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V.I. Ostrik, "Indentation of a Punch into an Elastic Strip with Friction and Adhesion," Mech. Solids. 46 (5), 755-765 (2011)
Year 2011 Volume 46 Number 5 Pages 755-765
DOI 10.3103/S0025654411050098
Title Indentation of a Punch into an Elastic Strip with Friction and Adhesion
Author(s) V.I. Ostrik (Institute of Applied Physics, National Academy of Sciences of Ukraine, Petropavlovskaya St., 58, Sumy, 40030 Ukraine, ostrik_v@rambler.ru)
Abstract In the contact interaction between elastic bodies with friction taken into account, the contact region splits, as a rule, into adhesion and sliding regions [1]. Contact with adhesion and sliding was first considered by L. A. Galin [2] in the problem of indentation of a punch with a rectilinear foundation into an elastic half-plane, who obtained an approximate solution of this problem [2, 3]. Galin's problem was further studied in [4-9].

The paper considers an analogue of Galin's problem for a strip, namely, the problem of contact with adhesion and sliding between a punch with a rectilinear foundation and a strip one of whose sides is fixed. The Wiener-Hopf method is used to reduce the system of integral equations of the problem to an infinite system of algebraic equations. The case of a half-infinite punch was considered earlier [10]. An asymptotic solution was obtained in [11-13].
Keywords contact with adhesion and sliding, elastic strip, stresses, factorization
References
1.  K. L. Johnson, Contact Mechanics (Cambridge University Press, Cambridge, 1987; Mir, Moscow, 1989).
2.  L. A. Galin, "Indentation of a Punch in the Presence of Friction and Adhesion," Prikl. Mat. Mekh. 9 (5), 413-424 (1945) [J. Appl. Math. Mech. (Engl. Transl.)].
3.  L. A. Galin, Contact Problems in Elasticity (Gostekhizdat, Moscow, 1953) [in Russian].
4.  V. I. Mossakovskii and A. G. Biskup, "Impression of a Stamp with Friction and Adhesion Present," Dokl. Akad. Nauk SSSR 206 (5), 1068-1070 (1972) [Sov. Phys. Dokl. (Engl. Transl.) 17, 984-986 (1973)].
5.  D. A. Spence, "An Eigenvalue Problem for Elastic Contact with Finite Friction," Proc. Cambridge Phil. Soc. 73, 249-268 (1973).
6.  V. I. Mossakovskii, A. G. Biskup, and L. V. Mossakovskaya, "Further Development of the Galin Problem with Friction and Adhesion," Dokl. Akad. Nauk SSSR 271 (51), 60-64 (1983) [Sov. Phys. Dokl. (Engl. Transl.) 28, 557-559 (1983)].
7.  I. I. Mishishin, "On a Mixed Problem," in Hydroaeromechanics and Elasticity, Vol. 32 (Izd-vo DGU, Dnepropetrovsk, 1984), pp. 93-100 [in Russian].
8.  Yu. A. Antipov and N. Kh. Arutyunyan, "Contact Problems of the Theory of Elasticity with Friction and Adhesion," Prikl. Mat. Mekh. 55 (6), 1005-1017 (1991) [J. Appl. Math. Mech. (Engl. Transl.) 55 (6), 887-901 (1991)].
9.  V. I. Ostrik, "Contact Interaction between a Punch and an Elastic Half-Plane with Friction and Adhesion," Teor. Prikl. Mekh., No. 39, 94-101 (2004).
10.  V. I. Ostrik, "Impression of a Semiinfinite Stamp in an Elastic Strip with Regard for Friction and Adhesion," Mat. Met. Fiz.-Mekh. Polya 51 (1), 138-149 (2008) [J. Math. Sci. (Engl. Transl.) 160 (4), 453-469 (2009)].
11.  L. I. Manevich and A. V. Pavlenko, "To the Solution of Contact Problems of Elasticity for an Orthotropic Strip with Friction Forces Taken into Account," Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela, No. 6, 72-80 (1974) [Mech. Solids (Engl. Transl.)].
12.  S. G. Koblik and L. I. Manevich, "Contact Problem for an Orthotropic Strip with Adhesion and Sliding Areas in the Contact Region," in Hydroaeromechanics and Elasticity, Vol. 20 (Izd-vo DGU, Dnepropetrovsk, 1976), pp. 106-110 [in Russian].
13.  L. I. Manevich, A. V. Pavlenko, and S. G. Koblik, Asymptotic Methods in Theory of Elasticity of Orthotropic Solids (Vishch. Shk., Kiev-Donetsk, 1982) [in Russian].
14.  A. F. Ulitko and V. I. Ostrik, "Mixed Problem of Elasticity for a Strip on a Rigid Foundation," in Proc. Third All-Russia Conf. on Elasticity with International Participation (Novaya kniga, Rostov-on-Don, 2004), pp. 372-375 [in Russian].
15.  V. I. Ostrik, "Sliding and Smooth Contacts of Punches of Different Cross-Sections with an Elastic Strip," Mat. Met. Fiz.-Mekh. Polya 49 (4), 133-146 (2006).
16.  M. P. Ganin, "On the Fredholm Integral Equation with an Argument Difference Dependent Kernel," Izv. Vyssh. Uchebn. Zaved. Mat., No. 2, 31-43 (1963) [Sov. Math. (Izv. VUZ)].
17.  M. M. Ignatenko and V. Kh. Kirillov, "On the Solution of Some Problems of Mathematical Physics," Differents. Uravn. 5 (7), 1296-1302 (1969) [Differ. Equations (Engl. Transl.)].
18.  Yu. A. Antipov, "Exact Solution of the Problem on the Indentation of an Annular Punch into a Half-Space," Dokl. Akad. Nauk Ukr. SSR. Ser. A. Fiz. Mat. Tekhn. Nauki, No. 7, 29-33 (1987).
19.  V. I. Ostrik, "Contact of Elastic and Rigid Wedges with Friction and Adhesion Taken into Account," Mat. Met. Fiz.-Mekh. Polya 48 (3), 88-100 (2005).
20.  B. Noble, Methods Based on the Wiener-Hopf Technique for the Solution of Partial Differential Equations (Pergamon Press, London etc., 1958; Izd-vo Inostr. Lit., Moscow, 1962).
21.  G. M. Zrazhevskii and V. I. Ostrik, "Asymptotics of Canonical Solutions," Mat. Met. Fiz.-Mekh. Polya 47 (3), 69-77 (2004).
22.  M. V. Fedoryuk, Asymptotics: Integrals and Series (Nauka, Moscow, 1987) [in Russian].
Received 27 April 2009
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