Equilibrium problem for a Timoshenko plate contacting by its lateral surface along a strip of a given width

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

Čelâbinskij fiziko-matematičeskij žurnal, Tome 9 (2024) no. 4, pp. 596-608.

Résumé

A new model of a transversally isotropic Timoshenko plate is justified, which may come into contact by its side surface with a non-deformable obstacle along a strip of a given width. The non-deformable obstacle restricts displacements and rotation angles of the plate along the outer side edge. The obstacle is defined by a cylindrical surface, the generatrices of which are perpendicular to the middle plane of the plate. A problem is formulated in variational form. A set of admissible displacements is specified in a suitable Sobolev space in the framework of a clamping condition and a non-penetration condition. The non-penetration condition is given as a system of two inequalities. The existence and uniqueness of a solution to the problem is proven. An equivalent differential formulation is found under the assumption of additional regularity of the solution to the variational problem. A qualitative connection has been established between the proposed model and a previously studied problem in which the plate is in contact over the entire side surface.

Détail

Keywords: contact problem, limit passage, 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.)

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

[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

[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

[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

[7] Lazarev N. P., Sharin E. F., Efimova E. S., “Equilibrium problem for an inhomogeneous kirchhoff–love plate contacting with a partially delaminated inclusion”, Lobachevskii Journal of Mathematics, 44:10 (2023), 4127–4134

[8] 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

[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

[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

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

[12] 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

[13] 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

[14] 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

[15] 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

[16] 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

[17] 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

[18] Furtsev A. I., “Problem of equilibrium for hyperelastic body with rigid inclusion and non-penetrating crack”, Siberian Electronic Mathematical Reports, 21:1 (2024), 17–40

[19] Rudoy E. M., Shcherbakov V. V., “Domain decomposition method for a membrane with a delaminated thin rigid inclusion”, Siberian Electronic Mathematical Reports, 13 (2016), 395–410

[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

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

[22] 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

[23] 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

[24] 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

[25] Lazarev N. P., Nikiforov D. Y., Romanova N. A., “Equilibrium problem for a Timoshenko plate contacting by the side and face surfaces”, Chelyabinsk Physical and Mathematical Journal, 8:4 (2023), 528–541

[26] Lazarev N. P., Semenova G. M., Fedotov E. D., “An equilibrium problem for a Kirchhoff–Love plate, contacting an obstacle by top and bottom edges”, Lobachevskii Journal of Mathematics, 44:2 (2023), 614–619

[27] 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

[28] 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

[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

[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”, Siberian Journal of Industrial Mathematics, 14:4 (2011), 32–43 (In Russ.)

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

[33] Rudoy E. M., Itou H., Lazarev N. P., “Asymptotic justification of the models of thin inclusions in an elastic body in the antiplane shear problem”, Journal of Applied and Industrial Mathematics, 15:1 (2021), 129–140

[34] Fankina I. V., Furtsev A. I., Rudoy E. M., Sazhenkov S.A., “The homogenized quasi-static model of a thermoelastic composite stitched with reinforcing threads”, Journal of Computational and Applied Mathematics, 434 (2023), 115346

[35] Lazarev N. P., Romanova N. A., “Optimal control of the angle between two rigid inclusions in an inhomogeneous 2d body”, Mathematical Notes of NEFU, 30:3 (2023), 38–57

[36] Khludnev A. M., “Shape control of thin rigid inclusions and cracks in elastic bodies”, Archive of Applied Mechanics, 83 (2013), 1493–1509

[37] Kovtunenko V. A., Kunisch K., “Shape derivative for penalty-constrained nonsmooth-nonconvex optimization: cohesive crack problem”, Journal of Optimization Theory and Applications, 194 (2022), 597–635