Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T06:00:59.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Revisit of Two Classical Elasticity Problems by using the Null-Field Boundary Integral Equations

Published online by Cambridge University Press:  05 May 2011

J. T. Chen ,
Y. T. Lee  and
K. H. Chou
J. T. Chen*
Affiliation:
Department of Harbor and River Engineering, Department of Mechanical and Mechatronics Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
Y. T. Lee*
Affiliation:
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
K. H. Chou*
Affiliation:
Department of Harbor and River Engineering, National Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
*
* Life-time Distinguished Professor, corresponding author
** Ph.D.
*** Master student
Get access

Abstract

In this paper, the two classical elasticity problems, Lamé problem and stress concentration factor, are revisited by using the null-field boundary integral equation (BIE). The null-field boundary integral formulation is utilized in conjunction with degenerate kernels and Fourier series. To fully utilize the circular geometry, the fundamental solutions and the boundary densities are expanded by using degenerate kernels and Fourier series, respectively. In the two classical problems of elasticity, the null-field BIE is employed to derive the exact solutions. The Kelvin solution is first separated to degenerate kernel in this paper. After employing the null-field BIE, not only the stress but also the displacement field are obtained at the same time. In a similar way, Lamé problem is solved without any difficulty.

Keywords

Degenerate kernelNull-field boundary integral equationStress concentration factorLamé problem
Type
Articles
Information
Journal of Mechanics , Volume 26 , Issue 3 , September 2010 , pp. 393 - 401
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

1.Barone, M. R. and Caulk, D. A., “Optimal Arrangement of Holes in a Two-Dimensional Heat Conductor by a Special Boundary Integral Method,” International Journal for Numerical Methods in Engineering, 18, pp. 675–685 (1982).CrossRefGoogle Scholar
2.Cheng, H. W. and Greengard, L., “On the Numerical Evaluation of Electrostatic Field in Dense Random Dispersions of Cylinders,” Journal for Computational Physics, 136, pp. 629–639 (1997).CrossRefGoogle Scholar
3.Caulk, D. A., “Analysis of Elastic Torsion in a Bar with Circular Holes by a Special Boundary Integral Method,” Journal for Applied Mechanics, ASME, 50, pp. 101–108 (1983).CrossRefGoogle Scholar
4.Hutchinson, J. R., An Alternative BEM Formulation Appliedto Membrane Vibrations, Brebbia, C. A.andMaier, G. Eds., Boundary Elements VII, Springer-Verlag, Berlin (1985).Google Scholar
5.Chen, J. T. and Chen, K. H., “Dual Integral Formulation for Determining the Acoustic Modes of a Two-Dimensional Cavity with a Degenerate Boundary,” Engineering Analysis with Boundary Elements, 21, pp. 105–116(1998).CrossRefGoogle Scholar
6.Chen, K. H., Chen, J. T., Chou, C. R. and Yueh, C. Y., “Dual Boundary Element Analysis of Oblique Incident Wave Passing a Thin Submerged Breakwater,” Engineering Analysis with Boundary Elements, 26, pp. 917–928 (2002).CrossRefGoogle Scholar
7.Hutchinson, J. R, Vibration of Plates, Brebbia, C. A. Ed., Boundary Elements X, Springer-Verlag, Berlin (1988).Google Scholar
8.Mills, R. D., “Computing Internal Viscous Flow Problems for the Circle by Integral Methods,” Journal for Fluid Mechanics, 73, pp. 609–624 (1977).CrossRefGoogle Scholar
9.Waterman, P. C, “Matrix Formulation of Electromagnetic Scattering,” Proceedings of the Institute of Electrical and Electronics Engineers, 53, pp. 805–812 (1965).CrossRefGoogle Scholar
10.Bates, R. H. T., “Modal Expansions for Electromagnetic Scattering From Perfectly Conducting Cylinders of Arbitrary Cross-Section,” Proceedings of the Institution of Electrical Engineers, 115, pp. 1443–1445 (1968).CrossRefGoogle Scholar
11.Waterman, P. C, “Matrix Theory of Elastic Wave Scattering,” Journal of the Acoustical Society of America, 60, pp. 567–580 (1967).CrossRefGoogle Scholar
12.Martin, P. A., “On the Null-Field Equations for Water-Wave Radiation Problems,” Journal of Fluid Mechanics, 113, pp. 315–332(1981).CrossRefGoogle Scholar
13.Boström, A., “Time-Dependent Scattering by a Bounded Obstacle in Three Dimensions,” Journal of Mathematical Physics, 23, pp. 1444–1450 (1982).CrossRefGoogle Scholar
14.Olsson, P., “Elastostatics as a Limit of Elastodynamics–A Matrix Formulation,” Applied Scientific Research, 41, pp. 125–151(1984).CrossRefGoogle Scholar
15.Sloan, I. H., Burn, B. J. and Datyner, N, “A New Approach to the Numerical Solution of Integral Equations,” Journal of Computational Physics, 18, pp. 92–105 (1975).CrossRefGoogle Scholar
16.Chen, J. T. and Hong, H.-K., “Dual Boundary Integral Equations at a Corner Using Contour Approach Around Singularity,” Advances in Engineering Software, 21, pp. 169–178(1994).CrossRefGoogle Scholar
17.Chen, J. T., Shen, W. C. and Wu, A. C, “Null-Field Integral Equations for Stress Field Around Circular Holes Under Anti-Plane Shear,” Engineering Analysis with Boundary Elements, 30, pp. 205–217 (2005).CrossRefGoogle Scholar
18.Chen, J. T., Chen, C. T., Chen, P. Y. and Chen, I. L., “A Semi-Analytical Approach for Radiation and Scattering Problems with Circular Boundaries,” Computer Methods in Applied Mechanics and Engineering, 196, pp. 2751–2764(2007).CrossRefGoogle Scholar
19.Chen, J. T., Hsiao, C. C. and Leu, S. Y., “Null-Field Integral Equation Approach for Plate Problems with Circular Boundaries,” Journal for Applied Mechanics, ASME, 73, pp. 679–693 (2006).CrossRefGoogle Scholar
20.Lee, W. M., Chen, J. T. and Lee, Y. T., “Free Vibration Analysis of Circular Plates with Multiple Circular Holes Using Indirect BiEMs,” Journal for Sound and Vibration, 304, pp. 811–830(2007).CrossRefGoogle Scholar
21.Chen, J. T. and Wu, A. C, “Null-Field Approach for the Multi-Inclusion Problem Under Anti-Plane Shears,” Journal for Applied Mechanics, ASME, 74, pp. 469–487 (2007).CrossRefGoogle Scholar
22.Chen, J. T., Kuo, S. R. and Lin, J. H., “Analytical Study and Numerical Experiments for Degenerate Scale Problems in the Boundary Element Method for Two-Dimensional Elasticity,” International Journal for Numerical Methods in Engineering, 54, pp. 1669–1681 (2002).CrossRefGoogle Scholar
23.Chen, J. T., Lin, S. R. and Chen, K. H., “Degenerate Scale Problem when Solving Laplace's Equation by BEM and its Treatment,” International Journal for Numerical Methods in Engineering, 62, pp. 233–261 (2005).CrossRefGoogle Scholar
24.Timoshenko, S. P. and Goodier, J. N, Theory of Elasticity, McGraw-Hill, New York (1970).Google Scholar
25.Banerjee, P. K. and Butterfield, R., Boundary Element Method in Engineering Science, McGraw-Hill, New York (1981).Google Scholar
26.Hong, H.-K. and Chen, J. T., “Derivations of Integral Equations of Elasticity,” Journal for Engineering Mechanics, ASCE, 114, pp. 1028–1044 (1988).CrossRefGoogle Scholar
27.Chen, J. T. and Hong, H.-K., “Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series,” Applied Mechanics Reviews, ASME, 52, pp. 17–33(1999).CrossRefGoogle Scholar
28.Lamé, G., Leçons Sur La Thèorie De L'Élasticité, Gauthier-Villars, Paris (1852).Google Scholar
29.Chen, J. T and Lee, Y. T., “Torsional Rigidity of a Circular Bar with Multiple Circular Inclusions Using the Null-Field Integral Approach,” Computational Mechanics, 44, pp. 221–232 (2009).CrossRefGoogle Scholar