Abstract
We study the problem of contact of an elastic body with a beam. The most attention is paid to describing boundary conditions on the possible contact set. Moreover, we study asymptotic properties of solutions and the energy functional as the rigidity parameters tend to infinity or the length of the beam (or the zone of possible contact) changes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P.V. Goldstein and N. M. Osipenko, “A Beam Approximation in Problems of Delamination of Thin Coatings,” Mekh. Tverd. Tela, No. 5, 154–163 (2003) [Mech. Solids. 38 (5), 127–135 (2003)].
P. V. Goldstein and N. M. Osipenko, “Delamination of Coating Under Thermoelastic Stresses (a Beam Approximation),” Vestnik Samarsk. Gos. Univ. Ser. Estestv. Nauk, No. 4, 66–83 (2007).
V. M. Entov and R. L. Salganik, “On the Beam Approximation in Crack Theory,” Izv. Akad. Nauk SSSR Ser. Mekh., No. 5, 95–102 (1965).
J. G. Williams, “On the Calculation of the Energy Release Rates for Cracked Laminates,” Internat. J. Fract. 36(2), 101–119 (1998).
A. V. Dyskin, L. N. Germanovich, and K. B. Ustinov, “Asymptotic Analysis of Crack Interaction with Free Boundary,” Internat. J. Solids Structures 37(6), 857–886 (2000).
V. V. Partsevskii, “Delaminations in Polimeric Composites: A Review,” Mekh. Tverd. Tela, No. 5, 62–94 (2003) [Mech. Solids 38 (5), 50–78 (2003)].
A. M. Khludnev and J. Sokolowski, “The Griffith Formula and the Rice-Cherepanov Integral for Crack Problems with Unilateral Conditions in Nonsmooth Domains,” European J. Appl. Math. 10(4), 379–394 (1999).
A. M. Khludnev, “Theory of Cracks with Possible Contact of Crack Faces,” Uspekhi Mekh. 3(4), 41–82 (2005).
E. M. Rudoi, “Differentiation of Energy Functionals in the Problem of a Curvilinear Crack in a Plate with Possible Contact of the Crack Faces,” Prikl. Mekh. Tekh. Fiz. 49(5), 153–168 (2008) [J. Appl. Mech. Tech. Phys. 49 (5), 832–845 (2008)].
V. A. Kovtunenko, “Shape Sensitivity of Curvilinear Cracks on Interface to Nonlinear Perturbations,” Z. Angew. Math. Phys. 54, 410–423 (2003).
E. M. Rudoi, “Differentiation of Energy Functionals in the Problem on a Curvilinear Crack with Possible Contact Between the Shores,” Mekh. Tverd. Tela, No. 6, 113–127 (2007) [Mech. Solids. 42 (6), 935–946 (2007)].
J. Sokolowski and J.-P. Zolesio, Introduction to Shape Optimization. Shape Sensitivity Analysis (Springer, Berlin, 1992).
A. M. Khludnev and J. Sokolowski, Modeling and Control in Solid Mechanics (Birkhauser, Basel, 1997).
A.M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT-Press, Southampton, 2000).
S. A. Nazarov and J. Sokolowski, “Asymptotic Analysis of Shape Functionals,” J. Math. Pures Appl. 82(2), 125–196 (2003).
A. M. Il’in, Asymptotic Expansion Matching for Solutions of Boundary Value Problems (Nauka, Moscow, 1989) [in Russian].
V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, Asymptotics of Solutions to Eliptic Bounndary Value Problems under Singular Perturbation of a Domain (Tbilis. Gos. Universitet, Tbilisi, 1981) [in Russian].
M. Van-Dyke, Perturbation Methods in Fluid Mechanics (Academic Press, New York, 1964; Mir, Moscow, 1967).
N. V. Banichuk, “Shape Definition of a Curvilinear Crack by the Perturbation Method,” Izv. Akad. Nauk SSSR, Ser. Mekh. Tverd. Tela, No. 2, 130–137 (1970).
P. V. Gol’dshtein and P. L. Salganik, “Brittle Fracture of Solids with Arbitrary Cracks,” in Progres in Mechanics of Deformable Media (Nauka, Moscow, 1975), pp. 156–171.
B. Cotterell and J. R. Rice, “Slightly Curved or Kinked Cracks,” Internat. J. Fract. 16(2), 155–169 (1980).
M. Amestoy and J. B. Leblond, “Crack Paths in Plane Situations. II. Detailed Form of the Expansion of the Stress Intensity Factors,” Internat. J. Solids Structures 29(4), 465–501 (1992).
J. B. Leblond, “Crack Paths in Three-Dimensional Elastic Solids. I. Two-Term Expansion of the Stress Intensity Factors-Application to Cracks Path Stability in Hydraulic Fracturing,” Internat. J. Solids Structures 36(1), 79–103 (1999).
D. Leguillon, “Asymptotic and Numerical Analysis of a Crack Branching in Non-Isotropic Materials,” European J. Mech. A/Solids 12(1), 33–51 (1993).
H. Gao and C.-H. Chiu, “Slightly Curved or Kinked Cracks in Anisotropic Elastic Solids,” Internat. J. Solids Structures 29(8), 947–972 (1992).
P. A. Martin, “Perturbed Cracks in Two-Dimensions: An Integral-Equation Approach,” Internat. J. Fract. 104(4), 317–327 (2000).
S. A. Nazarov and M. Specovius-Neugebauer, “Use of the Energy Criterion of Fracture to Determine the Shape of a Slightly Curved Crack,” Prikl. Mekh. Tekh. Fiz. 47(5), 119–130 (2006) [J. Appl. Mech. Tech. Phys. 47 (5), 714–723 (2006)].
S. A. Nazarov, “A Singorini Problem with Friction for Thin Elastic Solids,” Trudy Moskov. Mat. Obshch. 56, 262–302 (1993).
V. I. Feodos’ev, Resistance of Materials (Bauman Moskov. Gos. Tekh. Univ., Moscow, 2000) [in Russian].
F. Ziegler, Mechanics of Solids and Fluids (Springer, New York, 1998; Regulyarnaya i Khaoticheskaya Dinamika, Izhevsk, 2002).
A. S. Kravchuk, Variational and Semivariational Inequalities in Mechanics (Izd. MGAPI, Moscow, 1997) [in Russian].
G. Fichera, Boundary Value Problems of Elasticity with Unilateral Constraints: Handbuch der Physik, Band 6a/2 (Springer, Berlin, 1972; Mir, Moscow, 1974).
G. Duvaut and J.-L. Lions, Les inequations en mechanique et en physique (Dunod, Paris, 1972; Nauka, Moscow, 1980).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.M. Rudoi, A.M. Khludnev, 2009, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2009, Vol. XII, No. 2, pp. 120–130.
Rights and permissions
About this article
Cite this article
Rudoi, E.M., Khludnev, A.M. Unilateral contact of a plate with a thin elastic obstacle. J. Appl. Ind. Math. 4, 389–398 (2010). https://doi.org/10.1134/S1990478910030117
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478910030117