Geodesic
Equilibrium problem for a Kirchhoff--Love plate contacting by the side edge and the bottom boundary
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 3, pp. 355-364.

Voir la notice de l'article provenant de la source Math-Net.Ru

A new model of a Kirchhoff–Love plate which is in contact with a rigid obstacle of a certain given configuration is proposed in the paper. The plate is in contact either on the side edge or on the bottom surface. A corresponding variational problem is formulated as a minimization problem for an energy functional over a non-convex set of admissible displacements subject to a non-penetration condition. The inequality type non-penetration condition is given as a system of inequalities that describe two cases of possible contacts of the plate and the rigid obstacle. Namely, these two cases correspond to different types of contacts by the plate side edge and by the plate bottom. The solvability of the problem is established. In particular case, when contact zone is known equivalent differential statement is obtained under the assumption of additional regularity for the solution of the variational problem.
Keywords: contact problem, non-penetration condition, non-convex set, variational problem. 
@article{JSFU_2024_17_3_a6,
     author = {Nyurgun P. Lazarev and Evgeny M. Rudoy and Djulustan Ya. Nikiforov},
     title = {Equilibrium problem for a {Kirchhoff--Love} plate contacting by the side edge and the bottom boundary},
     journal = {\v{Z}urnal Sibirskogo federalʹnogo universiteta. Matematika i fizika},
     pages = {355--364},
     publisher = {mathdoc},
     volume = {17},
     number = {3},
     year = {2024},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JSFU_2024_17_3_a6/}
}
TY  - JOUR
AU  - Nyurgun P. Lazarev
AU  - Evgeny M. Rudoy
AU  - Djulustan Ya. Nikiforov
TI  - Equilibrium problem for a Kirchhoff--Love plate contacting by the side edge and the bottom boundary
JO  - Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
PY  - 2024
SP  - 355
EP  - 364
VL  - 17
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/JSFU_2024_17_3_a6/
LA  - en
ID  - JSFU_2024_17_3_a6
ER  - 
%0 Journal Article
%A Nyurgun P. Lazarev
%A Evgeny M. Rudoy
%A Djulustan Ya. Nikiforov
%T Equilibrium problem for a Kirchhoff--Love plate contacting by the side edge and the bottom boundary
%J Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika
%D 2024
%P 355-364
%V 17
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/JSFU_2024_17_3_a6/
%G en
%F JSFU_2024_17_3_a6
Nyurgun P. Lazarev; Evgeny M. Rudoy; Djulustan Ya. Nikiforov. Equilibrium problem for a Kirchhoff--Love plate contacting by the side edge and the bottom boundary. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 17 (2024) no. 3, pp. 355-364. http://geodesic.mathdoc.fr/item/JSFU_2024_17_3_a6/

[1] A.Signorini, “Questioni di elasticita non linearizzata e semilinearizzata”, Rend. Mat. e Appl., 18:5 (1959), 95–139 | MR | Zbl

[2] G.Fichera, “Existence Theorems in Elasticity”, Handbuch der Physik, VIa/2, ed. C. Truesdell, Springer, Berlin, 1972 | MR

[3] G.Dal Maso, G.Paderni, “Variational inequalities for the biharmonic operator with variable obstacles”, Annali di Matematica pura ed applicata, 153 (1988), 203–227 | DOI | MR | Zbl

[4] N.P.Lazarev, H.Itou, N.V.Neustroeva, “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 | DOI | MR | Zbl

[5] A.M.Khludnev, V.A.Kovtunenko, Analysis of Cracks in Solids, WIT-Press, Southampton, 2000

[6] A.I.Furtsev, “On contact between a thin obstacle and a plate containing a thin inclusion”, J. Math. Sci., 237:4 (2019), 530–545 | DOI | MR

[7] A.I.Furtsev, “The unilateral contact problem for a Timoshenko plate and a thin elastic obstacle”, Sib. Electron. Math. Rep., 17 (2020), 364–379 (in Russian) | DOI | MR | Zbl

[8] E.M.Rudoi, A.M.Khludnev, “Unilateral contact of a plate with a thin elastic obstacle”, J. Appl. Ind. Math., 4 (2010), 389–398 | DOI | MR

[9] A.M.Khludnev, K.Hoffmann, N.D.Botkin, “The variational contact problem for elastic objects of different dimensions”, Siberian Math. J., 47:3 (2006), 584–593 | DOI | MR | Zbl

[10] V.D.Stepanov, A.M.Khludnev, “The method of fictitious domains in the Signorini problem”, Siberian Math. J., 44:6 (2003), 1061–1074 | DOI | MR | Zbl

[11] N.A.Nikolaeva, “Method of fictitious domains for Signorini's problem in Kirchhoff–Love theory of plates”, J. Math. Sci., 221:6 (2017), 872–882 | DOI | MR

[12] N.P.Lazarev, V.V.Everstov, N.A.Romanova, “Fictitious domain method for equilibrium problems of the Kirchhoff-Love plates with nonpenetration conditions for known configurations of plate edges”, J. Sib. Fed. Univ. Math. Phys., 12:6 (2019), 1–13 | DOI | MR

[13] N.P.Lazarev, V.A.Kovtunenko, “Signorini-type problems over non-convex sets for composite bodies contacting by sharp edges of rigid inclusions”, Mathematics, 10:2 (2022), 250 | DOI | MR

[14] N.P.Lazarev, E.D.Fedotov, “Three-dimensional Signorini-type problem for composite bodies contacting with sharp edges of rigid inclusions”, Chelyab. Fiz.-Mat. Zh., 7:4 (2022), 412–423 | DOI | MR | Zbl

[15] C.Baiocchi, A.Capelo, Variational and quasivariational inequalities. Applications to free boundary problems, Wiley, Chichester, 1984 | MR | Zbl

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