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

  • J. T. Chen (a1), Y. T. Lee (a2) and K. H. Chou (a2)


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.


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* Life-time Distinguished Professor, corresponding author
** Ph.D.
*** Master student


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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. 675685 (1982).
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. 629639 (1997).
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. 101108 (1983).
4.Hutchinson, J. R., An Alternative BEM Formulation Appliedto Membrane Vibrations, Brebbia, C. A.andMaier, G. Eds., Boundary Elements VII, Springer-Verlag, Berlin (1985).
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. 105116(1998).
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. 917928 (2002).
7.Hutchinson, J. R, Vibration of Plates, Brebbia, C. A. Ed., Boundary Elements X, Springer-Verlag, Berlin (1988).
8.Mills, R. D., “Computing Internal Viscous Flow Problems for the Circle by Integral Methods,” Journal for Fluid Mechanics, 73, pp. 609624 (1977).
9.Waterman, P. C, “Matrix Formulation of Electromagnetic Scattering,” Proceedings of the Institute of Electrical and Electronics Engineers, 53, pp. 805812 (1965).
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. 14431445 (1968).
11.Waterman, P. C, “Matrix Theory of Elastic Wave Scattering,” Journal of the Acoustical Society of America, 60, pp. 567580 (1967).
12.Martin, P. A., “On the Null-Field Equations for Water-Wave Radiation Problems,” Journal of Fluid Mechanics, 113, pp. 315332(1981).
13.Boström, A., “Time-Dependent Scattering by a Bounded Obstacle in Three Dimensions,” Journal of Mathematical Physics, 23, pp. 14441450 (1982).
14.Olsson, P., “Elastostatics as a Limit of Elastodynamics–A Matrix Formulation,” Applied Scientific Research, 41, pp. 125151(1984).
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. 92105 (1975).
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. 169178(1994).
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. 205217 (2005).
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. 27512764(2007).
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. 679693 (2006).
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. 811830(2007).
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. 469487 (2007).
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. 16691681 (2002).
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. 233261 (2005).
24.Timoshenko, S. P. and Goodier, J. N, Theory of Elasticity, McGraw-Hill, New York (1970).
25.Banerjee, P. K. and Butterfield, R., Boundary Element Method in Engineering Science, McGraw-Hill, New York (1981).
26.Hong, H.-K. and Chen, J. T., “Derivations of Integral Equations of Elasticity,” Journal for Engineering Mechanics, ASCE, 114, pp. 10281044 (1988).
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. 1733(1999).
28.Lamé, G., Leçons Sur La Thèorie De L'Élasticité, Gauthier-Villars, Paris (1852).
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. 221232 (2009).


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

  • J. T. Chen (a1), Y. T. Lee (a2) and K. H. Chou (a2)


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