Additional Balance Laws Research Articles

Nonclassical continuum theories for fluent media incorporating rotation rates and their thermodynamic consistency

AbstractIn this paper, we present three micropolar nonclassical continuum theories (NCCT) for fluent medium in which a material point always has velocities as degrees of freedom, additionally, we consider (i) in the first NCCT, we consider classical or internal rotations rates (known) due to skew‐symmetric part of velocity gradient tensor; (ii) in the second NCCT, we consider rotation rates and Cosserat or microrotation rates (unknown degrees of freedom); (iii) in the third NCCT, we consider Cosserat or microrotation rates only, hence are neglected. Conservation and balance laws (CBL) and the constitutive theories are derived for all three NCCT. We examine consistent choice of kinematic variables, modification of the CBL of classical continuum theories (CCT) due to new physics and determine if the new physics of rotation rates requires additional balance law. All three NCCT derived here are examined for thermodynamic and/or mathematical consistency and are compared with published works. Model problem studies are also presented.

Relativistic Barotropic Fluids: A Godunov–Boillat Formulation for their Dynamics and a Discussion of Two Special Classes

A relativistic fluid is called barotropic if its internal energy $${\rho}$$ and its pressure p are one-to-one related; it is called isentropic if $${\rho}$$ and p are both one-to-one related to the fluid’s particle number density n. Deriving Godunov variables and 4-potentials for the relativistic Euler equations of perfect barotropic fluids, this paper first pursues ideas on symmetric hyperbolicity going back to Godunov and Boillat that Ruggeri and Strumia have elaborated as their theory of convex covariant density systems. The associated additional balance law (not a conservation law in the presence of shock waves) has different interpretations for different fluids. Among all barotropic fluids, we notably distinguish those for which this extra equation coincides with the second law of thermodynamics. We characterize these ‘thermobarotropic’ fluids, a class which comprises in particular the so-called pure radiation gas used in cosmology. The paper also shows that isentropic fluids are not thermobarotropic and that, like in the non-relativistic case, they cannot simultaneously satisfy the conservation laws of mass, momentum, and energy, as soon as shock waves occur. Curiously, relativistic isentropic flows which conserve energy and momentum generate mass across shock waves; this is a counterpart of the well-known fact that across shock waves, non-relativistic isentropic flows which conserve mass and momentum dissipate energy.

Ordered rate constitutive theories for non-classical thermoviscoelastic solids with memory incorporating internal and Cosserat rotations

This paper considers conservation and balance laws for non-classical solid continua in the presence of internal rotations ( $${}_i \pmb {\varvec{\varTheta }}$$ ) due to the Jacobian of deformation and Cosserat rotations ( $${}_e \pmb {\varvec{\varTheta }}$$ ) at each material point. In these balance laws, internal rotations are completely defined as functions of the displacement gradient tensor, but Cosserat rotations are additional three degrees of freedom at each material point. When these rotations are resisted by the deforming matter, conjugate moments are created. For thermoviscoelastic solids with memory, these result in additional energy storage, dissipation mechanism, and rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such solids based on the entropy inequality in conjunction with representation theorem. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as additional balance law for non-classical continuum theories, and the resulting theories are compared with classical continuum theories in which this balance law is not needed. Retardation moduli corresponding to the Cauchy stress tensor as well as the Cauchy moment tensor are derived. In this paper we only consider small strain, small deformation physics.

Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries

The mixed three-moment hydrodynamic description of fermionic radiation transport based on the Boltzmann entropy optimization procedure is considered for the case of one-dimensional flows. The conditions for realizability of the mixed three moments chosen as the energy density and two partial heat fluxes are established. The domain of admissible values of those moments is determined and the existence of the solution to the optimization problem is proved. Here, the standard approaches related to either the truncated Hausdorff or Markov moment problems do not apply because the non-negative fermionic distribution function, denoted \begin{document}$f$\end{document} , must satisfy the inequality \begin{document}$f≤q 1$\end{document} and, at the same time, there are three different intervals of integration in the integral formulae defining the mixed moments. The hydrodynamic equations are obtained in the form of the symmetric hyperbolic system for the Lagrange multipliers of the optimization problem with constraints. The potentials generating this system are explicitly determined as dilogarithm and trilogarithm functions of the Lagrange multipliers. The invertibility of the relation between moments and Lagrange multipliers is proved. However, the inverse relation cannot be determined in a closed analytic form. Using the \begin{document}$H$\end{document} -theorem for the radiative transfer equation, it is shown that the derived system of hydrodynamic radiation equations has as a consequence an additional balance law with a non-negative source term.

Non-linear theories of beams and plates accounting for moderate rotations and material length scales

The primary objective of this paper is to formulate the governing equations of shear deformable beams and plates that account for moderate rotations and microstructural material length scales. This is done using two different approaches: (1) a modified von Kármán non-linear theory with modified couple stress model and (2) a gradient elasticity theory of fully constrained finitely deforming hyperelastic cosserat continuum where the directors are constrained to rotate with the body rotation. Such theories would be useful in determining the response of elastic continua, for example, consisting of embedded stiff short fibers or inclusions and that accounts for certain longer range interactions. Unlike a conventional approach based on postulating additional balance laws or ad hoc addition of terms to the strain energy functional, the approaches presented here extend existing ideas to thermodynamically consistent models. Two major ideas introduced are: (1) inclusion of the same order terms in the strain–displacement relations as those in the conventional von Kármán non-linear strains and (2) the use of the polar decomposition theorem as a constraint and a representation for finite rotations in terms of displacement gradients for large deformation beam and plate theories. Classical couple stress theory is recovered for small strains from the ideas expressed in (1) and (2). As a part of this development, an overview of Eringen׳s non-local, Mindlin׳s modified couple stress theory, and the gradient elasticity theory of Srinivasa–Reddy is presented.

A nonlocal phase-field model of Ginzburg–Landau–Korteweg fluids

A thermodynamic model of Korteweg fluids undergoing phase transition and/or phase separation is developed within the framework of weakly nonlocal thermodynamics. Compatibility with second law of thermodynamics is investigated by applying a generalized Liu procedure recently introduced in the literature. Possible forms of the free energy and of the stress tensor, which generalize some earlier ones proposed by several authors in the last decades, are carried out. Owing to the new procedure applied for exploiting the entropy principle, the thermodynamic potentials are allowed to depend on the whole set of variables spanning the state space, including the gradients of the unknown fields, without postulating neither the presence of an energy or entropy extra-flux, nor an additional balance law for microforce.

A thermodynamic model of multiphase flows with moving interfaces and contact line

In this paper, we develop a general continuum description for thermodynamic multiphase flows with intersecting dividing surfaces, and three-phase common contact line, taking the contribution of the excess surface and line thermodynamic quantities into account. Starting with the standard postulates of continuum mechanics and the general global balance statement for an arbitrary physical quantity in a physical domain of three bulk phases including singular material or non-material phase interfaces and a three-phase contact line, we derive, in addition to the classical local balance equations for each bulk phase, the local conservation equations on the phase interfaces and at the contact line. Then, these additional interface and line balance laws are specified for excess surface and line physical quantities, e.g., excess mass, momentum, angular momentum, energy, and entropy, respectively. Some simplified forms of these balance laws are also presented and discussed.

Configurational balance and entropy sinks

For evolutionary processes of material remodelling and growth, a comparison is drawn between a conventional formulation and one that postulates the existence of additional balance laws for the configurational forces.

On the modeling of inhomogeneous incompressible fluid-like bodies

We use a thermodynamic framework that has been recently put into place to describe the dissipative response of materials to develop a model for the response of inhomogeneous incompressible fluid-like bodies whose stored energy depends on gradients of the density. Without appealing to additional balance laws, or concepts such as interstitial working, we are able to obtain models for such bodies that do not need boundary conditions for the density in order to have well posed governing equations. After obtaining a model for the response of inhomogeneous incompressible fluid-like body, we solve a simple boundary value problem and introduce the notion of (weak) solutions to the equations governing the flow of the body that has been developed to illustrate the issues concerning boundary conditions.

Modification of Goodman-Cowin theory and its Application to the Constitutive Models of Flowing Granular Materials

Goodman and Cowin proposed in 1972 a continuum theory of a dry cohesionless granular material in which the solid volume fraction ν is treated as an independent kinematical field for which an additional balance law of equilibrated force is postulated. In the derivation of said balance of equilibrated of force there exists some logical inconsistency and it results in the incorrect explanation of this balance equation and the incorrect balance of internal energy. It is demonstrated that the balance of equilibrated force can be modified by a simple dimensional analysis. The resulted modified Goodman-Cowin theory is then applied to investigate the consti- tutive models of flowing granular materials. A complete thermodynamic analysis based upon Muller-Liu approach is performed and the constitutive responses of a granular material in ther- modynamic equilibrium are obtained. From the theoretical investigations it shows that the results are more general than those obtained from the original theory and for simple shearing flow problems the current theory can reproduce all results from previous works based on the original theory.

A constitutive equation for the porosity field

This paper deals with the derivation of an evolution equation for the field here denoted by Δ n, which represents the difference between the porosity in an arbitrary state of a deformable porous medium and its equilibrium value. The equation is obtained assuming that Δ n is a constitutive quantity depending on the history of the fields describing macroscopically a porous material. Moreover, at least in the vicinity of the state of stable equilibrium, the equation obtained for Δ n can be regarded as a balance equation liable to various degrees of approximation, so including various models with additional balance laws known in the literature.

A variational approach to a micro-structured theory of solid-fluid mixtures

In this paper, we present a micro-structured model for describing global deformations of heterogeneous mixtures. In particular, for a saturated solid-fluid mixture, we regard the solid volume fraction as a microstructural parameter so as to enlarge the space of admissible deformations with respect to the classical theory of mixtures. According to the variational approach, the governing equations are obtained as the stationarity of a suitable action functional. The micro-structured model is then forced to establish a second-gradient mixture theory, by introducing among the considered state parameters a suitable internal constraint. Finally, we determine under which (integrability) conditions the additional balance laws, typically employed to close the theory of porous media endowed with the volume fraction, can fit the variational framework.

Toward a Field Theory for Elastic Bodies Undergoing Disarrangements

Structured deformations are used to refine the basic ingredients of continuum field theories and to derive a system of field equations for elastic bodies undergoing submacroscopically smooth geometrical changes as well as submacroscopically non-smooth geometrical changes (disarrangements). The constitutive assumptions employed in this derivation permit the body to store energy as well as to dissipate energy in smooth dynamical processes. Only one non-classical field G, the deformation without disarrangements, appears in the field equations, and a consistency relation based on a decomposition of the Piola-Kirchhoff stress circumvents the use of additional balance laws or phenomenological evolution laws to restrict G. The field equations are applied to an elastic body whose free energy depends only upon the volume fraction for the structured deformation. Existence is established of two universal phases, a spherical phase and an elongated phase, whose volume fractions are (1−γ0)3 and (1−γ0) respectively, with \( \gamma _0 : = (\sqrt 5 - 1)/2 \) the “golden mean”.

On configurational forces in the context of the finite element method

AbstractThe theory of configurational forces is briefly recast, together with the underlying balance laws. It is shown, that in the case of a homogeneous body without body forces the additional balance laws are identically satisfied if the standard force balance holds. In approximate solutions, for example obtained by finite elements, the equilibrium is not satisfied exactly, thus configurational forces occur on discretization nodes. An implementation of the configurational force balance into the finite element scheme is presented. The use of configurational forces is discussed with three main aspects. It is demonstrated how configurational forces can be used to check and to improve the finite element solution. Examples from fracture mechanics and problems with material inhomogeneities are discussed. Copyright © 2001 John Wiley & Sons, Ltd.

A continuum model accounting for defect and mass densities in solids with inelastic material behaviour

In this paper the analysis of structures with inelastic material behaviour is considered taking into account the evolution of defects and changes in mass density. The underlying kinematical concept of an oriented continuum is general enough to describe the micro- and macrobehaviour of material bodies appropriately. Based on the logical and consistent variational arguments for a Lagrangian functional the dynamic balance laws, boundary and transversality conditions, all related to the evolution of defect density and mass changes, are derived for macro- and microstresses of deformational as well as of configurational type. The adopted procedure, which formally leaves the balance laws unaltered, leads to the additional balance law for changes in defect density and additional boundary conditions for the changes in mass and defect densities. Driving forces or affinities, associated with the evolution of defect and mass densities, and a generalization of the J-integral representing the thermodynamic forces on defects are obtained. A nonlocal constitutive model accounting for changes in the defect density is presented.

The Debye Theory of Rotary Diffusion: History, Derivation, and Generalizations

Motivated by the Debye theory of rotary diffusion in a dipolar fluid, we systematically develop a continuum mechanical theory of rotary diffusion. This theory generalizes classical kinematics to include continuous rotary degrees of freedom and introduces an additional balance law associated with the rotary degrees of freedom. Various constitutive relations are proposed in accordance with standard procedures of nonlinear continuum mechanics. The resulting set of equations provides a properly invariant and thermodynamically consistent theory that allows for constitutive nonlinearities. In particular, the classical Debye theory along with the Nernst-Einstein relations are shown to follow from a special case of linear constitutive relations and an assumption of ideality in which the free energy consists only of a classical entropic contribution. Within our theory, the notion of osmotic pressure arises naturally as a consequence of accounting for forces that act conjugate to the rotary degrees of freedom and serves as the driving force for rotary diffusion.

SHEARING FLOWS IN A GOODMAN-COWIN TYPE GRANULAR MATERIAL—THEORY AND NUMERICAL RESULTS

ABSTRACT Goodman and Cowin (1972) proposed a continuum theory of a dry cohesionless granular material in which the solid volume fraction v is treated as an independent kinematic field for which an additional balance law of equilibrated forces is postulated. By adopting the Müller-Liu approach to the exploitation of the entropy inequality we show that in a constitutive model containing v1, v and grad v as independent variables results agree with the classical Coleman-Noll approach only provided the Helmholtz free energy does not depend on v, for which the Goodman-Cowin equations are reproduced. This reduced theory is then applied to analyses of steady fully-developed horizontal shearing flow and gravity flows of granular materials down an inclined plane and between vertical parallel plates. It is demonstrated that the equations and numerical results, presented by Passman et al. (1980) are false, and they are corrected. The results show that the dynamical behaviour of these materials is quite different from that of a viscous fluid. In some cases the dilatant shearing layers exist only in the narrow zones near the boundaries.

Original Article On the frame dependence and form invariance of the transport equations for the Reynolds stress tensor and the turbulent heat flux vector: its consequences on closure models in turbulence modelling

The present paper shows that the transport equations governing second order turbulent closures are form invariant, but remain frame dependent through the emergence of the body force; thus they do not fulfil the principle of material frame indifference as formulated by Truesdell & Noll (1965). However, this frame dependence corresponds to that first discussed by Muller (1972) and today developed in the framework of the new concept of extended thermodynamics. Following this new concept, these relations are consequently incorporated as additional basic balance laws. The results are: 1) in the case of the Reynolds-stress-transport equation, this eliminates the so-called constraints imposed in [15–17, 19] on turbulence models; 2) to ensure the closure of the new set of basic balance laws, closure assumptions can then be considered as proper constitutive equations which must be restricted by the well known constitutive theory principles in extended thermodynamics.

A unified procedure for construction of theories of deformable media. I. Classical continuum physics

A new unified procedure for constructing continuum theories of deformable media is presented and used in this and a companion paper. The procedure starts with a balance of energy and derives from it all the relevant balance laws that may also include those that are associated with thermal, electrical and magnetic effects; the basic energetic ingredients that are included in the balance of energy depend, of course, on the nature of the particular theory of material behaviour desired. The advantage of the new procedure becomes especially apparent when one considers formulation of a new theory of material behaviour for which additional balance laws (involving new kinetic quantities) are required to accompany any additional basic kinematic and thermal variables additional to those in the classical formulation. Indeed, in the formulation of such new theories, usually little or no previous information is available concerning properties of the new kinetic quantities in the additional balance laws; and, in this connection, the unified procedure of this paper provides a simple attractive setting for deriving the basic equations that are automatically consistent with the energy balance. In this paper, first the basic features of the new procedure are illustrated in the context of classical thermomechanics. Generalizations of this thermomechanical theory are then discussed in two cases: (1) in the presence of an additional kinematic variable and (2) in the presence of full electromagnetic effects. Both of these generalizations bring out some interesting novel features when new theories are being constructed.

Thermomechanical equations of magnetic fluids

As a general continuum introduction to the subject matter, the thermomechanical field and constitutive equations of magnetic fluids are formulated on the basis of the principle of virtual power (for finite virtual velocity fields) and the two laws of thermodynamics. This is probably the most “economic” approach to continuum physics while being also the most pregnant of fruitful generalizations and the closest to the spirit of modern applied mathematics. Using the thermodynamical framework as a constraint, we present successively the classical theory of viscous heat-conducting magnetic fluids with dynamical magnetization and more elaborate continuum descriptions involving either a finer mechanical description (higher-grade theory) or a true microstructure. The latter can be introduced either via additional degrees of freedom (requiring thus additional balance laws provided systematically by the principle of virtual power) or through so-called internal variables of state (requiring thus evolution equations subjected to the dissipation inequality). Such refined descriptions make full use of continuum thermodynamics while being most useful in studying effects such as the influence of rotating magnetic fields or the aggregation of magnetic particles and the formation of chains in the magnetic fluid.