Contact problems for elastic bodies with rigid inclusions

Author:
Alexander Khludnev

Journal:
Quart. Appl. Math. **70** (2012), 269-284

MSC (2010):
Primary 35J20, 74E30

DOI:
https://doi.org/10.1090/S0033-569X-2012-01233-3

Published electronically:
February 3, 2012

MathSciNet review:
2953103

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Abstract: This paper is concerned with a new type of free boundary problems for elastic bodies with a rigid inclusion being in contact with another rigid inclusion or with a non-deformable punch. We propose correct problem formulations with inequality type boundary conditions of a non-local type describing a mutual non-penetration between surfaces. Solution existence is proved for different types of inclusions and different geometries. Qualitative properties of solutions are analyzed provided that rigidity parameters are changed.

**1.**Fichera G. Boundary value problems of elasticity with unilateral constraints. In: Handbuch der Physik, Band 6a/2, Springer-Verlag, 1972.**2.**Khludnev A.M., Leugering G. On elastic bodies with thin rigid inclusions and cracks, Math. Meth. Appl. Sciences, 2010, v. 33, N16, pp. 1955-1967. MR**2744613****3.**Khludnev A.M., Kovtunenko V.A. Analysis of cracks in solids. Southampton-Boston, WIT Press, 2000.**4.**Khludnev A.M., Tani A. Unilateral contact problems for two inclined elastic bodies. European Journal of Mechanics A/Solids, 2008, v.27, N3, pp. 365 - 377. MR**2407924 (2009d:74066)****5.**Khludnev A.M., Novotny A.A., Sokolowski J., Zochowski A. Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions. Journal of the Mechanics and Physics of Solids, 2009, v. 57, N 10, pp. 1718-1732. MR**2567570****6.**Khludnev A.M., Sokolowski J. Modelling and control in solid mechanics. Basel-Boston-Berlin, Birkhäuser, 1997. MR**1433133 (98c:93004)****7.**Khludnev A.M., Leugering G. On the equilibrium of elastic bodies with thin rigid inclusions. Doklady Physics, 2010, v.430, N1, pp. 47-50. MR**2668827****8.**Khludnev A.M., Sokolowski J. Smooth domain method for crack problems. Quart. Appl. Math., 2004, v. 62, N3, pp. 401-422. MR**2086037 (2005d:35096)****9.**Khludnev A.M., Tani A. Overlapping domain problems in the crack theory with possible contact between crack faces. Quart. Appl. Math., 2008, v. 66, N 3, pp. 423-435. MR**2445521 (2009i:74084)****10.**Kovtunenko V.A. Invariant integrals in nonlinear problem for a crack with possible contact between crack faces. J. Appl. Math. Mech., 2003, v. 67, N. 1, pp. 109-123. MR**1997626 (2004e:74074)****11.**Mallick P. K. Fiber-reinforced composites. Materials, manufacturing, and design, Marcel Dekker, Inc., 1993.**12.**Neustroeva N.V. Contact problem for elastic bodies of different dimensions. Vestnik of Novosibirsk State University (math., mech., informatics), 2008, v. 8, N4, pp. 60-75.**13.**Neustroeva N.V. Unilateral contact of elastic plates with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2009, N4, pp. 51-64.**14.**Prechtel M., Leugering G., Steinmann P., Stingl M. Towards optimization of crack resistance of composite materials by adjusting of fiber shapes, Engineering fracture mechanics, 2011, v.78, N 6, pp. 944-960.**15.**Rudoĭ E.M. Differentiation of energy functionals in the problem of a curvilinear crack with possible contact between crack faces. Izvestiya RAN, Solid mechanics, 2007, N 6, pp. 113-127.**16.**Rudoĭ E.M. Griffith formula and Rice-Cherepanov integral for a plate with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2010, v.10, N 2, pp. 98-117.**17.**Rudoĭ E.M. Asymptotic behavior of energy functional for a three dimensional body with a rigid inclusion and a crack. J. Appl. Mech. Techn. Phys., 2011, v.52, N 2, pp. 252-263.**18.**Rudoĭ E.M. Asymptotics of energy functional for an elastic body with a rigid inclusion. 2D problem. J. Appl. Math. Mech., 2011, v. 75, N 5, pp. 719-729.**19.**Rotanova T.A. Unilateral contact problem for two plates with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2011, v. 11, N 1, pp. 87-98.

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Additional Information

**Alexander Khludnev**

Affiliation:
Lavrentyev Institute of Hydrodynamics of the Russian Academy of Sciences, and Novosibirsk State University, Novosibirsk 630090, Russia

Email:
khlud@hydro.nsc.ru

DOI:
https://doi.org/10.1090/S0033-569X-2012-01233-3

Keywords:
Rigid inclusion,
non-penetration condition,
delamination,
crack

Received by editor(s):
May 24, 2010

Published electronically:
February 3, 2012

Article copyright:
© Copyright 2012
Brown University

The copyright for this article reverts to public domain 28 years after publication.