# Homogenized Balance Equations for Nonlinear Poroelastic Composites
###### Abstract
In the literature, the homogenization of systems involving nonlinear mechanics is generally carried out by other homogenization techniques, such as average field techniques (see, e.g., [29,30] and the references therein). These other techniques do not provide a precise description of the model coefficients in the way that the asymptotic homogenization technique can and, instead, provide bounds for the model coefficients. There are currently only a few examples of using asymptotic homogenization in situations where the material undergoes large deformations. In [31], the effective poroelastic model of Biot was extended to a nonlinear Biot model that includes pore-scale deformation. In [32], a system of effective equations that describe the flow, elastic deformation and transport in an active medium was derived. The authors considered the spatial homogenization of a coupled transport and fluid-structure interaction model. In [33], the asymptotic homogenization technique was applied to the equations that describe the dynamics of a heterogeneous material with an evolving microstructure to obtain a set of effective equations. The heterogeneous body is assumed to be composed of two hyperelastic materials, and the evolution of the microstructure is through plastic-like distortions. We investigate materials that are subject to large deformations that have the underlying microstructure comprised of both a hyperelastic-fluid-filled porous matrix and a number of embedded hyperelastic subphases (fibres or inclusions). We then assume that both the matrix and the subphases interact with each other and the fluid at the pore-scale. This type of structure can be described as a poroelastic composite material that undergoes large deformations. For example, this is applicable to artery walls, which can be considered as a composite nonlinear elastic material consisting of a matrix with two families of symmetrically arranged embedded collagen and elastin fibres as well as fibroblast cells that interact with the fluid that is flowing in the pores [34, 35, 36, 12, 37]. If we wish to consider the artery walls in our framework, then the matrix can be identified with our matrix in the model, and the fibres and fibroblast cells could all be considered as the elastic subphases that are embedded in the matrix. The fluid in this setting would be the water that flows in the pores of the matrix. This modelling approach is applicable to the myocardium in the heart, which is a nonlinear elastic porous structure that consists of a matrix with muscle cells, fibroblasts, collagen fibres and embedded blood vessels [10; 11; 38]. Again, here, the various muscle cells, fibroblasts and fibres would all be the elastic subphases embedded in the matrix. By considering these systems as poroelastic composites, we are able to account for the mechanical contribution of each of the various phases individually. Here, we generalise [27] to account for nonlinear deformations by using the asymptotic homogenization technique to upscale the interaction between a hyperelastic porous matrix where there is an incompressible Newtonian fluid flowing in the pores and a number of embedded hyperelastic subphases. We make the assumption that both the hyperelastic matrix and subphases interact with the fluid that is flowing in the pores. We assume that the length scale where the individual hyperelastic subphases are clearly visible from the surrounding matrix is comparable to the pore size. We therefore determine that this scale will be the _pore-scale_ of the material, that is the distance between adjacent subphases is comparable with the size of the pores. This length is assumed to be much smaller than the size of the whole domain, which is denoted the _macroscale_. The upscaling process is then used taking into account the continuity of the stresses and elastic displacements across the interfaces between the matrix and the subphases, as well as the continuity of the stresses and velocities across the interfaces between the fluid and solid domains.