# On the solution regularity of an equilibrium problem for the Timoshenko plate having an inclined crack

• Lazarev Nyurgun P., nyurgun@ngs.ru M. K. Ammosov North-Eastern Federal University, Institute of Mathematics, 48 Kulakovsky Street, Yakutsk 677891, Russia
• Itou Hiromichi, h-itou@rs.tus.ac.jp Tokyo University of Science, Department of Mathematics, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan
• Sivtsev Petr V., sivkapetr@mail.ru M. K. Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677000, Russia
• Tikhonova Irina M., IrinaMikh3007@mail.ru M. K. Ammosov North-Eastern Federal University, 42 Kulakovsky Street, Yakutsk 677000, Russia
Keywords: variational inequality, the Timoshenko plate, crack, nonpenetration condition, solution regularity

### Abstract

The equilibrium problem for an transversely isotropic elastic plate (Timoshenko model) with an inclined crack is studied. It is supposed that the crack does not touch the external boundary. For initial state, we assume that opposite crack faces are in contact with each other on a frictionless crack surface. Herewith, the crack is described with the use of a surface satisfying certain assumptions. On the crack curve defining the crack in the middle plane, we impose a nonlinear boundary condition as an inequality describing the nonpenetration of the opposite crack faces. It is assumed that on the exterior boundary of the cracked elastic plate the homogeneous Dirichlet boundary conditions are prescribed. We establish additional smoothness of the solution in comparison with that given in the variational statement. We prove that the solution functions are infinitely smooth under additional assumptions on the function of external loads and the functions of displacements near the curve describing the inclined crack.

### References

 [1] Cherepanov G. P., Mechanics of Brittle Fracture, McGraw-Hill, New York, 1979 [2] Rabotnov Yu. N., Mechanics of a Deformable Rigid Body, Nauka, Moscow, 1988 [3] Levin V. A., Morozov E. M., Matvienko Yu. G., Selected Nonlinear Problems in Mechanics of Fracture, Fizmatlit, Moscow, 2004 [4] Slepyan L. I., Mechanics of Cracks, Sudostroenie, Leningrad, 1981 [5] Morozov N. F., Mathematical problems of the theory of cracks, Nauka, Moscow, 1984 [6] Khludnev A. M., “Equilibrium problem of an elastic plate with an oblique crack”, J. Appl. Mech. Tech. Phys., 38:5 (1997), 757–761 [7] Kovtunenko V. A., Leont'ev A. N., Khludnev A. M., “Equilibrium problem of a plate with an oblique cut”, J. Appl. Mech. Tech. Phys., 39:2 (1998), 302–311 [8] Khludnev A. M., Elasticity Problems in Nonsmooth Domains, Fizmatlit, Moscow, 2010 [9] Lazarev N. P., Rudoy E. M., “Optimal size of a rigid thin stiffener reinforcing an elastic plate on the outer edge”, Z. Angew. Math. Mech., 97:9 (2017), 1120–1127 [10] Lazarev N. P., “The equilibrium problem for a Timoshenko-type shallow shell containing a through crac”, J. Appl. Ind. Math., 7:1 (2013), 58–69 [11] Lazarev N. P., “An equilibrium problem for a Timoshenko plate with a through crack”, Sib. Zh. Ind. Mat., 14:4 (2011), 32–43 [12] Rudoy E. M., “Griffith's formula and Cherepanov–Rice's integral for a plate with a rigid inclusion and a crack”, J. Math. Sci., 186:3 (2012), 511-529 [13] Khludnev A. M., “Thin rigid inclusions with delaminations in elastic plates”, Europ. J. Mech. A Solids, 32:1 (2012), 69–75 [14] 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., 9:3 (2015), 402–411 [15] Shcherbakov V., “Shape optimization of rigid inclusions for elastic plates with cracks”, Z. Angew. Math. Phys., 67:3 (2016), 71 [16] Lazarev N. P., “Equilibrium problem for a Timoshenko plate with an oblique crack”, J. Appl. Mech. Tech. Phys., 54:4 (2013), 662–671 [17] Lazarev N. P., “An iterative penalty method for a nonlinear problem of equilibrium of a Timoshenko-type plate with a crack”, Numer. Anal. Appl., 4:4 (2011), 309–318 [18] Lazarev N. P., Rudoy E. M., “Shape sensitivity analysis of Timoshenko's plate with a crack under the nonpenetration condition”, Z. Angew. Math. Mech., 94:9 (2014), 730–739 [19] Lazarev N. P., Itou H., 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”, Jpn. J. Ind. Appl. Math., 33:1 (2016), 63–80 [20] Pelekh B. L., Shell Theory with Finite Shear Stiffness, Naukova Dumka, Kiev, 1973 [21] Mikhailov V. P., Partial Differential Equations, Mir, Moscow, 1978 [22] Lions J. L., Magenes E., Nonhomogeneous Boundary Value Problems and Applications, v. 1, Springer-Verlag, Berlin, New York, 1972 [23] Khludnev A. M., Sokolowski J., Modelling and control in solid mechanics, Birkhäuser-Verl., Basel, Boston, Berlin, 1997
How to Cite
Lazarev, N., Itou, H., Sivtsev, P. and Tikhonova, I. (&nbsp;) “On the solution regularity of an equilibrium problem for the Timoshenko plate having an inclined crack”, Mathematical notes of NEFU, 25(1), pp. 38-49. doi: https://doi.org/10.25587/SVFU.2018.1.12767.
Issue
Section
Mathematics