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Archive of Applied Mechanics

, Volume 88, Issue 11, pp 2081–2099 | Cite as

A least squares approach for effective shear properties in an \({{\varvec{n}}}\)-layered sphere model

  • Rolf Mahnken
  • Peter Lenz
  • Christian Dammann
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  • 40 Downloads

Abstract

This work presents the derivation of the effective shear modulus for a heterogeneous material composed of multilayered composite spheres embedded in a linear elastic matrix. It is based on the composite spheres model known from the literature. In contrast to Herve and Zaoui (Int J Eng Sci 31:1–10, 1993), the effective shear modulus is obtained by equating the results of two models: In the first model, a heterogeneous sphere is embedded in an equivalent homogeneous material, whereas in the second model, the heterogeneous sphere is replaced by an equivalent homogeneous sphere. In the context of both, a shear stress approach and a shear deformation approach, this results in an overdetermined system of equations which is solved with the least squares method. In a numerical study, our results are compared to effective moduli and bounds from the literature. Furthermore, a convincing agreement with experimental data for glass microspheres embedded in a polyester matrix is demonstrated.

Keywords

Heterogeneous materials Effective properties Composite sphere model RVE Homogenization 

Notes

Acknowledgements

This work is based on investigations of the “SPP 1712 - Intrinsische Hybridverbunde für Leichtbautragstrukturen,” which is kindly supported by the Deutsche Forschungsgemeinschaft (DFG).

References

  1. 1.
    Bulut, O., Kadioglu, N., Ataoglu, S.: Absolute effective elastic constants of composite materials. Struct. Eng. Mech. 57(5), 897–920 (2016)CrossRefGoogle Scholar
  2. 2.
    Christensen, R.M.: Mechanics of Composite Materials. Dover Publications, Mineola (1979)Google Scholar
  3. 3.
    Christensen, R.M., Lo, K.H.: Solutions for effective shear properties in three phase sphere and cylinder models. J. Mech. Phys. Solids 27(4), 315–330 (1979)CrossRefGoogle Scholar
  4. 4.
    Dvorak, G.: Micromechanics of Composite Materials. Springer, Berlin (2013)CrossRefGoogle Scholar
  5. 5.
    Gusev, A.: Effective coefficient of thermal expansion of n-layered composite sphere model: exact solution and its finite element validation. Int. J. Eng. Sci. 84, 54–61 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hashin,Z.: The elastic moduli of heterogeneous materials. Division of Engineering and Applied Physics, Harvard University, Contract Nonr 1866(2), TR 9, (1960)Google Scholar
  7. 7.
    Hashin, Z.: The elastic moduli of heterogeneous materials. J. Appl. Mech. 29(1), 143–150 (1962)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hashin, Z., Shtrikman, S.: A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11(2), 127140 (1963)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Herve, E., Zaoui, A.: n-layered inclusion-based micromechanical modelling. Int. J. Eng. Sci. 31, 1–10 (1993)CrossRefGoogle Scholar
  10. 10.
    Kerner, E.H.: The elastic and thermo-elastic properties of composite media. Proc. Phys. Soc. Lond. Sect. B 69(8), 808 (1956)CrossRefGoogle Scholar
  11. 11.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity, 4th edn. Dover Publications, Mineola (1944)zbMATHGoogle Scholar
  12. 12.
    Mahnken, R., Dammann, C.: A three-scale framework for fibre-reinforced-polymer curing Part I: microscopic modeling and mesoscopic effective properties. Int. J. Solids Struct. 100, 341–355 (2016)CrossRefGoogle Scholar
  13. 13.
    Mahnken, R., Dammann, C.: A three-scale framework for fibre-reinforced-polymer curing Part II: mesoscopic modeling and macroscopic effective properties. Int. J. Solids Struct. 100, 356–375 (2016)CrossRefGoogle Scholar
  14. 14.
    Mahnken, R., Dammann, C., Lenz, P.: (n)- and (n+1)-layered composit sphere models for thermo-chemo-mechanical effective properties. Int. J. Multiscale Comput. Eng. 15(4), 295–322 (2017)CrossRefGoogle Scholar
  15. 15.
    Richard, T.G.: The mechanical behavior of a solid microsphere filled composite. J. Compos. Mater. 9(2), 108–113 (1975)CrossRefGoogle Scholar
  16. 16.
    Slaughter, W.: The Linearized Theory of Elasticity, 1st edn. Birkhäuser, Basel (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Chair of Engineering MechanicsUniversity of PaderbornPaderbornGermany

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