Equilibrium problem for a Timoshenko plate contacting by the side and face surfaces

N. P. Lazarev ; D. Ya. Nikiforov ; N. A. Romanova

Čelâbinskij fiziko-matematičeskij žurnal, Tome 8 (2023) no. 4, pp. 528-541.

Résumé

A new model for a Timoshenko plate contacting by the side surface or the edge of the bottom surface (with respect to the chosen coordinate system) with a rigid obstacle of a given configuration is justified. The non-deformable obstacle is defined by a cylindrical surface, the generators of which are perpendicular to the middle plane of the plate, as well as by a part of the plane that is parallel to the middle plane of the plate. A corresponding variational problem is formulated as a minimization of an energy functional over a non-convex set of admissible displacements. The set of admissible displacements is defined taking into account a condition of fixing and a nonpenetration condition. The nonpenetration condition is given as a system of inequalities describing 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 edge of the plate bottom surface. The solvability of the problem is established. In particular case, when contact zones is previously known, an equivalent differential statement is found under the assumption of additional regularity for the solution to the variational problem.

Détail

Keywords: contact problem, non-convex set, variational inequality, nonpenetration condition.

Bibliographie

[1] Signorini A., “Questioni di elasticita non linearizzata e semilinearizzata”, Rendiconti di Matematica e delle sue Applicazioni, 18:5 (1959), 95–139 (In Italian.) | MR | Zbl

[2] Fichera G., “Boundary Value Problems of Elasticity with Unilateral Constraints”, Handbook der Physik, Springer-Verlag, Band 6a/2. Berlin, Heidelberg, New York, 1972 | MR

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

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

[5] Rademacher A., Rosin K., “Adaptive optimal control of Signorini's problem”, Computational Optimization and Applications, 70 (2018), 531–569 | DOI | MR | Zbl

[6] Lazarev N., Rudoy E., “Optimal location of a finite set of rigid inclusions in contact problems for inhomogeneous two-dimensional bodies”, Journal of Computational and Applied Mathematics, 403 (2022), 113710 | DOI | MR | Zbl

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

[8] Rudoi E.M., Khludnev A.M., “Unilateral contact of a plate with a thin elastic obstacle”, Journal of Applied and Industrial Mathematics, 4:3 (2010), 389–398 | DOI | MR | MR | Zbl

[9] Furtsev A.I., “On contact between a thin obstacle and a plate containing a thin inclusion”, Journal of Mathematical Sciences, 237:4 (2019), 530–545 | DOI | MR | Zbl

[10] Furtsev A.I., “The unilateral contact problem for a Timoshenko plate and a thin elastic obstacle”, Siberian Electronic Mathematical Reports, 17 (2020), 364–379 | MR | Zbl

[11] Pyatkina E. V., “A Contact of two elastic plates connected along a thin rigid inclusion”, Siberian Electronic Mathematical Reports, 17 (2020), 1797–1815 | MR | Zbl

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

[13] 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”, Japan Journal of Industrial and Applied Mathematics, 33:1 (2016), 63–80 | DOI | MR | Zbl

[14] Nikolaeva N.A., “Method of fictitious domains for Signorini's problem in Kirchhoff–Love theory of plates”, Journal of Mathematical Sciences, 221:6 (2017), 872–882 (In Russ.) | Zbl

[15] Lazarev N.P., Everstov V.V., Romanova N.A., “Fictitious domain method for equilibrium problems of the Kirchhoff-Love plates with nonpenetration conditions for known configurations of plate edges”, Journal of Siberian Federal University. Mathematics and Physics, 12:6 (2019), 674–686 | DOI | MR

[16] Knees D., Schroder A., “Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints”, Mathematical Methods in the Applied Sciences, 35 (2012), 1859–1884 | DOI | MR | Zbl

[17] Rudoy E. M., Shcherbakov V. V., “Domain decomposition method for a membrane with a delaminated thin rigid inclusion”, Sib. elektron. mat. izv., 13 (2016), 395–410 | MR | Zbl

[18] Itou H., Kovtunenko V. A., Rajagopal K. R., “Nonlinear elasticity with limiting small strain for cracks subject to non-penetration”, Mathematics and Mechanics of Solids, 22:6 (2017), 1334–1346 | DOI | MR | Zbl

[19] Itou H., Kovtunenko V. A., Rajagopal K. R., “Well-posedness of the problem of non-penetrating cracks in elastic bodies whose material moduli depend on the mean normal stress”, International Journal of Engineering Science, 136 (2019), 17–25 | DOI | MR | Zbl

[20] Furtsev A., Itou H., Rudoy E., “Modeling of bonded elastic structures by a variational method: Theoretical analysis and numerical simulation”, International Journal of Solids and Structures, 182–183 (2020), 100–111 | DOI

[21] Lazarev N.P., Semenova G.M., “Equilibrium problem for a Timoshenko plate with a geometrically nonlinear condition of nonpenetration for a vertical crack”, Journal of Applied and Industrial Mathematics, 14:3 (2020), 532–540 | DOI | MR | Zbl

[22] Khludnev A., “T-shape inclusion in elastic body with a damage parameter”, Journal of Computational and Applied Mathematics, 393 (2021), 113540 | DOI | MR | Zbl

[23] Khludnev A., Fankina I., “Equilibrium problem for elastic plate with thin rigid inclusion crossing an external boundary”, Zeitschrift für Angewandte Mathematik und Physik, 72:3 (2021), 121 | DOI | MR | Zbl

[24] Kovtunenko V. A., “Poroelastic medium with non-penetrating crack driven by hydraulic fracture: Variational inequality and its semidiscretization”, Journal of Computational and Applied Mathematics, 405 (2022), 113953 | DOI | MR | Zbl

[25] Kovtunenko V. A., Itou H., Khludnev A. M., et al., “Non-smooth variational problems and applications”, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 380 (2022), 20210364 | DOI | MR

[26] Lazarev N.P., Fedotov E.D., “Three-dimensional Signorini-type problem for composite bodies contacting with sharp edges of rigid inclusions”, Chelyabinsk Physical and Mathematical Journal, 7:4 (2022), 412–423 | DOI | MR | Zbl

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

[28] Lazarev N.P., “Equilibrium problem for a Timoshenko plate contacting with an inclined obstacle”, Mathematical Notes of NEFU, 21:3 (2014), 68–75 | MR | Zbl | Zbl

[29] Lazarev N.P., “Fictitious domain method in the equilibrium problem for a Timoshenko-type plate contacting with a rigid obstacle”, Journal of Mathematical Sciences, 203:4 (2014), 527–539 (In Russ.) | Zbl

[30] Pelekh B.L., Theory of shells with finite shear modulus, Naukova Dumka, Kiev, 1973 (In Russ.)

[31] Lazarev N.P., “An equilibrium problem for a Timoshenko plate with a through crack”, Journal of Applied and Industrial Mathematics, 14:4 (2011), 32–43 | MR | Zbl

[32] Baiocchi C., Capello A., Variational and Quasivariational Inequalities: Application to Free Boundary Problems, Wiley, New York, 1984 | MR