Keywords: large deformation, microstructure, homogenization, porous media, fluid-structure interaction, hereditary creep, numerical modelling, soft tissues.

In this paper we treat interactions between solid and fluid media at the microscopic level. This phenomenon is responsible for viscoelastic behaviour observed at the macroscopic scale where the material model is described in terms of the homogenized (effective) parameters. Our approach allows to study relationships between the two scales, pursuing detail interactions of a given porous material at the scale of the inhomogeneities. The so-called micromodel is based on that of Biot, cf. [1]; here we consider its quasi-static version extended for large deformation by considering the stress-equilibrium equation in terms of the incremental updated Lagrangian (UL) formulation. The micromodel is defined for finite inhomogeneities of the scale w.r.t. characteristic length of ; the weak formulation of the subproblem for unknown incremental fields of displacements , incremental pressure and actual perfusion velocity incorporates the quasi-static equilibrium equation, the Darcy law and the volume conservation condition (overall incompressibility of the porous medium).

The homogenization for is based on the two scale convergence method. We obtain the local "cell-problems" to be solved for local RVEs, denoted by ; these are related to specific points of the macroscopic domain . The fields (displacements, pressures and perfusion velocities) in the microscopic cells are computed by solving the associated boundary value problems. In the context of thermodynamics, in this way one introduces the internal variables which present the relationship between the macro- and microscopic responses and which account for history of deformation. In particular, for all the fields we define the characteristic and the particular responses (e.g. for displacement , and , respectively), so that the local cell is updated by , where is the local macroscopic deformation. In a similar sense we decompose also the microscopic response of pressures and velocities.

By virtue of the homogenization method, cf. [2], one obtains the following effective material coefficients: the homogenized tangent stiffness tensor as a function of the characteristic responses and of the time discretization step , the averaged Cauchy stress computed using microscopic stress, and the retardation stress as a function of the particular microscopic response.

As a particular microstructure we consider material with quasi-periodic distribution of fluid capsules embedded in the porous matrix. In this situation the macroscopic pressure is that in the microscopic capsules. The local RVE problems describe coupled deformation-diffusion processes in cell , accounting for the fluid flow in the matrix compartment and through membranes of the capsules. On the macroscale we compute and such that the UL equilibrium conditions hold. Due to large deformation, only local periodicity of the microstructure is preserved, so that the microscopic responses should be computed for many micro-cells associated with integration points of the macroscopic finite element model. We have dealt with two approaches to overcome this hurdle: 1) parallel algorithm for computation of micro-responses, which is discussed in this paper; 2) use of sensitivity analysis [3] for an approximate recovery of homogenized coefficients. The latter approach has proved its viability in application to hyperelastic materials [2].

The multiscale approach seems to be promising for application in biomechanics, where the coupling between dissipative processes related to the induced viscoelasticity and microflow is essential, and is responsible for redistribution of dissolved species with relationships to growth and tissue remodelling. In Figure IRefrohan:fig the deformation and perfusion through microstructure of smooth muscle cell is illustrated for two time instants of the contraction process. We discuss numerical aspects of the modelling. (The research has been supported generously by the project LN00B084 of The New Technologies Research Centre in Pilsen.)

1

Murad, A.M., Guerreiro, J.N., Loula, A.F.D., "Micromechanical computational modelling of secondary consolidation and hereditary creep in soils", Comput. Methods Appl. Mech. Engrg. 190, 1985-2016, 2001. doi:10.1016/S0045-7825(00)00218-8

2

Rohan, E., "Mathematical modelling of soft tissues", Habilitation thesis, University of West Bohemia, Plzen, 2002.

3

Rohan, E., "Sensitivity strategies in modelling heterogeneous media undergoing finite deformation", Math. and Computers in Simul., 61, 261-270, 2003. doi:10.1016/S0378-4754(02)00082-4

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