Poromechanics
Poromechanics is a branch of physics and specifically continuum mechanics that studies the behavior of fluid-saturated porous media.
In the simplest case, both the solid matrix and the pore space constitute two separate, continuously connected domains.
An archtypal example of such a porous material is the kitchen sponge, which is formed of two interpenetrating continua.
Alternatively, in the case of granular porous media, the solid phase may constitute disconnected domains, termed the "grains", which are load-bearing under compression, though can flow when sheared.
Natural substances including rocks,[2] soils,[3] biological tissues including heart[4] and cancellous bone,[5] and man-made materials such as foams, ceramics, and concrete[6] can be considered as porous media.
Porous media whose solid matrix is elastic and the fluid is viscous are called poroviscoelastic.
A poroviscoelastic medium is characterised by its porosity, permeability, and the properties of its constituents - solid matrix and fluid.
The distribution of pores, fluid pressure, and stress in the solid matrix gives rise to the viscoelastic behavior of the bulk.
[7] Porous media whose pore space is filled with a single fluid phase, typically a liquid, is considered to be saturated.
[8] 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.
The prediction of Biot’s slow wave generated controversy until Thomas Plona experimentally observed it in 1980.
[14] Other important early contributors to the theory of poroelasticity were Yakov Frenkel and Fritz Gassmann.
Recent applications of poroelasticity to biology, such as modeling 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.
A representative elementary volume (REV) of a porous medium and the superposition of the domains of the skeleton and connected pores is shown in Fig.
In tracking the material deformation, one must be careful to properly apportion sub-volumes that correspond to the solid matrix and pore space.
To do this, it is often convenient to introduce a porosity, which measures the fraction of the REV that constitutes pore space.
As such, the void ratio takes definition in an Eulerian frame of reference and is calculated as where
, is related to the deformed and undeformed total material volumes by where the definition of the Lagrangian porosity further requires
When linearizing the strain in a poroelastic solid body, several conditions should hold true.
Firstly, as is the requirement for a general continuum solid, displacement gradients should be small,
, should be small in comparison to the characteristic length scale defining the grain size (in case of a granular material) or solid matrix (in case of a continuous solid phase),
When measuring the linear elastic properties of porous solids, laboratory experiments are typically performed under one of two limit cases: Reinhard Woltman (1757-1837), a German hydraulic and geotechnical engineer, first introduced the concepts of volume fractions and angles of internal friction within porous media in his study on the connection between soil moisture and its apparent cohesion.
[18] His work addressed the calculation of earth pressure against retaining walls.
Achille Delesse (1817-1881), a French geologist and mineralogist, reasoned that the volume fraction of voids – otherwise termed the volumetric porosity – equals the surface fraction of voids – otherwise termed the areal porosity – when the size, shape, and orientation of the pores are randomly distributed.
[20] The first important concept related to saturated, deformable porous solids might be considered the principle of effective stress introduced by Karl von Terzaghi (1883-1963), an Austrian engineer.
Terzaghi postulated that the mean effective stress experienced by the solid skeleton of a porous medium with incompressible constituents,
[21] Terzaghi combined his effective stress concept with Darcy’s law for fluid flow and derived a one-dimensional consolidation theory explaining the time-dependent deformation of soils as the pore fluid drains, which might be the first mathematical treatise on coupled hydromechanical problems in porous media.
