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Models of Composite Materials and Mathematical Methods of Their Investigation

  • Igor V. AndrianovEmail author
  • Jan AwrejcewiczEmail author
  • Vladyslav V. DanishevskyyEmail author
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 77)

Abstract

The linear theory of elasticity yields the following relations between the displacements \({{u}_{i}}\), deformations \({{\varepsilon }_{ij}}\) and stress \({{\sigma }_{ij}}\) in a continuous matter.

References

  1. 1.
    Christensen, R.M. 2003. Theory of viscoelasticity. Mineola, New York: Dover Publications.Google Scholar
  2. 2.
    Rabotnov, Yu.N. 1980. Elements of hereditary solid mechanics. Moscow: Mir.Google Scholar
  3. 3.
    Hashin, Z. 1965. Viscoelastic behavior of heterogeneous media. Journal of Applied Mechanics 8: 630–636.CrossRefGoogle Scholar
  4. 4.
    Hashin, Z. 1966. Viscoelastic fiber reinforced materials. AIAA Journal 8: 1411–1417.CrossRefzbMATHGoogle Scholar
  5. 5.
    Selivanov, M.F., and Yu.A. Chernoivan. 2007. A combined approach of the Laplace transform and Padé approximation solving viscoelasticity problems. International Journal of Solids and Structures 44: 66–76.Google Scholar
  6. 6.
    Kaminskii, A.A. 2000. Study of the deformation of anisotropic viscoelasctic bodies. International Applied Mechanics 36 (11): 1434–1457.CrossRefGoogle Scholar
  7. 7.
    Kaminskii, A.A., and M.F. Selivanov. 2003. A method for solving boundary-value problems of linear viscoelastisity for anisotropic composites. International Applied Mechanics 39 (11): 1294–1304.CrossRefGoogle Scholar
  8. 8.
    Kaminskii, A.A., and M.F. Selivanov. 2005. A method for determining the viscoelastic characteristics of composites. International Applied Mechanics 41 (5): 469–480.CrossRefGoogle Scholar
  9. 9.
    Lur’e, A.I. 1990. Nonlinear theory of elasticity. Amsterdam: North-Holland.zbMATHGoogle Scholar
  10. 10.
    Murnaghan, A.D. 1951. Finite deformation of an elastic solid. New York: Wiley.zbMATHGoogle Scholar
  11. 11.
    Landau, L., and G. Rumer. 1937. Über Schallabsorption in festen Korpern. Physikalische Zeitschrift der Sowjetunion 11: 18–23.zbMATHGoogle Scholar
  12. 12.
    Voigt, W. 1893. Über eine anscheinend notwendige Erweiterung der Theorie der Elasticit, Nachrichten von der Koniglichen Gesellschaft der Wissenschaften und der Georg-Augusts-Universitat zu Gottingen, 534–552.Google Scholar
  13. 13.
    Catheline, S., J.-L. Gennisson, and M. Fink. 2003. Measurement of elastic nonlinearity of soft solid with transient elastography. JASA 114: 3087–3091.CrossRefGoogle Scholar
  14. 14.
    Egle, D.M., and D.T. Bray. 1976. Measurement of acoustoelastic and third-order elastic constants for rail steel. JASA 60: 741–744.CrossRefGoogle Scholar
  15. 15.
    Franzevich, I.N., F.F. Voronov, and S.A. Bakuta. 1982. Elastic constants and modules of elasticity of metals and nonmetals. Naukova Dumka, Kiev: Reference Book. (in Russian).Google Scholar
  16. 16.
    Huges, D.S., and I.L. Kelly. 1953. Second-order elastic deformation of solids. Physical Review 92: 1145–1156.CrossRefzbMATHGoogle Scholar
  17. 17.
    Porubov, A.V. 2009. Localization of nonlinear strain waves: Asymptotic and numerical methods. Moscow: Fizmatlit. (in Russian).Google Scholar
  18. 18.
    Ogden, R.W. 1997. Nonlinear elastic deformations. New York: Dover.zbMATHGoogle Scholar
  19. 19.
    Torquato, S. 1991. Random heterogeneous media: Microstructure and improved bounds on the effective properties. Applied Mechanics Reviews 44: 37–76.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Sahimi, M. 2003. Heterogeneous materials. New York: Springer.zbMATHGoogle Scholar
  21. 21.
    Stauffer, D., and A. Aharony. 1994. Introduction to percolation theory. London: Taylor and Francis.zbMATHGoogle Scholar
  22. 22.
    Torquato, S. 2002. Random heterogeneous materials. Microstructure and macroscopic properties. New York: Springer.CrossRefzbMATHGoogle Scholar
  23. 23.
    Mityushev, V.V., E. Pesetskaya, and S.V. Rogosin. 2008. Analytical methods for heat conduction in composites and porous media. In Cellular and porous materials: Thermal properties simulation and prediction, ed. A. Öchsner, G.E. Murch, and M.J.S. de Lemos, 121–164. Weinheim: Wiley-VCH.CrossRefGoogle Scholar
  24. 24.
    Snarskii, A.A. 2007. Did Maxwell know about the percolation threshold? (on the 15th anniversary of percolation theory). Physics-Uspekhi 50 (12): 1239–1242.CrossRefGoogle Scholar
  25. 25.
    Bergman, D.J. 2007. The self-consistent effective medium approximation (SEMA): New tricks from an old dog. Physica B 394: 344–350.CrossRefGoogle Scholar
  26. 26.
    Doetsch, G. 1974. Introduction to the theory and application of the Laplace-transformation. Berlin: Springer.CrossRefzbMATHGoogle Scholar
  27. 27.
    Tranter, C.J. 1971. Integral transforms in mathematical physics. London: Chapman and Hall.zbMATHGoogle Scholar
  28. 28.
    Litvinov, G.L. 1994. Approximate construction of rational approximations and the effect of autocorrection error. Russian Journal of Mathematical Physics 1 (3): 313–352.zbMATHGoogle Scholar
  29. 29.
    Luke, Y.L. 1980. Computations of coefficients in the polynomials of Padé approximants by solving systems of linear equations. Journal of Computational and Applied Mathematics 6 (3): 213–218.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Luke, Y.L. 1982. A note on evaluation of coefficients in the polynomials of Padé approximants by solving systems of linear equations. Journal of Computational and Applied Mathematics 8 (2): 93–99.CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Longman, I.M. 1973. On the generalization of rational function applied to Laplace transform inversions, with an application to viscoelasticity. SIAM Journal on Applied Mathematics 24: 429–440.CrossRefzbMATHGoogle Scholar
  32. 32.
    Bateman, H., and A. Erdélyi (eds.). 1954. Tables of integral transformations, vol. 1. New York: McGraw-Hill.Google Scholar
  33. 33.
    Sveshnikov, A.G., and A.N. Tikhonov. 1978. The theory of functions of a complex variable. Moscow: Mir.zbMATHGoogle Scholar
  34. 34.
    Abate, J., and W. Whitt. 2006. A unified framework for numerically inverting Laplace transforms. INFORMS Journal on Computing 18: 408–421.CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Nayfeh, A.H. 2000. Perturbation methods. New York: Wiley.CrossRefzbMATHGoogle Scholar
  36. 36.
    Lomov, S.A. 1992. Introduction to the general theory of singular perturbations. Providence, RI: AMS.zbMATHGoogle Scholar
  37. 37.
    Kantorovich, L.V., and V.I. Krylov. 1958. Approximate methods of higher analysis. Groningen: Noordhoff.zbMATHGoogle Scholar
  38. 38.
    Vishik, M.I., and L.A. Lyusternik. 1960. The asymptotic behaviour of solutions of linear differential equations with large or quickly changing coefficients and boundary conditions. Russian Mathematical Surveys 15 (4): 23–91.CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Bakhvalov, N., and G. Panasenko. 1989. Averaging processes in periodic media. Mathematical problems in mechanics of composite materials. Dordrecht: Kluwer.Google Scholar
  40. 40.
    Wiener, O. 1889. Die Theorie des Mischkörpers für das Feld der stationären Strömung. Erste Abhandlung die Mittelwertsätze für Kraft, Polarisation und Energie, Abhandlungen der Mathematisch-Physischen Klasse. der Königlich Sächsischen Gesellschaft der Wissenschaften 32 (6): 507–604.Google Scholar
  41. 41.
    Bourgat, J.F. 1979. Numerical experiments of the homogeneisation method for operators with periodic coefficients. Lectures Notes in Mathematics 704: 330–356.CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Nayfeh, A.H. 1981. Introduction to perturbation techniques. New York: Wiley.zbMATHGoogle Scholar
  43. 43.
    Abramowitz, M., and I.A. Stegun (eds.). 1965. Handbook of mathematical functions, with formulas, graphs, and mathematical tables. New York: Dover Publications.zbMATHGoogle Scholar
  44. 44.
    Bakhvalov, N.S., and M.E. Eglit. 1995. The limiting behavior of periodic media with soft media inclusions. Computational Mathematics and Mathematical Physics 35 (6): 719–730.zbMATHMathSciNetGoogle Scholar
  45. 45.
    Guz, A.N., and Yu.N. Nemish. 1987. Perturbation of boundary shape in continuum mechanics (review). Soviet Applied Mechanics 23 (9): 799–822.Google Scholar
  46. 46.
    Henry, D., and J. Hale. 2005. Perturbation of the boundary in boundary value problems of partial differential equations. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  47. 47.
    Hinch, E.J. 1991. Perturbation methods. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  48. 48.
    Christensen, R.M. 2005. Mechanics of composite materials. Mineola, New York: Dover Publications.Google Scholar
  49. 49.
    Tayler, A.B. 2001. Mathematical models in applied mechanics. Oxford: Clarendon Press.zbMATHGoogle Scholar
  50. 50.
    Fadle, J. 1940. Die Selbstspannungs-Eigenwertfunktionen der quadratischen Scheibe. Österreich Ingenieur-Archive 11 (2): 125–149.CrossRefzbMATHMathSciNetGoogle Scholar
  51. 51.
    Papkovich, P.F. 1940. On the form of solution of the plane problem of the theory of elasticity for a rectangular strip. Doklady Akademii Nauk SSSR 27: 335–339.Google Scholar
  52. 52.
    Van Dyke, M. 1975. Perturbation methods in fluid mechanics. Stanford: The Parabolic Press.zbMATHGoogle Scholar
  53. 53.
    Baker, G.A. 1975. Essential of padé approximants. N.Y.: Academic Press.zbMATHGoogle Scholar
  54. 54.
    Baker, G.A., and P. Graves-Morris. 1996. Padé approximants, 2nd ed. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar
  55. 55.
    Bender, C.M., and S.A. Orszag. 1978. Advanced mathematical methods for scientists and engineers. New York: McGraw-Hill.zbMATHGoogle Scholar
  56. 56.
    Suetin, S.P. 2002. Padé approximants and efficient analytic continuation of a power series. Russian Mathematical Surveys 57 (1): 43–141.CrossRefzbMATHMathSciNetGoogle Scholar
  57. 57.
    Vyatchin, A.V. 1982. On the convergence of Padé approximants. Moscow University Mathematics Bulletin 37 (4): 1–4.zbMATHMathSciNetGoogle Scholar
  58. 58.
    Vinogradov, V.N., E.V. Gay, and N.C. Rabotnov. 1987. Analytical Approximation of Data in Nuclear and Neutron Physics. Moscow: Energoatomizdat. (in Russian).Google Scholar
  59. 59.
    Andersen, C.M., M.B. Dadfar, and J.F. Geer. 1984. Perturbation analysis of the limit cycle of the Van der Pol equation. SIAM Journal on Applied Mathematics 44 (5): 881–895.CrossRefzbMATHMathSciNetGoogle Scholar
  60. 60.
    Slepyan, L.I., and Yu.S. Yakovlev. 1980. Integral transforms in the nonstationary problems of mechanics. Leningrad: Sudostroyenie. (in Russian).Google Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Institut für Allgemeine MechanikRWTH Aachen UniversityAachenGermany
  2. 2.Automation, Biomechanics and MechatronicsLodz University of TechnologyŁódźPoland
  3. 3.School of Computing and MathematicsKeele UniversityKeeleUK

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