# Homogenized Balance Equations for Nonlinear Poroelastic Composites
###### Abstract  
Within this work, we upscale the equations that describe the pore-scale behaviour of nonlinear porous elastic composites, using the asymptotic homogenization technique in order to derive the macroscale effective governing equations. A porous hyperelastic composite can be thought of as being comprised of a matrix interacting with a number of subphases and percolated by a fluid flowing in the pores (which is chosen to be Newtonian and incompressible here). A general nonlinear macroscale model is derived and is then specified for a particular choice of strain energy function, namely the de Saint-Venant function. This leads to a macroscale system of PDEs, which is of poroelastic type with additional terms and transformations to account for the nonlinear behaviour of the material. Our new porohyperelastic-type model describes the effective behaviour of nonlinear porous composites by prescribing the stress balance equations, the conservation of mass and Darcy's law. The coefficients of these macroscale equations encode the detailed microstructure of the material and are to be found by solving pore-scale differential problems. The model reduces to the following limit cases of (a) linear poroelastic composites when the deformation gradient approaches the identity, (b) nonlinear composites when there are no pores and (c) nonlinear poroelasticity when only the matrix-fluid interaction is considered. This model is applicable when the interactions between various hyperelastic solid phases occur at the pore-scale, as in biological tissues such as artery walls, the myocardium, lungs and liver. asymptotic homogenization; nonlinear elasticity; poroelastic composites; pore-scale 12 May 20  
a scale (called the _pore-scale_) where the interactions between the fluid and the solid take place, and this scale is much smaller than the scale at which we describe the whole material (which is called the _macroscale_). The coupled fluid-structure balance equations that describe the material on the pore-scale can be used in an upscaling process to obtain the macroscale governing equations of a poroelastic material. The upscaling process can be carried out by a variety of _homogenization_ techniques. These homogenization techniques include effective medium theory, mixture theory, volume averaging and asymptotic homogenization. These techniques were discussed and described in [18,19]. We develop our analysis by means of the asymptotic homogenization technique [20, 21, 22, 23]. This technique requires a sharp length-scale separation, between the pore-scale and the macroscale, to enable the decoupling of spatial variations that exist in the system. This technique expresses the relevant fields from the pore-scale fluid-structure interaction problem as power series of the ratio between the two different typical length scales. The balance equations obtained via this method describe the effective behaviour of the material in the homogenized macroscale domain. A key feature of this technique is that the coefficients of the macroscale model encode information about the microstructure of the material, and they are to be computed by solving differential problems at the pore-scale. In the literature, the asymptotic homogenization technique has been applied to various physical systems. For example, the asymptotic homogenization technique has been used to derive Biot's poroelasticity (see [24,25]) and extended to study vascularised poroelastic materials [26], poroelastic composites [27] and modelling of the bones [28]. In the asymptotic homogenization literature, the majority of the applications focus on linearised balance equations. This is due to the fact that in the linearised case, it is possible to fully decouple the pore-scale and the macroscale (under some simplifying assumptions). This decoupling then leads to a large reduction in the computational complexity of the system.