Showing papers on "Mixture theory published in 2019"

Theory and Applications of Macroscale Models in Porous Media

Abstract: Systems dominated by heterogeneity over a multiplicity of scales, like porous media, still challenge our modeling efforts. The presence of disparate length- and time-scales that control dynamical processes in porous media hinders not only models predictive capabilities, but also their computational efficiency. Macrosopic models, i.e., averaged representations of pore-scale processes, are computationally efficient alternatives to microscale models in the study of transport phenomena in porous media at the system, field or device scale (i.e., at a scale much larger than a characteristic pore size). We present an overview of common upscaling methods used to formally derive macroscale equations from pore-scale (mass, momentum and energy) conservation laws. This review includes the volume averaging method, mixture theory, thermodynamically constrained averaging, homogenization, and renormalization group techniques. We apply these methods to a number of specific problems ranging from food processing to human bronchial system, and from diffusion to multiphase flow, to demonstrate the methods generality and flexibility in handling different applications. The primary intent of such an overview is not to provide a thorough review of all currently available upscaling techniques, nor a complete mathematical treatment of the ones presented, but rather a primer on some of the tools available for upscaling, the basic principles they are based upon, and their specific advantages and drawbacks, so to guide the reader in the choice of the most appropriate method for particular applications and of the most relevant technical literature.

A continuum model of multi-phase reactive transport in igneous systems

Abstract: Multiphase reactive transport processes are ubiquitous in igneous systems A challenging aspect of modelling igneous phenomena is that they range from solid-dominated porous to liquid-dominated suspension flows and therefore entail a wide spectrum of rheological conditions, flow speeds and length scales Most previous models have been restricted to the two-phase limits of porous melt transport in deforming, partially molten rock and crystal settling in convecting magma bodies The goal of this paper is to develop a framework that can capture igneous system from source to surface at all phase proportions including not only rock and melt but also an exsolved volatile phase Here, we derive an n-phase reactive transport model building on the concepts of Mixture Theory, along with principles of Rational Thermodynamics and procedures of Non-equilibrium Thermodynamics Our model operates at the macroscopic system scale and requires constitutive relations for fluxes within and transfers between phases, which are the processes that together give rise to reactive transport phenomena We introduce a phase- and process-wise symmetrical formulation for fluxes and transfers of entropy, mass, momentum and volume, and propose phenomenological coefficient closures that determine how fluxes and transfers respond to mechanical and thermodynamic forces Finally, we demonstrate that the known limits of two-phase porous and suspension flow emerge as special cases of our general model and discuss some ramifications for modelling pertinent two- and three-phase flow problems in igneous systems

Mechanics of Poro-Elastic Media: A Review with Emphasis on Foundational State Variables

Abstract: We review fundamental aspects of linear poro-elasticity. In contrast to most available textbooks and review articles, our treatment of poro-elastic media is based on the continuum Mixture Theory. Kinematic state variables and dynamic variables are introduced and formally linearized before the fundamental constitutive relations, between pairs of these, are extensively discussed. The role of porosity in linear poro-elasticity is highlighted, and it is shown that porosity is one of the possible choices for one of the two kinematic state variables, and therefore, relations to alternative pairs of kinematic variables can be formulated. The treatment is concluded by the formulation of the governing set of partial differential equations that constitute the basis for analytical or numerical investigations of boundary value problems.

Constrained mixture modeling affects material parameter identification from planar biaxial tests

Abstract: The constrained mixture theory is an elegant way to incorporate the phenomenon of residual stresses in patient-specific finite element models of arteries. This theory assumes an in vivo reference geometry, obtained from medical imaging, and constituent-specific deposition stretches in the assumed reference state. It allows to model residual stresses and prestretches in arteries without the need for a stress-free reference configuration, most often unknown in patient-specific modeling. A finite element (FE) model requires material parameters, which are classically obtained by fitting the constitutive model to experimental data. The characterization of arterial tissue is often based on planar biaxial test data, to which nonlinear elastic fiber-reinforced material parameters are fitted. However, the introduction of the constrained mixture theory requires an adapted approach to parameter fitting. Therefore, we introduce an iterative fitting method, alternating between nonlinear least squares parameter optimization and an FE prestressing algorithm to obtain the correct constrained mixture material state during the mechanical test. We verify the method based on numerically constructed planar biaxial test data sets, containing ground truth sets of material parameters. The results show that the method converges to the correct parameter sets in just a few iterations. Next, the iterative fitting approach is applied to planar biaxial test data of ovine pulmonary artery tissue. The obtained results demonstrate a convergence towards constrained mixture compatible parameters, which differ significantly from classically obtained parameters. We show that this new modeling approach yields in vivo wall stresses similar to when using classically obtained parameters. However, due to the numerous advantages of constrained mixture modeling, our fitting method is relevant to obtain compatible material parameters, that may not be confused with parameters obtained in a classical way.

Homogenization of pseudoparabolic reaction-diffusion-mechanics systems: multiscale modeling, well-posedness and convergence rates

Abstract: In this dissertation, parabolic-pseudoparabolic equations are proposed to couple chemical reactions, diffusion, flow and mechanics in heterogeneous materials using the framework of mixture theory. ...

Two-Dimensional Numerical Model of the Fracture Process in Steel Fibre Reinforced Concrete with the Continuum Strong Discontinuity Approach and Functional Data Analysis

Abstract: This paper presents the formulation of a two-dimensional numeri-cal model able to describe the fracture process in structural mem-bers of steel fibre reinforced concrete (SFRC) from the volume ratio of the fibres and the mechanical properties of the compo-nents: a concrete matrix and a set of steel fibres with a random orientation. The relationship between the stress and the strain fields of the composite material is obtained using the mixture theory with a compatibility strain of its component materials. The concrete matrix is represented with a scalar damage constitutive model with a softening strain and a different strength in tension and compression. The mechanical strain of an insulated fibre and the slip between the fibre and the matrix are simultaneously de-scribed with a one-dimensional plasticity constitutive model. The cracking of the composite material indicates a jump in the dis-placement field and non-bounded values of the strain field, which are represented by the Continuum Strong Discontinuity Ap-proach. The model has been implemented in the framework of the nonlinear analysis with the Finite Element Method, using con-stant strain triangular elements. Moreover, the fibres distribution and orientation change randomly in each finite element and each simulation or observation. The structural responses of the simula-tions are treated as curves and analysed by tools from the Func-tional Data Analysis. Confidence intervals for the structural re-sponse are built using bootstrap methodology. Finally, experi-mental tests of SFRC members subjected to tension and bending are simulated. The structural response and the cracking patterns obtained from the numerical simulation are satisfactory.

Constitutive Relations in Electromagnetism and Ion Electrodynamics

Abstract: This chapter applies the theory of multicomponent mixtures to a charged mixture comprising ions, a fluid component, and, possibly, a solid component. In the first part of the chapter, the constitutive laws for production terms, total forces, and energy of each component of the mixture are derived on the basis of biophysically motivated assumptions. In the second part of the chapter, the system of equations modeling the electrodynamics of a charged mixture is derived. Two special instances of the general multicomponent charged mixture theory are considered, namely, the Poisson–Nernst–Planck model for ion electrodynamics and the drift-diffusion model for electron and hole charge transport in a semiconductor.

Extension of unsaturated soil mechanics and its applications

Abstract: Although the soil, water and air coupling theory has been deductively derived from the three-phase mixture theory, assumptions and interpretations inherent in soil mechanics are also conveniently i...

The two-domain model of solute transport in binary alloy

Abstract: A mixed model for micro-macroscopic computer simulation of binary alloy solidification is proposed. It involves a two-domain approach to solute conservation equations in the liquid and solid phases, whereas transport of momentum and energy in the two-phase region is modelled using the phase mixture theory. To distinguish regions of columnar and equiaxed crystal structures evolving in a cast during solidification, the special front tracking technique on non-structural triangular grids is included in the model. In this two-domain approach, solute conservation equations are averaged across solid and liquid phases, and the solute transport at the phase interface is included. Additionally, the microstructure evolution is modelled to capture the development of various complex grain structures and more accurately describe the solute transport between the phases. The accuracy of the proposed model is first verified by a grid refinement analysis, and then the model is used to predict the solute concentration and macro-segregation in the example problem of Pb-48%wt Sn alloy solidification in a 2D mould. The results obtained are next compared with the relevant ones predicted by the fully single-domain model, earlier developed by authors. Thus, the role of finite diffusion in liquid and solid phases is identified and discussed.