Refined Equations of Motion of Multilayered Shells with Transversely Soft Fillers under a Medium Bending

V. N. Paimushin   ; T. V. Polyakova

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 155 (2013) no. 2, pp. 167-183

Résumé

As a development of the earlier results, we constructed a refined two-dimensional mathematical model of the dynamic deformation of multilayered plates and shells with transversely soft fillers, based on the classic Kirchhoff–Love model for supporting layers and the hypothesis on similarity of the laws of variation of movements along the thickness of fillers under both static and dynamic loading process. On the ground of this hypothesis, for a transversely soft filler, we derived simplified quasi-static equations of the theory of elasticity, which allow transverse integrating. When integrating the equations to describe the stress-strain state (SSS), we introduced (as in the static problems) two-dimensional unknown functions representing transverse tangent stresses, constant in thickness. Based on the generalized variational Ostrogradskii–Hamilton principle for describing the dynamic processes of deformation with high variability of SSS parameters, we obtained two-dimensional motion equations of general form, where the inertial components have the same degree of accuracy in comparison with the other ones. We simplified the obtained equations for the case of low variability of SSS parameters and considered the problem of free oscillations of a small rectangular multilayered plate, which are characterized by a zero variability of the functions in the tangential directions and are realized in the plate without deformation of the supporting layers.

Détail

Keywords: orthotropic plate, refined theory, trigonometric functions, free oscillations, oscillation frequencies.
Mots-clés : longitudinal-transverse form

Bibliographie

[1] Norr A. K., Burton W. S., Bert Ch.W., “Computational models for sandwich panels and shells”, Appl. Mech. Rev., 49:3 (1996), 155–199 | DOI

[2] Paimushin V. N., “Teoriya ustoichivosti trekhsloinykh elementov konstruktsii. Analiz sovremennogo sostoyaniya i utochnennaya klassifikatsiya form poteri ustoichivosti”, Mekhanika kompozitnykh materialov, 35:6 (1999), 707–716

[3] Paimushin V. N., “Teoriya ustoichivosti trekhsloinykh plastin i obolochek (Etapy razvitiya, sovremennoe sostoyanie i napravleniya dalneishikh issledovanii)”, Izv. RAN. MTT, 2001, no. 2, 148–162

[4] Mushtari Kh. M., “O primenimosti razlichnykh teorii trekhsloinykh plastin i obolochek”, Izv. AN SSSR. OTN Mekhanika i mashinostroenie, 1960, no. 6, 163–165 | Zbl

[5] Mushtari Kh. M., “K obschei teorii pologikh obolochek s zapolnitelem”, Izv. AN SSSR. OTN. Mekhanika i mashinostroenie, 1961, no. 2, 24–29

[6] Bolotin V. V., Novichkov Yu. N., Mekhanika mnogosloinykh konstruktsii, Mashinostroenie, M., 1980, 375 pp.

[7] Galimov K. Z., Osnovy nelineinoi teorii tonkikh obolochek, Izd-vo Kazan. un-ta, Kazan, 1975, 326 pp.

[8] Paimushin V. N., “K variatsionnym metodam resheniya prostranstvennykh zadach sopryazheniya deformiruemykh tel”, Dokl. AN SSSR, 273:5 (1983), 1083–1086 | Zbl

[9] Paimushin V. N., “Utochnennaya nelineinaya teoriya srednego izgiba trekhsloinykh obolochek s transversalno-myagkim zapolnitelem pri termosilovykh vozdeistviyakh”, Izv. vuzov. Aviats. tekhnika, 1989, no. 4, 8–12

[10] Ivanov V. A., Paimushin V. N., “Utochnennaya teoriya ustoichivosti trekhsloinykh konstruktsii (nelineinye uravneniya dokriticheskogo ravnovesiya obolochek s transversalno-myagkim zapolnitelem)”, Izv. vuzov. Matem., 1994, no. 11, 29–42 | Zbl

[11] Paimushin V. N., Petrushenko Yu. Ya., “Variatsionnyi metod resheniya zadach mekhaniki prostranstvennykh sostavnykh tel. Obobschennyi variatsionnyi printsip Gamiltona–Ostrogradskogo”, Soobsch. AN Gruzinskoi SSR, 131:1 (1988), 130–135

[12] Ivanov V. A., Paimushin V. N., “Utochnennaya postanovka dinamicheskikh zadach trekhsloinykh obolochek s transversalno-myagkim zapolnitelem i chislenno-analiticheskii metod ikh resheniya”, Prikl. mekhanika i tekhn. fizika, 36:4 (1995), 137–151 | Zbl

[13] Ivanov V. A., Paimushin V. N., “Utochnenie uravnenii dinamiki mnogosloinykh obolochek s transversalno-myagkimi zapolnitelyami”, Izv. RAN. MTT, 1995, no. 3, 142–152

[14] Paimushin V. N., Khusainov V. R., “Utochnennaya teoriya trekhsloinykh plastin i obolochek dlya issledovaniya dinamicheskikh protsessov deformirovaniya s bolshimi pokazatelyami izmenyaemosti”, Mekhanika kompozitnykh materialov i konstruktsii, 7:2 (2001), 215–235

[15] Paimushin V. N., Khusainov V. R., “Uravneniya i klassifikatsiya svobodnykh i sobstvennykh kolebanii simmetrichnykh po tolschine trekhsloinykh plastin s transversalno-myagkim zapolnitelem”, Mekhanika kompozitnykh materialov i konstruktsii, 7:3 (2001), 310–317

[16] Paimushin V. N., Ivanov V. A., Khusainov V. R., “Analiz svobodnykh i sobstvennykh kolebanii trekhsloinoi plastiny s ispolzovaniem dlya zapolnitelya uravnenii teorii uprugosti”, Mekhanika kompozitnykh materialov i konstruktsii, 8:2 (2002), 197–213

[17] Paimushin V. N., Ivanov V. A., Khusainov V. R., “Analiz svobodnykh i sobstvennykh kolebanii trekhsloinoi plastiny na osnove uravnenii utochnennoi teorii”, Mekhanika kompozitnykh materialov i konstruktsii, 8:4 (2002), 543–554

[18] Paimushin V. N., Ivanov V. A., Khusainov V. R., “Analiz uravnenii i zadach o svobodnykh kolebaniyakh trekhsloinykh plastin s transversalno-myagkim zapolnitelem i simmetrichnym po tolschine stroeniem”, Izv. vuzov. Aviats. tekhnika, 2001, no. 4, 22–25