Compressible Constituents Research Articles

A Macro-microscopic Coupled Constitutive Model for Fluid-Saturated Porous Media with Compressible Constituents

The paper provides a macro-microscopic coupled constitutive model for fluid-saturated porous media with respect to the compressibility of the solid skeleton, the real solid material and the fluid phase. The derivation of the model is carried out based on the porous media theory and is consistent with the second law of thermodynamics. In the present paper, two different sets of independent variables are introduced to implement the coupled behavior between the compressibility of the solid skeleton and the real solid material. Altogether the proposed model exploits five independent variables, i.e. the deviatoric part of the right Cauchy-Green deformation tensor, the partial density of solid phase, the density of the real solid material, the density of the real fluid material and the relative velocity of the fluid phase. Subsequently, the linearized version of the proposed constitutive model is also presented and compared with some models by other authors. It is found that Biot's model can also be derived based on the linearized version of the proposed model, which indicates that the present work bridges the gap between the porous media theory and Biot's model. Compared with Biot's model, the present model can provide the evolution of the porosity by considering the volumetric strain of the solid skeleton and the volumetric strain of the real solid material.

A macro–microscopic coupled consolidation model for saturated porous media with compressible constituents

Porosity is an important variable in consolidation theory. However, porosity information is missing in Biot’s model when the solid constituent of the porous media is assumed to be compressible. This paper develops a macro–microscopic coupled consolidation model in order to track the evolution of the porosity for saturated porous media with compressible constituents. The proposed model exploits a u−εvsr−p form with the aim to incorporate the coupled behavior between the macroscopic and microscopic volumetric compressibility. Subsequently, a specific study on the analytical solutions for the case of a linear-loaded one-dimensional consolidation problem based on the proposed model is presented. Two examples are shown in order to demonstrate how to apply the proposed model. The proposed model paves the way to study the consolidation problem with porosity-dependent coefficients for saturated porous media with compressible constituents.

A Hybrid Biphasic Mixture Formulation for Modeling Dynamics in Porous Deformable Biological Tissues.

The primary aim of this study is to establish the theoretical foundations for a solid-fluid biphasic mixture domain that can accommodate inertial effects and a viscous interstitial fluid, which can interface with a dynamic viscous fluid domain. Most mixture formulations consist of constituents that are either all intrinsically incompressible or compressible, thereby introducing inherent limitations. In particular, mixtures with intrinsically incompressible constituents can only model wave propagation in the porous solid matrix, whereas those with compressible constituents require internal variables, and related evolution equations, to distinguish the compressibility of the solid and fluid under hydrostatic pressure. In this study, we propose a hybrid framework for a biphasic mixture where the skeleton of the porous solid is intrinsically incompressible but the interstitial fluid is compressible. We define a state variable as a measure of the fluid volumetric strain. Within an isothermal framework, the Clausius-Duhem inequality shows that a function of state arises for the fluid pressure as a function of this strain measure. We derive jump conditions across hybrid biphasic interfaces, which are suitable for modeling hydrated biological tissues. We then illustrate this framework using confined compression and dilatational wave propagation analyses. The governing equations for this hybrid biphasic framework reduce to those of the classical biphasic theory whenever the bulk modulus of the fluid is set to infinity and inertia terms and viscous fluid effects are neglected. The availability of this novel framework facilitates the implementation of finite element solvers for fluid-structure interactions at interfaces between viscous fluids and porous-deformable biphasic domains, which can include fluid exchanges across those interfaces.

Cell – extracellular matrix interaction in glioma growth. In silico model

The study aims to investigate the role of viscoelastic interactions between cells and extracellular matrix (ECM) in avascular tumor growth. Computer simulations of glioma multicellular tumor spheroid (MTS) growth are being carried out for various conditions. The calculations are based on a continuous model, which simulates oxygen transport into MTS; transitions between three cell phenotypes, cell transport, conditioned by hydrostatic forces in cell–ECM composite system, cell motility and cell adhesion. Visco-elastic cell aggregation and elastic ECM scaffold represent two compressible constituents of the composite. Cell–ECM interactions form a Transition Layer on the spheroid surface, where mechanical characteristics of tumor undergo rapid transition. This layer facilitates tumor progression to a great extent. The study demonstrates strong effects of ECM stiffness, mechanical deformations of the matrix and cell–cell adhesion on tumor progression. The simulations show in particular that at certain, rather high degrees of matrix stiffness a formation of distant multicellular clusters takes place, while at further increase of ECM stiffness subtumors do not form. The model also illustrates to what extent mere mechanical properties of cell–ECM system may contribute into variations of glioma invasion scenarios.

Vertical Vibration Analysis of a Pile Group in Multilayered Poroelastic Soils with Compressible Constituents

Abstract This paper will investigate the vertical vibration of a pile group connected to a rigid cap in multilayered poroelastic soils with compressible constituents. Individual piles in the pile g...

Consolidation analysis for layered transversely isotropic viscoelastic media with compressible constituents due to tangential circular loads

In soft soil area, the tangential load induced by the vehicle brake, tidal action and lateral loaded pile is inevitable for the consolidation problem. This article investigates the consolidation of a multilayered transversely isotropic viscoelastic media with compressible constituents subjected to tangential circular loads. A general solution scheme is established, in which the integration transform is firstly used to convert the governing equations into the ordinary differential equations, and then the extended precise integration method is employed to solve the aforementioned equations. The Merchant model is introduced to simulate the viscoelasticity of medium with the aid of the elastic-viscoelastic correspondence principle. After the verification implemented by existing analytical solutions and ABAQUS, numerical examples are given to further reveal the effects of viscoelastic parameters, compressibility of soil particles and the stratification on the consolidation behavior.

Quasi-static thermal analyses of layered compressible poroelastic materials with a finite depth or half-space

• The Laplace–Fourier transforms are applied to reduce partial differential equations to ordinary ones. • An extended precise integration solution of multilayered materials due to a heat source is derived. • The actual solution in the physical domain is further solved by inverting the Laplace-Fourier transforms. • The effects of compressible constituents and layered characteristic on thermal consolidation are investigated. Quasi-static thermal analyses of layered compressible poroelastic materials with a finite depth or half-space caused by a heat source are carried out via extending the precise integration method in this study. The Laplace-Fourier transforms are applied to reduce partial differential equations of three-dimensional coupled thermo-hydro-mechanical problem to ordinary ones in the transform domain. By means of the precise integration method, a precise solution of layered materials induced by a heat source is deduced, and then, the previous ordinary differential equations are solved by substituting these equations into the extended precise integration solution. The actual solution is solved by applying the Laplace–Fourier inverse transforms. And lastly, numerical computations are performed to validate the method, and to analyze the effects of compressible constituents and layered characteristic as well as the heat source's half-life on thermal consolidation.

Hydro-mechanical Behaviors of the Three-dimensional Consolidation of Multi-layered Soils with Compressible Constituents

Compressibility and permeability are two major coupled factors for the engineering properties of soils. Such hydro-mechanical behaviors of soil greatly rely on the composition and the structure of soil which is often expressed in terms of porosity. In this paper, the effect of soil porosity on the soil mechanisms is studied in the research of Biot's consolidation for soft soil. Assuming a constant porosity in a finite layer, the drained bulk modulus and the permeability are porosity-dependent, and the fluid and the soil skeleton of highly saturated soil are assumed to be compressible. The expressions of displacement, stress and pore pressure are obtained with the aid of Laplace-Hankel transforms for the axisymmetric problem. The continuity of multilayered soil yields the global stiffness matrix in the transformed domain. Meanwhile, by introducing the stiffness matrix of the soil base, the compressible and permeable boundary conditions are more acceptable in practical engineering and the global stiffness matrix is then solved in the transformed domain. Taking the inverse of the Laplace-Hankel transforms gives the real answers for Biot's consolidation. The results show that the increase in porosity leads to a higher compressibility and permeability of soil, and consequently, causes a higher settlement and accelerates the process of consolidation. However, the assumption of the constant porosity of soil has some limitations due to ignoring the nonlinear development of soil structure during consolidation and the further works about the variation modes of porosity need to be done for a better understanding of soil consolidation.

Thermal performance of stratified fluid-filled geomaterials with compressible constituents around a deep buried decaying heat source

This paper investigates the transient thermal performance of stratified fluid-filled geomaterials with compressible constituents due to a buried decaying heat source. Starting with the governing partial differential equations, the ordinary differential matrix equation is derived by means of Laplace transform and Hankel transform. Using the extended precise integration method, the ordinary differential matrix equation is solved in the transformed domain, and the actual solutions are acquired through the inversion of Laplace and Hankel transforms. Two numerical calculations are given to validate the accuracy and feasibility of the presented approach, and other calculations are performed to study the effects of the heat source’s half-life, compressible constituents, thermal expansion coefficients, and the stratified character on the transient thermal response of medium.

Analytical Solution of Plane-Strain Poroelasticity due to Surface Loading within a Finite Rectangular Domain

AbstractAn analytical solution is presented for plane-strain poroelasticity induced by surface-normal loading within a finite rectangular fluid-saturated domain. In this work, the porous medium is assumed to be isotropic, homogeneous, and composed of compressible constituents. Appropriate Fourier and Laplace transforms and inversions are used to derive the analytical solution. In particular, lateral boundary conditions for the displacement field are specifically prescribed with care to match the Fourier and Laplace transforms. The analytical solution is presented in terms of summations of multiple series. Then, a case study for the consolidation of a water-saturated soft clay layer is carried out. The analytical solution is compared to the finite-element (FE) solution through a nonlinear transient FE analysis software program. The excellent agreement between the FE solution and the analytical solution demonstrates the accuracy and reliability of the analytical solution presented in this paper. Meanwhile, ...

Homogeneous equilibrium model for geomechanical multi-material flow with compressible constituents

Multi-material flow generally describes a situation where several distinct materials separated by sharp material interfaces undergo large deformations. In order to model such flow situations in the context of geomechanics and geotechnical engineering, a theoretical framework is presented which introduces a possible two-phase coupled saturated granular material behavior among the different materials. This is achieved by extending the technique of local volume averaging to a hierarchy of three spatial scales, based on a product of two indicator functions. A homogeneous equilibrium mixture model is subsequently derived for an example flow consisting of bulk solid, bulk fluid, and undrained granular material with compressible constituents. The closure relations are provided at the macroscale, including those describing granular behavior covering the full frictional-collisional flow regime and bulk material volume fraction evolution. The paper discusses the advantages and restrictions of the proposed mixture model and addresses its application and full-scale numerical implementation.

Note on axisymmetric consolidation of multi-layered poroelastic soils with compressible constituents

ABSTRACTIn the conventional analysis of Biot’s axisymmetric consolidation, the solid phase and the surrounding fluid were often assumed incompressible for simplicity. Such assumption in soil engineering ignored the effect of compressibility of constituents on the consolidation. In this article, the compressible fluid and solid in soil were taken into account for 3-D consolidation. The pore pressure, the displacements, and the stresses were expressed by two displacement functions, and the Laplace–Hankel transform was applied to set up the stiffness matrix between the generalized displacement and stress. The stiffness matrix consisted of negative exponential functions, ensuring that the computation is efficient and stable. Then the global stiffness matrix is extended by embracing the continuity of the interfaces and boundary conditions of soil base. The relationship between the generalized displacement and stress of the soil base reduces the number of unknowns of the global matrix. Such consideration reduced the number of unknowns of the global matrix and brought in more acceptable boundary conditions where the stiffness and the permeability of semi-infinite soil base can be taken into account. After the inversion of the Laplace–Hankel transform, the real solutions were obtained. The results show that the stiffness and permeability of the soil base can change the development consolidation and that the compressibility of solid particles has a great effect on the settlement in the beginning of consolidation.

A mixture model of poroelastic materials

Abstract Chemically-reacting porous materials are investigated within the theory of mixtures. The key idea is that for incompressible constituents, the volume fractions satisfy balance equations strictly analogous to those of mass densities. As a consequence, upon compatibility with thermodynamics, constitutive models are established where the evolution of the volume fractions is governed by the conservation of mass and need not be given by additional balance equations. Next an analogous scheme is developed for materials with voids where chemical reactions no longer occur. Compressible constituents are also considered and the concentrations are used as the variables characterizing the immiscibility. Some comments are given about the approaches where further fields occur, such as equilibrated forces and stresses, and are associated with additional balance equations.

Dynamical effects during compaction band formation affecting their spatial periodicity

AbstractCompaction bands (CBs) are responsible for significant anisotropy alterations of permeability in geological materials; hence, understanding their formation conditions appears of key importance to all applications involving fluid extraction/injection from/into the ground. While most of the available models to understand CB formation are focused on interpreting the onset of a single CB, little effort has been so far dedicated to understand the documented periodicity of CBs. In this paper, the role of dynamical effects in inducing the post onset evolution of CBs is analyzed by means of a dedicated model for porous media with compressible constituents, with reference to a horizontal layer of sandy, water‐saturated material. Elastic waves are generated as a first CB occurs due to sudden, localized volumetric collapse. If the waves are reflected at the interface with a softer material or with a previously formed CB, they produce significant local effective stress concentrations, which can promote the formation of further CBs in a cascade fashion, according to a regular geometric pattern. The spatial distribution of dynamically generated CBs, as well as the extent of the phenomenon, depends on the geometry of the domain and on the material's permeability. Sensitivity analysis is also performed to assess the key properties that promote dynamical CB in situ formation, identifying as the most influential conditions large stratum stiffness (increasing with depth) and the presence of softer layers. In contrast, the presence of less permeable and/or stiffer layers is not believed to play a major role in the proposed mechanism.

An integral-free expression for short-term changes in fault stability due to pore pressure induced when a point load is placed on the pervious boundary of a porous elastic half space containing a fault

A concentrated load with step-function time behaviour is placed normal to the planar, pervious boundary of a porous elastic half space (PEHS) with compressible constituents. A planar fault exists in the PEHS in such a way that the poroelastic behaviour of the medium is unhindered. We derive an approximate but integral-free expression for CFSCPP, i.e., changes in fault stability due to changes in pore pressure, at a point not too far off the line along which the load acts. But, in the interest of simplicity, the main discussion is focussed on a consideration of CFSCPP at a point P located on the fault at depth z directly beneath the load. It is convenient to introduce dimensionless time t D directly proportional to real time t. The constant of proportionality is 4c/z 2, where c is hydraulic diffusivity. The derived approximate expression gives results with an accuracy of greater than 99% for limited values of t D after the load is imposed. We learn from the derived expression that, for a given z, fault stability undergoes an initial sudden decrease commensurate with the undrained pore pressure induced in the PEHS. This is followed by a more gradual decrease in fault stability with increasing t D until a minimum is reached. The real time t to minimum fault stability increases with z. The magnitude of CFSCPP decreases with z as z −2 for a given t D in the permissible range. The derived expression and the inferences based on it should be useful during earth science investigations of the possible hazards due to reactivation of a pre-existing shallow fault when a civil engineering project involving imposition of a heavy load on the earth’s surface is to be executed nearby. They should be useful also for investigations if a shallow earthquake occurs near such a project soon after its execution.

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

SUMMARYAn analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.

Non-axisymmetric consolidation of poroelastic multilayered materials with anisotropic permeability and compressible constituents

This paper presents an analytical layer-element solution to non-axisymmetric consolidation of multilayered poroelastic materials with anisotropic permeability and compressible constituents. By applying Fourier expansions, Hankel transforms and Laplace transforms to the state variables involved in the governing equations of poroelasticity with respect to the circumferential, radial and time coordinates, respectively, the analytical layer-element (i.e. a symmetric stiffness matrix) is derived, which describes the relationship between the transformed generalized stresses and displacements at the surface (z=0) and those at an arbitrary depth z, considering the corresponding boundary and continuity conditions at the layer interfaces, the global stiffness matrix of a multilayered system is assembled in the transformed domain. The actual solutions in the physical domain are acquired by applying numerical quadrature schemes for the inversion of the Laplace–Hankel transform. Finally, numerical calculation is presented to investigate the influence of layering and poroelastic material parameters on consolidation process.

A Rational Procedure for the Experimental Evaluation of the Elastic Coefficients in a Linearized Formulation of Biphasic Media with Compressible Constituents

This article illustrates the specialization to infinitesimal perturbations of the geometrically nonlinear formulation describing the macroscopic dynamic behavior of biphasic media presented in (Serpieri and Rosati J Mech Phys Solids 59(4):841–862, 2011) where, based on Lagrangian mechanics arguments, a consistent mathematical theory of fluid-saturated poroelastic material is developed accounting for the property that both constituent materials are microscopically compressible. The primary contribution of this work is the employment of the linearized model in the development of a general procedure for the determination of the relevant elastic coefficients on the basis of the data provided by a set of experimental measures analogous to the one originally considered by Biot and Willis consisting of a shear test, an unjacketed compressibility test and two jacketed tests in which drainage is either completely allowed or prevented. The linearization of the formulation is first presented for a generic configuration in which both phases are undergoing a generic motion. Subsequently, a specific study is presented on the case that the media is motionless, homogeneous, and isotropic in the reference configuration, which is typical of common experimental set-ups for static tests. A numerical example is shown and a general relation between the elastic coefficients and Biot’s constant is derived which consistently accounts for the compressibility of all materials.

A general approach for defining the macroscopic free energy density of saturated porous media at finite strains under non-isothermal conditions

Abstract A general approach is proposed for defining the macroscopic free energy density function (and its complement, the free enthalpy) of a saturated porous medium submitted to finite deformations under non-isothermal conditions, in the case of compressible fluid and solid constituents. Reference is made to an elementary volume treated as an ‘open system’, moving with the solid skeleton. The proposed free energy depends on the generalised strains (namely an appropriate measure of the strain of the solid skeleton and the variation in fluid mass content) and the absolute temperatures of the solid and fluid phases (which are assumed to differ from each other for the sake of generality). This macroscopic energy proves to be a potential for the generalised stresses (namely the associated measure of the total stress and the free enthalpy of the pore fluid per unit mass) and the entropies of the solid and fluid phases. In contrast with mixture theories, the resulting free energy is not the simple sum of the free energies of the single constituents. Two simplified cases are examined in detail, i.e. the semilinear theory (originally proposed for isothermal conditions and extended here to non-isothermal problems) and the linear theory. The proposed approach paves the way to the consistent non-isothermal-hyperelastic-plastic modelling of saturated porous media with a compressible fluid and solid constituents.

Finite strain hyperelastoplastic modelling of saturated porous media with compressible constituents

Starting from the general expression of the free-energy function of a saturated porous medium at finite deformations in the case of compressible fluid and solid constituents, and from the internal dissipation increment, the general expressions of the plastic potential and flow rule are deduced together with the general form of the consistency condition. Reference is made to an elementary volume moving with the solid skeleton in a Lagrangian description, which is treated as an open system from which the pore fluid can flow freely in and out. As a result, a generalisation is provided of the classical Prandtl–Reuss relationship of small strain elastoplasticity in single-phase media to finite strain multiplicative (for F ) and additive (for the fluid mass content) elastoplasticity in saturated porous media with compressible constituents. The following particular cases are analysed in detail: null plastic volume change of the solid constituent, incompressibility of the solid constituent, incompressibility of both fluid and solid constituents, quasi-linear theory (in which the solid constituent is assumed to be nearly incompressible, and therefore undergoing small volume changes), and geometrically linearised theory. The simplified approaches previously presented in the literature are thus recovered within a unified framework and new, simplified constitutive assumptions are made.