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Poromechanics

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Poromechanics is a branch of physics and specifically continuum mechanics and acoustics that studies the behaviour of fluid-saturated porous media.[1] A porous medium or a porous material is a solid referred to as matrix) permeated by an interconnected network of pores (voids) filled with a fluid (liquid or gas). Usually both solid matrix and the pore network, or pore space, are assumed to be continuous, so as to form two interpenetrating continua such as in a sponge. Natural substances including rocks,[2] soils,[3] biological tissues including heart[4] and cancellous bone,[5] 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 distribution of pores across multiple scales as well as the pressure of the fluid with which they are filled give rise to distinct elastic behaviour of the bulk. [6]

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.[7] 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 1962 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.[8][9][10][11][12] Biot's equations of the linear theory of poroelasticity are derived from the equations of linear elasticity for a solid matrix, the Navier–Stokes equations for a viscous fluid, and Darcy's law for a flow of fluid through a 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.[13] Other important early contributors to the theory of poroelasticity were Yakov Frenkel and Fritz Gassmann.[14][15][16]

Conversion of energy from fast compressional and shear waves into the highly attenuating slow compressional wave is a significant cause of elastic wave attenuation in porous media.

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

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References

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  1. ^ Coussy O (2004). Poromechanics. Hoboken: John Wiley & Sons.
  2. ^ Müller TM, Gurevich B, Lebedev M (2010). "Seismic wave attenuation and dispersion resulting from wave-induced flow in porous rocks: a review". Geophysics. 75 (5): 75A147–75A164. Bibcode:2010Geop...75A.147M. doi:10.1190/1.3463417. hdl:20.500.11937/35921.
  3. ^ Wang HF (2000). Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton: Princeton University Press. ISBN 9780691037462.
  4. ^ Chapelle D, Gerbeau JF, Sainte-Marie J, Vignon-Clementel I (2010). "A poroelastic model valid in large strains with applications to perfusion in cardiac modeling". Computational Mechanics. 46: 91–101. Bibcode:2010CompM..46..101C. doi:10.1007/s00466-009-0452-x. S2CID 18226623.
  5. ^ Aygün H, Attenborough K, Postema M, Lauriks W, Langton C (2009). "Predictions of angle dependent tortuosity and elasticity effects on sound propagation in cancellous bone" (PDF). Journal of the Acoustical Society of America. 126 (6): 3286–3290. doi:10.1121/1.3242358. PMID 20000942. S2CID 36340512.
  6. ^ Multiscale modeling of effective elastic properties of fluid-filled porous materials International Journal of Solids and Structures (2019) 162, 36-44
  7. ^ Terzaghi K (1943). Theoretical Soil Mechanics. New York: Wiley. doi:10.1002/9780470172766. ISBN 9780471853053.
  8. ^ Biot MA (1941). "General theory of three dimensional consolidation" (PDF). Journal of Applied Physics. 12 (2): 155–164. Bibcode:1941JAP....12..155B. doi:10.1063/1.1712886.
  9. ^ Biot MA (1956). "Theory of propagation of elastic waves in a fluid saturated porous solid. I Low frequency range" (PDF). The Journal of the Acoustical Society of America. 28 (2): 168–178. Bibcode:1956ASAJ...28..168B. doi:10.1121/1.1908239.
  10. ^ Biot MA (1956). "Theory of propagation of elastic waves in a fluid saturated porous solid. II Higher frequency range" (PDF). The Journal of the Acoustical Society of America. 28 (2): 179–191. Bibcode:1956ASAJ...28..179B. doi:10.1121/1.1908241.
  11. ^ Biot MA, Willis DG (1957). "The elastic coefficients of the theory of consolidation". Journal of Applied Mechanics. 24 (4): 594–601. Bibcode:1957JAM....24..594B. doi:10.1115/1.4011606.
  12. ^ Biot MA (1962). "Mechanics of deformation and acoustic propagation in porous media". Journal of Applied Physics. 33 (4): 1482–1498. Bibcode:1962JAP....33.1482B. doi:10.1063/1.1728759. S2CID 58914453.
  13. ^ Plona T (1980). "Observation of a Second Bulk Compressional Wave in a Porous Medium at Ultrasonic Frequencies". Applied Physics Letters. 36 (4): 259. Bibcode:1980ApPhL..36..259P. doi:10.1063/1.91445.
  14. ^ Frenkel J (1944). "On the theory of seismic and seismoelectric phenomena in moist soil" (PDF). Journal of Physics. 3 (4): 230–241. Republished as Frenkel J (2005). "On the Theory of Seismic and Seismoelectric Phenomena in a Moist Soil". Journal of Engineering Mechanics. 131 (9): 879–887. doi:10.1061/(ASCE)0733-9399(2005)131:9(879).
  15. ^ Gassmann F (1951). "Über die Elastizität poröser Medien". Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich. 96: 1–23. (English translation available as pdf here)
  16. ^ Gassmann F (1951). "Elastic waves through a packing of spheres". Geophysics. 16 (4): 673–685. Bibcode:1951Geop...16..673G. doi:10.1190/1.1437718.

Further reading

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