Showing papers on "Mixture theory published in 2015"

Coupled Axisymmetric Thermo-Poro-Mechanical Finite Element Analysis of Energy Foundation Centrifuge Experiments in Partially Saturated Silt

Abstract: The paper presents an axisymmetric, small strain, fully-coupled, thermo-poro-mechanical (TPM) finite element analysis (FEA) of soil–structure interaction (SSI) between energy foundations and partially saturated silt. To account for the coupled processes involving the mechanical response, gas flow, water species flow, and heat flow, nonlinear governing equations are obtained from the fundamental laws of continuum mechanics, based on mixture theory of porous media at small strain. Constitutive relations consist of the effective stress concept, Fourier’s law for heat conduction, Darcy’s law and Fick’s law for pore liquid and gas flow, and an elasto-plastic constitutive model for the soil solid skeleton based on a critical state soil mechanics framework. The constitutive parameters employed in the thermo-poro-mechanical FEA are mostly fitted with experimental data. To validate the TPM model, the modeling results are compared with the observations of centrifuge-scale tests on semi-floating energy foundations in compacted silt. Variables measured include the thermal axial strains and temperature in the foundations, surface settlements, and volumetric water contents in the surrounding soil. Good agreement is obtained between the experimental and modeling results. Thermally-induced liquid water and water vapor flow inside the soil were found to have an impact on SSI. With further improvements (including interface elements at the foundation-soil interface), FEA with the validated TPM model can be used to predict performance and SSI mechanisms for energy foundations.

Resistivity coefficients for body composition analysis using bioimpedance spectroscopy: effects of body dominance and mixture theory algorithm.

Abstract: Body composition is commonly predicted from bioelectrical impedance spectroscopy using mixture theory algorithms. Mixture theory algorithms require the input of values for the resistivities of intra-and extracellular water of body tissues. Various derivations of these algorithms have been published, individually requiring resistivity values specific for each algorithm. This study determined apparent resistivity values in 85 healthy males and 66 healthy females for each of the four published mixture theory algorithms. The resistivity coefficients determined here are compared to published values and the inter-individual (biological) variation discussed with particular reference to consequential error in prediction of body fluid volumes. In addition, the relationships between the four algorithmic approaches are derived and methods for the inter-conversion of coefficients between algorithms presented.

From discrete elements to continuum fields: Extension to bidisperse systems

Abstract: To develop, calibrate and/or validate continuum models from experimental or numerical data, micro-macro transition methods are required. These methods are used to obtain the continuum fields (such as density, momentum, stress) from the discrete data (positions, velocities, forces). This is especially challenging for non-uniform and dynamic situations in the presence of multiple components. Here, we present a general method to perform this micro-macro transition, but for simplicity we restrict our attention to two-component scenarios, e.g. particulate mixtures containing two types of particles. We present an extension to the micro-macro transition method, called \emph{coarse-graining}, for unsteady two-component flows. By construction, this novel averaging method is advantageous, e.g. when compared to binning methods, because the obtained macroscopic fields are consistent with the continuum equations of mass, momentum, and energy balance. Additionally, boundary interaction forces can be taken into account in a self-consistent way and thus allow for the construction of continuous stress fields even within one particle radius of the boundaries. Similarly, stress and drag forces can also be determined for individual constituents of a multi-component mixture, which is essential for several continuum applications, \textit{e.g.} mixture theory segregation models. Moreover, the method does not require ensemble-averaging and thus can be efficiently exploited to investigate static, steady, and time-dependent flows. The method presented in this paper is valid for any discrete data, \textit{e.g.} particle simulations, molecular dynamics, experimental data, etc.

Analysis and numerical solution of stochastic phase-field models of tumor growth

Abstract: Carcinogenesis, as every biological process, is not purely deterministic as all systems are subject to random perturbations from the environment. In tumor growth models, the values of the parameters are subjected to many uncertainties that can arise from experimental variations or due to patient-specific data. The present work is devoted to the development and analysis of numerical methods for the solution of a system of stochastic partial differential equations governing a six-species tumor growth model. The model system simulates the stochastic behavior of cellular and macrocellular events affecting the evolution of avascular cancerous tissue. It is a continuous phase-field model that incorporates several key features in tumor dynamics. A sensitivity analysis is performed to identify the more influential parameters. A mixed finite element method and a stochastic collocation scheme are introduced to approximate random-variables components of the solution. The results of numerous numerical experiments are also presented and discussed. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 552–574, 2015

Diffusion of chemically reacting fluids through nonlinear elastic solids: Mixture model and stabilized methods

Abstract: This paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve an anisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixture theory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy production relations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint of maximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of mass and balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multiplier. Present implementation considers a simplification of the full system of governing equations in the context of isothermal problems, although anisothermal studies are bein...

Continuum Mechanics: Constitutive Modeling Of Structural And Biological Materials

Abstract: 1. Mathematics 2. Kinematics 3. The stress tensor 4. Introduction to material modeling 5. Ideal gas 6. Fluids 7. Elastic material models 8. Continuum mixture theory 9. Growth models 10. Parameter estimation and curve fitting 11. Finite element method 12. Appendix.

Polydisperse granular flows over inclined channels

Abstract: This thesis focuses on modelling the dynamics of dense granular materials flowing over an inclined channel, utilising a continuum description. In the process of understanding and developing this, besides continuum modelling, the thesis also exhibits the utility of discrete particle simulations (DPMs), and the need for developing an accurate micro-macro mapping technique. As most of these dense gravity-driven flows are shallow in nature, for monodisperse mixtures, Chapter 2 illustrates the formulation of a novel one -- dimensional (width- and depth-averaged) shallow granular model. Using this model, we not only predict the flow dynamics -- flow height, velocity and granular jumps or shocks -- but also shows that one can forecast the existence of multiple steady states for a given a range of channel inclinations. However, in reality, the majority of flowing particulate mixtures are known to comprise of particles with varied physical attributes, i.e. they are bidisperse or polydisperse. Thereby, as a step towards understanding the associated flow dynamics, and developing improved continuum models, several studies presented in this thesis have utilised discrete particle method. DPMs provide a plethora of information at a particle scale, such as particle position, velocity, interaction forces or stresses. In order to accurately map the particle scale mechanics onto a macroscopic continuum scale, Chapter 3 comprehensively presents a generic framework for an efficient and accurate micro-macro mapping technique for polydisperse mixtures comprising of convex shaped particles, e.g. spheres. More importantly, the method presented is valid for any discrete data, e.g. particle simulations, molecular dynamics and experimental data, for both steady and unsteady configurations. Before employing the efficient mapping technique of Chapter 3 to its full capacity, based on the current understanding of bidisperse segregation dynamics, we formulate in Chapter 4 a mixture theory segregation model for bidisperse mixtures varying both in size and density. The developed formulation is an extension to an already existing size-segregation model, and is applicable to both shallow (linear velocity profile) and thick (Bagnold profile) flows. Besides predicting the extent of segregation, the theory also predicts zero or weak segregation for a range of size and density ratios, which was further benchmarked using DPMs. Although, we developed an efficient continuum size- and density-segregation model, a detailed study is to be implemented in order to determine more accurate pressure scalings and further extend it to polydisperse mixtures. As a stepping stone, towards determining these pressure scalings, in Chapter 5 we give an example application of the micro-macro mapping technique (illustrated in Chapter 3). For simplicity, we consider a purely size-based segregation model, which was built upon a pressure scaling function containing an unknown parameter. Not only did we determine this unknown material parameter but, more importantly, we also found out that the complete size- and density-based segregation in any flowing particulate mixture is an effect of the generated kinetic stress, rather than the contact stress. The current form of the existing scaling functions is, however, still an active area of research, which definitely needs further attention and care. Chapters 3, 4 and 5, show how one can mix and match continuum models with DPMs using an efficient coarse-graining method. However, it is still vital to see if the DPMs can actually emulate reality. As a consequence, we illustrate in Chapter 6, how DPMs can be used as a suitable alternative to experiments using two commonly used DPM experiments.

Investigations of Gravity-Driven Two-Phase Debris Flows

Abstract: A depth-integrated theory is derived for the gravity-driven two-phase debris flows over complex shallow topography. The mixture theory is adopted to describe the mass and momentum conservation of each phase. The model employs the Mohr-Coulomb plasticity for the solid rheology, and assumes the Newtonian fluid for the fluid phase. The interactive forces assumed here consist of viscous drag force linear to velocity difference between the both phases, and buoyancy force. The well-established governing equations are built in 3D topography; as a result, they are expressed in the curvilinear coordinate system. Considering the characteristics of flows, a shallow layer assumption is made to simplify the depth-integrated equations. The final resulting equations are solved numerically by a high-resolution TVD scheme. The dynamic behaviors of the mixture are investigated. Numerical results indicate that the model can adequately describe the flows of dry granular material, the pure water and general two-phase debris flows.

Framework for Coupling Flow and Deformation of a Porous Solid

Abstract: In this paper, the flow of an incompressible fluid in a deformable porous solid is considered. A mathematical model using the framework offered by the theory of interacting continua is presented. In its most general form, this framework provides a mechanism for capturing multiphase flow, deformation, chemical reactions, and thermal processes, as well as interactions between the various physics, in a conveniently implemented fashion. To simplify the presentation of the framework, the results are presented for a particular model, which can be seen as an extension of Darcy’s equation (which assumes that the porous solid is rigid) and that takes into account the elastic deformation of the porous solid. The model also considers the effect of deformation on porosity. It is shown that by using this model identical results can be recovered as in the framework proposed in the literature. Some salient features of the framework are as follows: (1) it is a consistent mixture theory model, and adheres to the l...

Vulcanization and the mechanical response of rubber

Abstract: Hyperelastic models are widely used to describe the mechanical response of rubber. However, purely mechanical models cannot account for changes in the material due to chemical reactions such as those that take place during vulcanization. Here, we present a model developed within a thermodynamic framework accounting for chemical reactions. A mixture theory approach that allows for the existence of multiple species and their interconversion is followed. The existence of a Helmholtz potential and a rate of entropy production function for the mixture as a whole are posited. Following the multiple natural configuration approach, the rate of entropy production is maximized to obtain constitutive equations. The viscoelastic model is then specialized to the elastic case. The model is calibrated using data available in the literature for rubber. A simulation of the stress–strain curve of rubber as vulcanization progresses is presented.

From discrete to continuum fields in bidisperse granular mixtures

Abstract: Micro–macro transition methods can be used to, both, calibrate and validate continuum models from discrete data obtained via experiments or simulations. These methods generate continuum fields such as density, momentum, stress, etc., from discrete data, i.e. positions, velocity, orientations and forces of individual elements. Performing this micro–macro transition step is especially challenging for non-uniform or dynamic situations. Here, we present a general method of performing this transition, but for simplicity we will restrict our attention to two-component scenarios. The mapping technique, presented here, is an extension to the micro–macro transition method, called coarse-graining, for unsteady two-component flows and can be easily extended to multi-component systems without any loss of generality. This novel method is advantageous; because, by construction the obtained macroscopic fields are consistent with the continuum equations of mass, momentum and energy balance. Additionally, boundary interaction forces can be taken into account in a self-consistent way and thus allow for the construction of continuous stress fields even within one element radius of the boundaries. Similarly, stress and drag forces can also be determined for individual constituents of a multi-component mixture, which is critical for several continuum applications, e.g. mixture theory-based segregation models. Moreover, the method does not require ensemble-averaging and thus can be efficiently exploited to investigate static, steady and time-dependent flows. The method presented in this paper is valid for any discrete data, e.g. particle simulations, molecular dynamics, experimental data, etc.; however, for the purpose of illustration we consider data generated from discrete particle simulations of bidisperse granular mixtures flowing over rough inclined channels. We show how to practically use our coarse-graining extension for both steady and unsteady flows using our open-source coarse-graining tool MercuryCG. The tool is available as a part of an efficient discrete particle solver MercuryDPM (www.​MercuryDPM.​org).

Boundary-Value Problems of Statics in the Two-Temperature Elastic Mixture Theory for a Half-Space

Abstract: The static case of the two-temperature elastic mixture theory is considered when partial displacements of the elastic components of the mixture are equal to each other. The formula obtained for the representation of a general solution of a homogeneous system of differential equations is expressed in terms of four harmonic functions and one metaharmonic function. The uniqueness theorem for a solution is proved. Solutions are obtained in quadratures by means of boundary functions.

Mixed finite element formulation for non-isothermal porous media in dynamics

Abstract: We present a mixed finite element formulation for the spatial discretization in dynamic analysis of non-isothermal variably saturated porous media using different order of approximating functions for solid displacements and fluid pressures/temperature. It is known in fact that there are limitations on the approximating functions N and N for displacements and pressures if the Babuska–Brezzi convergence conditions or their equivalent [5] are to be satisfied. Although this formulation complicates the numerical implementation compared to equal order interpolation, it provides competitive advantages e.g. in speed of computation, accuracy and convergence. A fully coupled mathematical [1] and numerical model for the analysis of the thermo-hydromechanical dynamic behaviour of multiphase geomaterials is reduced to a computationally efficient formulation by neglecting the relative acceleration of the fluid phases and the convective terms [2], [3]. The resulting mathematical model is based on an average procedure following Lewis and Schrefler [1] within the Hybrid Mixture theory. Small strains and dynamic loading conditions are assumed. The porous medium is treated as a multiphase system composed of a solid skeleton with open pores, filled with liquid water and gas. All the fluids are in contact with the solid phase. The constituents are assumed to be isotropic, homogeneous, immiscible, except for dry air and water vapour and chemically non-reacting. Local thermal equilibrium between the solid matrix, gas and liquid phases is assumed. Heat conduction, vapour diffusion, heat convection, and liquid water flow due to pressure gradients or capillary effects and water phase change (evaporation and condensation) inside pores are all taken into account. The model has been implemented in the finite element code COMES-GEO, [1], [4]. The numerical examples will show the effectiveness of the implementation by comparison with analytical or finite element solutions for quasi-static and dynamic problems. Acknowledgement. The authors would like to thank the 7 Framework Programme of the European Union (ITN MuMoLaDe project 289911) for the financial support.

Compositional Flow in Fractured Porous Media: Mathematical Background and Basic Physics

Abstract: This chapter presents an overview of the equations describing the flow of multiphase and multicomponent fluids through fractured and unfractured porous media using the framework of continuum mixture theory. The model equations and constraint relationships are described by steps of increasing level of complexity. We first describe the governing equations for multiphase flow in both undeformable and deformable porous media. This model is extended to include the transport of chemical species by first describing the flow of a multicomponent, single-phase fluid and then of a compositional (multiphase and multicomponent) fluid in a porous medium. Finally, the equations governing the flow of compositional fluids in fractured porous media are described. The proposed methodology is suitable for modelling any type of fractured media, including dual-, triple-, and multiple-continuum conceptual models.

A Mixture Theory Approach for a Power-Law Fluid Flow through a Porous Channel

Abstract: The steady-state saturated flow of an incompressible power-law fluid through a porous channel limited by two impermeable flat plates is modeled using a mixture theory, which considers fluid and porous matrix as superimposed continuous constituents of a binary mixture. After some simplifying assumptions, the mechanical model gives rise to a coupled system of ordinary differential equations that is simulated by employing a Runge-Kutta method coupled with a shooting strategy. Despite the strong nonlinearity of the problem, this simple methodology provides stable and accurate results, for both shear-thinning and shear-thickening behaviors.