Incompatible deformations of elastic plates

S. A. Lychev

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 165 (2023) no. 4, pp. 361-388

Résumé

This article considers the methods for mathematical modeling of incompatible finite deformations of elastic plates by using the principles of the differential geometry theory underlying continuously distributed defects. Equilibrium equations were derived by asymptotic expansions of the finite strain measures with respect to two small parameters. One parameter defines the order of smallness of displacements from the reference shape (self-stressed state), while the other specifies the thickness. Asymptotic orders were different for the deflections and displacements in the plate plane, as well as for their derivatives. They were selected in such a way that, with additional assumptions on the possibility of ignoring certain terms in the resulting expressions and the compatibility of deformations, the equations could be reduced to the system of Föppl–von Kármán equations.

Détail

Keywords: theory of thin-walled elastic structures, incompatible finite deformations of elastic plates, mathematical modeling methods, theory of continuously distributed defects, numerical and analytical method, iteration algorithm.
Mots-clés : system of Föppl–von Kármán equations

Bibliographie

[1] Zorman C., Mehregany M., “Material aspects of micro- and nanoelectromechanical systems”, Springer Handbook of Nanotechnology, Springer Handbooks, ed. Bhushan B., Springer, Berlin–Heidelberg, 2010, 333–356 | DOI

[2] Ciarlet P.G., “A justification of the von Kármán equations”, Arch. Ration. Mech. Anal, 73:4 (1980), 349–389 | DOI | MR | Zbl

[3] Dedkova A.A., Glagolev P.Yu., Gusev E.E., Djuzhev N.A., Kireev V.Yu., Lychev S.A., Tovarnov D.A., “Peculiarities of deformation of round thin-film membranes and experimental determination of their effective characteristics”, Zh. Tekh. Fiz., 91:10 (2021), 1454–1465 (In Russian) | DOI

[4] Bychkov P.S., Lychev S.A., Bout D.K., “Experimental methodology for determining the evolution of the bending shape of a thin substrate during electrocrystallization of copper in complex-shaped areas”, Vestn. Samar. Univ. Estestvennonauchn. Ser., 25:4 (2019), 48–73 (In Russian) | MR

[5] Manzhirov A.V., Lychev S.A., “On the equilibrium of accreted plates”, Topical Problems in Solid and Fluid Mechanics, eds. A.V. Manzhirov, N.K. Gupta, D.A. Indeitsev, Elite Publ. House Pvt Ltd., Delhi, 2011, 294–300

[6] Lychev S.A., Lycheva T.N., Manzhirov A.V., “Unsteady vibration of a growing circular plate”, Mech. Solids, 46:2 (2011), 325–333 | DOI

[7] Lychev S., “Equilibrium equations for transversely accreted shells”, ZAMM, 94:1–2 (2014), 118–129 | DOI | MR | Zbl

[8] Lychev S., Koifman K., Geometry of Incompatible Deformations: Differential Geometry in Continuum Mechanics, De Gruyter, 2019, 388 pp. | DOI | MR

[9] Vol'mir A.S., Flexible Plates and Shells, Gos. Izd. Tekh.-Teor. Lit., M., 1956, 210 pp. (In Russian)

[10] Timoshenko S., Woinowsky-Krieger S., Theory of Plates and Shells, McGraw-Hill, New York, 1959, 580 pp. | MR

[11] Freund L.B., Suresh S., Thin Film Materials. Stress, Defect Formation and Surface Evolution, Cambridge Univ. Press, 2004, 750 pp. | MR | Zbl

[12] Föppl A., Vorlesungen über Technische Mechanik, v. 5, B.G. Teubner Verlag, Leipzig, 1907 (In German) | MR

[13] Kármán T., Festigkeitsprobleme im Maschinenbau, Encyklopädie der mathematischen Wissenschaften, 4, B.G. Teubner Verlag, Leipzig, 1910 (In German)

[14] Biderman V.L., Mechanics of Thin-Walled Structures, Mashinostroenie, M., 1977, 488 pp. (In Russian)

[15] Feodos'ev V.I., “Axisymmetric flexible shells”, Strength Calculations in Mechanical Engineering, v. 2, MAShGIZ, M., 1958 (In Russian)

[16] Andreeva L.E., Elastic Elements of Instruments, MAShGIZ, M., 1962, 455 pp. (In Russian)

[17] Knightly G.H., “An existence theorem for the von Kármán equations”, Arch. Ration. Mech. Anal, 27:3 (1967), 233–242 | DOI | MR | Zbl

[18] Naumann J., “An existence theorem for the v. Kármán equations under the condition of free boundary”, Apl. Mat., 19:1 (1974), 17–27 | DOI | MR | Zbl

[19] Bilbao S., Thomas O., Touzé C., Ducceschi M., “Conservative numerical methods for the Full von Kármán plate equations”, Numer. Methods Partial Differ. Equations, 31:6 (2015), 1948–1970 | DOI | MR | Zbl

[20] Ciarlet P.G., Rabie P., Les équations de von Kármán, Springer, 2006, 181 pp. (In French) | MR

[21] Ciarlet P.G., Mathematical Elasticity: Theory of Shells, Classics in Applied Mathematics, Soc. Ind. Appl. Math., 2022 | MR | Zbl

[22] Borisovich A., Janczewska J., “Stable and unstable bifurcation in the von Kármán problem for a circular plate”, Abstr. Appl. Anal, 2005:8 (2005), 845303 | DOI | MR

[23] Van Gorder R.A., “Asymptotic solutions for the Föppl–von Kármán equations governing deflections of thin axisymmetric annular plates”, Int. J. Non-Linear Mech., 91 (2017), 8–21 | DOI

[24] Yu Q., Xu H., Liao S., “Coiflets solutions for Föppl–von Kármán equations governing large deflection of a thin flat plate by a novel wavelet-homotopy approach”, Numer. Algorithms, 79 (2018), 993–1020 | DOI | MR | Zbl

[25] Frakes J.P., Simmonds J.G., “Asymptotic solutions of the von Karman equations for a circular plate under a concentrated load”, J. Appl. Mech., 52:2 (1985), 326–330 | DOI | MR | Zbl

[26] Dickey R.W., “Nonlinear bending of circular plates”, SIAM J. Appl. Math., 30:1 (1976), 1–9 | DOI | MR | Zbl

[27] Michell J.H., “The flexure of a circular plate”, Proc. London Math. Soc., s1-34:1 (1902), 223–228 | DOI | MR

[28] Polyanin A.D., Nazaikinskii V.E., Handbook of Linear Partial Differential Equations for Engineers and Scientists, 2nd ed., Chapman and Hall/CRC Press, New York, 2016, 1643 pp. | DOI | MR

[29] Van Gorder R.A., “Analytical method for the construction of solutions to the Föppl–von Kármán equations governing deflections of a thin flat plate”, Int. J. Non-Linear Mech., 47:3 (2012), 1–6 | DOI

[30] Reissner E., “On finite deflections of circular plates”, Proc. Symp. Appl. Math., 1 (1949), 213–219 | DOI | MR | Zbl

[31] Zhang Y., “Large deflection of clamped circular plate and accuracy of its approximate analytical solutions”, Sci. China: Phys., Mech. Astron., 59:2 (2016), 624602 | DOI