Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-23T02:42:38.594Z Has data issue: false hasContentIssue false
  • English
  • Français

Scattering of Elastic Waves by a Buried Tunnel Under Obliquely Incident Waves Using T Matrix

Published online by Cambridge University Press:  05 May 2011

W.-I. Liao ,
C.-S. Yeh  and
T.-J. Teng
W.-I. Liao*
Affiliation:
Department of Civil Engineering, National Taipei University of Technology, Taipei, Taiwan 10608, R.O.C.
C.-S. Yeh*
Affiliation:
Dep. of Civil Engineering, Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
T.-J. Teng*
Affiliation:
National Center for Research on Earthquake Engineering, Taipei, Taiwan 10668, R.O.C.
*
* Associate Professor
** Professor
*** Research Fellow
Get access

Abstract

This paper first studies the transition matrix formulation for the analysis of responses of an elastic halfspace with a buried tunnel subjected to obliquely incident waves. The basis functions are constructed using the moving P-, SV-, and SH-wave source potentials and to represent the scattered and refracted wave fields in series forms. The associated T-matrix expression of elastic inclusion is derived using Betti's third identity. Second, this study proposes a technique for calculating the integral representation of basis functions in the wave-number domain using the method of steepest descent. Finally, typical numerical results obtained under incident plane waves are presented for verification.

Keywords

T-matrixElastic waveSteepest descentTunnel
Type
Articles
Information
Journal of Mechanics , Volume 24 , Issue 4 , December 2008 , pp. 405 - 418
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2008

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.Waterman, P. C., “New Formulation of Acoustic Scattering,” Journal of the Acoustical Society of America, 45, pp. 1417–1429(1969).Google Scholar
2.Waterman, P. C., “Matrix Theory of Elastic Wave Scattering,” Journal of the Acoustical Society of America, 60, pp. 567–580 (1976).Google Scholar
3.Pao, Y. H., “Betti's Identity and Transition Matrix for Elastic Waves,” Journal of the Acoustical Society of America, 64, pp. 302–310 (1978).CrossRefGoogle Scholar
4.Pao, Y. H., “The Transition Matrix for the Scattering of Acoustic Waves and Elastic Waves,” Proceeding of the IUTAM Symposium on Modern Problems in Elastic Wave Propagation, Miklowitz, J. and Achenbach, J., Ed., Wiley, New York, pp. 123–144 (1978).Google Scholar
5.Varatharajulu, V. and Pao, Y. H., “Scattering Matrix for Elastic Wave, I. Theory,” Journal of the Acoustical Society of America, 60, pp. 556–566 (1976).Google Scholar
6.Varadan, V. V. and Varadan, V. K., “Scattering Matrix for Elastic Wave, III. Application to Spheroids,” Journal of the Acoustical Society of America, 65, pp. 896–905 (1979).Google Scholar
7.Yeh, Y. K. and Pao, Y. H., “On the Transition Matrix for Acoustic Waves Scattered by a Multilayered Inclusion,” J. Acoust. Soc. Am., 81, pp. 1683–1687 (1987).CrossRefGoogle Scholar
8.Chai, J.-F., Teng, T.-J. and Yeh, C.-S., “Determination of Resonance Frequency of Two-Dimensional Alluvial Valley by Background Subtraction Method,” The Chinese Journal of Mechanics, 14, pp. 1–15 (1998).Google Scholar
9.Lee, V. W. and Trifunac, M. D., “Response of Tunnels to Incident SH-waves,” J. Engng. Mech. Div., ASCE, 105, pp. 643–659(1979).CrossRefGoogle Scholar
10.El-Akily, N. and Datta, S. K., “Response of Circular Cylindrical Shell to Disturbances in a Half-Space,” Earthq. Engng. Struct. Dynam., 8, pp. 469–477 (1980).Google Scholar
11.Wong, K. C., Shah, A. H. and Datta, S. K., “Dynamic Stresses and Displacements in Buried Tunnel,” J. Engng. Mech. Div., ASCE, 111, pp. 218–234 (1985).CrossRefGoogle Scholar
12.Luco, J. E. and de Barros, F. C. P., “Seismic Response of Cylindrical Shell Embedded in a Layered Viscoelastic Half- Space, I. Formulation, II. Validation and Numerical Results,” Earthq. Engng. Struct. Dynam., 23, pp. 553–580 (1994).CrossRefGoogle Scholar
13.Yeh, C. S., Teng, T. J., Shyu, W. S. and Tsai, I. C., “A Hybrid Method for Analyzing the Dynamics Responses of Cavities of Shells Buried in an Elastic Half-Plane,” The Chinese Journal of Mechanics, 18, pp. 75–87 (2002).CrossRefGoogle Scholar
14.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).Google Scholar
15.Chen, J. T., Shen, W. C. and Wu, A. C., “Null-Field Integral Equations for Stress Field Around Circular Holes under Antiplane Shear,” Engineering Analysis with Boundary Elements, 30, pp. 205–217 (2006).CrossRefGoogle Scholar
16.Chen, J. T. and Chen, P. Y., “A Semi-Analytical Approach for Stress Concentration of Cantilever Beams with Holes Under Bending,” Journal of Mechanics, 23, pp. 211–222(2007).Google Scholar
17.Apsel, R. J. and Luco, J. E., “On the Green's Function for a Layered Half-Space: Part II,” Bull. Seism. Soc. Am. 73, pp. 931–951 (1983).Google Scholar
18.Kundu, T. and Mal, A. K., “Elastic Waves in Multilayered Solids Due to a Dislocation Source,” Wave Motions, 7, pp. 459–471 (1985).CrossRefGoogle Scholar
19.Xu, P. C. and Mal, A. K., “An Adaptive Scheme for Irregularly Oscillatory Functions,” Wave Motions, 7, pp. 235–243 (1985).Google Scholar
20.Yeh, C. S., Teng, T. J. and Liao, W. I., “On Evaluation of Lamb's Integrals for Waves in a Two-Dimensional Elastic Half-Space,” The Chinese Journal of Mechanics, 16, pp. 109–124 (2000).CrossRefGoogle Scholar
21.Liao, W. I., Teng, T. J. and Yeh, C. S., “A Series Solution and Numerical Technique for Wave Diffraction by a Three-Dimensional Canyon,” Wave Motion, 39, pp. 129–142 (2004).Google Scholar
22.Achenbach, J. D., Wave Propagation in Elastic Solids, North-Holland, New York (1973).Google Scholar
23.Yeh, C. S., Teng, T. J., Liao, W. I. and Tsai, I. C., “A Series Solution for Dynamic Response of a Cylindrical Shell in an Elastic Half-Plane,” The 3rd National Conference on Structural Engineering, Kenting, Taiwan, pp. 1553–1562(1996).Google Scholar
24.Reddy, P. M. and Tajuddin, M., “Cylindrical Stress Waves in Poroelastic Flat Slabs,” Journal of Mechanics, 22, pp. 161–165(2006).CrossRefGoogle Scholar
25.Gregory, R. D., “An Expansion Theorem Applicable to Problems of Wave Propagation in an Elastic Half-Space Containing a Cavity,” Proc. Comb. Phil. Soc., 63, pp. 1341–1367(1967).CrossRefGoogle Scholar
26.Gregory, R. D., “The Propagation of Waves in an Elastic Half-Space Containing a Cavity,” Proc. Comb. Phil. Soc., 67, pp. 689–709(1970).Google Scholar