Strain porous media

Abstract A general theoretical and finite element model (FEM) for soft tissue structures is described including arbitrary constitutive laws based upon a continuum view of the material as a mixture or porous medium saturated by an incompressible fluid and containing charged mobile species. Example problems demonstrate coupled electro-mechano-chemical transport and deformations in FEMs of layered materials subjected to mechanical, electrical and chemical loading while undergoing small or large strains. 

Soft biological structures exhibit finite strains and nonlinear anisotropic material response. The hydrated tissue can be viewed as a fluid-saturated porous medium or a continuum mixture of incompressible solid (s), mobile incompressible fluid (f), and three (or an arbitrary number) mobile charged species a, (3 = p,m, b). A mixed Electro-Mechano-Chemical-Porous-Media-Transport or EMCPMT theory (previously denoted as the LMPHETS theory) is presented with (a) primary fields (continuous at material interfaces) displacements, Ui and generalized potentials, ifi ( , r/ = /, e, to, b) and (b) secondary fields (discontinuous) pore fluid pressure, pf electrical potential, /7e and species concentration (molarity), ca = dna/dVf or apparent concentration, ca = nca and c = Jnca = dna/dVo. The porosity, n = 1 — J-1(l — no) and no = no(Xi) = dVj/dVo for a fluid-saturated solid. Fixed charge density (FCD) in the solid is defined as cF = dnF/dV , cF = ncF, and cF = cF (Xf = JncF = dnF/d o. 

Having obtained the elastic equations in terms of shifted entities, and reverting to total entities, the constitutive equations express the total stress cr, the chemical potentials of the extrafibrillar water p,wE and of the salt psE, and the hydration potential of the intrafibrillar water //hydl . in terms of the generalized strains, namely the strain of the porous medium e, the mass-contents of the extrafibrillar water mWE and of the cations sodium mNae, and the mass-content of intrafibrillar water mwi. The interested reader is directed to [3]. 

INTERACTION BETWEEN AQUEOUS SOLUTION TRANSPORT AND STRESS/STRAIN IN A DEFORMABLE POROUS MEDIUM... 

Richefeu, V., El Youssoufi, M. S. and, Benet, J.-C. (2002) Saturated porous medium strain under osmotic actions. Poromechanics II, Auriault et el. (ed.), BaLkema, Rotterdam, 533-537... 

Abstract When subjected to a mechanical loading, the solid phase of a saturated porous medium undergoes a dissolution due to strain-stress concentration effects along the fluid-solid interface. Through a micromechanical analysis, the mechanical affinity is shown to be the driving force of the local dissolution. For cracked porous media, the elastic free energy is a dominant component of this driving force. This allows to predict dissolution-induced creep in such materials. 

To develop an understanding of the emulsion flow in porous media, it is useful to consider differences and similarities between the flow of an OAV emulsion and simultaneous flow of oil and water in a porous medium. As discussed in the preceding section, in simultaneous flow of oil and water, both fluid phases are likely to occupy continuous, and to a large extent, separate networks of flow channels. Assuming the porous medium to be water-wet, the oil phase becomes discontinuous only at the residual saturation of oil, where the oil ceases to flow. Even at its residual saturation, the oil may remain continuous on a scale much larger than pores. In the flow of an OAV emulsion, the oil exists as tiny dispersed droplets that are comparable in size to pore sizes. Therefore, the oil and water are much more likely to occupy the same flow channels. Consequently, at the same water saturation the relative permeabilities to water and oil are likely to be quite different in emulsion flow. In normal flow of oil and water, oil droplets or ganglia become trapped in the porous medium by the process of snap-off of oil filament at pore throats (8). In the flow of an OAV emulsion, an oil droplet is likely to become trapped by the mechanism of straining capture at a pore throat smaller than the drop. 

Soo and Radke (11) confirmed that the transient permeability reduction observed by McAuliffe (9) mainly arises from the retention of drops in pores, which they termed as straining capture of the oil droplets. They also observed that droplets smaller than pore throats were captured in crevices or pockets and sometimes on the surface of the porous medium. They concluded, on the basis of their experiments in sand packs and visual glass micromodel observations, that stable OAV emulsions do not flow in the porous medium as a continuum viscous liquid, nor do they flow by squeezing through pore constrictions, but rather by the capture of the oil droplets with subsequent permeability reduction. They used deep-bed filtration principles (i2, 13) to model this phenomenon, which is discussed in detail later in this chapter. 

Filtration Model. A model based on deep-bed filtration principles was proposed by Soo and Radke (12), who suggested that the emulsion droplets are not only retarded, but they are also captured in the pore constrictions. These droplets are captured in the porous medium by two types of capture mechanisms straining and interception. These were discussed earlier and are shown schematically in Figure 22. Straining capture occurs when an emulsion droplet gets trapped in a pore constriction of size smaller than its own diameter. Emulsion droplets can also attach themselves onto the rock surface and pore walls due to van der Waals, electrical, gravitational, and hydrodynamic forces. This mode of capture is denoted as interception. Capture of emulsion droplets reduces the effective pore diameter, diverts flow to the larger pores, and thereby effectively reduces permeability. 

This mechanism is similar to that of a deep-bed filtration process with some differences (12). In the filtration process the particle-size to pore-size ratio is small, and the particles are mostly captured on the media surface. Thus interceptive capture dominates, and this capture does not alter the flow distribution in the porous medium. Permeability reduction is not significant and is ignored. On the other hand, the emulsion droplet size is generally of the same order of the pore size, and the droplets are captured both by straining and interception. This capture blocks pores and results in flow redistribution and a reduced permeability. 

Forces applied to a water-saturated porous medium will cause stresses which result in strain (deformation). The stress, strain and groundwater pressure in a water-saturated porous medium are coupled, as first recognized by Biot (1941). Under the assumed stress conditions, the vertical normal component of total stress (o ) that acts downwards on a horizontal plane at any depth is caused by the weight of the overlying water-saturated rock. This stress is born by the solid matrix of the porous medium (o ) and by the pressure of the groundwater in the pores (p ) (e.g. Hubbert andRubey, 1959)... 

An important means by which small particles in suspension are separated from solutions is through capture by collectors, which may be larger particles, or granular, porous, or fibrous media. An example of such collection is filtration. The separated solids may be collected as a cake on the surface of the filter medium (much like ultrafiltration), and this is termed cake filtration. Alternatively, the solids may be retained within the pores of the medium, and this is termed depth filtration. It is important to recognize that particle collection in a porous medium is not simply a matter of straining that is, the capture is not purely steric, since, in filtration, particles are captured that are much smaller than pores of the medium. The capture of small suspended particles from fluids in laminar flow by a collector is a consequence of the simultaneous action of fluid mechanical forces and forces between the particle and collector, such as van der Waals or electrostatic forces. It is the combined forces, at least close to the collector, that govern the particle trajectories and determine whether a particle will be transported to and retained at the surface of a collector that is fixed in the flow (Spielman 1977). 

Here, we present in suminary form the equations that describe solid strain, solid stress, and solid internal energy of the porous medium. First, the internal energy statement ... 

Analysis of time-dependent consolidation requires the solution of Biot s consolidation equations coupled to the equations describing flow. The transient hydro-mechanical coupling between pore pressure and volumetric strain for a linear elastic, mechanically isotropic porous medium and fully saturated with a single fluid phase (i.e. water), is given by the fluid continuity equation ... 

Figure 2 shows the basic physical idea of the microstructure of the continuum rheologicS model we proposed earlier (2). The layers can be idealized as separated by porous slabs, which are connected by elastic springs. Liquid crystals may flow parallel to the planes in the usual Newtonian manner, as if the slabs were not there. In the direction normal to the layers, liquid crystals encounter resistance through the porous medium, proportional to the normal pressure gradient, which is known as permeation. The permeation is characterized by a body force which in turn causes elastic compression and splay of the layers. Applied strain from the compression causes dislocations to move into the sample from the side in order to relax the net force on the layers. When the compression stops and the applied stress is relaxed the permeation characteristic has no influence on stress strain field. 

In this section we reconstruct the theory of consolidation by introducing the concept of a finite strain and a nominal stress rate, which are given in Chap. 2. Note that in Chap. 5 a mixture theory was developed for a porous medium with multiple... 

ABSTRACT In the present paper a multiphase model including a hypoplastic formulation of the solid phase is presented and its application to earthquake engineering problems discussed. The macroscopic soil model, which is based on the theory of porous media, comprises three distinct phases namely, solid, fluid and gas phase. For each of these the compressibility of the respective medium is taken into account in the mathematical formulation of the model. The solid phase is modelled using the hypoplastic constitutive equation including intergranular strain to allow for a realistic description of material behaviour of cohesionless soils even under cyclic loading. The model was implemented into the finite element package ANSYS via the user interface and also allows the simulation of soil-structure interaction problems. 

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