Multiscale FE simulation of diffusion-deformation processes in homogenized dual-porous media

AbstractThe paper deals with a model of the homogenized fluid saturated porous material which recently was obtained by the authors using the asymptotic analysis of the Biot type medium characterized by the double porosity. The homogenized macroscopic model is featured by the fading memory effects arising from the microflow in the dual porosity. We derive the steady state formulations and discuss several topics related to the numerical implementation of the model, namely the solution procedure of the discretized microscopic problems, evaluation of the homogenized coefficients and an approximation of the convolution integrals of the macroscopic model, so that the fading memory effects are computationally tractable. Numerical examples are presented to illustrate the approximation schemes discussed in the paper. All computations were performed using the in-house developed finite element code SfePy allowing the multiscale simulations. Besides various potential engineering applications, the present model is intended for simulations of compact bone poroelasticity.

[1]  Claude Boutin,et al.  Deformable porous media with double porosity III: Acoustics , 1994 .

[2]  Claude Boutin,et al.  Deformable porous media with double porosity. Quasi-statics. I: Coupling effects , 1992 .

[3]  R. E. Showaltera,et al.  Double-diffusion models from a highly-heterogeneous medium ✩ , 2004 .

[4]  Eduard Rohan,et al.  Numerical modelling and homogenized constitutive law of large deforming fluid saturated heterogeneous solids , 2006 .

[5]  Vladimír Lukes,et al.  Microstructure based two-scale modelling of soft tissues , 2010, Math. Comput. Simul..

[6]  S. Cowin.  Bone mechanics handbook , 2001 .

[7]  U. Hornung.  Homogenization and porous media , 1996 .

[8]  Eduard Rohan,et al.  Sensitivity strategies in modelling heterogeneous media undergoing finite deformation , 2003, Math. Comput. Simul..

[9]  H. I. Ene,et al.  Model of diffusion in partially fissured media , 2002 .

[10]  Michal Šejnoha,et al.  From random microstructures to representative volume elements , 2007 .

[11]  A F Mak,et al.  Numerical simulation of streaming potentials due to deformation-induced hierarchical flows in cortical bone. , 2001, Journal of biomechanical engineering.

[12]  S. Cowin,et al.  On the calculation of bone pore water pressure due to mechanical loading , 1998 .

[13]  Malgorzata Peszynska.  Finite element approximation of diffusion equations with convolution terms , 1996, Math. Comput..

[14]  S. Cowin,et al.  Bone Mechanics Handbook, 2nd Edition. - , 2003 .

[15]  Salah Naili,et al.  Multiscale modeling of a fluid saturated medium with double porosity: Relevance to the compact bone , 2012 .

[16]  Todd Arbogast,et al.  Derivation of the double porosity model of single phase flow via homogenization theory , 1990 .

[17]  Claude Boutin,et al.  Deformable porous media with double porosity. Quasi-statics. II: Memory effects , 1993 .

[18]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[19]  E. Sanchez-Palencia.  Non-Homogeneous Media and Vibration Theory , 1980 .

[20]  R. A. Brand,et al.  Modeling Deformation-Induced Fluid Flow in Cortical Bone’s Canalicular–Lacunar System , 2005, Annals of Biomedical Engineering.

[21]  R. de Boer,et al.  Theory of Porous Media , 2020, Encyclopedia of Continuum Mechanics.

[22]  Eduard Rohan,et al.  Modeling Large-deformation-induced Microflow in Soft Biological Tissues , 2006 .

[23]  Eduard Rohan,et al.  Two-Scale Modeling of Tissue Perfusion Problem Using Homogenization of Dual Porous Media , 2010 .

[24]  Doina Cioranescu,et al.  The Periodic Unfolding Method in Homogenization , 2008, SIAM J. Math. Anal..

[25]  Thibault Lemaire,et al.  Multiscale analysis of the coupled effects governing the movement of interstitial fluid in cortical bone , 2006, Biomechanics and modeling in mechanobiology.

[26]  P. Donato,et al.  An introduction to homogenization , 2000 .

[27]  Georges Griso,et al.  On the homogenization of a diffusion–deformation problem in strongly heterogeneous media , 2007 .

[28]  Salah Naili,et al.  Transverse isotropic poroelastic osteon model under cyclic loading , 2005 .

[29]  Ajh Arjan Frijns,et al.  A four-component mixture theory applied to cartilaginous tissues : numerical modelling and experiments , 2000 .

[30]  Ralph E. Showalter,et al.  Double-diffusion models from a highly-heterogeneous medium☆ , 2004 .