Porous Fibre-Reinforced Materials Under Large Deformations Salvatore Federico Department of Mechanical and Manufacturing Engineering, The University of Calgary 2500 University Drive NW, Calgary, Alberta, T2N 1N4, Canada Phone: +1-403-220-5790; Facsimile: +1-403-282-8406; E-mail: salvatore.federico@ucalgary.ca Keywords: permeability, porous material, fibre-reinforced, large deformation. SUMMARY. Soft biological tissues can be represented by a porous matrix saturated by a fluid and reinforced by a network of statistically oriented, impermeable collagen fibres. A homogenisation method has been developed for porous fibre-reinforced materials with an isotropic matrix, under small deformations [2], and its application to articular cartilage correctly predicted some specific as- pects of the anisotropy and inhomogeneity of the permeability of the tissue [3]. The aim of this work is to generalise this model to the case of large deformations, and of matrix with an anisotropic per- meability. The whole framework is set in general coordinates, with the employment of the material and spatial metric tensors. 1 THEORETICAL BACKGROUND The Continuum Mechanics notation adopted here is almost identical to that in Marsden and Hughes [5]. With the exception of two-point tensors (an example of which is the deformation gra- dient F ), uppercase letters are reserved to material quantities, and lowercase letters to spatial quan- tities. In order to respect this customary use in modern Continuum Mechanics, a notation at times substantially different from that in [2, 3] has been adopted in the development of the theory. 1.1 Basic Continuum Mechanics The deformation is described by the configuration χ, mapping material points X =(X 1 ,X 2 ,X 3 ) in the reference configuration B⊆ R 3 into spatial points x =(x 1 ,x 2 ,x 3 ) in the natural Euclidean space S = R 3 . At each point X ∈B, the two-point tensor F (X): T X B→ T x S is the deformation gradient, mapping vectors in the tangent space T X B at X into vectors in the tangent space T x S at x = χ(X). F has components F i I = χ i ,I = ∂χ i /∂X I , and its determinant J = detF is the volumetric deformation ratio. The fully material tensor C = F T F , with components C I K = g ij F j J G JI F i K is the right Cauchy-Green deformation tensor, with inverse B = C −1 = F −1 F −T . The fully spatial tensor b = FF T , with components b i k = F i I g kj F j J G JI is the left Cauchy-Green deformation tensor, with inverse c = b −1 = F −T F −1 . The material identity tensor is I , with components I I J = δ I J . The material covariant metric tensor G has components G IJ , and its inverse G −1 , the material contravariant metric tensor, with components G IJ , is denoted G ♯ . Tensors G and G ♯ = G −1 , lower contravariant indices and rise covariant indices, respectively. For example, if W is a vector with (contravariant) components W I , its associated covector is W ♭ , with (covariant) components W I = G IJ W J . Conversely, if Q is a covector with (covariant) components Q I , its associated vector is Q ♯ , with (contravariant) components Q I = G IJ Q J . The definitions of the spatial identity tensor i, spatial covariant metric tensor g, and spatial contravariant metric tensor g ♯ = g −1 , as well as the use of the metric tensors to lower and rise indices, are analogous to the material case. 1