## Abstract

Biological materials typically consist of elastic fibers immersed in an incompressible aqueous milieu. We consider the generality of an elastic material expressed as a fiber-reinforced incompressible fluid. We show that, in the linear regime, any (possibly inhomogeneous and/or anisotropic) incompressible elastic material can be represented as a collection of fifteen families of straight, parallel elastic fibers embedded in an incompressible medium. We can choose these fiber directions to correspond to the fifteen diagonals of an icosahedron that connect the midpoints of its antipodal edges. This fiber architecture, together with the incompressible medium in which it is immersed, is universal and programmable in the sense that its elastic constants can be chosen to model any linear incompressible elastic material, without having to adapt the fiber architecture to the actual microstructure of the material. An explicit algorithm is given to compute the local elastic constants for each fiber direction in terms of the local components of the elasticity tensor. Optimality properties of the icosahedral fiber architecture are conjectured, and numerical evidence in support of these conjectures is presented.

Original language | English (US) |
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Pages (from-to) | 75-100 |

Number of pages | 26 |

Journal | Advances in Applied Mathematics |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - Jul 2009 |

Externally published | Yes |

### Bibliographical note

Funding Information:We thank Luca Heltai and Oliver Bühler for helpful discussions. We especially thank Oliver Büh-ler for pointing out the dimensional considerations mentioned after the statement of Theorem 1. We also thank the anonymous reviewer for providing insightful comments and pointing to important references in the theory of composites and their elastic properties. Y.M. was supported by the Mc-Cracken fellowship and the Dissertation fellowship of New York University during the course of this work. Y.M. also acknowledges the support by the Mathematics of Information Technology and Complex Systems (MITACS, Canada) team grand (Leah Keshet) and by the Natural Sciences and Engineering Research Council (NSERC) discovery grant (Leah Keshet).

## Keywords

- Biological fluids
- Composite materials
- Fiber-reinforced fluid
- Icosahedron
- Incompressible elasticity