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.

Finite strain poro-hyperelasticity: an asymptotic multi-scale ALE-FSI approach supported by ANNs

“…This was not solved at the homogenised level although it was simplified by neglecting the effects of macroscopic deformation on RVE mechanical response. In [7] this problem is approached theoretically using asymptotic homogenisation of the Lagrangian description of the fluid-solid interaction (FSI) at the microscale. However, since the fluid phase can introduce extremely large "deformations", this approach might be suffering from inaccuracy and numerical instability.…”

“…Assuming a sharp length scale separation (between micro and macro scales) and initial local periodicity, the problem can be regularised, thus standard homogenisation approaches can be adopted [15]. Here, the resulting dimensionless ALE-FSI formulation is analysed using two-scale asymptotic homogenisation techniques [6,7,16,17]. This includes differential operator decoupling and power series representation of the relevant fields which leads to the expanded form of the equations.…”

Investigating the effects of microstructural changes induced by myocardial infarction on the elastic parameters of the heart

“…The theory has since been extended to model a vast range of scenarios including growth of poroelastic materials (Penta et al 2014), vascularised poroelastic materials (Penta and Merodio 2017) and poroelastic composites (Miller and Penta 2020). Recently there has also been a development of the theory for nonlinear poroelastic materials (Collis et al 2017;Brown et al 2014;Ramírez-Torres et al 2018) and nonlinear poroelastic composites (Miller and Penta 2021a). The theory has also been investigated with various additional scales such as poroelastic with inclusion (Royer et al 2019) and Chen et al (2018) and double poroelastic (Miller and Penta 2021b).…”

“… 2014 ; Ramírez-Torres et al. 2018 ) and nonlinear poroelastic composites (Miller and Penta 2021a ). The theory has also been investigated with various additional scales such as poroelastic with inclusion (Royer et al.…”

Multi-Scale Modeling and Simulation of Transport Processes in an Elastically Deformable Perforated Medium

“…However, the explicit representation of the effective coefficients was not specified. This was overcome for the pure fluid-structure interaction problem for porous media (without transport) in Miller and Penta (2020) for linear elastic and in Miller and Penta (2021) for hyperelastic multi-composite media. Further results related to our investigations deal with processes in porous media with an evolving microstructure.…”