an Elastic Body Characterized by Transverse Isotropy,” Bulletin De L'Acadbmie Polonaise Des Sciences, 4, pp. 21–25 (1954).Google Scholar
19. Lodge, A. S., “The Transformation to Isotropic form of the Equilibrium Equations for a Class of Anisotropic Elastic Solids,” Quarterly Journal of Mechanics Applied Mathematics, 8, pp. 211–225 (1955).CrossRefGoogle Scholar
20. Alexsandrov, I. Y. and Soloviev, U. I., Three-Dimensional Problems of Elastic Theory. Science, Moscow (1978).Google Scholar
21. Ding, H. J. and Xu, B. H., “General Solutions of Axisymmetric Problems in Transversely Isotropic Body,” Applied Mathematics Mechanics, 9, pp. 135–142 (1988).Google Scholar
22. Horgan, C. O. and Simmonds, J. G., “Asymptotic Analysis of an End-Loaded, Transversely Isotropic, Elastic, Semi-Infinite Strip Weak in Shear,” International Journal of Solids and Structures, 27, pp. 1895–1914 (1991).Google Scholar
23. Wang, M. Z. and Wang, W., “Completeness and Non-uniqueness of General Solutions of Transversely Iso-tropic Elasticity,” International Journal of Solids and Structures, 32, pp. 501–513 (1995).CrossRefGoogle Scholar
24. Fabrikant, V. I., “Complete Solution to the Problem of an External Circular Crack in a Transversely Isotropic Body Subjected to Arbitrary Shear Loading,” International Journal of Solids and Structures, 33, pp. 167–191 (1996).Google Scholar
25. Tarn, J.-Q. and Wang, Y.-M., “A Fundamental Solution for a Transversely Isotropic Elastic Space,” Journal of the Chinese Institute of Engineers, 10, pp. 13–21 (1987).Google Scholar
26. Chen, J. T., Liao, H. Z. and Lee, W. M., “An Analytical Approach for the Green's Functions of Biharmonic Problems with Circular and Annular Domains,” Journal of Mechanics, 25, pp. 59–74 (2009).Google Scholar
27. Shiah, Y. C., Lin, Y.-S. and Chen, Y. H., “Analysis for Thermal Conductance Effect on the Interfacial Thermal Stresses of Anisotropic Composites,” Journal of Thermal Stresses, 31, pp. 991–1005 (2008).Google Scholar
28. Hong, H.-K. and Chen, J. T., “Derivations of Integral Equations of Elasticity,” Journal of Engineering Mechanics, ASCE, 114, pp. 1028–1044 (1988).Google Scholar
29. Rizzo, F. J. and Shippy, D. J., “An Advanced Boundary Integral Equation Method for Three-Dimensional Thermoelasticity,” International Journal of Numerical Methods Engineering, 11, pp. 1753–1768 (1977).Google Scholar
30. Lenitskii, S. G., Theory of Elasticity of an Anisotropic Body. Holden-Day, San Francisco (1963).Google Scholar