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The present volume offers a state-of-the-art report on the various recent sci­ entific developments in the Theory of Porous Media (TPM) comprehending the basic theoretical concepts in continuum mechanics on porous and mul­ tiphasic materials as well as the wide range of experimental and numerical applications. Following this, the volume does not only address the sophisti­ cated reader but also the interested beginner in the area of Porous Media by presenting a collection of articles. These articles written by experts in the field concern the fundamental approaches to multiphasic and porous materials as well as various applications to engineering problems. In many branches of engineering just as in applied natural sciences like bio- and chemomechanics, one often has to deal with continuum mechanical problems which cannot be uniquely classified within the well-known disci­ plines of either "solid mechanics" or "fluid mechanics". These problems, characterized by the fact that they require a unified treatment of volumetri­ cally coupled solid-fluid aggregates; basically fall into the categories of either mixtures or porous media. Following this, there is a broad variety of problems ranging in this category as for example the investigation of reacting fluid mix­ tures or solid-fluid suspensions as well as the investigation of the coupled solid deformation and pore-fluid flow behaviour of liquid- and gas-saturated porous solid skeleton materials like geomaterials (soil, rock, concrete, etc. ), polymeric and metallic foams or biomaterials (hard and soft tissues, etc).





Foundations of multiphasic and porous materials

Miscible multiphasic materials like classical mixtures as well as immiscible materials like saturated and partially saturated porous media can be successfully described on the common basis of the well-founded Theory of Mixtures (TM) or the Theory of Porous Media (TPM). In particular, both the TM and the TPM provide an excellent frame for a macroscopic description of a broad variety of engineering applications and further problems in applied natural sciences. The present article portrays both the standard and the micropolar approaches to multiphasic materials reflecting their mechanical and their thermodynamical frameworks. Including some constitutive models and various illustrative numerical examples, the article can be understood as a reference paper to all the following articles of this volume on theoretical, experimental and numerical investigations in the Theory of Porous Media.
Wolfgang Ehlers

Modelling of saturated thermo-elastic porous solids with different phase temperatures

Based on the Theory of Porous Media (TPM), a binary model for the description of saturated thermo-elastic porous solids with different phase temperatures will be presented. The constituents solid and fluid can be compressible or incompressible, i. e. the binary model discussed here includes the compressible model, the hybrid models of first and second type and the incompressible model. For the four different binary models the field equations, the constitutive relations and the dissipation mechanism will be developed and discussed.
Joachim Bluhm

Theory of Porous Media


Micropolar mixture models on the basis of the Theory of Porous Media

The behaviour of porous media can be described in a continuum mechanical setting by the Theory of Porous Media, i. e. by a mixture theory extended by the concept of volume fractions. In addition to the volume fractions, micropolarity is taken into account to model the internal structure of porous media on the macroscopic scale. After a microscopic motivation of the approach, which shows that it is physically motivated to deal with micropolar mixture models, the kinematics, the balance relations, and the constitutive framing of such a theory are discussed. A set of model equations is formulated within the presented frame and applied to some boundary value problems showing the evidence of the theoretical approach.
Stefan Diebels

Elasto-plastic behaviour of a granular material with an additional scalar degree of freedom

In granular material theories the introduction of internal variables is often useful or even necessary to adequately capture the material behaviour. Yet no common “rules” exist according to which equations governing the evolution of these internal variables should be formulated. In this article, 3 different approaches to model the evolution of such scalar valued internal variables are investigated in a continuum-thermodynamical framework, exploiting the entropy principle according to Müller and Liu. For all three models, a so-called generalized Gibbs equation is obtained, relating the differential of the entropy (which is a constitutive quantity) to that of the internal energy and an additional (model-specific) contribution. The main focus is then on the Poincaré conditions, the satisfaction of which provides a powerful tool to restrict the constitutive quantities in such a way that entropy is in fact a well defined scalar potential. The results emerging from this analysis performed for all three modelling approaches shed light on their ability and limitation, respectively. We arrive at reduced or even explicit forms of the constitutive equations and the Lagrange multipliers introduced to exploit the entropy principle.
Nina P. Kirchner, Kolumban Hutter

Mechanical aspect on drying of wet porous media

An aspect of thermomechanics of fluid saturated capillary-porous media concerning the mechanical phenomena accompanying the heat and mass transfer during drying processes is presented. The wet materials tend to shrink during drying and the shrinkage generates internal stresses which may cause fracturing of the dried body. The constitutive relations and the heat and mass rate equations are developed. The forces that produce shrinkage and the mechanisms responsible for transport of heat and moisture are examined. The considerations are based on the thermodynamics of irreversible processes and the continuum mechanics of porous media. The studies are quite general but the final form of the drying model is simplified by the assumptions admissible in practical applications. The example of convective drying of a ceramic cylinder is presented for illustration of the theory.
Stefan J. Kowalski

Coupling between the evolution of a deformable porous medium and the motion of fluids in the connected porosity

This contribution is aimed at presenting a consistent mathematical description of porous media. Basic assumptions concerned with the representation of a porous medium as a continuum are discussed in detail, making a clear distinction between two geometric scales: at microscale each constituent occupies a specific domain, while at macroscale soil and fluid particles are superimposed at the same geometric point. At macroscale, a unified Lagrangean formulation is given, by assuming the soil skeleton as a material reference volume and by referring the fluid motion to the soil skeleton Finally, two problems are analysed, both of relevant interest in soil mechanics, i. e. the propagation of body waves in undrained conditions and the consolidation of a soft clay stratum.
Renato Lancellotta

Fixed negative charges modulate mechanical behaviours and electrical signals in articular cartilage under unconfined compression — a triphasic paradigm

The unconfined compression test has been frequently used to study the mechanical behaviour of articular cartilage. Recently, it has also been used in explant and gel-cell-complex studies in tissue engineering. Mechanical responses in these experiments have been analyzed using the biphasic theory as well as fibril reinforced poroelastic theory (Armstrong et al. [1], Brown and Singerman [5], Spilker et al. [45, 46], Soulhat et al. [44], Li et al. [30, 31], Fortin et al. [12], DiSilvestro et al. [10, 11]). Using an optical technique and testing cartilage samples in unconfined compression, the apparent Poisson’s ratio of articular cartilage has also been determined (Jurvelin et al. [23]). In the biphasic and poroelastic theory, the effect of fixed charges is embodied in the apparent compressive Young’s modulus and Poisson’s ratio of the tissue, and the fluid pressure is considered to be that which is over and above the osmotic pressure. In order to understand the effects of fixed charges on the mechanical behaviours of articular cartilage, and in order to predict the osmotic pressure and electric fields inside the tissue in this experimental configuration, it is necessary to use a model that explicitly takes into account the charged nature of the tissue and ion flow. In this paper, the triphasic theory is used to study how the fixed charges within a porous-permeable soft tissue modulate its mechanical and electrochemical responses under a step displacement load. The results showed that: 1) A charged tissue always supports a larger load than an uncharged tissue of the same intrinsic elastic moduli. 2) The apparent Young’s modulus (ratio of equilibrium axial stress to axial strain) is always more than the intrinsic Young’s modulus of an uncharged tissue. 3) The apparent Poisson’s ratio (negative ratio of lateral strain to axial strain) is always larger than the intrinsic Poisson’s ratio of an uncharged tissue. 4) Load support derives from three sources: intrinsic matrix stiffness, hydraulic pressure and osmotic pressure. Under unconfined compression, the Donnan osmotic pressure can constitute between 13–22% of the total load support at equilibrium. 5) During the stress-relaxation and recoiling processes following the initial instant of loading, diffusion potential (due to the gradient of the fixed charge density or FCD, and the gradient of ion concentrations) and streaming potential (due to fluid pressure gradient) compete against each other. Within physiological range of material parameters, the polarity of the electric potential depends on both the mechanical properties and FCD of the tissue. For softer tissue, the diffusion potential dominates while the streaming potential dominates in a stiffer tissue. 6) Fixed charges do not affect the instantaneous strain field relative to the initial equilibrium state. However, there is a sudden increase in the fluid pressure above the initial equilibrium state. These new findings are relevant and necessary for the understanding of cartilage mechanics, cartilage biosynthesis, electromechanical signal transduction by chondrocytes, and tissue engineering.
Van C. Mow, Daniel D. Sun, X. Edward Guo, Morakot Likhitpanichkul, W. Michael Lai

Experiments and Numerical Applications


Time adaptive analysis of saturated soil by a discontinuous-Galerkin method

The topic of this presentation is the numerical analysis of saturated soil by the finite element method. As the solution procedure should be extended to describe the flow of the pore fluid through the deforming solid skeleton, a time dependency is introduced into the problem. Therefore, the time coordinate has to be discretized and treated by an appropriate integration scheme. In contrast to adaptive mesh refinement strategies in the spatial domain which are well founded for elasticity and have also been successfully applied to elastic-plastic problems only few papers deal with time adaptive procedures for the quasi-static analysis of consolidation.
Therefore, in this presentation a time discretization dependent on the specific problem is emphasized. For this purpose the time-discontinuous-Galerkin method is applied to the differential equations of first order in time. It is based on a variational form permitting jumps in the temporal evolution of the field variables, where the continuity is satisfied in a weak sense. It can be shown that these jumps may then be used to define a natural error indicator for the temporal discretization error. On the other hand, attention is drawn to another error which arises from the numerical integration of the rate equations of plasticity. In this context, an indicator is derived from the residual of the Kuhn-Tucker conditions within the time interval.
The numerical examples of a one dimensional consolidation problem and a strip footing on a half space demonstrate the applicability of the method to problems in geomechanics. Both indicators are combined to improve the efficiency of the time stepping scheme.
Harald Cramer, Rudolf Findeiß, Walter Wunderlich

Biphasic description of viscoelastic foams by use of an extended Ogden-type formulation

Soft polymeric foams exhibit distinct relaxation and creep phenomena which in combination with the cellular micro structure result in the outstanding mechanical characteristics of this type of porous materials. It is the goal of this contribution to present an appropriate biphasic continuum mechanical model based on the Theory of Porous Media (TPM) which allows the description of viscoelastic foams at a suitable means of computational costs. To reproduce the complex behaviour of the cellular polymer skeleton an extended Ogden-type viscoelasticity formulation is embedded into the porous media concept. Thus, the macroscopic model accounts for all relevant physical properties, i. e. the porous cell structure, the moving and interacting pore-fluid, and the intrinsic viscoelasticity of the polymeric matrix material.
Wolfgang Ehlers, Bernd Markert, Oliver Klar

Experimental measurement of electrical conductivity and electro-osmotic permeability of ionised porous media

Fine grained porous media typically exhibit non-Darcyan behaviour. One of them is electro-osmotic flow. The present paper presents an electro-osmotic flow experiment on hydrogel samples. The electric current density is prescribed. The electric potential drop across the sample and the volume flow through the sample are measured. The results show a linear relationship between potential and current and between flow and current. As diffusional properties are highly dependent on deformation, these linear relationships in a highly deformable medium support the theoretical finding that viscous drag is annihilated by opposite electrical forces in an electro-osmotic flow experiment.
Jacques M. Huyghe, Charles F. Janssen, Yoram Lanir, Corrinus C. van Donkelaar, Alice Maroudas, Dick H. van Campen

Theory and numerics of localization in a fluid-saturated elasto-plastic porous medium

In this contribution, we outline a theoretical and numerical approach for describing deformation localization due to hydro-mechanical coupling. In the localization analysis, the concept of “regularized strong discontinuity” is extensively used at the application to the conservation laws of momentum and mass. At the onset of localization, the displacement and pore pressure fields are assumed to contain regularized discontinuities that are superposed on the continuous fields. As a result, we obtain a coupled localization condition, whereby the partly drained situation is discussed and compared to the drained and undrained situations. As to the finite element modelling, it is proposed to capture the development of regularized discontinuities in the displacement and pressure fields it is proposed to use a finite element procedure for the mixture of soil and pore fluid based on the “embedded band approach”, where the finite element interpolation allows for discontinuities within the elements. The procedure is based on the enhanced assumed strain concept, and from the pertinent orthogonality condition a coupled set of finite element equations are obtained, where the coupling between continuous and discontinuous response is obtained at the element level. Under certain circumstances, the coupled localization condition may be shown to be preserved by the finite element formulation, and the element response may be characterized like in the continuum situation. It is shown that the algorithm is capable of capturing the onset of localization as well as the post-localized response. In a numerical example, we study the influence of the internal friction angle on the development of a slip surface within a soil slope.
Ragnar Larsson, Jonas Larsson, Kenneth Runesson

Geometrical and material non-linear analysis of fully and partially saturated porous media

A formulation for a partially saturated porous medium undergoing large elastic or elasto-plastic deformations is presented. The porous material is treated as a multiphase continuum with the pores of the solid skeleton filled by water and air, this last one at constant pressure. This pressure may either be the atmospheric pressure or the cavitation pressure. The governing equations at macroscopic level are derived in a spatial and a material setting. Solid grains and water are assumed to be incompressible at the microscopic level. The elasto-plastic behaviour of the solid skeleton is described by the multiplicative decomposition of the deformation gradient into an elastic and a plastic part. The effective stress state is limited by the Drucker-Prager yield surface. The water is assumed to obey Darcy’s law. Numerical examples of the Liakopoulos’ test and of strain localization of dense or loose sand and of clay under undrained conditions conclude the paper.
Lorenzo Sanavia, Bernhard A. Schrefler, Paul Steinmann

Waves in poroelastic half space: Boundary element analyses

For simulating wave propagation in fluid-saturated semi-infinite porous continua, a boundary element formulation in time domain is presented. Usually, for such an integral formulation, the respective time domain fundamental solution is needed. This solution does only exist for Biot’s theory of poroelastic continua and, moreover, it is not given in closed form contrary to its Laplace domain representation. The recently developed “Convolution Quadrature Method”, proposed by Lubich, utilizes this Laplace transformed fundamental solution. Hence, applying this quadrature formula to the time dependent boundary integral equation, a time stepping procedure is obtained based only on the Laplace domain fundamental solution and a linear multistep method.
Finally, as an application, wave propagation in poroelastic half space is considered. Especially, the Rayleigh wave and the Love wave is studied. Also, the so-called slow compressional wave, a wave only existing in two-phase materials as poroelastic is, is confirmed by numerical studies.
Martin Schanz, Heinz Antes

Multicomponent reactive transport modelling: Applications to ore body genesis and environmental hazards

The interplay of chemical and mechanical effects is of crucial importance for understanding and solving key problems of geology, engineering and environmental science. We begin with an outline of the governing equations for a coupled multicomponent reactive transport model. Thermal coupling and heat flow is also considered. The reaction part of the model is kept very general and applies to a wide variety of problems including simulation of in situ leaching, mineralization (ore body genesis), environmental hazards and calcite leaching to name just a few. We briefly introduce the CSIRO symbolic finite element solver FASTFLO (http://​www.​cmis.​csiro.​au/fastflo) and illustrate the theory by a number of finite element solutions. While focussing initially on reactive transport dominated problems (deformation and damage negligible) we close with a limit load problem involving chemically induced reduction of the yield strength. We consider a stiff, smooth strip foundation on an infinite half plane (Prandtl case). The substrate deforms and partially plastifies (von Mises yield, Prandtl-Reuss flow rule). We then prescribe a constant concentration of a leachant on the surface of the half plane. The concentration of the leachant diffuses through the half plane, lowering the yield strength and ultimately causing unacceptably large deformation of the structure.
Reem Freij-Ayoub, Hans-Bernd Mühlhaus, Laurent Probst

A numerical model and its finite element solution for multiphase flow: Application to pulp and paper processing

This paper presents a general framework for the large scale finite element simulation of rolling and compression of multiphase deformable porous media. The main objective is the simulation of pulp and paper processing operations. On the theoretical side, the generalized Biot theory is extended and modified to derive the governing equations of general multiphase flow through porous media. These equations are specialised to the case of two-fluid flow having the velocity of the solid skeleton, the pressure, and saturation of the wetting fluid as the primary unknowns. On the numerical side, the finite element method is used for spatial discretization of the relevant equations. This is combined with a Newmark scheme discretisation in time. A direct solution scheme is adopted in the treatment of the coupling between the solid, liquid, and gas phases. To demonstrate the effectiveness of the developed framework, the simulation of a compression and a rolling process of relevance to the paper industry are presented.
D. R. J. Owen, S. Y. Zhao, E. A. de Souza Neto


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