 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| N.A. Bazarenko, "Contact Problems for a Circular Plate with Sliding Fixation at the End Face," Mech. Solids. 49 (3), 314-325 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
3 |
Pages |
314-325 |
| DOI |
10.3103/S0025654414030078 |
| Title |
Contact Problems for a Circular Plate with Sliding Fixation at the End Face |
| Author(s) |
N.A. Bazarenko (Don State Technical University, pl. Gagarina 1, Rostov-on-Don, 344000 Russia, n_bazarenko@rambler.ru) |
| Abstract |
Two mixed elasticity problems of punch indentation into a circular plate placed without clearance in a rigid cylindrical holder with smooth walls are considered. In the first problem, the plate lies without friction on a rigid base, and in the second problem, the plate is rigidly fixed to the base. The problems are solved by a method that was developed for bodies of finite dimensions and is based on the properties of closed systems of orthogonal functions. Each of the problems is reduced to two integral equations, namely, a Volterra integral equation of the first kind for the contact pressure function and a Fredholm integral equation of the first kind for the derivatives of the displacement of the plate upper surface outside the punch. The displacement function is sought as the sum of a trigonometric series and a power function with a root singularity. After truncation, the obtained ill-posed system of linear algebraic equation has a stable solution. A method for solving Volterra integral equations is given. The contact pressure distribution function and the dimensionless indentation force are determined. Examples of calculation of the plate interaction with the plane punch are given. Contact problems were earlier studied for a rectangle and a circular plate with a stress-free end both without taking account of their fixation [1, 2] and with regard for their fixation [3, 4]. The solution method described here was used to study the interaction of elastic hollow cylinder of finite length with a rigid bandage and a rigid insert [5, 6]. Other papers dealing with contact problems for bodies of finite dimensions, in particular, for a circular plate, should also be mentioned. In these papers, the problems under study were solved by the method of homogeneous solutions [7, 8] and by the method of coupled series-equations [9]. |
| Keywords |
distinguishing of the root singularity, SLAE regularization, equivalent boundary conditions, series summation |
| References |
| 1. | V. M. Alexandrov and N. A. Bazarenko,
"The Contact Problem for a Rectangle with Stress-Free Side Faces,"
Prikl. Mat. Mekh.
71 (2), 340-351 (2007)
[J. Appl. Math. Mech. (Engl. Transl.)
71 (2), 305-317 (2007)]. |
| 2. | N. A. Bazarenko,
"The Contact Problem for a Circular Plate with a Stress-Free End Face,"
Prikl. Mat. Mekh.
74 (6), 980-993 (2010)
[J. Appl. Math. Mech. (Engl. Transl.)
74 (6), 699-709 (2010)]. |
| 3. | N. A. Bazarenko,
"Interaction of a Rigid Punch with a Base-Secured Elastic Rectangle with Stress-Free Sides,"
Prikl. Mat. Mekh.
74 (4), 667-680 (2010)
[J. Appl. Math. Mech. (Engl. Transl.)
74 (4), 476-485 (2010)]. |
| 4. | N. A. Bazarenko,
"Interaction of a Rigid Punch and a Circular Plate with a Fixed Side and a Stress-Free Face,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 3, 78-95 (2013)
[Mech. Solids (Engl. Transl.)
48 (3), 301-316 (2013)]. |
| 5. | N. A. Bazarenko,
"The Contact Problem for Hollow and Solid Cylinders with Stress-Free Faces,"
Prikl. Mat. Mekh.
72 (2), 328-341 (2008)
[J. Appl. Math. Mech. (Engl. Transl.)
72 (2), 214-225 (2008)]. |
| 6. | N. A. Bazarenko,
"Interaction of a Hollow Cylinder of Finite Length and
a Plate with a Cylindrical Cavity with a Rigid Insert,"
Prikl. Mat. Mekh.
74 (3), 126-139 (2010)
[J. Appl. Math. Mech. (Engl. Transl.)
74 (3), 323-333 (2010)]. |
| 7. | V. M. Alexandrov,
"Method of Homogeneous Solutions in Contact Problems of Elasticity for Bodies of Finite Dimensions,"
Izv. SKNTs VSch. Ser. Estestv. Nauk,
No. 4, 12-15 (1974). |
| 8. | A. N. Tsvetkov,
Method of Homogeneous Solutions in Contact Problems of Elasticity for Bodies of Noncanonical Shape,
Doctoral Dissertation in Physics and Mathematics (Rostov-on-Don, 1991)
[in Russian]. |
| 9. | V. M. Alexandrov and M. I. Chebakov,
"The Method of Dual Series in Terms of Bessel Functions in Mixed Problems
of the Theory of Elasticity for a Circular Plate,"
Prikl. Mat. Mekh.
41 (3), 486-492 (1977)
[J. Appl. Math. Mech. (Engl. Transl.)
41 (3), 492-499 (1977)]. |
| 10. | V. M. Alexandrov and D. A. Pozharskii,
Nonclassical Spatial Problems in Mechanics of Contact Interactions of Elastic Bodies
(Faktorial, Moscow, 1998)
[in Russian]. |
| 11. | D. S. Kuznetsov,
Special Functions
(Vysshaya Shkola, Moscow, 1977)
[in Russian] |
| 12. | M. Abramowitz and I. Stegun (Editors),
Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables,
(Gov. Print off., Wasgington, 1964; Nauka, Moscow, 1979). |
| 13. | M. V. Fedoryuk,
Saddle-Point Method
(Nauka, Moscow, 1977)
[in Russian]. |
| 14. | I. S. Gradshtein and I. M. Ryzhik,
Tables of Integrals, Sums, Series, and Products
(Nauka, Moscow, 1971)
[in Russian]. |
| 15. | N. N. Kalitkin,
Numerical Methods
(Nauka, Moscow, 1978)
[in Russian]. |
| 16. | H. Bateman and A. Erdélyi,
Higher Transcendental Functions
(NcGraw-Hill, New York, 1955; Nauka, Moscow, 1965). |
|
| Received |
15 November 2011 |
| Link to Fulltext |
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