Energetic properties of the elastic half-space loaded by a periodic distribution of vibrating punches: The study of the phase shift influence
Authors:
Edoardo Scarpetta and Mezhlum A. Sumbatyan
Journal:
Quart. Appl. Math. 69 (2011), 15-38
MSC (2010):
Primary 74J99
DOI:
https://doi.org/10.1090/S0033-569X-2010-01198-3
Published electronically:
December 9, 2010
MathSciNet review:
2807975
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Abstract:
We develop an analytical approach to study the wave properties of an elastic half-space subjected to harmonic vibrations applied on its free surface by a periodic array of rigid punches. Contrary to previous investigations, it is assumed that any neighbouring pair of these oscillates with a (common) phase shift, thus implying rather specific behaviours of the structure. Starting from integral equations for the contact stress and representation formulas for the wave field in both the anti-plane and in-plane problems, suitable mild approximations on the kernels allow analytical solutions of some related (auxiliary) integral equations in a given range of (not too high) frequency.
The explicit formulas thus obtained for the wave field are reflected through some figures and enable us to investigate the energetic properties of the structure with respect to different phase shifts. A direct numerical solution of the original integral equations confirms the precision of the analytical solution.
References
- K. Aki, P. G. Richards, Quantitative Seismology. Theory and Methods (1,2), Freeman: San Francisco, 1980.
- J. Brouwer, K. Helbig, Shallow High-Resolution Reflection Seismics (Handbook of Geophysical Exploration: Seismic Exploration), Pergamon: New York, 1998.
- United States Patent 6028818 : Method and apparatus for multiple seismic vibratory surveys. (http://www.freepatentsonline.com/6028818.html)
- United States Patent 6807508 : Seismic prospecting method and device using simultaneous emission of seismic signals based on pseudo-random sequences. (http://www.freepatentsonline.com/6807508.html)
- United States Patent 6714867 : Method for seismic monitoring of an underground zone by simultaneous use of several vibroseismic sources. (http://www.freepatentsonline.com/6714867.html)
- J. J. Postel, E. Gillot, M. Larroque, Review of specific parameters in high-resolution seismic, EAGE 66th Conf., Paris, 2004, Z99. (http://www.freepatentsonline.com/6714867.html)
- E. Scarpetta, M. A. Sumbatyan, Wave Properties of the Elastic Half-space loaded by a Periodic Array of Vibrating Punches: an Analytical Approach, Adv. Theor. Appl. Mech.1 (6) 2008, 281-300.
- J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland: Amsterdam, 1973.
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 2, Gordon & Breach Science Publishers, New York, 1986. Special functions; Translated from the Russian by N. M. Queen. MR 874987
- F. D. Gakhov, Boundary value problems, Pergamon Press, Oxford-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1966. Translation edited by I. N. Sneddon. MR 0198152
References
- K. Aki, P. G. Richards, Quantitative Seismology. Theory and Methods (1,2), Freeman: San Francisco, 1980.
- J. Brouwer, K. Helbig, Shallow High-Resolution Reflection Seismics (Handbook of Geophysical Exploration: Seismic Exploration), Pergamon: New York, 1998.
- United States Patent 6028818 : Method and apparatus for multiple seismic vibratory surveys. (http://www.freepatentsonline.com/6028818.html)
- United States Patent 6807508 : Seismic prospecting method and device using simultaneous emission of seismic signals based on pseudo-random sequences. (http://www.freepatentsonline.com/6807508.html)
- United States Patent 6714867 : Method for seismic monitoring of an underground zone by simultaneous use of several vibroseismic sources. (http://www.freepatentsonline.com/6714867.html)
- J. J. Postel, E. Gillot, M. Larroque, Review of specific parameters in high-resolution seismic, EAGE 66th Conf., Paris, 2004, Z99. (http://www.freepatentsonline.com/6714867.html)
- E. Scarpetta, M. A. Sumbatyan, Wave Properties of the Elastic Half-space loaded by a Periodic Array of Vibrating Punches: an Analytical Approach, Adv. Theor. Appl. Mech.1 (6) 2008, 281-300.
- J. D. Achenbach, Wave Propagation in Elastic Solids, North-Holland: Amsterdam, 1973.
- A. P. Prudnikov, Y. A. Brychkov, O. I. Marichev. Integrals and Series (vol.2), Gordon and Breach Science Publishers: Amsterdam, 1986. MR 874987 (88f:00014)
- F. D. Gakhov, Boundary Value Problems, Pergamon Press: Oxford, 1966. MR 0198152 (33:6311)
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Additional Information
Edoardo Scarpetta
Affiliation:
D.I.I.M.A., University of Salerno, 84084 Fisciano (SA), Italy
Mezhlum A. Sumbatyan
Affiliation:
Faculty of Mechanics and Mathematics, Rostov State University, 344090 Rostov-on-Don, Russia
Received by editor(s):
April 2, 2009
Published electronically:
December 9, 2010
Article copyright:
© Copyright 2010
Brown University
The copyright for this article reverts to public domain 28 years after publication.