Phase change interactions and singular fronts

The balances of mass, linear momentum and energy for a continuum provide jump relations between values of the physical variables on the two sides of a singular surface, either a boundary of the medium or an interior surface. In the case of a mixture, an overlap of interacting continua, there are jump relations for each constituent. While an elementary phase change front across which one phase of a constituent is transformed completely to a different phase can be treated as a single constituent, more general situations have co-existing phases on one side of the front, each with their own density, velocity, stress and internal energy fields, which must be treated as separate constituents. The phase change is now a mass transfer between constituents which becomes a surface production term in the mass balance jump relation for each constituent. In turn this implies surface production contributions to the momentum and energy relations associated with the surface mass transfer, including interaction body force and energy transfer contributions as well as the direct transfer terms. The general jump relations with such surface production contributions are formulated, and are illustrated for a number of situations arising in polythermal ice sheets and wet snow packs.     

On continuum thermodynamics

Within the scope of classical continuum thermodynamics, we elaborate on the basic concepts and adopt a different approach from usual to the formulation of conservation laws and an entropy production inequality, both for a single phase continuum and for a mixture of any number of constituents. These conservation laws and the entropy inequality can be regarded as applicable to both local and nonlocal problems. In the case of a single phase continuum and for a simple material which is homogeneous in its reference configuration, under fairly mild smoothness assumptions, we prove that all the conservation laws reduce to the usual classical ones and the entropy production inequality reduces to the Clausius-Duhem inequality. Some attention is given to possible redundancies in the basic concepts, as well as to alternative forms of the energy equation and the entropy inequality. The latter is particularly significant in regard to different but equivalent formulations of mixture theory.     

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…     

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

The nonlinear constitutive equations and field equations of unsaturated soils were constructed on the basis of mixture theory. The soils were treated as the mixture composed of three constituents. First, from the researches of soil mechanics, some basic assumptions about the unsaturated soil mixture were made, and the entropy inequality of unsaturated soil mixture was derived. Then, with the common method usually used to deal with the constitutive problems in mixture theory, the nonlinear constitutive equations were obtained. Finally, putting the constitutive equations of constituents into the balance equations of momentum, the nonlinear field equations of constituents were set up. The balance equation of energy of unsaturated soil was also given, and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed. Foundation items: the National Natural Science Foundation of China (59678003); Special Research Plan of the Education Department of Shaanxi Province (01JK178) Biographies: HUANG Yi (1936-) ZHANG Yin-ke (1964-)     

Balance relations for classical mixtures containing a moving non-material surface with application to phase transitions

This work is concerned with an extension of classical mixture theory to the case in which the mixture contains an evolving non-material surface on which the constituents may interact, as well as be created and/or annihilated. The formulation of constituent and mixture jump balance relations on/across such a non-material surface proceed by analogy with the standard volume or bulk constituent and mixture balance relations. On this basis, we derive various forms of the constituent mass, momentum, energy and entropy balances assuming (1), that the constituent in question is present on both sides of the moving, non-material surface, and (2), that it is created or annihilated on this surface, as would be the case in a phase transition. In particular, we apply the latter model to the transition between cold and temperate ice found in polythermal ice masses, obtaining in the process the conditions under which melting or freezing takes place at this boundary. On a more general level, one of the most interesting aspects of this formulation is that it gives rise to certain combinations of the limits of constituent and mixture volume fields on the moving mixture interface which can be interpreted as the corresponding surface form of these fields, leading to the possibility of exploiting the surface entropy inequality to obtain restrictions on surface constitutive relations.     

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.     

A simple mixture theory for ν Newtonian and generalized Newtonian constituents

This work presents the development of mathematical models based on conservation laws for a saturated mixture of ν homogeneous, isotropic, and incompressible constituents for isothermal flows. The constituents and the mixture are assumed to be Newtonian or generalized Newtonian fluids. Power law and Carreau–Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a ν constituent mixture with volume fractions ${\phi_\alpha}$ using principles of continuum mechanics: conservation of mass, balance of momenta, first and second laws of thermodynamics, and principles of mixture theory yielding continuity equations, momentum equations, energy equation, and constitutive theories for mechanical pressures and deviatoric Cauchy stress tensors in terms of the dependent variables related to the constituents. It is shown that for Newtonian fluids with constant transport properties, the mathematical models for constituents are decoupled. In this case, one could use individual constituent models to obtain constituent deformation fields, and then use mixture theory to obtain the deformation field for the mixture. In the case of generalized Newtonian fluids, the dependence of viscosities on deformation field does not permit decoupling. Numerical studies are also presented to demonstrate this aspect. Using fully developed flow of Newtonian and generalized Newtonian fluids between parallel plates as a model problem, it is shown that partial pressures p _α of the constituents must be expressed in terms of the mixture pressure p. In this work, we propose ${p_\alpha=\phi_\alpha p}$ and ${\sum_\alpha^\nu p_\alpha = p}$ which implies ${\sum_\alpha^\nu \phi_\alpha = 1}$ which obviously holds. This rule for partial pressure is shown to be valid for a mixture of Newtonian and generalized Newtonian constituents yielding Newtonian and generalized Newtonian mixture. Modifications of the currently used constitutive theories for deviatoric Cauchy stress tensor are proposed. These modifications are demonstrated to be essential in order for the mixture theory for ν constituents to yield a valid mathematical model when the constituents are the same. Dimensionless form of the mathematical models is derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional, that is, least squares finite element processes in which local approximations are considered in ${H^{k,p}\left(\bar{\Omega}^e\right)}$ scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step is used as model problems for a mixture of two constituents.     

Non-equilibrium thermodynamics of heterogeneous systems and history dependent materials

A non-equilibrium thermodynamic theory due to Bree and Beevers [1] has been applied by Bree [2] to a certain class of mixtures, referred to in [2] as simple mixtures where it is found that global and local entropy inequalities can be obtained for the whole mixture, for each of its constituents or for any combination of its constituents, each of these entropy inequalities being derived from the single statement of the second law of thermodynamics adopted in [1]. The present work is primarily concerned with developing the theory further in order to include the more complex class of mixtures referred to in [2] as heterogeneous systems. In contrast with the thermodynamics of simple mixtures, it is found that for heterogeneous systems global and local entropy inequalities can be obtained only for the whole mixture and, in connection with this, it is felt that an early controversy in the subject is resolved. It is shown how the results obtained for heterogeneous systems may be used to provide a thermodynamic foundation for two methods of representing history dependent materials.     

On the equations of balance for mixtures constructed by means of classical statistical mechanics

Summary We use the techniques of an earlier paper to construct gross quantities and equations of balance for each constituent of a mixture as well as for the whole mixture. The equations of balance for mass, linear momentum and energy for the constituents are similar to those proposed by TRUESDELL, but here growth terms are absent and the peculiar stress tensor and heat flux vector have different interpretations than TRUESDELL's. We also show that the gross fields for the mixture, and the equations they satisfy, can be obtained by suitably adding their analogues for the constituents. In spite of the fact that the internal forces on the ^th constituent are described by means of a peculiar stress tensor T, we show that this theory does not give rise to the paradox considered and resolved by GURTIN, OLIVER & WILLIAMS.

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Sommario Mediante tecniche presentate in un precedente lavoro costruisco campi macroscopici ed equazioni di bilancio per ogni componente di una miscela cosi come per la miscela nel suo complesso. Le equazioni di bilancio per la massa, la quantità di moto e l'energia per i costituenti sono simili a quelle proposte da TRUESDELL, ma qui mancano i termini di sorgente, inoltre il tensore degli sforzi e il vettore flusso termico hanno interpretazioni diverse da quelle di TRUESDELL. Mostro che i campi macroscopici, e le equazioni che essi soddisfano, si ottengono sommando opportunamente gli analoghi costruiti per i componenti. Malgrado le forze interne su un componente vengano rappresentate mediante un tensore degli sforzi, mostro che questa teoria non da luogo al paradosso analizzato e risolto da GURTIN, OLIVER & WILLIAMS.

     

Thermo-Chemo-Electro-Mechanical Formulation of Saturated Charged Porous Solids

A thermo-chemo-electro-mechanical formulation of quasi-static finite deformation of swelling incompressible porous media is derived from a mixture theory including the volume fraction concept. The model consists of an electrically charged porous solid saturated with an ionic solution. Incompressible deformation is assumed. The mixture as a whole is assumed locally electroneutral. Different constituents following different kinematic paths are defined: solid, fluid, anions, cations and neutral solutes. Balance laws are derived for each constituent and for the mixture as a whole. A Lagrangian form of the second law of thermodynamics for incompressible porous media is used to derive the constitutive restrictions of the medium. The material properties are shown to be contained in one strain energy function and a matrix of frictional tensors. A principle of reversibility results from the constitutive restrictions. Existing theories of swelling media should be evaluated with respect to this principle.     

A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle

A link is shown to exist between the so-called residual-based strain gradient plasticity theory and the analogous theories based on the (extended) virtual work principle (VWP). To this aim, the former theory is reformulated and cast in a residual-free form, whereby the insulation condition and the (nonlocal) Clausius–Duhem inequality, on which the theory is grounded, are substituted with equivalent residual-free ingredients, namely the energy balance condition and the residual-free form of the Clausius–Duhem inequality. The equivalence of the residual-free formulation to the original one is shown, also in their ability to cope with energetic size effects and interfacial energy ones. It emerges that the residual-free form of the Clausius–Duhem inequality coicides with the way the second thermodynamics principle is enforced within the VWP-based theories, and also that the energy balance condition amounts to the extended VWP enforced in the whole body. This makes the residual-free formulation possess strong similarities with the more general VWP-based theories, such as it constitutes an assessment of the existing link, which can be synthetized by the statement: the insulation condition is equivalent to the extended VWP deprived by the content of the standard principle.     

A local superposed-constituent volume-fraction mixture theory based on relative motion

This work presents a local formulation of kinematics and balance relations for a mixture composed of superposed-constituents as based on their motion relative to a model of the mixture as a moving region, with the motion of this region being determined by that of all the constituents. The relative motion of each constituent with respect to this moving mixture region can be interpreted as its diffusive motion in the mixture. A number of constituent kinematic properties in the mixture, in particular the constituent volume density (i.e. the infinitesimal volume-fraction) and constituent diffusion velocity, are determined by the constituent diffusive motion, and arise naturally in a formulation based upon it. In addition, this formulation yields in a natural fashion an evolution relation for the constituent volume density relative to the moving mixture region. The consequences of this kinematic model for constituent and mixture local balance relations, as well as the sum relations, are investigated in the last part of the work.     

A unified thermodynamic theory of elasticity and plasticity

A thermodynamics is developed for a unified theory of elasticity and plasticity in infinitesmal strain. The constitutive equations which relate stress and strain deviators are rate type differential equations. When they satisfy a Lipschitz condition, uniqueness for the initial value problem dictates that the stress and strain will be related through elastic relations. Failure of the Lipschitz condition occurs when a von Mises yield condition is achieved: Plastic yield then occurs and the deviator relations turn into the Prandtl-Reuss equations. The plastic yield solution is stable during loading and unstable during unloading. The requirement that the solution followed during unloading be stable dictates entry into an elastic regime. Appropriate thermodynamic functions are constructed. It then appears that stress deviator (not strain deviator) is a viable state variable, and the thermodynamic relations are constructed in terms of a Gibbs function. The energy balance leads to satisfaction of the Clausius-Duhem inequality (and thus the second law of thermodynamics) in an elastic regime because it is shown that in an elastic regime entropy production is caused only by heat flux. During yield, the proper method of differentiating yields entropy production terms in addition to those arising from heat flux. These terms are positive during loading, whence it is concluded that the requirement that a stable solution be followed leads to satisfaction of the Clausius-Duhem inequality during plastic as well as elastic behavior.     

Thermomechanics of Porous Media - I: Thermohydraulic Model for Compacted Bentonite

A general thermomechanical model is derived for a mixture. The model describes the behavior of the mixture via proper choices of free energy and dissipation function. A model for any combination of the mixture constituents can be reduced from the general model. The theory is applied to a thermohydraulic model for a mixture of compacted bentonite, liquid water, vapor, and air with the assumption of rigid skeleton and constant uniform porosity. The free energy of the system is chosen to take into account the individual nondissipative behaviors of the constituents and their mutual interactions, namely, adsorption and mixing of the gaseous constituents. The choices for the interaction terms are based on the equilibrium conditions for the water species in different combinations of the constituents. The resulting thermodynamically consistent macroscopic model is fitted to a suction experiment and applied to a simple one-dimensional thermohydraulic simulation of the bentonite buffer of the Febex in situ test. The results calculated with finite element method are successfully compared to measurements.     

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.     

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.     

Non-equilibrium relativistic M.H.D. with finite electrical conductivity

A system of balance laws for relativistic m.h.d, with finite eIectrical conductivity, heat flux and viscosity is proposed, starting from the properties of the systems of conservation laws compatible with a supplementary balance law (entropy balance). Adopting a two-fluid scheme the plasma is treated as a mixture of a neutral fluid and a charged fluid. Following the approach ofextended thermodynamics heat flux, viscous stress and electric current density are considered as new field variables contributing to non equilibrium entropy density and flux.     

A virtual power format for thermomechanics

It is shown that a virtual power format slightly more general than usual may be employed to deduce all balance and imbalance laws of thermomechanics. An essential role is played by the notion of thermal displacement; the basic balance laws turn out to be those for momentum and entropy. In consequence of these balances and of two axioms of thermodynamical nature—namely, conservation of internal action in cyclic processes and dissipative nature of ordinary processes—balance of energy and inbalance of entropy are arrived at. Dedicated to Tommaso Ruggeri, on the occasion of his 60th birthday.