Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Singular invariant integrals for elastic body with delaminated thin elastic inclusion

Author: A. M. Khludnev
Journal: Quart. Appl. Math. 72 (2014), 719-730
MSC (2010): Primary 35J50, 35J55, 35J40
DOI: https://doi.org/10.1090/S0033-569X-2014-01355-9
Published electronically: October 17, 2014
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Abstract: We consider an equilibrium problem for a $ 2D$ elastic body with a thin elastic inclusion. It is assumed that the inclusion is partially delaminated, therefore providing the presence of a crack. Inequality type boundary conditions are imposed at the crack faces to prevent a mutual penetration of the faces. Differentiability properties of the energy functional with respect to the crack length are analyzed. We prove an existence of the derivative and find a formula for this derivative. It is shown that the formula for the derivative can be written in the form of a singular invariant integral.

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  • [1] P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris, 1992. MR 1173209 (93h:35004)
  • [2] A.M. Khludnev and V.A. Kovtunenko, Analysis of cracks in solids, WIT Press, Southampton-Boston, 2000.
  • [3] A.M. Khludnev, Elasticity problems in non-smooth domains, Fizmatlit, Moscow, 2010.
  • [4] 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), https://doi.org/10.1002/mma.1308
  • [5] Alexander Khludnev, Contact problems for elastic bodies with rigid inclusions, Quart. Appl. Math. 70 (2012), no. 2, 269-284. MR 2953103, https://doi.org/10.1090/S0033-569X-2012-01233-3
  • [6] 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)
  • [7] A.M. Khludnev and M. Negri, Crack on the boundary of a thin elastic inclusion inside an elastic body, ZAMM 92 (2012), no. 5, 341-354.
  • [8] Victor A. Kovtunenko, Shape sensitivity of curvilinear cracks on interface to non-linear perturbations, Z. Angew. Math. Phys. 54 (2003), no. 3, 410-423. MR 2048661 (2005a:35111), https://doi.org/10.1007/s00033-003-0143-y
  • [9] 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), https://doi.org/10.1016/S0021-8928(03)00021-2
  • [10] Victor A. Kovtunenko, Nonconvex problem for crack with nonpenetration, ZAMM Z. Angew. Math. Mech. 85 (2005), no. 4, 242-251. MR 2131983 (2005k:49005), https://doi.org/10.1002/zamm.200210176
  • [11] V. A. Kozlov, V. G. Mazya, and J. Rossmann, Spectral problems associated with corner singularities of solutions to elliptic equations, Mathematical Surveys and Monographs, vol. 85, American Mathematical Society, Providence, RI, 2001. MR 1788991 (2001i:35069)
  • [12] Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387 (95h:35001)
  • [13] N.V. Neustroeva, Unilateral contact of elastic plates with rigid inclusion, Vestnik of Novosibirsk State University (math., mech., informatics) 9 (2009), no. 4, 51-64.
  • [14] T. A. Rotanova, On unilateral contact between two plates one of which has a rigid inclusion, Vestnik of Novosibirsk State University (math., mech., informatics) 11 (2011), no. 1, 87-98.
  • [15] E.M. Rudoy, Differentiation of energy functionals in the problem on a curvilinear crack with possible contact between the shores, Mechanics of Solids 42 (2007), no. 6, 935-946.
  • [16] E.M. Rudoy and A.M. Khludnev, Unilateral contact of a plate with a thin elastic obstacle, J. Appl. Industr. Math. 4 (2010), no. 3, 389-398.

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Additional Information

A. M. 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-2014-01355-9
Keywords: Thin inclusion, crack, non-linear boundary conditions, non-penetration, variational inequality, derivative of energy functional, invariant integral
Received by editor(s): November 22, 2012
Published electronically: October 17, 2014
Article copyright: © Copyright 2014 Brown University

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