Modeling of viscoelastoplastic deformation of flexible shallow shells with spatial-reinforcements structures

A. P. Yankovskii

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 3, pp. 506-527.

Résumé

Based on the procedure of time steps, a mathematical model of the viscoelastoplastic behavior of shallow shells with spatial reinforcement structures is constructed. Plastic deformation of the components of the composition is described by flow theory with isotropic hardening; viscoelastic deformation by the equations of the Maxwell–Boltzmann model. The possible weakened resistance of composite curved panels to transverse shear is taken into account in the framework of the hypotheses of Reddy's theory, and the geometric nonlinearity of the problem is taken into account in the Karman approximation. The solution of the formulated initial-boundary value problem is constructed using an explicit numerical scheme of the “cross” type. The elastoplastic and viscoelastoplastic flexural dynamic behavior of “flat” and spatially reinforced fiberglass cylindrical panels under the action of explosive loads has been investigated. Using the example of relatively thin composite structures, it is shown that, depending on which of the front surface (convex or concave), a load is applied, replacing the traditional “flat” reinforcement structure with a spatial one can lead to both an increase and a decrease in the residual deflection. However, in both cases, such a replacement can significantly reduce the intensity of residual deformations of the binder material and fibers of some families. It was demonstrated that the amplitudes of oscillations of curved composite panels in the neighborhood of the initial moment of time significantly exceed the maximum absolute values of the residual deflections. In this case, the residual deflections are rather complicated. It is shown that the calculations carried out within the framework of the elastoplastic deformation theory of the composition components do not even allow an approximate the magnitude determination of the residual deformations of the materials making up the composition.

Détail

Keywords: shallow shells, “flat” reinforcement, dynamic deformation, viscoelastoplastic deformation, Reddy's theory, Maxwell–Boltzmann model, “cross” type scheme.
Mots-clés : spatial reinforcement

Bibliographie

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[24] Reissner E., “On transverse vibrations of thin, shallow elastic shells”, Quart. Appl. Math., 13:2 (1955), 169–176 | DOI

[25] Houlston R., DesRochers C. G., “Nonlinear structural response of ship panels subjected to air blast loading”, Comput. Struct., 26:1–2 (1987), 1–15 | DOI

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[27] Dekker K., Verwer J. G., Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North‐Holland, Amsterdam, New York, 1984, x+308 pp.

[28] Khazhinskii G. M., Modeli deformirovaniia i razrusheniia metallov [Deformation and Long-Term Strength of Metals], Nauchnyi Mir, Moscow, 2011, 231 pp. (In Russian)

[29] Yankovskii A. P., “Using of explicit time-central difference method for numerical simulation of dynamic behavior of elasto-plastic flexible reinforced plates”, Computational Continuum Mechanics, 9:3 (2016), 279–297 (In Russian) | DOI

[30] Handbook of composites, ed. G. Lubin, Van Nostrand Reinhold Company Inc., New York, 1982, 786 pp.

[31] Kompozitsionnye materialy [Composite Materials], ed. D. M. Karpinos, Naukova Dumka, Kiev, 1985, 592 pp. (In Russian)

[1] Qatu M. S., Sullivan R. W., Wang W., “Recent research advances on the dynamic analysis of composite shells: 2000–2009”, Compos. Struct., 93:1 (2010), 14–31 | DOI

[2] Kazanci Z., “Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses”, Int. J. Nonlin. Mech., 46:5 (2011), 807–817 | DOI

[3] Gill S. K., Gupta M., Satsangi P., “Prediction of cutting forces in machining of unidirectional glass-fiber-reinforced plastic composites”, Front. Mech. Eng., 8:2 (2013), 187–200 | DOI

[4] Vasiliev V. V., Morozov E., Advanced Mechanics of Composite Materials and Structural Elements, Amsterdam, Elsever, 2013, xii+412 pp. | DOI

[5] Solomonov Yu. S., Georgievskii V. P., Nedbai A. Ya., Andryushin V. A., Prikladnye zadachi mekhaniki kompozitnykh tsilindricheskikh obolochek [Applied Problems of Mechanics of Composite Cylindrical Shells], Fizmatlit, Moscow, 2014, 408 pp. (In Russian)

[6] Gibson R. F., Principles of Composite Material Mechanics, CRC Press, Boca Raton, 2016 | DOI

[7] Malmeister A. K., Tamuzh V. P., Teters G. A., Soprotivlenie zhestkikh polimernykh materialov [The Strength of Polymeric and Composite Materials], Zinatne, Riga, 1972, 500 pp. (In Russian)

[8] Ambartsumyan S. A., Theory of Anisotropic Plates: Strength, Stability, and Vibration, v. 2, Progress in Materials Science Series, Technomic, Stamford, 1970, 248 pp.

[9] Bogdanovich A. E., Nelineinye zadachi dinamiki tsilindricheskikh kompozitnykh obolochek [Non-Linear Dynamic Problems for Composite Cylindrical Shells], Zinatne, Riga, 1987, 295 pp. (In Russian)

[10] Abrosimov N. A., Bazhenov V. G., Nelineinye zadachi dinamiki kompozitnykh konstruktsii [Nonlinear Problems of Dynamics of Composite Structures], Nizhni Novgorod State Univ., Nizhni Novgorod, 2002, 400 pp. (In Russian)

[11] Reddy J. N., Mechanics of laminated composite plates. Theory and analysis, CRC Press, Boca Raton, 2004, xxiii+831 pp. | DOI

[12] Kaledin V. O., Aul'chenko S. M., Mitkevich A. B., et. al., Modelirovanie statiki i dinamiki obolochechnykh konstruktsii iz kompozitsionnykh materialov [Modeling Statics and Dynamics of Shell Structures Made of Composite Materials], Fizmatlit, Moscow, 2014, 196 pp. (In Russian)

[13] Yankovskii A. P., “Modeling of dynamic elastic-plastic behavior of flexible reinforced shallow shells”, Composite Materials Constructions, 2018, no. 2, 3–14 (In Russian)

[14] Zhigun I. G., Dushin M. I., Polyakov V. A., Yakushin V. A., “Composites reinforced with a system of three straight mutually orthogonal fibers. 2. Experimental study”, Polymer Mechanics, 9:6 (1973), 895–900 | DOI

[15] Tarnopol'skii Yu. M., Zhigun I. G., Polyakov V. A., Prostranstvenno-armirovannye kompozitsionnye materialy [Spatially Reinforced Composite Materials], Mashinostroenie, Moscow, 1987, 224 pp. (In Russian)

[16] Mohamed M. H., Bogdanovich A. E., Dickinson L. C., Singletary J. N., Lienhart R. R., “A new generation of 3D woven fabric preforms and composites”, Sampe J., 37:3 (2001), 3–17

[17] Schuster J., Heider D., Sharp K., Glowania M., “Measuring and modeling the thermal conductivities of three-dimensionally woven fabric composites”, Mech. Compos. Mater., 45:2 (2009), 241–254 | DOI

[18] Tarnopol'skii Y. M., Polyakov V. A., Zhigun I. G., “Composite materials reinforced with a system of three straight, mutually orthogonal fibers. 1. Calculation of the elastic characteristics”, Polymer Mechanics, 9:5 (1973), 754–759 | DOI

[19] Kregers A. F. Teters G. A., “Structural model of deformation of anisotropic three-dimensionally reinforced composites”, Mech. Compos. Mater., 18:1 (1982), 10–17 | DOI

[20] Yankovskii A. P., “Determination of the thermoelastic characteristics of spatially reinforced fibrous media in the case of general anisotropy of their components. 1. Structural model”, Mech. Compos. Mater., 46:5 (2010), 451–460 | DOI

[21] Yankovskii A. P., “Elastic-plastic deformation of flexible plates with spatial reinforcement structures”, J. Appl. Mech. Tech. Phys., 59:6 (2018), 1058–1066 | DOI | DOI

[22] Pisarenko G. S., Yakovlev A. P., Matveev V. V., Vibropogloshchaiushchie svoistva konstruktsionnykh materialov: Spravochnik [Vibration-Absorbing Properties of Structural Materials: A Handbook], Naukova dumka, Kiev, 1971, 375 pp. (In Russian)

[23] Freudenthal A. M., Geiringer H., “The Mathematical Theories of the Inelastic Continuum”, Elasticity and Plasticity. Encyclopedia of Physics, ed. S. Flügge, Springer, Berlin, Heidelberg, 1958, 229–433 | DOI

[24] Reissner E., “On transverse vibrations of thin, shallow elastic shells”, Quart. Appl. Math., 13:2 (1955), 169–176 | DOI

[25] Houlston R., DesRochers C. G., “Nonlinear structural response of ship panels subjected to air blast loading”, Comput. Struct., 26:1–2 (1987), 1–15 | DOI

[26] Zeinkiewicz O. C., Taylor R. L., The Finite Element Method, Butterworth-Heinemann, Oxford, 2000, 707 pp.

[27] Dekker K., Verwer J. G., Stability of Runge–Kutta Methods for Stiff Nonlinear Differential Equations, North-Holland, Amsterdam, New York, 1984, x+308 pp.

[28] Khazhinskii G. M., Modeli deformirovaniia i razrusheniia metallov [Deformation and Long-Term Strength of Metals], Nauchnyi Mir, Moscow, 2011, 231 pp. (In Russian)

[29] Yankovskii A. P., “Using of explicit time-central difference method for numerical simulation of dynamic behavior of elasto-plastic flexible reinforced plates”, Computational Continuum Mechanics, 9:3 (2016), 279–297 (In Russian) | DOI

[30] Handbook of composites, ed. G. Lubin, Van Nostrand Reinhold Company Inc., New York, 1982, 786 pp.

[31] Kompozitsionnye materialy [Composite Materials], ed. D. M. Karpinos, Naukova Dumka, Kiev, 1985, 592 pp. (In Russian)