# Equilibrium problems for Kirchhoff–Love plates with nonpenetration conditions for known configurations of crack edges

• Lazarev Nyurgun P., nyurgun@ngs.ru Ammosov North-Eastern Federal University, 48 Kulakovsky Street, Yakutsk 677000, Russia
• Itou Hiromichi, h-itou@rs.tus.ac.jp Tokyo University of Science, Department of Mathematics, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162–8601
Keywords: variational inequality, nonpenetration condition, crack, Kirchhoff-Love plate

### Abstract

The paper focuses on nonlinear problems describing the equilibrium of Kirchhoff–Love plates with cracks. We assume that under an appropriate load, plates have special deformations with previously known configurations of edges near a crack. Owing to this particular case, we propose two types of new nonpenetration conditions that allow us to more precisely describe the possibility of contact interaction of crack faces. These conditions correspond to two special cases of configurations of plate edges. In each case, the nonpenetration conditions are given in the form of a system of equalities and inequalities. For initial variational statements, we prove the existence and uniqueness of solutions in an appropriate Sobolev space. Assuming that the solutions are sufficiently smooth, we have found differential statements that are equivalent to the corresponding variational formulations. The relations of the obtained differential statements are compared with the well-known setting of an equilibrium problem for a Kirchhoff–Love plate with the general nonpenetration condition on crack faces.

### References

[1]
Grisvard P., Elliptic Problems in Nonsmooth Domains, Pitman, Boston (1985).

[2]
Morozov N. F., Mathematical Problems of Crack Theory [in Russian], Nauka, Moscow (1984).

[3]
Nazarov S. A. and Plamenevski B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries [in Russian], Nauka, Moscow (1991).

[4]
Ohtsuka K., “Mathematics of brittle fracture,” in: Theoretical Studies on Fracture Mechanics in Japan, pp. 99–172, Hiroshima-Denki Inst. Technol., Hiroshima (1997).

[5]
Itou H., Kovtunenko V. A., and Rajagopal K. R., “Nonlinear elasticity with limiting small strain for cracks subject to nonpenetration,” Math. Mech. Solids, 22, No. 6, 1334–1346 (2017).

[6]
Kazarinov N. A., Rudoy E. M., Slesarenko V. Y., and Shcherbakov V. V., “Mathematical and numerical simulation of equilibrium of an elastic body reinforced by a thin elastic inclusion,” Comput. Math. Math. Phys., 58, No. 5, 761–774 (2018).

[7]
Khludnev A. M., Elasticity Problems in Nonsmooth Domains [in Russian], Fizmatlit, Moscow (2010).

[8]
Khludnev A. M., “Equilibrium problem of an elastic plate with an oblique crack,” J. Appl. Mech. Tech. Phys., 38, 757–761 (1997).

[9]
Khludnev A. M., “On modeling thin inclusions in elastic bodies with a damage parameter,” Math. Mech. Solids, 24, No. 9, 2742–2753 (2019).

[10]
Khludnev A., “Thin rigid inclusions with delaminations in elastic plates,” Eur. J. Mech. A Solid, 32, 69–75 (2012).

[11]
Khludnev A. M., Faella L., and Perugia C., “Optimal control of rigidity parameters of thin inclusions in composite materials,” Z. Angew. Math. Mech., 68, No. 2, paper No. 47 (2017).

[12]
Khludnev A. M. and Kovtunenko V. A., Analysis of Cracks in Solids, WIT-Press, Southampton (2000).

[13]
Khludnev A. M. and Shcherbakov V. V., “A note on crack propagation paths inside elastic bodies,” Appl. Math. Lett., 79, No. 1, 80–84 (2018).

[14]
Kovtunenko V. A., Leont’ev A. N., and Khludnev A. M., “The problem of the equilibrium of a plate with an oblique cut,” Appl. Mech. Tech. Phys., 39, No. 2, 302–311 (1998).

[15]
Lazarev N., “Existence of an optimal size of a delaminated rigid inclusion embedded in the Kirchhoff–Love plate,” Bound. Value Probl., paper No. 180 (2015).

[16]
Lazarev N. P., Itou H., and Neustroeva N. V., “Fictitious domain method for an equilibrium problem of the Timoshenko-type plate with a crack crossing the external boundary at zero angle,” Jap. J. Ind. Appl. Math., 33, 63–80 (2016).

[17]
Lazarev N. P. and Popova T. S., “Variational problem of the equilibrium of a plate with geometrically nonlinear nonpenetration conditions on a vertical crack,” J. Math. Sci., 188, No. 4, 398–409 (2013).

[18]
Lazarev N. P., Popova T. S., and Rogerson G. A., “Optimal control of the radius of a rigid circular inclusion in inhomogeneous two-dimensional bodies with cracks,” Z. Angew. Math. Phys., 69, No. 3, paper No. 53 (2018).

[19]
Lazarev N. P. and Rudoy E. M., “Optimal size of a rigid thin stiffener reinforcing an elastic plate on the outer edge,” Z. Angew. Math. Mech., 97, No. 9, 1120–1127 (2017).

[20]
Lazarev N. and Semenova G., “An optimal size of a rigid thin stiffener reinforcing an elastic two-dimensional body on the outer edge,” J. Optim. Theory Appl., 178, No. 2, 614–626 (2018).

[21]
Nikolaeva N. A., “Method of fictitious domains for Signorini’s problem in Kirchhoff–Love theory of plates,” J. Math. Sci., 221, No. 6, 872–882 (2017).

[22]
Fichera G., Boundary Value Problems of Elasticity with Unilateral Constraints, in: Handbuch der Physik, Band 6a/2, Springer-Verl., Berlin (1972).

[23]
Rudoy E. M., “Asymptotics of the energy functional for a fourth-order mixed boundary value problem in a domain with a cut,” Sib. Math. J., 50, No. 2, 341–354 (2009).

[24]
Shcherbakov V. V., “Existence of an optimal shape of the thin rigid inclusions in the Kirchhoff–Love plate,” J. Appl. Ind. Math., 8, 97–105 (2014).

[25]
Shcherbakov V. V., “Shape optimization of rigid inclusions for elastic plates with cracks,” Z. Angew. Math. Phys., 67, No. 3, paper No. 71 (2016).

[26]
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, No. 3, 402–411 (2015).

[27]
Reddy J. N., Theory and Analysis of Elastic Plates and Shells, 2nd ed., CRC Press/Taylor and Francis, Boca Raton (2007).
How to Cite
Lazarev, N. and Itou, H. (2020) “Equilibrium problems for Kirchhoff–Love plates with nonpenetration conditions for known configurations of crack edges”, Mathematical notes of NEFU, 27(3), pp. 52-65. doi: https://doi.org/10.25587/10.25587/SVFU.2020.75.68.005.
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Mathematics