A universal programmable fiber architecture for the representation of a general incompressible linearly elastic material as a fiber-reinforced fluid

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.