Longitudinal and transverse bending on the cylindrical shape of a sandwich plate reinforced with absolutely rigid bodies in the front sections

I. B. Badriev   ; M. V. Makarov   ; V. N. Paimushin

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 2, pp. 174-190

Résumé

A geometrically nonlinear problem of longitudinal and transverse bending on the cylindrical shape of a sandwich plate with transversally soft core reinforced in the front sections by absolutely rigid bodies intended to ensure the transfer of load to the carrier layers when they interact with other structural elements has been considered. The equations of the refined geometrically nonlinear theory, which allow one to describe the process of their precritical deformation and to reveal all possible buckling forms of the carrier layers (in-phase, antiphase, mixed bending and mixed bending-shear, and also arbitrary, including all of the above) have been used. These equations have been derived by introducing the contact forces of the interaction of the outer layers with the filler, as well as those of the outer layers and the filler with the reinforcing bodies at all points of the surfaces of their conjugation, as unknown parameters. A numerical method for solving the formulated problem has been developed. The method has been constructed by preliminary reduction of the problem to a system of integro-algebraic equations solved with the help of the finite-sum method. A method has been developed for studying the precritical geometrically nonlinear behavior of the plate as a result of its front compression through a reinforcing body. The results of the numerical experiments have been presented and analyzed.

Détail

Keywords: sandwich plate, transversely soft core, contour reinforcing body, middle plate bending, refined model of core, contact stresses, integro-algebraic equations, finite sums method, geometrically nonlinear deformation, precritical behavior.

Bibliographie

[1] Kobelev V. N., Calculation of Sandwich Structures, Mashinostroenie, Moscow, 1984, 303 pp. (In Russian)

[2] Bank L. C., Composites for Construction: Structural Design with FRP Materials, John Wiley Sons, Inc., New Jersey, 2006, 552 pp.

[3] Badriev I. B., Makarov M. V., Paimushin V. N., “On the interaction of composite plate having a vibration-absorbing covering with incident acoustic wave”, Russ. Math., 59:3 (2015), 66–71 | DOI | MR | Zbl

[4] Frostig Y., “Elastica of sandwich panels with a transversely flexible core – A high-order theory approach”, Int. J. Solids Struct., 46 (2009), 2043–2059 | DOI | Zbl

[5] Hollaway L. C., “Polymers, fibres, composites and the civil engineering environment: A personal experience”, Adv. Struct. Eng., 13:5 (2010), 927–960 | DOI

[6] Dyatchenko S. V., Ivanov A. P., “A Technology for Manufacturing Ship Hulls out of Polymer Composites”, Izd. Kaliningr. Tekh. Univ., Kaliningrad, 2007, 156 pp. (In Russian)

[7] Prokhorov B. F., Kobelev V. N., Sandwich Constructions in Shipbuilding, Sudostroenie, Leningrad, 1972, 344 pp. (In Russian)

[8] Raicu A., “The advantages of the composite materials used in shipbuilding and marine structure”, J. Marine Technol. Environ., 1 (2012), 99–102

[9] J. Graham-Jones, J. Summerscales (Eds.), Marine Applications of Advanced Fibre-reinforced Composites, Woodhead Publ., 2015, 360 pp.

[10] Vasil'ev V. V., Dobryakov A. A., Dudchenko A. A., Fundamentals of the Planning and Production of Aircraft Structures Made of Composite Materials, MAI, Moscow, 1985, 218 pp. (In Russian)

[11] Krysin V. N., Layered Laminated Constructions in Aircraft Engineering, Mashinostroenie, Moscow, 1980, 232 pp. (In Russian)

[12] Pavlov N. A., Constructions of Rockets and Space Vehicles, Mashinostroenie, Moscow, 1993, 148 pp. (In Russian)

[13] Mangalgiri P. D., “Composite materials for aerospace applications”, Bull. Mater. Sci., 22:3 (1999), 657–664 | DOI

[14] Bouvet C., Mechanics of Aeronautical Composite Materials, John Wiley Sons, Inc., N.Y., 2017, 309 pp.

[15] Grigolyuk E. I., Chulkov P. P., “Stability and Vibrations of Sandwich Shells”, Mashinostroenie, Moscow, 1973, 168 pp. (In Russian)

[16] Bolotin V. V., Novichkov Y. N., Mechanics of Multilayered Structures, Mashinostroenie, Moscow, 1980, 375 pp. (In Russian)

[17] Grigolyuk E. I., Kogan F. A., “Present state of the theory of multilayered shells”, Prikl. Mekh., 8:6 (1972), 5–17 (In Russian)

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

[19] Paimushin V. N., “Nonlinear theory of the central bending of three-layer shells with defects in the form of sections of bonding failure”, Sov. Appl. Mech., 23:11 (1987), 1038–1043 | DOI

[20] Ivanov V. A., Paimushin V. N., “An improved theory of the stability of three-layer structures (non-linear equations for the subcritical equilibrium of shells with a transversely soft filler)”, Russ. Math., 38:11 (1994), 26–39 | MR | Zbl

[21] Paimushin V. N., Bobrov S. N., “Refined geometric nonlinear theory of sandwich shells with a transversely soft core of medium thickness for investigation of mixed buckling forms”, Mech. Compos. Mater., 36:1 (2000), 59–66 | DOI

[22] Badriev I. B., Banderov V. V., Makarov M. V., Paimushin V. N., “Determination of stress-strain state of geometrically nonlinear sandwich plate”, Appl. Math. Sci., 9:78 (2015), 3887–3895 | DOI

[23] Badriev I. B., Banderov V. V., Garipova G. Z., Makarov M. V., Shagidullin R. R., “On the solvability of geometrically nonlinear problem of sandwich plate theory”, Appl. Math. Sci., 9:82 (2015), 4095–4102 | DOI

[24] Badriev I. B., Garipova G. Z., Makarov M. V., Paymushin V. N., “Numerical solution of the issue about geometrically nonlinear behavior of sandwich plate with transversal soft filler”, Res. J. Appl. Sci., 10:8 (2015), 428–435 | MR

[25] Badriev I. B., Makarov M. V., Paimushin V. N., “Geometrically nonlinear problem of longitudinal and transverse bending of a sandwich plate with transversally soft core”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 158, no. 4, 2016, 453–468 (In Russian)

[26] Badriev I. B., Makarov M. V., Paimushin V. N., “Solvability of physically and geometrically nonlinear problem of the theory of sandwich plates with transversally-soft core”, Russ. Math., 59:10 (2015), 57–60 | DOI | MR | Zbl

[27] Badriev I. B., Garipova G. Z., Makarov M. V., Paimushin V. N., Khabibullin R. F., “Solving physically nonlinear equilibrium problems for sandwich plates with a transversally soft core”, Lobachevskii J. Math., 369:4 (2015), 474–481 | DOI | MR

[28] Badriev I. B., Makarov M. V., Paimushin V. N., “Numerical investigation of physically nonlinear problem of sandwich plate bending”, Proc. Eng., 150 (2016), 1050–1055 | DOI

[29] Badriev I. B., Makarov M. V., Paimushin V. N., “Numerical investigation of a physically nonlinear problem of longitudinal bending of sandwich plate with transversal-soft core”, PNRPU Mech. Bull., 2017, no. 1, 39–51 | DOI

[30] Paimushin V. N., “Theory of moderately large deflections of sandwich shells having a transversely soft core and reinforced along their contour”, Mech. Compos. Mater., 53:1 (2017), 1–16 | DOI

[31] Paimushin V. N., “Contact formulation of non-linear problems in the mechanics of shells with their end sections connected by a plane curvilinear rod”, J. Appl. Math. Mech., 78:1 (2014), 84–89 | DOI | MR

[32] Lukankin S. A., Paimushin V. N., Kholmogorov S. A., “Non-classical forms of loss stability of cylindrical shells joined by a stiffening ring for certain forms of loading”, J. Appl. Math. Mech., 78:4 (2014), 395–408 | DOI | MR

[33] Paimushin V. N., “Variational methods for solving non-linear spatial problems of the joining of deformable bodies”, Dokl. Akad. Nauk SSSR, 273:5 (1983), 1083–1086 (In Russian) | MR

[34] Karchevskii M. M., Lyashko A. D., Difference Schemes for Nonlinear Problems of Mathematical Physics, Izd. Kazan. Univ., Kazan, 1976, 156 pp. (In Russian)

[35] Samarskii A. A., The theory of difference schemes, Marcel Dekker, Inc., N.Y.–Basel, 2001, 761 pp. | MR | Zbl

[36] Dautov R. Z., Paimushin V. N., “On the method of integrating matrices for the solution of boundary value problems for fourth-order ordinary equations”, Russ. Math., 40:10 (1996), 11–23 | MR | Zbl

[37] Dautov R. Z., Karchevskii M. M., Paimushin V. N., “On the method of integrating matrices for systems of ordinary differential equations”, Russ. Math., 47:7 (2003), 16–24 | MR | Zbl

[38] Karchevskii M. M., “Iteration schemes for equations with monotone operators”, Izv. Vyssh. Uchebn. Zaved. Mat., 1971, no. 5, 32–37 (In Russian) | MR | Zbl

[39] Badriev I. B., Karchevskii M. M., “Convergence of an iterative process in a Banach space”, J. Math. Sci., 71:6 (1994), 2727–2735 | DOI | MR | Zbl

[40] Makarov M. V., Badriev I. B., Paimushin V. N., “Nonlinear problems on mixed buckling of sandwich plates under longitudinal and transverse bending”, Vestn. Tambov. Univ. Ser. Est. Tekh. Nauki, 20:5 (2015), 1275–1278 (In Russian)

[41] Badriev I. B., Banderov V. V., Makarov M. V., Paimushin V. N., “On the solvability of nonlinear problems within the theory of sandwich shells with a transversely soft filler”, Mesh Methods for Boundary-Value Problems and Applications, Proc. 10th Int. Conf., Izd. Kazan. Univ., Kazan, 2014, 103–107 (In Russian)