# Homogenized Balance Equations for Nonlinear Poroelastic Composites
## 5 Concluding Remarks  
We derived a novel framework consisting of partial differential equations that describe the effective mechanical behaviour of nonlinear poroelastic composites. These structures are comprised of a porous hyperelastic matrix with embedded hyperelastic subphases, both of which interact with the fluid flowing in the pores. This type of structure is applicable to many real-world situations, including modelling of soft biological tissues. We began by considering the multiphase fluid-structure interaction (FSI) problem among all the constituents. The problem is closed by appropriate interface conditions arising from the continuity of stresses, displacements and velocities. We also performed a coordinate transformation on certain quantities in the FSI problem in order to obtain a formulation in the reference configuration. We exploited the length scale separation between the pore-scale (where the pores and elastic subphases are clearly visible) and the macroscale (average size of the material domain) to apply the asymptotic homogenization technique to the nondimensionalised system of PDEs in order to obtain a macroscale system of governing equations. We were able to recover previously known results in the literature by considering particular limit cases of our model. The new model encodes the detailed properties of the microstructure in its coefficients, that is the microstructural details are encoded in the effective hydraulic conductivity tensor, the Biot modulus and Biot's tensor of coefficients. These are computed by solving the arising differential problems on the periodic cell. The model obtained here is a generalisation of the formulations for multiphase elastoplastic composites [33] in the limit of no plastic distortions and the formulations of hyperelastic porous media [31]. The model is also a natural extension to the formulation for linear poroelastic composites [27]. All three of these models are recovered as particular cases of our new macroscale model. The key novelty of this work is the ability to describe a scenario where the hyperelastic matrix is inhomogeneous at the pore-scale, that is we are able to account for the interactions between various hyperelastic phases and the fluid flowing in the pores. This is generally the case in biological tissues. This means that this model is applicable to a wide range of biological scenarios including modelling lungs. The lungs have previously been approached in a biphasic (tissue and air) manner [9]. However, the lung microstructure is more complex, and there exist collagen and elastin fibres embedded in the matrix and in the fluid, so it could therefore be beneficial to use a nonlinear poroelastic composite approach to modelling. Another example is [12], where the interaction between pulsatile blood flow and the arterial wall mechanics was modelled. The blood flow was modelled as an incompressible viscous fluid, confined by [PERSON]'s equations of poroelasticity for the artery wall. This model could be generalised by considering the wall as a nonlinear poroelastic composite of the type we modelled in this work. The formulation of standard nonlinear poroelasticity is applicable when the solid phase can be approximated as a homogeneous matrix. The linear formulation of poroelastic composites is applicable to situations where the deformations are small such as in hard hierarchical material such as bones (see, e.g., [28; 47]). Our novel model provides a formulation that bridges a gap that has not previously been considered and can be grouped with those found in [17] for strength homogenization, nonsaturated microporous, microporoplasticity and microporofracture and microporodamage theory as an extension to the nonlinear homogenization of porous media. There are some limitations of the current model, and there are a number possible theoretical extensions that could potentially improve its applicability to certain biological systems.