Theory of multiphase mixtures—A thermomechanical formulation

The paper presents a theory of mixtures with the nonzero interfacial area between the constituents of the mixture. The conservation laws are physically motivated by utilizing a volume averaging procedure and by the definition of a mapping transformation. It is shown that the theory constructed in this manner is consistent with the theory of mixtures with a vanishingly small interfacial area and that a second law of thermodynamics can be assigned for each phase of the mixture. The conservation laws are examined for invariance properties with the principle of the material frame indifference, and a particular constitutive assumption is discussed. Also presented in the paper are the conservation laws in the integral form and the jump conditions for the singular surfaces in the multiphase mixture.     

On the theory of mixtures of elastic solids

In this paper we derive a theory for binary mixtures of elastic solids in which the independent constitutive variables are the displacement gradients, displacement fields, volume fractions and volume fraction gradients. The theory is linearized and a uniqueness theorem with no definiteness assumption on the elasticities and no restriction on the initial stresses is presented.     

Coupled thermo-hygro-mechanical damage model for concrete subjected to high temperatures

Based on the theory of mixtures, a coupled thermo-hygro-mechanical (THM) damage model for concrete subjected to high temperatures is presented in this paper. Concrete is considered as a mixture composed of solid skeletons, liquid water, water vapor, dry air, and dissolved air. The macroscopic balance equations of the model consist of the mass conservation equations of each component and the momentum and energy conservation equations of the whole medium mixture. The state equations and the constitutive model used in the model are given. Four final governing equations are given in terms of four primary variables, i.e., the displacement components of soil skeletons, the gas pressure, the capillary pressure, and the temperature. The processes involved in the coupled model include evaporation, dehydration, heat and mass transfer, etc. Through the process of deformation failure and the energy properties, the mechanics damage evolution equations are established based on the principle of conversation of energy and the Lemaitre equivalent strain assumption. Then, the influence of thermal damage on the mechanical property is considered.     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

Toward the thermodynamic modeling of reacting ionic mixtures

Based on the Müller–Liu entropy principle and the axioms of constitutive theory, a continuum model for reacting ionic mixtures is presented. The influence of microscopic structure on the mixture dynamics is taken into account through the thermodynamics of polar materials. Moreover, mechanical balance laws for classical mixtures under influence of electromagnetic fields and quasi-electrostatic Maxwell’s equations are briefly shown. With an appropriate constitutive model for a diluted and isotropic mixture of non-volatile solutes and by considering the same temperature field for all constituents, constraints on constitutive quantities are imposed, and the conditions for the thermodynamic equilibrium are established from the entropy principle. Furthermore, the nonlinear nature of chemical reactions as well as the reciprocal nature of some irreversible processes is highlighted. Unlike the classical approach for electrolyte solutions, the current constitutive model incorporates thermoelectric and electro-kinetic phenomena into the phenomenological equations, providing a more comprehensive approach of electrolyte solutions dynamics.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.     

A two-fluid model for reactive dilute solid–liquid mixtures with phase changes

Based on the Eulerian spatial averaging theory and the Müller–Liu entropy principle, a two-fluid model for reactive dilute solid–liquid mixtures is presented. Initially, some averaging theorems and properties of average quantities are discussed and, then, averaged balance equations including interfacial source terms are postulated. Moreover, constitutive equations are proposed for a reactive dilute solid–liquid mixture, where the formation of the solid phase is due to a precipitation chemical reaction that involves ions dissolved in the liquid phase. To this end, principles of constitutive theory are used to propose linearized constitutive equations that account for diffusion, heat conduction, viscous and drag effects, and interfacial deformations. A particularity of the model is that the mass interfacial source term is regarded as an independent constitutive variable. The obtained results show that the inclusion of the mass interfacial source term into the set of independent constitutive variables permits to easily describe the phase changes associated with precipitation chemical reactions.     

A Model for Multiphase and Multicomponent Flow in Porous Media,Built on the Diffuse-Interface Assumption

This article presents a new theory for flow in porous media of a mixture of nonreacting chemical components. In the examples considered, these components are hydrocarbons and water. The model presented assumes that porosity is constant and uniform, and that the wetting properties of the medium are nearly neutral. The flow equations are obtained by starting with the balance equations (mass, momentum, and energy) at pore level, and averaging them over a large number of pores, using the diffuse interface assumption then the methods of irreversible thermodynamics, thus obtaining, among other things, the collective convective velocity and the component-wise diffusive velocities as functions of the component densities. When the simplification of uniform temperature is introduced, the flow equations are of the Cahn–Hilliard type (with an extra term accounting for gravitation) where the thermodynamic function is the Helmholtz free energy per unit volume of the mixture. There are no relative permeabilities. Also, the set of equations is complete in the sense that no flash calculations are necessary, phase segregation being part of the calculation. The numerical examples considered are: (i) phase segregation in a gravitational field and (ii) coning where the initial state is fully segregated.     

微极介质混合物的等效本构方程

对微极介质混合物引入代表性体积的等效均匀体,用代表性体积边界上的面力和面力偶定义等效应力和等效偶应力,提出了建立微极介质混合物的等效本构方程的一般原理和方法.讨论了以十字形框架为胞元的多胞材料面内变形问题的等效本构方程,给出合理的解析结果.     

Diffuse interface model for high speed cavitating underwater systems

High speed underwater systems involve many modelling and simulation difficulties related to shocks, expansion waves and evaporation fronts. Modern propulsion systems like underwater missiles also involve extra difficulties related to non-condensable high speed gas flows. Such flows involve many continuous and discontinuous waves or fronts and the difficulty is to model and compute correctly jump conditions across them, particularly in unsteady regime and in multi-dimensions. To this end a new theory has been built that considers the various transformation fronts as ‘diffuse interfaces’. Inside these diffuse interfaces relaxation effects are solved in order to reproduce the correct jump conditions. For example, an interface separating a compressible non-condensable gas and compressible water is solved as a multiphase mixture where stiff mechanical relaxation effects are solved in order to match the jump conditions of equal pressure and equal normal velocities. When an interface separates a metastable liquid and its vapor, the situation becomes more complex as jump conditions involve pressure, velocity, temperature and entropy jumps. However, the same type of multiphase mixture can be considered in the diffuse interface and stiff velocity, pressure, temperature and Gibbs free energy relaxation are used to reproduce the dynamics of such fronts and corresponding jump conditions. A general model, based on multiphase flow theory is thus built. It involves mixture energy and mixture momentum equations together with mass and volume fraction equations for each phase or constituent. For example, in high velocity flows around underwater missiles, three phases (or constituents) have to be considered: liquid, vapor and propulsion gas products. It results in a flow model with 8 partial differential equations. The model is strictly hyperbolic and involves waves speeds that vary under the degree of metastability. When none of the phase is metastable, the non-monotonic sound speed is recovered. When phase transition occurs, the sound speed decreases and phase transition fronts become expansion waves of the equilibrium system. The model is built on the basis of asymptotic analysis of a hyperbolic total non-equilibrium multiphase flow model, in the limit of stiff mechanical relaxation. Closure relations regarding heat and mass transfer are built under the examination of entropy production. The mixture equation of state (EOS) is based on energy conservation and mechanical equilibrium of the mixture. Pure phases EOS are used in the mixture EOS instead of cubic one in order to prevent loss of hyperbolicity in the spinodal zone of the phase diagram. The corresponding model is able to deal with metastable states without using Van der Waals representation.     

A Micropolar Theory of Porous Media: Constitutive Modelling

The extension of the classical mixture theory by the concept of volume fractions leads to the theory of porous media. In this article, the theory of porous media is generalised to micropolar constituents. The kinematic relations and the balance equations for a porous medium are developed without restricting the number of constituents. Based on the entropy inequality, the general form of the constitutive equations are derived for a binary medium consisting of a porous elastic skeleton saturated by a viscous pore-fluid. Both constituents are assumed to be compressible. Handling the saturation constraint by a Lagrangian multiplier leads to a compatibility of the proposed model to so-called hybrid and incompressible models.     

Numerical simulation of multiphase cavitating flows around an underwater projectile

The present simulation investigates the multiphase cavitating flow around an underwater projectile. Based on the Homogeneous Equilibrium Flow assumption, a mixture model is applied to simulate the multiphase cavitating flow including ventilated cavitation caused by air injection as well as natural cavitation that forms in a region where the pressure of liquid falls below its vapor pressure. The transport equation cavitating model is applied. The calculations are executed based on a suite of CFD code. The hydrodynamics characteristics of flow field under the interaction of natural cavitation and ventilated cavitation is analyzed. The results indicate that the ventilated cavitation number is under a combined effect of the natural cavitation number and gas flow rate in the multiphase cavitating flows.     

Inelastic constitutive equations of polycrystalline metals subjected to finite deformation part I: Effect of grain rotation

In this article, a set of inelastic constitutive equations of polycrystalline metals is derived by combining a finite deformation kinematics of single crystal component, and a shear stress-shear strain relation of slip system based on a thermoactivated motion of dislocation. Interactions among grains are incorporated by “constant deformation gradient assumption.” The forms of these equations are rather simple internal variable theory types. By using these equations, some fundamental effects of grain rotations on inelastic behaviors of polycrystalline metals in a finite deformation range under complex loading and elevated temperature conditions are demonstrated. Some comments are given on a problem of plastic spin tensor.     

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY(Ⅰ)—NONLINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

IntroductionSoilisthemostcommonlyusedconstructionmaterialincivilengineeringandhydraulicengineering .Thecharacteristicsofsoilhavebeeninvestigatingfornearlyonehundredyears.Butbecauseofitscomplexstructure,changeableenvironmentandbeingsensitivetotheoutsideconditions,thesoiloftenshowsvariedproperties[1,2 ].Themaindifficultytothedevelopmentofgeotechnicalmechanicsishowtosetupconstitutiveequationswhichcouldsatisfactorilyaccountforengineeringpropertiesofsoil[3].Manyconstitutivemodelshavebeenformedinth…     

A micropolar mixture theory of multi-component porous media

A mixture theory is developed for multi-component micropolar porous media with a combination of the hybrid mixture theory and the micropolar continuum theory. The system is modeled as multi-component micropolar elastic solids saturated with multi- component micropolar viscous fluids. Balance equations are given through the mixture theory. Constitutive equations are developed based on the second law of thermodynamics and constitutive assumptions. Taking account of compressibility of solid phases, the volume fraction of fluid as an independent state variable is introduced in the free energy function, and the dynamic compatibility condition is obtained to restrict the change of pressure difference on the solid-fluid interface. The constructed constitutive equations are used to close the field equations. The linear field equations are obtained using a linearization procedure, and the micropolar thermo-hydro-mechanical component transport model is established. This model can be applied to practical problems, such as contaminant, drug, and pesticide transport. When the proposed model is supposed to be porous media, and both fluid and solid are single-component, it will almost agree with Eringen＇s model.     

THE CONSTITUTIVE EQUATION RESEMBLANCE OF ELASTIC-PLASTIC MULTIPHASE SOLID

Based on composite materials,the equivalent elastic-plastic constitutive equations ofmultiphase solid are researched.According to the suggested definition of.constitutiveequivalence it is demonstrated that the multiphase solid,composed of several kinds ofhomogeneous elastic-plastic media that conform to the generalized normality rule,has thesame type of constitutive equations as its constituents have that also conform to thegeneralized normality rule.     

Modeling of gravity-induced shape distortions during sintering of cylindrical specimens

A finite element approach for modeling the gravity-affected sintering process is presented. The constitutive equations are based on the continuum theory of sintering in the framework of a linear viscous material behavior. The model describes the gravity influence on porosity evolution and shrinkage inhomogeneity. Simulations of densification, shape distortion, and porosity gradients are presented. The results are compared with a previously developed analytical model of sintering under the influence of gravity. First time a direct assessment of the impact of the densification inhomogeneity on the gravity-induced shape distortion during sintering is provided in a generic form similar to the master sintering curve approach.