Michael Eden, Hari Shankar Mahato
We study the multiscale behavior of a coupled visco-poroelastic system arising in the modelling of fibre-reinforced hydrogels (FIHs) used in tissue engineering scaffolds. The composite material consists of a periodic fibre scaffold, which is governed by quasi-static linear elasticity, and a hydrogel phase saturating the interstitial space, which is modelled as a Biot linear poroelastic medium enhanced with Kelvin--Voigt structural damping. The two phases are coupled through continuity of displacement and traction across their shared interface. Both the fibre scaffold and the hydrogel phase are connected, so that mechanical forces can be transmitted through the composite and interstitial fluid can flow directly through the hydrogel network. Starting from a microscopic ($\varepsilon$-scale) model, we derive uniform a-priori estimates and establish well-posedness via a Rothe time-discretisation argument for both the case of standard Biot fluid content $η=0$ and the case of viscous fluid content $η=αδ>0$. We then perform a rigorous two-scale homogenization in the limit $\varepsilon \to 0$ using periodic unfolding. In addition to the usual effective elasticity, storage, coupling, and permeability coefficients, the homogenized constitutive laws contain nonlocal-in-time memory terms generated by the microscopic viscoelastic relaxation. All effective coefficients and memory kernels are explicitly characterized in terms of the microscale geometry and material parameters.