A Review of Mixture Theory for Deformable Porous Media and Applications

Mixture theory provides a continuum framework to model a multi-phase system. The basic assumption is, at any instant of time all phases are present at every material point and momentum and mass balance equations are postulated. This paper reviews the recent developments in mixture theory and focuses on the applications of the theory in particular areas of biomechanics, composite manufacturing and infiltration into deformable porous materials. The complexity based upon different permeability and stress functions is also addressed. The review covers the literature presented in the past fifty years and summarizes applications of mixture theory in specific areas of interest, for the sake of brevity, only necessary details are provided rather than complete modeling and simulation.

[1]  Angiolo Farina,et al.  Flow in deformable porous media: Modelling and simulations of compression moulding processes , 1997 .

[2]  Francesco dell’Isola,et al.  A visco-poroelastic model of functional adaptation in bones reconstructed with bio-resorbable materials , 2016, Biomechanics and Modeling in Mechanobiology.

[3]  J. Humphrey.  Continuum biomechanics of soft biological tissues , 2003 .

[4]  Luigi Preziosi,et al.  Multiphase and Multiscale Trends in Cancer Modelling , 2009 .

[5]  X. Peng,et al.  A mixture theory based constitutive model for SMA , 2000 .

[6]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[7]  Jay D. Humphrey,et al.  Review Paper: Continuum biomechanics of soft biological tissues , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  Reza Masoodi,et al.  Wicking in Porous Materials : Traditional and Modern Modeling Approaches , 2012 .

[9]  Francesco dell’Isola,et al.  “Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal , 2017 .

[10]  R. Decker,et al.  A continuum mixture theory with an application to turbulent snow, air flows and sedimentation , 1990 .

[11]  Boško P. Cvetković,et al.  An Application of Continuum Theory on the Case of One-Dimensional Sedimentation , 2011 .

[12]  Donald M. Anderson,et al.  CAPILLARY RISE OF A NON-NEWTONIAN LIQUID INTO A DEFORMABLE POROUS MATERIAL , 2011 .

[13]  Mark H. Holmes,et al.  COMPARISON THEOREMS AND SIMILARITY SOLUTION APPROXIMATIONS FOR A NONLINEAR DIFFUSION EQUATION ARISING IN THE STUDY OF SOFT TISSUE , 1984 .

[14]  Adnan H. Nayfeh,et al.  A Continuum Mixture Theory of Heat Conduction in Laminated Composites , 1975 .

[15]  S. I. Barry,et al.  Radial flow through deformable porous shells , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[16]  Donald M. Anderson,et al.  Imbibition of a liquid droplet on a deformable porous substrate , 2005 .

[17]  J. Tarbell,et al.  Modeling water flow through arterial tissue. , 1987, Bulletin of mathematical biology.

[18]  W M Lai,et al.  A triphasic theory for the swelling and deformation behaviors of articular cartilage. , 1991, Journal of biomechanical engineering.

[19]  W M Lai,et al.  Drag-induced compression of articular cartilage during a permeation experiment. , 1980, Biorheology.

[20]  M. H. Friedman,et al.  General theory of tissue swelling with application to the corneal stroma. , 1971, Journal of theoretical biology.

[21]  Javed Siddique,et al.  Capillary Rise of Magnetohydrodynamics Liquid into Deformable Porous Material , 2016 .

[22]  Andreas Mortensen,et al.  Forced unidirectional infiltration of deformable porous media , 1996, Journal of Fluid Mechanics.

[23]  JD Jan Janssen,et al.  A validation of the quadriphasic mixture theory for intervertebral disc tissue , 1997 .

[24]  S Naili,et al.  Ion induced deformation of soft tissue. , 1995, Bulletin of mathematical biology.

[25]  Francesco dell’Isola,et al.  A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio‐resorbable materials , 2012 .

[26]  Matthew R. Glucksberg.  Mechanics of the perialveolar interstitium of the lung , 1990 .

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

[28]  Joachim Bluhm,et al.  A Triphasic Theory for Growth in Biological Tissue – Basics and Applications , 2006 .

[29]  Kumbakonam R. Rajagopal,et al.  Modeling electro-rheological materials through mixture theory , 1994 .

[30]  Frank Diederich,et al.  The Non Linear Field Theories Of Mechanics , 2016 .

[31]  W M Lai,et al.  Fluid transport and mechanical properties of articular cartilage: a review. , 1984, Journal of biomechanics.

[32]  A. Cemal Eringen.  A mixture theory of electromagnetism and superconductivity , 1998 .

[33]  Van C. Mow,et al.  A Finite Deformation Theory for Nonlinearly Permeable Soft Hydrated Biological Tissues , 1986 .

[34]  R. de Boer,et al.  Trends in Continuum Mechanics of Porous Media , 2005 .

[35]  A. D. Kirwan,et al.  A review of mixture theory with applications in physical oceanography and meteorology , 1985 .

[36]  D. S. Drumheller,et al.  Theories of immiscible and structured mixtures , 1983 .

[37]  M. Biot.  MECHANICS OF DEFORMATION AND ACOUSTIC PROPAGATION IN POROUS MEDIA , 1962 .

[38]  R. M. Bowen,et al.  Incompressible porous media models by use of the theory of mixtures , 1980 .

[39]  C. Lakshmana Rao,et al.  Permeability and bleeding of asphalt concrete using mixture theory , 2001 .

[40]  S. I. Barry,et al.  Unsteady flow induced deformation of porous materials , 1991 .

[41]  G Jayaraman,et al.  Water transport in the arterial wall--a theoretical study. , 1983, Journal of biomechanics.

[42]  Louay N. Mohammad,et al.  Application of Mixture Theory in the Evaluation of Mechanical Properties of Asphalt Concrete , 2004 .

[43]  L. Preziosi,et al.  ON THE CLOSURE OF MASS BALANCE MODELS FOR TUMOR GROWTH , 2002 .

[44]  G. Aldis,et al.  Comparison of models for flow induced deformation of soft biological tissue. , 1990, Journal of biomechanics.

[45]  V. Mow,et al.  Biphasic creep and stress relaxation of articular cartilage in compression? Theory and experiments. , 1980, Journal of biomechanical engineering.

[46]  D. E. Kenyon,et al.  The theory of an incompressible solid-fluid mixture , 1976 .

[47]  C. Lakshmana Rao,et al.  Mechanics of air voids reduction of asphalt concrete using mixture theory , 2000 .

[48]  Donald M. Anderson,et al.  Capillary rise of a liquid into a deformable porous material , 2009 .

[49]  Luigi Preziosi,et al.  The theory of deformable porous media and its application to composite materials manufacturing , 1996 .

[50]  Van C. Mow,et al.  Mechanics of Animal Joints , 1979 .

[51]  D. AMBROSI,et al.  Modeling Injection Molding Processes with Deformable Porous Preforms , 2000, SIAM J. Appl. Math..

[52]  M. Biot.  THEORY OF ELASTICITY AND CONSOLIDATION FOR A POROUS ANISOTROPIC SOLID , 1955 .

[53]  Peter J. Chen.  A coupled solid/fluids mixture theory that suffices for diffusion problems , 1996 .

[54]  G. Aldis,et al.  Flow-induced deformation from pressurized cavities in absorbing porous tissues. , 1992, Bulletin of mathematical biology.

[55]  van Dh Dick Campen,et al.  A mixture approach to the mechanics of skin. , 1987, Journal of biomechanics.

[56]  Mark H. Holmes.  A nonlinear diffusion equation arising in the study of soft tissue , 1983 .

[57]  R. E. Craine,et al.  CONTINUUM THEORIES OF MIXTURES: BASIC THEORY AND HISTORICAL DEVELOPMENT , 1976 .

[58]  Luca Placidi,et al.  A mixture theory framework for modeling the mechanical actuation of ionic polymer metal composites , 2008 .

[59]  George Z. Voyiadjis,et al.  The coupled theory of mixtures in geomechanics with applications , 2006 .

[60]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

[61]  L. Billi,et al.  Unidirectional infiltration in deformable porous media: Mathematical modeling and self-similar solution , 2000 .

[62]  Luigi Preziosi,et al.  Infiltration of initially dry, deformable porous media , 1996 .

[63]  Luigi Preziosi,et al.  The insight of mixtures theory for growth and remodeling , 2010 .

[64]  G. Aldis,et al.  Fluid flow over a thin deformable porous layer , 1991 .

[65]  L. Preziosi,et al.  Modelling Solid Tumor Growth Using the Theory of Mixtures , 2001, Mathematical medicine and biology : a journal of the IMA.