Contact problems for elastic bodies with rigid inclusions
Author:
Alexander Khludnev
Journal:
Quart. Appl. Math. 70 (2012), 269284
MSC (2010):
Primary 35J20, 74E30
Published electronically:
February 3, 2012
MathSciNet review:
2953103
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Abstract 
References 
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Additional Information
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 nondeformable punch. We propose correct problem formulations with inequality type boundary conditions of a nonlocal type describing a mutual nonpenetration 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.
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 2.
Alexander
Khludnev and Günter
Leugering, On elastic bodies with thin rigid inclusions and
cracks, Math. Methods Appl. Sci. 33 (2010),
no. 16, 1955–1967. MR 2744613
(2011k:74103), 10.1002/mma.1308
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Khludnev A.M., Kovtunenko V.A. Analysis of cracks in solids. SouthamptonBoston, WIT Press, 2000.
 4.
Alexander
Khludnev and Atusi
Tani, Unilateral contact problem for two inclined elastic
bodies, Eur. J. Mech. A Solids 27 (2008), no. 3,
365–377. MR 2407924
(2009d:74066), 10.1016/j.euromechsol.2007.08.001
 5.
A.
M. Khludnev, A.
A. Novotny, J.
Sokołowski, and A.
Żochowski, Shape and topology sensitivity analysis for
cracks in elastic bodies on boundaries of rigid inclusions, J. Mech.
Phys. Solids 57 (2009), no. 10, 1718–1732. MR 2567570
(2010k:49090), 10.1016/j.jmps.2009.07.003
 6.
A.
M. Khludnev and J.
Sokolowski, Modelling and control in solid mechanics,
International Series of Numerical Mathematics, vol. 122,
Birkhäuser Verlag, Basel, 1997. MR 1433133
(98c:93004)
 7.
G.
Lo&ibreve;gering and A.
M. Khludnev, On the equilibrium of elastic bodies with thin rigid
inclusions, Dokl. Akad. Nauk 430 (2010), no. 1,
47–50 (Russian). MR
2668827
 8.
Alexander
M. Khludnev and Jan
Sokołowski, Smooth domain method for crack problems,
Quart. Appl. Math. 62 (2004), no. 3, 401–422.
MR
2086037 (2005d:35096)
 9.
Alexander
Khludnev and Atusi
Tani, Overlapping domain problems in the
crack theory with possible contact between crack faces, Quart. Appl. Math. 66 (2008), no. 3, 423–435. MR 2445521
(2009i:74084), 10.1090/S0033569X08011187
 10.
V.
A. Kovtunenko, Invariant energy integrals for a nonlinear crack
problem with possible contact of the crack faces, Prikl. Mat. Mekh.
67 (2003), no. 1, 109–123 (Russian, with
Russian summary); English transl., J. Appl. Math. Mech.
67 (2003), no. 1, 99–110. MR 1997626
(2004e:74074), 10.1016/S00218928(03)000212
 11.
Mallick P. K. Fiberreinforced 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. 6075.
 13.
Neustroeva N.V. Unilateral contact of elastic plates with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2009, N4, pp. 5164.
 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. 944960.
 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. 113127.
 16.
Rudoĭ E.M. Griffith formula and RiceCherepanov integral for a plate with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2010, v.10, N 2, pp. 98117.
 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. 252263.
 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. 719729.
 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. 8798.
 1.
 Fichera G. Boundary value problems of elasticity with unilateral constraints. In: Handbuch der Physik, Band 6a/2, SpringerVerlag, 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. 19551967. MR 2744613
 3.
 Khludnev A.M., Kovtunenko V.A. Analysis of cracks in solids. SouthamptonBoston, 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. 17181732. MR 2567570
 6.
 Khludnev A.M., Sokolowski J. Modelling and control in solid mechanics. BaselBostonBerlin, 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. 4750. MR 2668827
 8.
 Khludnev A.M., Sokolowski J. Smooth domain method for crack problems. Quart. Appl. Math., 2004, v. 62, N3, pp. 401422. 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. 423435. 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. 109123. MR 1997626 (2004e:74074)
 11.
 Mallick P. K. Fiberreinforced 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. 6075.
 13.
 Neustroeva N.V. Unilateral contact of elastic plates with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2009, N4, pp. 5164.
 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. 944960.
 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. 113127.
 16.
 Rudoĭ E.M. Griffith formula and RiceCherepanov integral for a plate with a rigid inclusion. Vestnik of Novosibirsk State University (math., mech., informatics), 2010, v.10, N 2, pp. 98117.
 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. 252263.
 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. 719729.
 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. 8798.
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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:
http://dx.doi.org/10.1090/S0033569X2012012333
PII:
S 0033569X(2012)012333
Keywords:
Rigid inclusion,
nonpenetration 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.
