 | | 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 |
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| I. A. Savichev and A. D. Chernyshov, "Application of the angular superposition method to the contact problem on the compression of an elastic cylinder," Mech. Solids. 44 (3), 463-472 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
3 |
Pages |
463-472 |
| DOI |
10.3103/S0025654409030157 |
| Title |
Application of the angular superposition method to the contact problem on the compression of an elastic cylinder |
| Author(s) |
I. A. Savichev (Voronezh State Technological Academy, pr-t Revolyutsii 19, Voronezh, 394000 Russia, kolbax@mail.ru)
A. D. Chernyshov (Voronezh State Technological Academy, pr-t Revolyutsii 19, Voronezh, 394000 Russia, chernyshovad@mail.ru) |
| Abstract |
The angular superposition method is used to construct an
approximate solution of the contact problem on the compression of an elastic cylinder by two rigid plates. The solution thus obtained has a closed-form analytic expression and can be used in the entire domain of the cylinder cross-section. We analyze the absolute error, which takes the largest value near the points of contact between the plates and the cylinder, where the boundary conditions are discontinuous. According to the von Mises criterion, when moving into the depth of the cylinder from the contact site along the symmetry axis, the second invariant J2 of the stress deviator tensor first decreases and then, after attaining a minimum, increases and attains the largest value at a small depth, which agrees with Johnson's photoelastic experiments and Dinnik's computations. We present the graphs of the displacement and normal stress distributions over the contact site, the dependence of the compressing force on the displacements of rigid plates, and the dependence of the invariant J2 on the coordinate along the symmetry axis. If 640 computation points are chosen on the cylinder boundary and the Hertz law for the normal pressure on the contact site is used, then the error in the approximate solution near the endpoint of the contact site is approximately 55%, and if the proposed two-parameter normal law is used, then the error is of the order of 4%. On the free lateral surface of the cylinder boundary, we find the critical point M*, which separates the cylinder contraction and extension parts.
The contact problems are the most difficult problems, and their solution is complicated by the discontinuous boundary conditions [1-5]. In [6], the contact problem is solved by the Fourier method, which can be used only for bodies of classical shapes. In such cases, the problem can be reduced to solving coupled integral equations [7]. The interaction between the bandage and a cylindrical body is considered in [2, 6, 7]. In [8], the possibility of using the finite element method is investigated in the case of contact problems for a differential wheel with roughness of the contacting surfaces taken into account. In [9, 10], the method of homogeneous solutions is used to consider contact problems for a finite-dimensional elastic cylinder loaded on its end surfaces. Note that only error estimates are given in the literature cited above; the absolute error over the entire domain of the elastic body is not studied, although this is one of the important characteristics of the obtained approximate solution. A sufficiently complete survey of the literature in the field of contact interactions of elastic bodies is given in [3-5].
In what follows, we propose to solve contact problems by the
angular superposition method [11]. This method can be used for bodies of nonclassical shapes, which can be multiply connected, and the friction on the contact site can be taken into account. In the present paper, as a first example of applied character, we show how this method can be used in the simplest case. The multiple connectedness and the curvilinearity of the shape of the body, as well as taking into account the friction on the boundary, do not create new essential difficulties in this method. |
| References |
| 1. | L.A. Galin,
Contact problems of Elasticity and Viscoelasticity
(Nauka, Moscow, 1980) [in Russian]. |
| 2. | V.M. Alexandrov and D.A. Pozharskii,
Nonclassical Spatial Problems in Mechanics of Contact Interactions between Elastic Bodies
(Factorial, Moscow, 1998) [in Russian]. |
| 3. | I.I. Argatov,
"Solution of the Plane Hertz Problem,"
Zh. Prikl. Mekh. Tekhn. Fiz. 42 (6), 166-176 (2001)
[J. Appl. Mech. Tech. Phys. (Engl. Transl.) 42 (6), 1064-1072 (2001)]. |
| 4. | I.I. Argatov,
"An Approximate Solution of the Axisymmetric Contact Problem for an Elastic Sphere,"
Prikl. Mat. Mekh. 69 (2), 303-314 (2005)
[J. Appl. Math. Mech. (Engl. Transl.) 69 (2), 275-286 (2005)]. |
| 5. | K.L. Johnson,
Contact Mechanics
(Univ. Press, Cambridge, 1987; Mir, Moscow, 1989). |
| 6. | O.V. Bochkova,
"Axially Symmetric Mixed Problem for a Short Circular Cylinder Loaded on the Lateral Surface,"
in Studies in Mechanics of Materials and Constructions
(Peterb. Univ. Putei Soobshch., St. Petersburg, 1988),
pp. 84-87 [in Russian]. |
| 7. | S.M. Aizikovich and I.S. Trubchik,
"Contact Problem for an Elastic Cylinder Inhomogeneous in Radius,"
in Modern Problems of Continuum Mechanics
(Rostov-on-Don, 2002), pp. 9-13 [in Russian]. |
| 8. | A.A. Ol'shevskii, L.V. Vinnik, A.M. Fridberg, and V.I. Sakalo,
"Solution of Normal Contact Problem for the Differential Wheel with the Contact Surface Roughness,"
in Dynamics, Strength, and Reliability of Transport Machines. Collection of Scientific Works
(Bryansk, 2002), pp. 109-118 [in Russian]. |
| 9. | M.I. Chebakov,
"Method of Homogeneous Solutions in the Mixed Problem
for a Finite Circular Cylinder,"
Prikl. Mat. Mekh. 43 (6), 1073-1081 (1979)
[J. Appl. Math. Mech. (Engl. Transl.) 43 (6), 1160-1169 (1979)]. |
| 10. | M.I. Chebakov,
"Certain Dynamic and Static Contact Problems of the Theory
of Elasticity for a Circular Cylinder of Finite Size,"
Prikl. Mat. Mekh. 44 (5), 923-933 (1980)
[J. Appl. Math. Mech. (Engl. Transl.) 44 (5), 651-658 (1980)]. |
| 11. | A.D. Chernyshov,
"On a Method of Solving Linear Dynamical Problems in Elasticity,"
Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 131-142 (2000)
[Mech. Solids (Engl. Transl.) 35 (5), 107-115 (2000)]. |
| 12. | A.D. Chernyshov, A.A. Dan'shin, and N.A. Chernyshov,
"Evaluation of the Error of the Method of Superpositions of One-Dimensional Solutions in Nonstationary Heat-Conduction Problems,"
Inzh.-Fiz. Zh. 77 (4), 27-30 (2004)
[J. Engng Phys. Thermophys. (Engl. Transl.) 77 (4), 707-711 (2004)]. |
| 13. | I.K. Kikoin (Editor),
Tables of Physical Quantities. Handbook
(Atomizdat, Moscow, 1976) [in Russian]. |
| 14. | D.D. Ivlev,
Mechanics of Plastic Media, Vol. 1
(Fizmatlit, Moscow, 2001) [in Russian]. |
| 15. | A.I. Lurie,
The Theory of Elasticity
(Nauka, Moscow, 1970) [in Russian]. |
| 16. | A.D. Chernyshov and O.A. Chernyshov,
"Approximation of Solutions of Boundary Value Problems for the Poisson Equation by Solutions of the Euler-Lagrange Equations,"
Differents. Uravn. 43 (3), 423-428 (2007)
[Differ. Equations (Engl. Transl.) 43 (3), 439-444 (2007)]. |
|
| Received |
28 September 2006 |
| Link to Fulltext |
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