539.3 Incompressible layered composites with finite deformations on the basis of the asymptotic averaging method

Dimitrienko Y. I. (Bauman Moscow State Technical University), Gubareva E. A. (Bauman Moscow State Technical University), Kolzhanova D. Y. (Bauman Moscow State Technical University), Karimov S. B. (Bauman Moscow State Technical University)

LAYERED COMPOSITES, FINITE DEFORMATIONS, INCOMPRESSIBLE MEDIA, ASYMPTOTIC AVERAGING METHOD, PIOLA — KIRCHHOFF STRESS TENSOR, STRAIN GRADIENT, UNIVERSAL MODELS OF DETERMINING RELATIONS, DEFORMATION DIAGRAMS


doi: 10.18698/2309-3684-2017-1-3254


The article considers the modeling results of incompressible layered composites with finite strains deformation according to the individual layers characteristics. The article proposes an asymptotic averaging method version for layered nonlinearly elastic incompressible composites with finite deformations and periodic structure. We are using a universal representation of the defining relations for incompressible composite layers, proposed by Yu.I. Dimitrienko, which allows us to simulate simultaneously for a complex of various nonlinear elastic media models characterizedby the choice of a pair of energy tensors. We proved that if all composite layers are incompressible, the composite as a whole is also an incompressible, but anisotropic, medium. The article considers the problem of laminated plate uniaxial stretching from incompressible layers with finite deformations. Using the developed method, we calculated the effective deformation diagrams connecting the averaged Piola — Kirchhoff stress tensors components and the strain gradient, as well as the stress distribution in the composite layers.
The developed method for calculating effective deformation diagrams and stresses in composite layers can be used in the design of elastomeric composites with specified properties.


[1] Christensen R.M. Mechanics of composite materials. New York, John Wiley & Sons, 1979, 324 p.
[2] Bakhvalov N.S., Panasenko G.P. Osrednenie protsessov v periodicheskikh sredakh [Averaging processes in periodic media]. Moscow, Nauka Publ., 1984, 356 p.
[3] Pobedrya B.E. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow, Lomonosov Moscow State University Publ., 1984, 336 p.
[4] Dimitrienko Yu.I. Composites Science and Technology, 1999, vol. 59, issue 7, pp. 1041–1053.
[5] Dimitrienko Yu.I., Gubareva E.A., Sborschikov S.V. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no. 2, pp. 28–48.
[6] Dimitrienko Yu.I., Kashkarov A.I. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki — Herald of Bauman Moscow State Technical University. Series Natural Sciences, 2002, no. 2, pp. 95–108.
[7] Dimitrienko Yu. I., Kashkarov A.I., Makashov A.A. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki — Herald of Bauman Moscow State Technical University. Series Natural Sciences, 2007, no.1, pp. 102–116.
[8] Dimitrienko Yu.I., Gubareva E.A., Kolzhanova D.Yu. Inzhenerny zhurnal: nauka i innovatsii — Engineering Journal: Science and Innovation, 2015, no. 5. Available at: http://engjournal.ru/catalog/msm/pmcm/1405.html (accessed May 24, 2017). DOI 10.18698/2308-6033-2015-5-1405
[9] Dimitrienko Yu.I. Izvestiya vyshikh uchebnykh zavedeny. Mashinostroenie — Proceedings of Higher Educational Institutions. Machine Building, 2015, no. 11, pp. 68–77. DOI 10.18698/0536-1044-2015-11-68-77
[10] Yang Q., Xu F. Frontiers of Mechanical Engineering in China, 2009, vol. 4, no. 3, pp. 284–288.
[11] Aboudi J. International Journal Multiscale Computational Engineering, 2008, vol. 6, no. 5, pp. 411–434.
[12] Zhang B., Yu X., Gu B. Polymer Composites, 2015. DOI 10.1002/pc.23596
[13] Ge Q., Luo X., Iversen C.B., Nejad H.B., Mather P.T., Dunn M.L., Qi H.J. International Journal of Solids and Structures, 2014, vol. 51, pp. 2777–2790.
[14] Dimitrienko Yu.I. Mekhanika sploshnoy sredy. V 4 tomakh. Tom 4. Osnovy mekaniki tverdykh sred [Continuum mechanics. In 4 vols. Vol. 4. Fundamentals of mechanics of solid mechanics]. Moscow, BMSTU Publ., 2013, 624 p.
[15] Dimitrienko Yu.I., Dashtiev I.Z. Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki — Herald of Bauman Moscow State Technical University. Series. Natural Sciences, 2001, no. 1, pp. 21–41.
[16] Dimitrienko Yu.I. Nelineynaya mekhanika sploshnoy sredy [Nonlinear mechanics of continuous medium]. Moscow, Fizmatlit Publ., 2009, 610 p.
[17] Dimitrienko Yu.I., Yakovlev N.O., Erasov V.S., Fedonyuk N.N., Sborschikov S.V., Gubareva E.A., Krylov V.D., Grigorev M.M., Prozorovsky A.A. Kompozity i nanostruktury — Composites and Nanostructures, 2014, vol. 6, no. 1, pp. 32–48.
[18] Dimitrienko Yu.I., Gubareva E.A., Sborschikov S.V. Matematicheskoe modelirovanie i chislennye metody — Mathematical Modeling and Computational Methods, 2014, no. 1, pp. 36–57.
[19] Dimitrienko Yu.I., Yakovlev D.O. Mekhanika kompozitsionnykh materialov i konstruktsiy — Mathematical modeling and Computational Methods, 2014, vol. 20, no. 2, pp. 260–282.


Dimitrienko Y., Gubareva E., Kolzhanova D., Karimov S. Incompressible layered composites with finite deformations on the basis of the asymptotic averaging method. Маthematical Modeling and Coтputational Methods, 2017, №1 (13), pp. 32-54



Download article

Количество скачиваний: 770