Abstract
The problem of equilibrium of a Kirchhoff–Love elastic plate with an inclined crack on the boundary of a rigid inclusion is considered. On the crack faces, the nonpenetration conditions are set in the form of equations and inequalities. On the boundary of the rigid inclusion, an identity holds that describes the impact of external forces on the rigid part of the plate. Some variational formulation of the problem is studied, and also an equivalent boundary value problem is formulated. The passage to the limit is considered for a family of problems about a plate with an inclined crack on the boundary of an elastic inclusion as the parameter of inclusion rigidity tends to infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. M. Khludnev, “Problem of a Crack on the Boundary of a Rigid Inclusion in an Elastic Plate,” Izv. Ross. Akad. Nauk.Mekh. Tverd. Tela No. 5, 98–110 (2010) [Mechanics of Solids 45 (5), 733–742 (2010)].
G. Leugering and A. M. Khludnev, “On the Equilibriumof Elastic Bodies Containing Thin Rigid Inclusions,” Dokl. Akad. Nauk, Ross. Akad. Nauk 430 (1), 47–50 (2010) [Doklady Phys. 55 (1), 18–22 (2010)].
V. A. Kovtunenko, A. N. Leont’ev, and A. M. Khludnev, “A Problem of Equilibrium of a Plate with an Inclined Cut,” Prikl. Melh. i Tekhn. Fiz. 39 (2), 164–174 (1998).
A. M. Khludnev, “An Equilibrium Problem Concerned a Plate with an Inclined Crack,” Prikl. Melh. i Tekhn. Fiz. 38 (5), 117–121 (1997).
N. P. Lazarev, “An Equilibrium Problem Concerned a Timoshenko Plate with an Incline Crack,” Prikl. Melh. i Tekhn. Fiz. 54 (4), 171–181 (2013).
A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids (WIT Press, Southampton, Boston, 2000).
A. M. Khludnev, Problems of Elasticity Theory in Nonsmooth Domains (Fizmatlit, Moscow, 2010) [in Russian].
N. V. Neustroeva, “A Rigid Inclusion in the Contact Problem for Elastic Plates,” Sibirsk. Zh. Indust. Mat. 12 (4), 92–105 (2009) [J. Appl. Indust.Math. 4 (4), 526–538 (2010)].
N. V. Neustroeva, “One-Face Contact of Elastic Plates with a Rigid Inclusion,” Vestnik Novosib. Gos. Univ. Ser. Mat. Mekh. Informat. 9 (4), 51–64 (2009).
E. M. Rudoi, “The Griffith Formula and the Cherepanov-Rice Integral for a Plate with a Rigid Inclusion and a Crack,” Vestnik Novosib. Gos. Univ. Ser. Mat. Mekh. Informat. 10 (2), 98–117 (2010).
T. A. Rotanova, “On Statement and Solvability of a Contact Problem of Two Plates with Rigid Inclusions,” Sibirsk. Zh. Indust. Mat. 15 (2), 107–118 (2012).
N. P. Lazarev, “An Equilibrium Problem for the Timoshenko-Type Plate Containing a Crack on the Boundary of a Rigid Inclusion,” J. Siberian Federal Univ. Math. Phys. 6 (1), 53–62 (2013).
M. Negri and A. M. Khludnev, “About Equilibrium of Elastic Bodies with Thin Elastic Inclusions,” Dokl. Akad. Nauk 443 (1), 53–62 (2012).
N. P. Lazareva, “An Equilibrium Problem of a Timoshenko Plate with a Crack along a Thin Rigid Inclusion,” Vestnik Udmurt.Gos.Univ. Ser. Mat.Mekh.Comput. Sci. No. 1, 32–45 (2014).
T. S. Popova, “An Equilibrium Problem for a Viscoelastic Body with a Thin Rigid Inclusion,” Mat. Zametki Severo-Vostochn. Federal. Univ. 21 (1), 47–55 (2014).
A. Morassi and E. Rosset, “Detecting Rigid Inclusions, or Cavities, in an Elastic Body,” J. Elasticity 73 (1–3), 101–126 (2003).
A. S. Volmir, Nonlinear Dynamics of Plates and Shells (Nauka, Moscow, 1972) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N.V. Neustroeva, 2015, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2015, Vol. XVIII, No. 2, pp. 74–84.
Rights and permissions
About this article
Cite this article
Neustroeva, N.V. An equilibrium problem for an elastic plate with an inclined crack on the boundary of a rigid inclusion. J. Appl. Ind. Math. 9, 402–411 (2015). https://doi.org/10.1134/S1990478915030114
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1990478915030114