Consistent Thermodynamic Framework Research Articles

Thermodynamic properties of seawater, ice and humid air: TEOS-10, before and beyond

Abstract. In the terrestrial climate system, water is a key player in the form of its different ambient phases of ice, liquid and vapour, admixed with sea salt in the ocean and with dry air in the atmosphere. For proper balances of climatic energy and entropy fluxes in models and observations, a highly accurate, consistent and comprehensive thermodynamic standard framework is requisite in geophysics and climate research. The new Thermodynamic Equation of Seawater – 2010 (TEOS-10) constitutes such a standard for properties of water in its various manifestations in the hydrological cycle. TEOS-10 was recommended internationally in 2009 by the Intergovernmental Oceanographic Commission (IOC) to replace the previous 1980 seawater standard, EOS-80, and in 2011 by the International Union of Geodesy and Geophysics (IUGG) as the official description for the properties of seawater, of ice and of humid air. This paper briefly reviews the development of TEOS-10, its novel axiomatic properties, the new oceanographic tools it offers and the important tasks that still await solutions by ongoing research. Among the latter are new definitions and measurement standards for seawater salinity and pH in order to establish their metrological traceability to the International System of Units (SI) for the first time after a century of widespread use. Of similar climatological relevance is the development and recommendation of a uniform standard definition of atmospheric relative humidity that is unambiguous and rigorously based on physical principles.The leading thermodynamic properties of a fluid are determined by the relations which exist between volume, pressure, temperature, energy, and entropy … But all the relations existing between these five quantities for any substance …may be deduced from the single relation existing for that substance between volume, energy, and entropy. Josiah Willard Gibbs, 1873b

Equilibrium States in Two-Temperature Systems.

Systems characterized by more than one temperature usually appear in nonequilibrium statistical mechanics. In some cases, e.g., glasses, there is a temperature at which fast variables become thermalized, and another case associated with modes that evolve towards an equilibrium state in a very slow way. Recently, it was shown that a system of vortices interacting repulsively, considered as an appropriate model for type-II superconductors, presents an equilibrium state characterized by two temperatures. The main novelty concerns the fact that apart from the usual temperature T, related to fluctuations in particle velocities, an additional temperature was introduced, associated with fluctuations in particle positions. Since they present physically distinct characteristics, the system may reach an equilibrium state, characterized by finite and different values of these temperatures. In the application of type-II superconductors, it was shown that , so that thermal effects could be neglected, leading to a consistent thermodynamic framework based solely on the temperature . In the present work, a more general situation, concerning a system characterized by two distinct temperatures and , which may be of the same order of magnitude, is discussed. These temperatures appear as coefficients of different diffusion contributions of a nonlinear Fokker-Planck equation. An H-theorem is proven, relating such a Fokker-Planck equation to a sum of two entropic forms, each of them associated with a given diffusion term; as a consequence, the corresponding stationary state may be considered as an equilibrium state, characterized by two temperatures. One of the conditions for such a state to occur is that the different temperature parameters, and , should be thermodynamically conjugated to distinct entropic forms, and , respectively. A functional is introduced, which presents properties characteristic of an entropic form; moreover, a thermodynamically conjugated temperature parameter can be consistently defined, so that an alternative physical description is proposed in terms of these pairs of variables. The physical consequences, and particularly, the fact that the equilibrium-state distribution, obtained from the Fokker-Planck equation, should coincide with the one from entropy extremization, are discussed.

A consistent framework of material configurational mechanics in piezoelectric materials

The fundamental concepts of material configurational mechanics are formulated in piezoelectric materials. A consistent thermodynamic framework is outlined to develop the corresponding theory of material configurational stresses and forces associated with the \(J_{k}\)-, M- and L-integrals by the gradient, divergence and curl operation of the electric enthalpy density, respectively. The physical interpretation of material configurational stresses is explored, and they can be explained as the energy release rates due to the translation of material point along \(x_{k}\)-direction, the self-similar expansion, the rotation of material element, respectively. The path independence or path dependence of variant integrals such as the \(J_{k}\)-, M- and L-integrals is examined in piezoelectric material. As an application of material configurational mechanics in piezoelectric materials, an explicit method is derived to calculate the change of the J-integral as a dominant fracture parameter for a crack interaction with domain switching near the crack tip. It is concluded that domain switching has an obvious shielding effect on the fracture toughness in piezoelectrics by the present explicit forms. The present framework of material configurational mechanics is expected to provide an effective and convenient tool to deal with various crack or damage problems in piezoelectric materials.

Thermodynamic framework for compact q-Gaussian distributions

Recent works have associated systems of particles, characterized by short-range repulsive interactions and evolving under overdamped motion, to a nonlinear Fokker–Planck equation within the class of nonextensive statistical mechanics, with a nonlinear diffusion contribution whose exponent is given by ν=2−q. The particular case ν=2 applies to interacting vortices in type-II superconductors, whereas ν>2 covers systems of particles characterized by short-range power-law interactions, where correlations among particles are taken into account. In the former case, several studies presented a consistent thermodynamic framework based on the definition of an effective temperature θ (presenting experimental values much higher than typical room temperatures T, so that thermal noise could be neglected), conjugated to a generalized entropy sν (with ν=2). Herein, the whole thermodynamic scheme is revisited and extended to systems of particles interacting repulsively, through short-ranged potentials, described by an entropy sν, with ν>1, covering the ν=2 (vortices in type-II superconductors) and ν>2 (short-range power-law interactions) physical examples. One basic requirement concerns a cutoff in the equilibrium distribution Peq(x), approached due to a confining external harmonic potential, ϕ(x)=αx2∕2 (α>0). The main results achieved are: (a) The definition of an effective temperature θ conjugated to the entropy sν; (b) The construction of a Carnot cycle, whose efficiency is shown to be η=1−(θ2∕θ1), where θ1 and θ2 are the effective temperatures associated with two isothermal transformations, with θ1>θ2; (c) Thermodynamic potentials, Maxwell relations, and response functions. The present thermodynamic framework, for a system of interacting particles under the above-mentioned conditions, and associated to an entropy sν, with ν>1, certainly enlarges the possibility of experimental verifications.

Thermodynamic framework for the ground state of a simple quantum system.

The ground state of a two-level system (associated with probabilities p and 1-p, respectively) defined by a general Hamiltonian H[over ̂]=H[over ̂]_{0}+λV[over ̂] is studied. The simple case characterized by λ=0, whose Hamiltonian H[over ̂]_{0} is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (S_{BG}=0) at zero temperature (T=0). Herein it is shown that the introduction of a perturbation λV[over ̂] (λ>0) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy S[p] (with S[p]≠S_{BG}[p]), if the Hermitian operator V[over ̂] is represented by a 2×2 matrix, defined by nonzero off-diagonal elements V_{12}=V_{21}=-z, where z is a real positive number. Hence, this new term in the Hamiltonian, presenting V_{12}≠0, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature θ (θ∝λz), which is shown to be a parameter conjugated to the entropy S. Based on this, one introduces an infinitesimal heatlike quantity, δQ=θdS, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification T↔θ, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover, S also follows a behavior typical of a third law, i.e., S→0, when θ→0.

Repulsive particles under a general external potential: Thermodynamics by neglecting thermal noise.

A recent proposal of an effective temperature θ, conjugated to a generalized entropy s_{q}, typical of nonextensive statistical mechanics, has led to a consistent thermodynamic framework in the case q=2. The proposal was explored for repulsively interacting vortices, currently used for modeling type-II superconductors. In these systems, the variable θ presents values much higher than those of typical room temperatures T, so that the thermal noise can be neglected (T/θ≃0). The whole procedure was developed for an equilibrium state obtained after a sufficiently long-time evolution, associated with a nonlinear Fokker-Planck equation and approached due to a confining external harmonic potential, ϕ(x)=αx^{2}/2 (α>0). Herein, the thermodynamic framework is extended to a quite general confining potential, namely ϕ(x)=α|x|^{z}/z (z>1). It is shown that the main results of the previous analyses hold for any z>1: (i) The definition of the effective temperature θ conjugated to the entropy s_{2}. (ii) The construction of a Carnot cycle, whose efficiency is shown to be η=1-(θ_{2}/θ_{1}), where θ_{1} and θ_{2} are the effective temperatures associated with two isothermal transformations, with θ_{1}>θ_{2}. The special character of the Carnot cycle is indicated by analyzing another cycle that presents an efficiency depending on z. (iii) Applying Legendre transformations for a distinct pair of variables, different thermodynamic potentials are obtained, and furthermore, Maxwell relations and response functions are derived. The present approach shows a consistent thermodynamic framework, suggesting that these results should hold for a general confining potential ϕ(x), increasing the possibility of experimental verifications.

Nonequilibrium thermodynamics in the strong coupling and non-Markovian regime based on a reaction coordinate mapping

We propose a method to study the thermodynamic behaviour of small systems beyond the weak coupling and Markovian approximation, which is different in spirit from conventional approaches. The idea is to redefine the system and environment such that the effective, redefined system is again coupled weakly to Markovian residual baths and thus, allows to derive a consistent thermodynamic framework for this new system–environment partition. To achieve this goal we make use of the reaction coordinate (RC) mapping, which is a general method in the sense that it can be applied to an arbitrary (quantum or classical and even time-dependent) system coupled linearly to an arbitrary number of harmonic oscillator reservoirs. The core of the method relies on an appropriate identification of a part of the environment (the RC), which is subsequently included as a part of the system. We demonstrate the power of this concept by showing that non-Markovian effects can significantly enhance the steady state efficiency of a three-level-maser heat engine, even in the regime of weak system–bath coupling. Furthermore, we show for a single electron transistor coupled to vibrations that our method allows one to justify master equations derived in a polaron transformed reference frame.

Consistent thermodynamic framework for interacting particles by neglecting thermal noise

An effective temperature θ, conjugated to a generalized entropy s(q), was introduced recently for a system of interacting particles. Since θ presents values much higher than those of typical room temperatures T≪θ, the thermal noise can be neglected (T/θ≃0) in these systems. Moreover, the consistency of this definition, as well as of a form analogous to the first law of thermodynamics, du=θds(q)+δW, were verified lately by means of a Carnot cycle, whose efficiency was shown to present the usual form, η=1-(θ(2)/θ(1)). Herein we explore further the heat contribution δQ=θds(q) by proposing a way for a heat exchange between two such systems, as well as its associated thermal equilibrium. As a consequence, the zeroth principle is also established. Moreover, we consolidate the first-law proposal by following the usual procedure for obtaining different potentials, i.e., applying Legendre transformations for distinct pairs of independent variables. From these potentials we derive the equation of state, Maxwell relations, and define response functions. All results presented are shown to be consistent with those of standard thermodynamics for T>0.

An implicit adaptive node‐splitting algorithm to assess the failure mechanism of inelastic elastomeric continua

SUMMARYThis contribution presents a mesh adaptive crack propagation scheme for the evaluation of the viscoelastic fracture response of elastomers at large strains and up to high loading rates. The approach accounts for micromechanical based features of both elastic and viscoelastic bulk responses of idealized polymer networks. To this end, the Bergstörm–Boyce model is considered to introduce hyperelastic and nonlinear finite viscoelastic responses. Moreover, the crack driving force and the crack driving direction are predicted by the material force approach. A consistent thermodynamic framework for the combined configurational motion in viscoelastic continua at finite strain regime is discussed.The fracture toughness of non‐strain‐crystallizing elastomers shows strong rate dependency and the energy release rate versus the rate of tearing to be a fundamental material property. Therefore, in this contribution, a dynamic fracture criterion, which is a function of the rate of crack growth, is shown to be adequate in numerical simulations. The use of the presented method enables to study fracture behaviour of any material nonlinearity within the implicit time integration. Main feature of the proposed algorithm is restructuring the overall discrete system by duplication of crack front DOFs based on minimization of the overall energy via the Griffith criterion. Copyright © 2014 John Wiley & Sons, Ltd.

Evaluation of energy contributions in elasto-plastic fracture: A review of the configurational force approach

This paper discusses the role of the material or rather configurational force approach in elastic–plastic materials with a pre-cracked configuration and gives an overview of some recent numerical investigations of the crack tip field. On the theoretical side, a consistent thermodynamic framework for the combined configurational and deformational motion in elasto-plastic continua at small and finite strains is discussed. Furthermore, the study researches the separation of the total dissipation in terms of the change in elastic energy and in terms of the material dissipation by a configurational change obtained from the global energy momentum balance. On the numerical side, an equivalent general expression of the vectorial material forces is derived from the weak form of the energy momentum balance. For the sake of simplicity, all results are obtained neglecting dynamic and thermo-mechanical phenomena. The computations are applied to a stationary crack in a circular pre-cracked domain and a compact tension specimen under plasticity condition. The results show that the material force approach remains path independent only if all components of the momentum balance equation are properly included into the corresponding variational formulation. In addition, the cohesive fracture theory is combined with the material force approach in order to increase the clarity of the interpretation of the approach in engineering applications. Correspondingly, the results obtained from the compact tension specimen with three different free energy functions are compared to the conventional J-integral method and to experimental results available from a previous study. The contributions of this study are threefold. First, the path dependency of the material force approach in elasto-plastic continua is found to be considerably depending on the so-called material body forces. Secondly, interpretation of the induced material dissipation forces in the definition of the crack driving forces is not explicitly clear but they play an important role in case of path independency. Finally, with further analyses on compact tension examples, it is shown that the introduced energy functions in the material momentum balance yield a difference for the evaluation of the material force approach and the traditional J-integral.

Modeling of release and absorption of gas in liquid–gas flows within a consistent thermodynamic framework

This work presents a consistent thermodynamic model to describe the gas release and gas absorption phenomena in homogeneous liquid–gas flows undergoing isothermal transformations. The liquid–gas mixture is regarded as a pseudo-fluid whose constitutive behavior is obtained from two thermodynamic potentials: the Helmholtz free energy and a pseudo-potential of dissipation. Thanks to the inclusion of the concentration of dissolved gas in the liquid in the list of state variables, along with the gas volume fraction and the mass densities of the liquid and gas constituents, a suitable and simple expression for the rate of mass transfer of gas is derived. Besides of unconditionally satisfying the Second Law of the Thermodynamics, the proposed expression is capable to properly describe experimental data available in the literature with great accuracy for a relatively wide range of saturation pressures. Numerical simulations carried out for an water–air mixture, subjected to a single expansion and cyclic expansion–contraction loadings, illustrate the influence of the gas release on the mechanical response of the fluid, which exhibits dispersion and attenuation and also hysteresis.

Modelling of Shockwave Propagation in Orthotropic Materials

Modelling of shockwave propagation in orthotropic materials requires an appropriate description of material behaviour within elastic and plastic regimes. To deal with this issues, a finite strain constitutive model for orthotropic materials was developed within a consistent thermodynamic framework of irreversible process in this paper. The important features of this material model are the multiplicative decomposition of the deformation gradient and a Mandel stress tensor combined with the new stress tensor decomposition generalised for orthotropic materials. The elastic free energy function and the yield function are defined within an invariant theory by means of the introduction of the structural tensors. The plastic behaviour is characterised within the associative plasticity framework using the Hills yield criterion. The complexity was further extended by coupling the formulation with the equation of state (EOS) to control the response of the material to shock loading. This material model which was developed and integrated in the isoclinic configuration provides a unique treatment for elastic and plastic anisotropy. The effects of elastic anisotropy are taken into account through the stress tensor decomposition and plastic anisotropy through yield surface defined in the generalized deviatoric plane perpendicular to the generalised pressure. To test its ability to describe shockwave propagation, the new material model was implemented into the LLNL-DYNA3D code. The results generated by the proposed material model were compared against the experimental Plate Impact test data of Aluminium Alloy 7010. A good agreement between experimental and simulation was obtained for two principal directions of material orthotropy.

Localization in stainless steel using microstructural based viscoplastic model

The plastic deformation behavior of AL-6XN stainless steel over a wide range of strain rates and temperatures is modeled using a combination of recently developed constitutive models VA-BCC [3,52] and VA-FCC [4,53]. In this regard, a general consistent thermodynamic framework for thermo-viscoplastic deformations of steel alloys is presented here along with an appropriate definition of the dissipation potentials and free energy, after considering the strain rate effect imbedded through the flow stress and hardening definition. Moreover, the additive combination of thermal and athermal yield function definitions are utilized in this framework for dynamic deformations of AL-6XN stainless steel.Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS. Numerical implementation for a simple tensile problem is used for validating the material parameters of the AL-6XN under low and high strain rates and temperatures. The numerical results of the adiabatic true stress–true strain curves compare very well with the experimental data. The effectiveness of the present approach is also tested by studying strain localization in plane strain problems. Results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh-independent results.

Partial Solvation Parameters and Mixture Thermodynamics

The recently introduced partial solvation parameters (PSPs) are molecular descriptors that combine elements from quantum mechanics with the QSPR/LSER/solvatochromic and solubility parameter approaches. Basic regularities and universalities exhibited by PSPs are examined in this work and the concepts of homosolvation, heterosolvation and solvation energy density are quantified. A simple consistent thermodynamic framework is developed, through which the validity of the PSP approach is tested. The predictions are compared with experimental phase equilibrium data that span the full composition range from the pure fluid state to infinite dilution. They include vapor-liquid equilibria of fluids interacting with strong specific forces, dissolution of solids/liquids in various solvents and probe/oligomer or probe/polymer interactions as typically determined by inverse gas-chromatography. These applications show the potential of the PSP approach not only to reasonably predict a variety of properties of classes of complex systems but, also, to shed light to challenging aspects of intermolecular interactions. The perspectives of this unified approach to solution thermodynamics are discussed.

Pattern study of thermal phase separation for binary fluid mixtures

PurposeThe purpose of this paper is to present numerical results about phase separation of binary fluid mixtures quenched by contact with cold walls.Design/methodology/approachThe thermal phase separation is simulated by using a hybrid lattice Boltzmann method that solves the continuity and the Navier‐Stokes equations. The equations for energy and concentration are solved by using a finite‐difference scheme. This approach provides a complete description of the thermo‐hydrodynamic effects in the mixture.FindingsA rich variety of domain patterns are found depending on the viscosity and on the heat conductivity of the mixture. Ordered lamellar structures are observed at high viscosity while domains rounded in shape dominate the phase separation at low viscosity, where two scales characterize the growth of domains.Research limitations/implicationsThe present approach provides a numerical method that can be extended to other systems such as liquid‐vapor or lamellar systems. Moreover, a three‐dimensional study can give a complete picture of thermo‐hydrodynamic effects.Originality/valueThis paper provides a consistent thermodynamic theoretical framework for a binary fluid mixture and a numerically stable method to simulate them.

A thermodynamic framework for a gradient theory of continuum damage

In this paper, we present a formulation of state variable based gradient theory to model damage evolution and alleviate numerical instability associated within the post-bifurcation regime. This proposed theory is developed using basic microforce balance laws and appropriate state variables within a consistent thermodynamic framework. The proposed theory provides a strong coupling and consistent framework to prescribe energy storage and dissipation associated with internal damage. Moreover, the temporal evolution equation derived here naturally shows the effect of damage—nucleation, growth and coalescence. In addition, the theoretical framework presented here is easily extendable to the addition of other defects (not shown here), and can be generalized to the development of consistent coupled transport equations for species, such as hydrogen (Bammann et al. in JMPS, 2009, submitted), as well as providing a consistent structure for modeling events at diverse length scales.

Formulation of strain gradient plasticity with interface energy in a consistent thermodynamic framework

In this work, the strain gradient formulation is used within the context of the thermodynamic principle, internal state variables, and thermodynamic and dissipation potentials. These in turn provide balance of momentum, boundary conditions, yield condition and flow rule, and free energy and dissipative energies. This new formulation contributes to the following important related issues: (i) the effects of interface energy that are incorporated into the formulation to address various boundary conditions, strengthening and formation of the boundary layers, (ii) nonlocal temperature effects that are crucial, for instance, for simulating the behavior of high speed machining for metallic materials using the strain gradient plasticity models, (iii) a new form of the nonlocal flow rule, (iv) physical bases of the length scale parameter and its identification using nano-indentation experiments and (v) a wide range of applications of the theory. Applications to thin films on thick substrates for various loading conditions and torsion of thin wires are investigated here along with the appropriate length scale effect on the behavior of these structures. Numerical issues of the theory are discussed and results are obtained using Matlab and Mathematica for the nonlinear ordinary differential equations (NODE) which constitute the boundary value problem. This study reveals that the micro-stress term has an important effect on the development of the boundary layers and hardening of the material at both hard and soft interface boundary conditions in thin films. The interface boundary conditions are described by the interfacial length scale and interfacial strength parameters. These parameters are important to control the size effect and hardening of the material. For more complex geometries the generalized form of the boundary value problem using the nonlocal finite element formulation is required to address the problems involved.

The Prigogine–Defay ratio and the microscopic theory of supercooled liquids

Prigogine-Defay ratios and, more recently, their frequency extension have been proposed to be a measure of the number of nonmacroscopic processes involved in the relaxation dynamics of supercooled liquids. We show that the microscopic theory of the Navier-Stokes equations of those liquids provides a consistent thermodynamic framework in which all possible dynamical Prigogine-Defay ratios can be expressed in terms of the same relaxation functions and that these ratios provide less information than the microscopic theory itself. The latter shows that more than one relaxation process is certainly always involved in this relaxation dynamics, whatever is the molecular dynamics, or experimental, technique used to determine the latter.

A thermodynamic approach to non-local damage modelling of concrete

This paper focuses on the development of a thermodynamic approach to constitutive modelling of concrete materials, with emphasis on the use of non-local damage models. Effort is put on the construction of a consistent and rigorous thermodynamic framework, which readily allows the incorporation of non-local features into the constitutive modelling. This is an important feature in developing non-local constitutive models based on thermodynamics. Examples of non-local constitutive models derived from this framework and numerical examples are given to demonstrate the promising features of the proposed approach.

A computational framework of configurational-force-driven brittle fracture based on incremental energy minimization

A variational formulation of quasi-static brittle fracture in elastic solids at small strains is proposed and an associated finite element implementation is presented. On the theoretical side, a consistent thermodynamic framework for brittle crack propagation is outlined. It is shown that both the elastic equilibrium response as well as the local crack evolution follow in a natural format by exploitation of a global Clausius–Planck inequality. Here, the canonical direction of the crack propagation associated with the classical Griffith criterion is the direction of the material configurational force which maximizes the local dissipation at the crack tip. On the numerical side, we first consider a standard finite element discretization in the two-dimensional space which yields a discrete formulation of the global dissipation in terms of configurational nodal forces. Next, consistent with the node-based setting, the discretization of the evolving crack discontinuity for two-dimensional problems is performed by the doubling of critical nodes and interface segments of the mesh. A crucial step for the success of this procedure is its embedding into a r-adaptive crack-segment re-orientation algorithm governed by configurational-force-based directional indicators. Here, successive crack propagation is performed by a staggered loading-release algorithm of energy minimization at frozen crack state followed by nodal releases at frozen deformation. We compare results obtained by the proposed formulation with other crack propagation criteria. The computational method proposed is extremely robust and shows an excellent performance for representative numerical simulations.