Transmission of axial-symmetric superficial disturbances in the elastic-porous half-space

Аuthors

Tarlakovsky D. V.^1*, Dang Q. G.^2**

1. Institute of Mechanics Lomonosov Moscow State University, 1, Michurinsky prospect, Moscow, 119192, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: tdvhome@mail.ru
**e-mail: giang_1986_5@yahoo.com

Abstract

Mathematical modeling of non-stationary processes in saturated environment on model-based two-component environment is an important and urgent problem. Its urgency is dictated by pressing request of practice (extraction of groundwater, oil and gas, construction of earthen weirs, dams and earthworks, stability of slope, underground construction, etc.) and the need for further development of the general theory of multi-component environment, including questions of construction of mathematical models and basis of analytical and numerical methods for solving concrete boundary problems. Similar problems also find application in various areas of new technology, including problems of different apparatus landing of aviation, rocket and space technology.
Currently, in spite of major successes in this area, a lot of unsolved problems are remained. Especially, the problems of non-stationary interaction of deformable objects with grounds, elastic and multi-component environments are little examined. In connection with considerable mathematical difficulties in this area, there is a little number of analytical solutions, relating generally to the areas of canonical forms.
For modeling of dynamic processes in some grounds the model of elastic-porous Biot environment is often used. For the half-space filled by this environment, currently, the most of plane non-stationary problems are particularly researched. Thus, practically, there aren’t any analytical solutions to the corresponding axial-symmetric problems that are examined in this article.
The Hankel transform in radius and the Laplace transform in time are applied for solving. Originals are found by the theorems on the connection of plane axial-symmetric problems with use of known solutions to the first of them. Corresponding integrals are numerically found by use of their canonical regularization. Some results of the calculations are presented in graphical form.

Keywords:

porous elastic environment, model Bio, half-space, surface influence function, the Laplace and Hankel integral transforms, communication and flat axial-symmetric tasks

References

1. Bio M.A. Mechanics. Sbornik statei, 1963, no. 6, pp. 103-134.
2. Narimov Sh.N. Volnovye protsessy v nasyshchennykh poristykh sredakh (Wave processes in saturated porous media), Tashkent, Mehnat, 1988, 304 p.
3. Gorshkov A.G., Tarlakovskii D.V. Nestatsionarnaya aerogidrouprugost’ tel sfericheskoi formy (Unsteady aero-hydro-elasticity bodies spherical shape), Moscow, Nauka, 1990, 264 p.
4. Belov A.A, Igumnov LA , Karelin I.S , Lytvynchuk S.Y., Elektronnyi zhurnal «Trudy MAI», 2010, no. 40, available at: http://www.mai.ru/science/trudy/eng/published.php?ID=22862 (accessed 01.10.2010).
5. Hoa Nguyen Ngoc, Tarlakovskii D.V. Mechanics of Composite Materials and Structures, 2011, vol. 17, no. 4. pp. 483-492.
6. Gorshkov A.G., Medvedskii A.L., Rabinskii L.N., Tarlakovskii D.V. Volny v sploshnykh sredakh (Waves in continuous media), Moscow, Fizmatlit, 2004, 472 p.
7. Giang Dang Quang, Tarlakovskii D.V. Mechanics of Composite Materials and Structures, 2014, no. 1, pp. 148-158.
8. Gorshkov A.G., Tarlakovskii D.V. Dinamicheskie kontaktnye zadachi s podvizhnymi granichnymi (Dynamic contact problems with moving boundary), Moscow, Fizmatlit, 1995, 350 p.
9. Gel’fand I.M., Shilov G.E. Obobshchennye funktsii i deistviya nad nimi (Generalized functions over them), Moscow, Fizmatlit, 1959, 470 p.

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