Coupling among deformation, fluid flow, structural reorganisation and fibre reorientation in fibre-reinforced, transversely isotropic biological tissues

We highlight some mechanical aspects of the coupling among deformation, fluid flow, structural evolution, and reorientation of fibres in fibre-reinforced, hydrated, soft biological tissues. For our purposes, we elaborate a model in which the tissue’s interstitial fluid is inviscid and obeys Darcy’s law, and the solid constituents are transversely isotropic, hyperelastic materials. Within this setting, we consider two different types of remodelling: One consists of the reorientation of the fibres, while the other one is the manifestation, at the tissue scale, of structural rearrangements representable in terms of inelastic distortions. Our focus is on the interplay between the latter ones and the fibre reorientation. In our model, such interplay is a consequence of the constitutive framework, which resolves explicitly the space variability of a parameter, the “fibre mean angle”, that determines the direction along which the fibres tend to align themselves. Our main results concern the description of a Mandel-like stress tensor, which drives the inelastic distortions when the fibre mean angle is distributed inhomogeneously throughout the tissue, and of a diffusion-like tensor depending on the inelastic distortions, which guides the evolution of the fibre mean angle