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Continuum Mechanics and Thermodynamics

, Volume 31, Issue 1, pp 101–132 | Cite as

Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates

  • Patrick FranciosiEmail author
  • Mario Spagnuolo
  • Oguz Umut Salman
Original Article

Abstract

Composites comprising included phases in a continuous matrix constitute a huge class of meta-materials, whose effective properties, whether they be mechanical, physical or coupled, can be selectively optimized by using appropriate phase arrangements and architectures. An important subclass is represented by “network-reinforced matrices,” say those materials in which one or more of the embedded phases are co-continuous with the matrix in one or more directions. In this article, we present a method to study effective properties of simple such structures from which more complex ones can be accessible. Effective properties are shown, in the framework of linear elasticity, estimable by using the global mean Green operator for the entire embedded fiber network which is by definition through sample spanning. This network operator is obtained from one of infinite planar alignments of infinite fibers, which the network can be seen as an interpenetrated set of, with the fiber interactions being fully accounted for in the alignments. The mean operator of such alignments is given in exact closed form for isotropic elastic-like or dielectric-like matrices. We first exemplify how these operators relevantly provide, from classic homogenization frameworks, effective properties in the case of 1D fiber bundles embedded in an isotropic elastic-like medium. It is also shown that using infinite patterns with fully interacting elements over their whole influence range at any element concentration suppresses the dilute approximation limit of these frameworks. We finally present a construction method for a global operator of fiber networks described as interpenetrated such bundles.

Keywords

Composites Fiber networks Green operator Effective properties Phase co-continuity 

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Notes

Acknowledgements

Mario Spagnuolo has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 665850.

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

Authors and Affiliations

  1. 1.LSPM UPR 3407 CNRS, University of Paris13, USPCVilletaneuseFrance

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