Consistent equations of nonlinear rectilinear laminated bars theory in quadratic approximation

V. N. Paimushin ; S. A. Kholmogorov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 75-87

Résumé

Two versions of one-dimensional equilibrium equations for rectilinear laminated bars on basis of S. P. Timoshenko's model subject to transversal compression for each layer and describing geometrical nonlinear deformation by arbitrary displacements and small strain have been derived. The equations are based on the earlier proposed consistent theory of elasticity relations, the usage of which does not lead to spurious bifurcation solutions. The first version corresponds to contact problem statement, when contact stresses are introduced in the coupling points of layers as unknown parameters. The second version corresponds to preliminary satisfaction to the kinematic coupling conditions of layers with respect to displacements.

Détail

Keywords: rectilinear bar, laminated structure, geometrically nonlinearity, arbitrary displacements, small strain, S. P. Timoshenko's model, contact stresses, kinematic coupling conditions.

Bibliographie

[1] Paimushin V. N., Shalashilin V. I., “Consistent variant of continuum deformation theory in the quadratic approximation”, Dokl. Ross. Akad. Nauk, 396:4 (2004), 492–495 (In Russian) | MR

[2] Paimushin V. N., Shalashilin V. I., “The relations of deformation theory in the quadratic approximation and the problems of constructing improved versions of the geometrically non-linear theory of laminated structures”, J. Appl. Math. Mech., 69:5 (2005), 773–791 | DOI | MR

[3] Paimushin V. N., “The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements”, J. Appl. Math. Mech., 72:5 (2008), 597–610 | DOI | MR

[4] Novozhilov V. V., Fundamentals of Nonlinear Theory of Elasticity, Gostekhizdat, Leningrad–Moscow, 1948, 211 pp. (In Russian) | MR

[5] Donnell L. H., Beams, Plates and Shells, McGraw-Hill, New York, 1976, 453 pp. | Zbl

[6] Shklyarchuk F. N., “Analysis of the strain state and stability of geometrically nonlinear elastic systems”, Mech. Solids, 33:1 (1998), 114–119 | MR

[7] Berezhnoi D. V., Paimushin V. N., Shalashilin V. I., “Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements”, Mech. Solids, 44:6 (2009), 837–851 | DOI | MR

[8] Guz A. N., Stability of Elastic Bodies under Uniform Compression, Naukova Dumka, Kiev, 1973, 270 pp. (In Russian) | MR

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

[10] Rosen B. W., “Mechanics of composite strengthening”, Fibre Composite Materials, American Society for Metals, 1965

[11] Budiansky B., Fleck N. A., “Compressive failure of fibre composites”, J. Mech. Phys. Solids, 41:1 (1993), 183–211 | DOI

[12] Xu Y. L., Reifsnider K. L., “Micromechanical modeling of composite compressive strength”, J. Composite Materials, 27:6 (1993), 572–588 | DOI

[13] Zhang G., Latour R. A. (Jr.), “FRP composite compressive strength and its dependence upon interfacial bond strength, fiber misalignment, and matrix nonlinearity”, J. Thermoplastic Composite Materials, 6:4 (1993), 298–311 | DOI

[14] Zhang G., Latour R. A. (Jr.), “An analytical and numerical study of fiber microbuckling”, Composites Sci. Technol., 51:1 (1994), 95–109 | DOI

[15] Naik N. K., Kumar R. S., “Compressive strength of unidirectional composites: Evaluation and comparison of prediction models”, Composite Structures, 46:3 (1999), 299–308 | DOI

[16] Jumahat A., Soutis C., Jones F. R., Hodzic A., “Fracture mechanisms and failure analysis of carbon fibre/toughened epoxy composites subjected to compressive loading”, Composite Structures, 92:2 (2010), 295–305 | DOI

[17] Paimushin V. N., Kholmogorov S. A., Badriev I. B., Makarov M. V., “Geometrically and physically nonlinear task on the three-point bending of test composite samples”, Proc. XXII Int. Symp. “Dynamical and Technological Problems of Structural and Continuum Mechanics”, TRP, Moscow, 2016, 197–202 (In Russian)

[18] Paimushin V. N., “Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns”, Prikl. Mat. Mekh., 71:5 (2007), 855–893 (In Russian) | MR

[19] Paimushin V. N., Gyunal I. Sh., Lukankin S. A., Firsov V. A., “Investigation of the quality of nonlinear equations of elasticity theory based on the problems of stability of flat curved bars with laminated structure (problem formulation)”, Izv. Vyssh. Uchebn. Zaved., Aviats. Tekh., 2010, no. 2, 34–37 (In Russian)

[20] Paimushin V. N., Gyunal I. Sh., Lukankin S. A., Firsov V. A., “Investigation of the quality of nonlinear equations of elasticity theory based on the problems of stability of flat curved bars with laminated structure (algorithm and results of calculation)”, Izv. Vyssh. Uchebn. Zaved., Aviats. Tekh., 2010, no. 3, 16–19 (In Russian)

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

[22] Paimushin V. N., “Variational formulation of mechanics problems of composite bodies of piecewise-homogeneous structure”, Prikl. Mekh., 21:1 (1985), 27–34 (In Russian) | MR | Zbl

[23] Paimushin V. N., Firsov V. A., Kholmogorov S. A., “Nonlinear behavior of the fiber composite based on carbon fibre under the conditions of displacement”, Proc. XXII Int. Symp. “Dynamical and Technological Problems of Structural and Continuum Mechanics”, TRP, Moscow, 2016, 143–145 (In Russian)

[24] Badriev I. B., Makarov M. V., Paimushin V. N., “Mathematical simulation of nonlinear problem of three-point composite sample bending test”, Procedia Eng., 150 (2016), 1056–1062 | DOI