Some results on the energy transmission through an elastic half-space loaded by a periodic distribution of vibrating punches. (English) Zbl 1293.74239

Summary: We develop an analytical approach to study the wave process arising in an elastic half-space because of harmonic vibrations applied on its free surface by a (periodic) distribution of rigid punches. By assuming perfect coupling between punches and half-space, the (in-plane) propagation problem is firstly reduced to a \(2\times 2\) system of integral equations for the contact stresses. Then, in the frequency range implying the so-called one-mode (far-field) propagation, suitable mild approximations on the kernels lead to some related auxiliary systems of integral equations, which are independent on frequency and can be solved analytically. The explicit formulas thus obtained are reflected through some figures and enable us to discuss the energetic properties of the wave process with respect to frequency. A direct numerical treatment of the original system of (exact) integral equations confirms the precision of the analytical solution.


74J10 Bulk waves in solid mechanics
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[1] Aki K, Richards PG (1980) Quantitative seismology. Theory and methods, vols. 1, 2. Freeman, San Francisco
[2] Brouwer J, Helbig K (1998) Shallow high-resolution reflection seismics. Handbook of geophysical exploration: seismic exploration. Pergamon, New York
[3] Sheriff ER, Geldart LP (1995) Exploration seismology. Cambridge University Press, Cambridge
[4] Krautkramer J, Krautkramer H (1983) Ultrasonic testing of materials, 3rd edn. Springer, New York
[5] Postel JJ, Gillot E, Larroque M (2004) Review of specific parameters in high-resolution seismic, EAGE 66th Conf., Paris, 2004, Z99. http://www.freepatentsonline.com/6714867.html
[6] Goryacheva IG (1998) The periodic contact problem for an elastic half-space. J Appl Math Mech 62:959–966
[7] Galin LA (1945) Indentation of a punch in the presence of friction and adhesion. J Appl Math Mech 9:413–424 · Zbl 0060.42003
[8] Habraken AM, Cescotto S (1998) Contact between deformable solids: the fully coupled approach. Math Comput Model 28:153–169
[9] Scarpetta E, Sumbatyan MA (2009) Wave properties of the elastic half-space loaded by a periodic distribution of vibrating punches: an analytical approach. Adv Theor Appl Mech 1(6):281–300 · Zbl 1195.74092
[10] Lavrov NA, Pavlovskaya EE (2000) Transient behavior of a few dies on an elastic half-space. Acta Mech 144:185–195 · Zbl 0967.74046
[11] Argatov II (2009) Slow vertical motions of a system of punches on an elastic half-space. Mech Res Commun 36:199–206 · Zbl 1258.74160
[12] Achenbach JD (1973) Wave propagation in elastic solids. North-Holland, Amsterdam · Zbl 0268.73005
[13] Papoulis A (1962) The Fourier integral and its applications. McGraw-Hill, New York · Zbl 0108.11101
[14] Scarpetta E, Sumbatyan MA (2008) One-mode wave propagation through a periodic array of interface cracks: explicit analytical results. J Math Anal Appl 337:576–593 · Zbl 1293.74250
[15] Pierce AD (1991) Acoustics. An introduction to its physical principles and applications. ASA Publisher, New York
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