Poromechanics
Poromechanics is a branch of physics and specifically continuum mechanics and acoustics that studies the behaviour of fluid-saturated porous media. A porous medium or a porous material is a solid (often called matrix) permeated by an interconnected network of pores (voids) filled with a fluid (liquid or gas). Usually both solid matrix and the pore network (also known as the pore space) are assumed to be continuous, so as to form two interpenetrating continua such as in a sponge. Many natural substances such as rocks, soils, biological tissues, and man made materials such as foams and ceramics can be considered as porous media. Porous media whose solid matrix is elastic and the fluid is viscous are called poroelastic. A poroelastic medium is characterised by its porosity, permeability as well as the properties of its constituents (solid matrix and fluid).
The concept of a porous medium originally emerged in soil mechanics, and in particular in the works of Karl von Terzaghi, the father of soil mechanics. However a more general concept of a poroelastic medium, independent of its nature or application, is usually attributed to Maurice Anthony Biot (1905–1985), a Belgian-American engineer. In a series of papers published between 1935 and 1957 Biot developed the theory of dynamic poroelasticity (now known as Biot theory) which gives a complete and general description of the mechanical behaviour of a poroelastic medium. Biot's equations of the linear theory of poroelasticity are derived from
- Equations of linear elasticity for the solid matrix,
- Navier–Stokes equations for the viscous fluid, and
- Darcy's law for the flow of fluid through the porous matrix.
One of the key findings of the theory of poroelasticity is that in poroelastic media there exist three types of elastic waves: a shear or transverse wave, and two types of longitudinal or compressional waves, which Biot called type I and type II waves. The transverse and type I (or fast) longitudinal wave are similar to the transverse and longitudinal waves in an elastic solid, respectively. The slow compressional wave, (also known as Biot’s slow wave) is unique to poroelastic materials. The prediction of the Biot’s slow wave generated some controversy, until it was experimentally observed by Thomas Plona in 1980. Other important early contributors to the theory of poroelasticity were Yakov Frenkel and Fritz Gassmann.
Recent applications of poroelasticity to biology such as modeling of blood flows through the beating myocardium have also required an extension of the equations to nonlinear (large deformation) elasticity and the inclusion of inertia forces.
See also
References
- Terzaghi, K., 1943, Theoretical Soil Mechanics, John Wiley and Sons, New York
- Frenkel, J. (1944). "On the theory of seismic and seismoelectric phenomena in moist soil" (pdf). Journal of Physics III (4): 230–241. doi:10.1061/(ASCE)0733-9399(2005)131:9(879).
- Gassmann, F., 1951. Über die elastizität poröser medien. Viertel. Naturforsch. Ges. Zürich, 96, 1 – 23. (English translation available as pdf here).
- Gassmann, Fritz (1951). "Elastic waves through a packing of spheres". Geophysics 16 (4): 673–685. Bibcode:1951Geop...16..673G. doi:10.1190/1.1437718.
- Biot, M.A. (1941). "General theory of three dimensional consolidation". Journal of Applied Physics 12: 155–164. Bibcode:1941JAP....12..155B. doi:10.1063/1.1712886.
- Biot, M.A. (1956). "Theory of propagation of elastic waves in a fluid saturated porous solid. I Low frequency range". The Journal of the Acoustical Society of America (28): 168–178. Bibcode:1956ASAJ...28..168B. doi:10.1121/1.1908239.
- Biot, M.A. (1956). "Theory of propagation of elastic waves in a fluid saturated porous solid. II Higher frequency range". The Journal of the Acoustical Society of America (28): 179–191. Bibcode:1956ASAJ...28..179B. doi:10.1121/1.1908241.
- Biot, M.A. and Willis, D.G. (1957). "The elastic coefficients of the theory of consolidation". Journal of Applied Mechanics (Trans. ASME) 24: 594–601.
- Biot, M.A. (1962). "Mechanics of deformation and acoustic propagation in porous media". Journal of Applied Physics (33): 1482–1498. Bibcode:1962JAP....33.1482B. doi:10.1063/1.1728759.
- Rice, J.R., and Cleary, M.P. (1976). "Some basic stress diffusion solutions for fluid-saturated elastic porous media with compressible constituents". Reviews of Geophysics and Space Physics (14): 227–241. Bibcode:1976RvGSP..14..227R. doi:10.1029/RG014i002p00227.
- Plona, T. (1980). "Observation of a Second Bulk Compressional Wave in a Porous Medium at Ultrasonic Frequencies". Applied Physics Letters (36): 259–251. Bibcode:1980ApPhL..36..259P. doi:10.1063/1.91445.
- Coussy, O., 2004, Poromechanics, John Wiley & Sons.
- Bourbie, T., Coussy, O., Zinszner, B., 1987, Acoustics of Porous Media, Gulf Pub. Co.; Editions Technip.
- Nigmatulin, R.I., 1990, Dynamics of Multiphase Media, Hemisphere.
- Wang, H.F., 2000, Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press.
- Allard, J. F., 1993, Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials, Chapman & Hall.
- Chapelle, D., Gerbeau, J.-F., Sainte-Marie, J. and Vignon-Clementel, I. (2010). "A poroelastic model valid in large strains with applications to perfusion in cardiac modeling". Computational Mechanics 46: 91–101. doi:10.1007/s00466-009-0452-x.
- Chapelle, D. and Moireau, P. (2014). "General coupling of porous flows and hyperelastic formulations - From thermodynamics principles to energy balance and compatible time schemes". European Journal of Mechanics - B/Fluids 46: 82–96. doi:10.1016/j.euromechflu.2014.02.009.