Nonlinear axisymmetric thermomechanical response of piezo-FGM shallow spherical shells

Thermomechanical instability of shallow spherical shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators is studied in this paper. The governing equations are based on the classical shell theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the property of the FGMs varies continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials. The constituent materials of the functionally graded shell are assumed to be mixture of ceramic and metal. The analytical solutions are obtained for uniform external pressure, thermal loading, and constant applied actuator voltage.     

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 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 theory of mixtures

This paper is concerned with a dynamical theory of mixtures, composed of n reactive constituents in relative motion to each other. The theory is developed in terms of the constituent ingredients using a balance of energy and an entropy production inequality for each constituent of the mixture, together with invariance requirements under superposed rigid body motions of the whole mixture. The balance of energy and the entropy production inequality for each of the constituents, which include contributions arising from interactions, combine to yield a single energy equation and a single entropy production inequality in terms of the ingredients of the mixture as a whole; the relations between the thermodynamical variables of the mixture and those of its constituents depend, in general, on the past history of the temperature and the kinematic variables. Full thermodynamical restrictions are deduced, and the theory is applied to the special case of a mixture of two ideal fluids.     

Recasting theory of elasticity with micro-finite elements

In the classical theory of elasticity,a body is initially modeled as a homogeneous and dense assemblage of constituent "material particles".The kernel concept of elastic deformation is the displacement of the particle that associates the current configuration with the reference one.In this paper,we exploit an alternative constituent "micro-finite element",and use the stretch of the element as the essential quality to recast the theory of elasticity.It should be realized that such a treatment means that the elastic body can be modeled as a finite covering of elements and consequently characterized by a manifold.The recasting of the elasticity theory becomes more feasible for dealing with defects and topological evolution.     

Dielectric properties of heterogeneous mixtures with a polar constituent

Summary After defining the boundaries for the dielectric constant of a heterogeneous mixture, the behaviour of such a mixture is studied as a function of the frequency, when one of its components is polar. Deviations from a semicircle are to be expected for the function _m =f( _m ) even when the dielectric properties of the polar constituent can be described with a semicircular Cole-Cole-arc. The relaxation time of the mixture is shorter than that of the polar constituent.     

Thermal buckling of piezo-FGM shallow spherical shells

Thermal instability of shallow spherical shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators is studied in this paper. The governing equations are based on the first order theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the property of the functionally graded materials vary continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials. The constituent material of the functionally graded shell is assumed to be a mixture of ceramic and metal. The analytical solutions are obtained for three types of thermal loadings and constant applied actuator voltage. Results for simpler states are validated with the known data in literature.     

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.     

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.     

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.     

Light scattering in binary mixtures of monatomic gases from an extended kinetic description

$14\nu$   fields of an ideal gas mixtures of constituents. It is shown that for binary mixtures with disparate masses - like the Helium-Xenon mixture - the light scattering spectrum changes from the hydrodynamic regime to the free molecular regime by increasing the molar fraction of the light constituent. Received October 20, 2001 / Published online February 4, 2002     

Mechanics and thermodynamics of fluid-saturated highly elastic materials

We consider a mixture that consists of a highly elastic material and a liquid dissolved in this material. Using the laws of classical thermodynamics, we state a variational principle describing the mixture equilibrium under static loading conditions. From this principle, we derive equilibrium equations and a system of constitutive relations characterizing the mixture elastic and thermodynamic properties. We state problems describing the stress-strain state of a swollen material and a statically loaded material in thermodynamic equilibrium with the liquid. We consider the case of incompressible mixture. The general theory is illustrated by examples concerned with the deformation behavior of inhomogeneously swollen cross-linked polymers and with their thermodynamics of strains and swelling in solvent media.     

A Local Model for the Energy Transfer in a Saturated Static Mixture

In the present work the transient energy transfer in a nonsaturated porous medium is studied, using a mixture theory viewpoint. The porous matrix is assumed homogeneous, rigid and isotropic, while the fluid is a Newtonian incompressible one and both are assumed static. Since the homogeneous matrix is not saturated, gradients of concentration are present. The porous medium and the fluid (a liquid) will be regarded as continuous constituents of a mixture that will have also a third constituent, an inert gas, assumed with zero mass density and thermal conductivity. The problem is described by a set of two partial differential equations which represent the energy balances for the fluid and the solid constituents. Isovalues for these two constituents are plotted, considering representative time instants and selected values for the energy equations coefficients and for the saturation.     

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.     

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.     

Generalized Forchheimer Equation for Two-Phase Flow Based on Hybrid Mixture Theory

In this paper, we derive a Forchheimer-type equation for two-phase flow through an isotropic porous medium using hybrid mixture theory. Hybrid mixture theory consists of classical mixture theory applied to a multiphase system with volume averaged equations. It applies to media in which the characteristic length of each phase is small relative to the extent of the mixture. The derivation of a Forchheimer equation for single phase flow has been obtained elsewhere. These results are extended to include multiphase swelling materials which have nonnegligible interfacial thermodynamic properties.