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Elastic properties of a unidirectional composite reinforced with hexagonal array of fibers

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Języki publikacji
EN
Abstrakty
EN
It is difficult to measure the transverse shear modulus of the fibrous composites. Thus, theoretical investigations by means of analytical and numerical techniques are paramount. In particular, they are important for the regime with high-concentration of fibers. We apply general techniques to study the mechanical properties of unidirectional fibers with a circular section embedded into the matrix and organized into the hexagonal array. Our theoretical considerations are designed to include two regimes, of low and high concentrations of inclusions. The former regime is controlled by Hashin–Shtrikman lower bounds, while the latter is controlled by square-root singularity. We derived the analytical formulae for the effective shear, Young and bulk moduli in the form of the rational expressions valid up to O(f7) by the method of functional equations. The obtained formulae contains elastic constants of components in a symbolic form as well as the concentration f. The general scheme based on the asymptotically equivalent transformations is developed to extend the obtained analytical formulae to the critical concentration of touching fibers. A comparison with the numerical FEM is performed for all concentrations of inclusions. Good agreement is achieved for all available concentrations.
Rocznik
Strony
207--239
Opis fizyczny
Bibliogr. 43 poz., rys.
Twórcy
  • Department of Mechanics, Materials Science and Engineering Faculty of Mechanicsl Engineering Wrocław University of Science and Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland, tomasz.czaplinski@nobosolutions.com
autor
  • Department of Differential Equations and Statistics Faculty of Mathematics and Natural Sciences University of Rzeszów Pigonia 1, 35-959 Rzeszów, Poland, drygaspi@ur.edu.pl
autor
autor
  • Faculty of Mathematics, Physics and Technology Pedagogical University Podchorążych 2, 30-084 Kraków, Poland, mityu@up.krakow.pl
  • Faculty of Mathematics, Physics and Technology Pedagogical University Podchorążych 2, 30-084 Kraków, Poland, wnawalaniec@gmail.com
autor
  • Department of Mechanics, Materials Science and Engineering Faculty of Mechanicsl Engineering Wrocław University of Science and Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland, grazyna.zietek@pwr.edu.pl
Bibliografia
  • 1. V.V. Vasiliev, E.V. Morozov, Advanced Mechanics of Composite Materials, Elsevier, Amsterdam, 2007.
  • 2. M. Majewski, M. Kursa, P. Holobut, K. Kowalczyk-Gajewska, Micromechanical and numerical analysis of packing and size effects in elastic particulate composites, Composites Part B: Engineering, 124, 158–174, 2017, doi.org/10.1016/j.compositesb.2017.05.004.
  • 3. N.I. Muskhelishvili, Some Mathematical Problems of the Plane Theory of Elasticity, Nauka, Moscow, 1966.
  • 4. E.I. Grigolyuk, L.A. Filishtinskii, Perforated plates and shells, Nauka, 556, 1970 [in Russian].
  • 5. E.I. Grigolyuk, L.A. Filishtinskii, Periodical Piece-Homogeneous Elastic Structures, Nauka, Moscow, 1992 [in Russian].
  • 6. E.I. Grigolyuk, L.A. Filishtinskii, Regular Piece-Homogeneous Structures with defects, Fiziko-Matematicheskaja Literatura, Moscow, 1994 [in Russian].
  • 7. J. Helsing, An integral equation method for elastostatics of periodic composites, Journal of the Mechanics and Physics of Solids, 43, 815–828, 1995.
  • 8. E.J. Barbero, Finite Element Analysis of Composite Materials, CRC Press, Boca Raton, 2008.
  • 9. J.W. Eischen, S. Torquato, Determining elastic behavior of composites by the boundary element method, Journal of Applied Physics, 74, 159–170, 1993.
  • 10. A.P.S. Selvadurai, H. Nikopour, Transverse elasticity of a unidirectionally reinforced composite with an irregular fibre arrangement: experiments, theory and computations, Composite Structures, 94, 1973–1981, 2012.
  • 11. S.B.R. Devireddy, S. Biswas, Effect of fiber geometry and representative volume element on elastic and thermal properties of unidirectional fiber-reinforced composites, Journal of Composites, 2014, Article ID 629175.
  • 12. I. Andrianov, V. Mityushev, Exact and “exact” formulae in the theory of composites, [in:] Modern Problems in Applied Analysis, P. Drygas, S. Rogosin [eds.], Birkhäuser Basel, 2018; arXiv:1708.02137v1
  • 13. R. Hill, Theory of mechanical properties of fibre-strengthened materials: I. Elastic behaviour, Journal of the Mechanics and Physics of Solids, 12, 199–212, 1964.
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  • 15. V.I. Kushch, I. Sevostianov, Maxwell homogenization scheme as a rigorous method of micromechanics: application to effective conductivity of a composite with spheroidal particles, International Journal of Engineering Science, 98, 36–50, 2016.
  • 16. V. Mityushev, N. Rylko, it Maxwell’s approach to effective conductivity and its limitations, The Quarterly Journal of Mechanics and Applied Mathematics, 66, 241–251, 2013,doi: 10.1093/qjmam/hbt003.
  • 17. P. Drygas, V. Mityushev, Effective elastic properties of random two-dimensional composites, International Journal of Solids and Structures, 97-98, 543—553, 2016, doi.org/10.1016/j.ijsolstr.2016.06.034.
  • 18. V. Mityushev, Cluster method in composites and its convergence, Applied Mathematics Letters, 77, 44–48, 2018, doi:org/10.1016/j.aml.2017.10.001
  • 19. S. Gluzman, V. Mityushev, W. Nawalaniec, G.A. Starushenko, Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders, [in:] Contributions in Mathematics and Engineering. In Honor of Constantin Caratheodory, P.M. Pardalos, Th.M. Rassias [eds.], Springer, Switzerland, 255–297, 2016.
  • 20. P. Wall, A Comparison of homogenization, Hashin–Shtrikman bounds and the Halpin–Tsai equations, Applications of Mathematics, 42, 245–257, 1997.
  • 21. I.V. Andrianov, G.A. Starushenko, V.V. Danishevskyy, S. Tokarzewski, Homogenization procedure and Padé approximants for the effective heat conductivity of composite materials with cylindrical inclusions having square cross-section, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, A455, 3401–3413, 1999.
  • 22. Z. Hashin, The elastic moduli of heterogeneous materials, Journal of Applied Mechanics, 29, 143–150, 1962.
  • 23. S.D. Ryan, V. Mityushev, V. Vinokur, L. Berlyand, Rayleigh approximation to ground state of the Bose and Coulomb glasses, Scientific Reports, 5, 7821, 2015.
  • 24. S. Gluzman, V. Mityushev, Series, index and threshold for random 2D composite, Archives of Mechanics, 67, 1, 75–93, 2015.
  • 25. S. Gluzman, V. Mityushev, W. Nawalaniec, G. Sokal, Random composite: stirred or shaken?, Archives of Mechanics, 68, 3, 229–241, 2016.
  • 26. V.Ya. Natanson, On the stresses in a stretched plate weakened by identical holes located in chessboard arrangement, Matematiceskij Sbornik, 42(5), 616–636, 1935 [in Russian].
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  • 28. R. Guinovart-Diaz, J. Bravo-Castillero, R. Rodriguez-Ramos, F.J. Sabina, Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents-I. Elastic and hexagonal symmetry, Journal of the Mechanics and Physics of Solids, 49, 1445–1462, 2001.
  • 29. A.B. Movchan, N.A. Nicorovici, R.C. McPhedran, Green’s tensors and lattice sums for elastostatics and elastodynamics, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 453, 643–662, 1997.
  • 30. R.C. McPhedran, A.B. Movchan, The Rayleigh multipole method for linear elasticity, Journal of the Mechanics and Physics of Solids, 42, 5, 711–727, 1994.
  • 31. P. Drygaś, V. Mityushev, Contrast expansion method for elastic incompressible fibrous composites advances in mathematical physics, 2017, 2017, Article ID 4780928, 11 pp., https://doi.org/10.1155/2017/4780928.
  • 32. S.G. Lekhnitskii, Theory of Elasticity of an Anisotropic Elastic Body, Mir Publishers, Moscow, 1981.
  • 33. L.A. Filshtinsky, V. Mityushev, Mathematical Models of Elastic and Piezoelectric Fields in Two-Dimensional Composites, Mathematics Without Boundaries, Springer New York, 217–262, 2014.
  • 34. V. Mityushev, S. Rogosin, Constructive Methods for Linear and Nonlinear. Boundary Value Problems for Analytic Functions: Theory and Applications, Chapman & Hall / CRC, Monographs and Surveys in Pure and Applied Mathematics, Boca Raton, 2000.
  • 35. V. Mityushev, Thermoelastic plane problem for material with circular inclusions, Archives of Mechanics, 52, 6, 915–932, 2000.
  • 36. S. Yakubovich, P. Drygas, V. Mityushev, Closed-form evaluation of two-dimensional static lattice sums, Proceedings Mathematical, Physical, and Engineering Sciences. 2016;472(2195):20160510. doi:10.1098/rspa.2016.0510.
  • 37. S. Gluzman, V. Mityushev, W. Nawalaniec, Computational Analysis of Structured Media, Elsevier, 2017.
  • 38. S. Jun, I. Jasiuk, Elastic moduli of two-dimensional composites with sliding inclusions –a comparison of effective medium theories, International Journal of Solids and Structures, 30, 18, 2501–2523, 1993.
  • 39. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Springer, New York, 2002.
  • 40. S. Gluzman, D. Karpeev, Perturbative Expansions and Critical Phenomena in Random Structured Media, [in:] Modern Problems in Applied Analysis, P. Drygas and S. Rogosin [eds.], Birkhäuser, 2017 (to appear).
  • 41. S. Gluzman, V.I. Yukalov, Additive self-similar approximants, Journal of Mathematical Chemistry, 55, 607–622, 2017, doi:10.1007/s10910-016-0698-4.
  • 42. G.A. Baker, P. Graves-Moris, Padé Approximants, Cambridge University, Cambridge, 1996.
  • 43. P. Drygas, S. Gluzman, V. Mityushev, W. Nawalaniec, Effective elastic constants of hexagonal array of soft fibers, Computational Materials Science, 139, 395–405, 2017.
Uwagi
PL
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-b9da7ea4-ce88-4188-bc8c-ffda29d6158c
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