Thermodynamic Theory Of Irreversible Processes Research Articles

Thermodynamic theory of irreversible processes and generalized hydrodynamics of fluids

In this article, a review is presented of the thermodynamic theory of irreversible processes, based on the Clausius inequality representative of the literal forms of the second law of thermodynamics as stated by Kelvin and Clausius. Generalized hydrodynamic equations in conformation to the law are presented for transport processes in fluids removed far from equilibrium. They generalize the Navier–Stokes–Fourier hydrodynamics to flows of nonlinear irreversible processes.

Analysis of Heat Transfer of Cu-Water Nanofluid Flow Past a Moving Wedge

In this paper, heat transfer of a steady, two-dimensional, incompressible Cu-water nanofluid flow over a moving wedge in the presence of thermal radiation effect are investigated. Gyarmati's variational principle developed on the thermodynamic theory of irreversible processes is employed to solve the problem numerically. The governing boundary layer equations are approximated as simple polynomial functions, and the functional of the variational principle is constructed. The Euler-Langrange equations are reduced to simple polynomial equations in terms of boundary layer thicknesses. The velocity and temperature profiles as well as skin friction and heat transfer are analyzed for various parameters. The obtained numerical solutions are compared with the previously published results and are found to be in good agreement.

Entropy production and Newton's cooling law

The thermodynamic theory of irreversible processes was applied to studying heat exchange described by Newton's cooling law. It is shown that the steady states described by Newton's law are not states having minimum entropy production; it is then proved that Newton's law is not a consequence of a linear relationship between fluxes and thermodynamic forces. Systems are often constrained and heat exchange can be optimised if entropy production is minimised according to the second law.

Relative Boltzmann entropy, evolution equations for fluctuations of thermodynamic intensive variables, and a statistical mechanical representation of the zeroth law of thermodynamics

Generalized thermodynamics or extended irreversible thermodynamics presumes the existence of thermodynamic intensive variables (e.g., temperature, pressure, chemical potentials, generalized potentials) even if the system is removed from equilibrium. It is necessary to properly understand the nature of such intensive variables and, in particular, of their fluctuations, that is, their deviations from those defined in the extended irreversible thermodynamic sense. The meaning of temperature is examined by means of a kinetic theory of macroscopic irreversible processes to assess the validity of the generalized (or extended) thermodynamic method applied to nonequilibrium phenomena. The Boltzmann equation is used for the purpose. Since the relative Boltzmann entropy has been known to be intimately related to the evolution of the aforementioned fluctuations in the intensive thermodynamic variables, we derive the evolution equations for such fluctuations of intensive variables to lay the foundation for investigating the physical implications and evolution of the relative Boltzmann entropy, so that the range of validity of the thermodynamic theory of irreversible processes can be elucidated. Within the framework of this work, we examine a special case of the evolution equations for the aforementioned fluctuations of intensive variables, which also facilitate investigation of the molecular theory meaning of the zeroth law of thermodynamics. We derive an evolution equation describing the relaxation of temperature fluctuations from its local value and present a formula for the temperature relaxation time.

Non-Equilibrium Critical Behavior: An Extended Irreversible Thermodynamics Approach

Critical phenomena in non-equilibrium systems have been studied by means of a wide variety of theoretical and experimental approaches. Mode-coupling, renormalization group, complex Lie algebras and diagrammatic techniques are some of the usual theoretical tools. Experimental studies include light and inelastic neutron scattering, X-ray photon correlation spectroscopy, microwave interferometry and several other techniques. Nevertheless no conclusive reatment has been developed from the basic principles of a thermodynamic theory of irreversible processes. We have developed a formalism in which we obtain correlation functions as field averages of the associated functions. By applying such formalism we attempt to find out if the resulting correlation functions will inherit the mathematical properties (integrability, generalized homogeneity, scaling laws) of its parent potentials, and we will also use these correlation functions to study the behavior of macroscopic systems far from equilibrium, specially in the neighborhood of critical points or dynamic phase transitions. As a working example we will consider the mono-critical behavior of a non-equilibrium binary fluid mixture close to its consolute point.

Thermodynamic conditions for stability in materials with rate-independent dissipation

A distinctive feature of the examined class of solids is that a part of the entropy production is due to rate-independent dissipation, as in models of plasticity, damage or martensitic transformations. The standard condition for thermodynamic stability is shown to be too restrictive for such solids and, therefore, an extended condition for stability of equilibrium is developed. The classical thermodynamic theory of irreversible processes is used along with the internal variable approach, with the emphasis on the macroscopic effects of micro-scale instabilities in the presence of two different scales of time. Specific conditions for material stability against internal structural rearrangements under deformation-sensitive loading are derived within the incremental constitutive framework of multi-mode inelasticity. Application to spontaneous formation of deformation bands in a continuum is presented. Conditions for stability or instability of a quasi-static process induced by varying loading are given under additional constitutive postulates of normality and symmetry. As illustration of the theory, the stability of equilibrium or a deformation path under uniaxial tension is analysed for a class of inelastic constitutive laws for a metal crystal deformed plastically by multi-slip or undergoing stress-induced martensitic transformation.

Irreversible thermodynamics, parabolic law and self-similar state in grain growth

The formalism of the thermodynamic theory of irreversible processes is applied to grain growth to investigate the nature of the self-similar state and its corresponding parabolic law. Grain growth does not reach a steady state in the sense that the entropy production remains constant. However, the entropy production can be written as a product of two factors: a scale factor that tends to zero for long times and a scaled entropy production. It is suggested that the parabolic law and the self-similar state may be associated with the minimum of this scaled entropy production. This result implies that the parabolic law and the self-similar state have a sound irreversible thermodynamical basis.

Irreversible processes in inflationary cosmological models

By using the thermodynamic theory of irreversible processes and Einstein general relativity, a cosmological model is proposed where the early universe is considered as a mixture of a scalar field with a matter field. The scalar field refers to the inflaton while the matter field to the classical particles. The irreversibility is related to a particle production process at the expense of the gravitational energy and of the inflaton energy. The particle production process is represented by a nonequilibrium pressure in the energy-momentum tensor. The nonequilibrium pressure is proportional to the Hubble parameter and its proportionality factor is identified with the coefficient of bulk viscosity. The dynamic equations of the inflaton and the Einstein field equations determine the time evolution of the cosmic scale factor, the Hubble parameter, the acceleration and of the energy densities of the inflaton and matter. Among other results it is shown that in some regimes the acceleration is positive which simulates inflation. Moreover, the acceleration decreases and tends to zero in the instant of time where the energy density of matter attains its maximum value.

An irreversible thermodynamic approach to normal grain growth with a pinning force

The formalism of the Thermodynamic Theory of Irreversible Processes is applied to grain growth in the presence of a pinning force due to a particle dispersion. The Irreversible Thermodynamics treatment is able to show that both Burke's law and Zener type expressions have a sound thermodynamical basis. Particle pinning models are interpreted in terms of the irreversible entropy production that takes place as the grain boundary bypasses a particle.

Damage mechanics characterization on the fatigue behaviour of a solder joint material

This paper presents the first part of a comprehensive mechanics approach capable of predicting the integrity and reliability of solder joint material under fatigue loading without viscoplastic damage considerations. A separate report will be made to present the comprehensive damage model describing life prediction of the solder material under thermomechanical fatigue (TMF) loading. The method is based on the theory of damage mechanics, which makes possible a macroscopic description of the successive material deterioration caused by the presence of microcracks/voids in engineering materials. A damage mechanics model based on the thermodynamic theory of irreversible processes with internal state variables is proposed and used to provide a unified approach in characterizing the cyclic behaviour of a typical solder material. With the introduction of a damage effect tensor, the constitutive equations are derived to enable the formulation of a fatigue damage dissipative potential function and a fatigue damage criterion. The fatigue evolution is subsequently developed on the basis of the hypothesis that the overall damage is induced by the accumulation of fatigue and plastic damage. This damage mechanics approach offers a systematic and versatile means that is effective in modelling the entire process of material failure, ranging from damage initiation and propagation leading eventually to macrocrack initiation and growth. As the model takes into account the load history effect and the interaction between plasticity damage and fatigue damage, with the aid of a modified general-purpose finite element program, the method can readily be applied to estimate the fatigue life of solder joints under different loading conditions.

Damage mechanics characterization on the fatigue behaviour of a solder joint material

This paper presents the first part of a comprehensive mechanics approach capable of predicting the integrity and reliability of solder joint material under fatigue loading without viscoplastic damage considerations. A separate report will be made to present the comprehensive damage model describing life prediction of the solder material under thermomechanical fatigue (TMF) loading. The method is based on the theory of damage mechanics, which makes possible a macroscopic description of the successive material deterioration caused by the presence of microcracks/voids in engineering materials. A damage mechanics model based on the thermodynamic theory of irreversible processes with internal state variables is proposed and used to provide a unified approach in characterizing the cyclic behaviour of a typical solder material. With the introduction of a damage effect tensor, the constitutive equations are derived to enable the formulation of a fatigue damage dissipative potential function and a fatigue damage criterion. The fatigue evolution is subsequently developed on the basis of the hypothesis that the overall damage is induced by the accumulation of fatigue and plastic damage. This damage mechanics approach offers a systematic and versatile means that is effective in modelling the entire process of material failure, ranging from damage initiation and propagation leading eventually to macrocrack initiation and growth. As the model takes into account the load history effect and the interaction between plasticity damage and fatigue damage, with the aid of a modified general-purpose finite element program, the method can readily be applied to estimate the fatigue life of solder joints under different loading conditions.

Application of the Thermodynamic Theory of irreversible processes to normal grain growth

Application of the Thermodynamic Theory of irreversible processes to normal grain growth

A Method of Damage Mechanics Analysis for Solder Material

This paper presents as a method of damage mechanics analysis for solder joint material stressed to extensive plastic deformation. The material chosen for the current work is the 60Sn-40Pb eutectic alloy due to its wide use. The analysis is based on the thermodynamic theory of irreversible processes. With the introduction of a set of internal state variables, known as damage variables, and a damage effect tensor, a damage dissipative potential function is proposed to enable the formulation of the constitutive equations of elasticity and plasticity coupled with damage. The equations of damage evolution are also derived to monitor damage initiation and growth. Before a damage analysis can be performed with a finite element analysis, the mechanical properties of the chosen solder joint material and its damage variables must first be determined. A method of experimental analysis was developed and used to successfully measure the highly strain sensitive 60Sn-40Pb solder material. The measured properties are presented and various characteristics of the solder material are examined and discussed. 7 refs., 8 figs.

A kinetic equation for damage during superplastic deformation

In this paper a kinetic equation for the isotropic damage during superplastic deformation is proposed, based on the thermodynamic theory of irreversible processes with internal state variables and the normality criterion. The damage variable D during superplastic deformation is non-linear with the equivalent plastic strain ξ P, and it is shown that D is affected strongly by the triaxiality stress ratio (σ m/σ eq). Furthermore, in order to demonstrate the efficiency to the proposed equation, superplastic deformation experiments were performed on aluminum alloy LY12CZ and brass H62 without any pre-treatment. The comparision with the experimental results is presented on the influence of the triaxiality stress ratio on the equivalent strain to rupture: good correlation is found between the theoretical and experimental results. Therefore, this equation for damage may reflect successfully superplastic damage behaviour during superplastic deformation processes.

Boltzmann entropy, relative entropy, and related quantities in thermodynamic space

The conventional solution methods for the Boltzmann kinetic equation such as the Chapman–Enskog method or the moment method provide a thermodynamic branch of the distribution function evolving through macroscopic variables under the functional hypothesis. Such a distribution function is different in nature from the phase-space distribution function obtained by directly solving the Boltzmann kinetic equation subject to initial and boundary conditions in the phase space without the functional hypothesis. The Boltzmann entropy is an information entropy enumerated with the phase-space distribution function in the phase space, whereas the compensation function introduced in the previous work is a representation of the former in thermodynamic space. The two quantities are not generally the same. By using the concept of relative entropy which is the difference between the two quantities, we examine their relations and significance for the mathematical structure of thermodynamics of irreversible processes. By using the balance equations for the compensation function and the relative entropy, we investigate the limiting behavior of the rate of relative entropy as the thermodynamic branch of the distribution function becomes convergent in the sense of means (i.e., weakly converges) to the phase-space distribution function. The time derivative of the relative entropy does not vanish in the limit, but tends to a limit associated with energy dissipation. Such a limit represents a contraction of information as the description of irreversible processes is made in the thermodynamic space contracted from the phase space of 1023 particles. On the basis of such results, it is indicated that, as was found in earlier work, a thermodynamic theory of irreversible processes can be erected on the compensation function since it is an integral of a one-form in the thermodynamic space whereas the Boltzmann entropy is not.

Fatigue Crack Propagation in Metals and Polymers with a Unified Formulation

Abstract As the traditional ΔK-based fatigue crack propagation (FCP) law has recently been elucidated by Chow and his group [1–7] to be inadequate for a complete description of fatigue cracking, we examine in this investigation the applicability of a unified formulation for characterizing FCP of both metals and nonmetals. We start by defining a new cyclic J-integral, ΔJ, which, under the confines of linear elastic fracture mechanics (LEFM), is equivalent to the range of elastic strain energy release rate, ΔG. ΔJ (or ΔG) is used as a proper criterion for FCP so long as the crack does not extend during the unloading portion of one load reversal. ΔJ (or ΔG) as such is also interpreted as the source supplier for the energy flow rate dissipated on crack-tip cyclic plastic deformations. It is then assumed that, under fixed test conditions, the minimum specific work of fracture required for fatigue crack initiation, ΔJth, is a material property independent of the load ratio R. Fatigue threshold values thus predicted are correlated favorably with experimental measurements. For fatigue crack propagation, a unified formulation is subsequently derived from the thermodynamic theory of irreversible processes with the cracked surface area taken as an internal variable together with the assumption that the rate of energy dissipated on FCP depends on ΔJ only. It is noted that, in addition to ΔJ, the role played by (Jc − Jmax) is also required in the unified FCP law. Applicability of the law for producing a master FCP diagram is finally examined using FCP data of both metals and polymers, including mild steel, aluminium alloys, PMMA, PVC, short fiber composites, and adhesives.

Characterization of Fatigue Crack Propagation in Short Fiber Reinforced Thermoplastics—A Unified Approach

The paper presents a study conducted to examme the applicability of a unified approach of fatigue crack propagation (FCP) to include a short fiber composite (SFC), which is subjected to tensile cyclic stressing, undergoing generalized plasticity. The proposed approach expressed in the form of da/dN = C( ΔG) m/( Gc - Gmax) has recently been shown capable of characterizing a wide range of materials including mild steel, aluminium alloys, PMMA and PVC with a master FCP diagram. For plasticially loaded cracked members, the above formulation is generalized to be as da/dN = C(Δ J) m/( Jc - Jmax) and the difference between the J-integral range, Δ J, and the con ventional cyclic J-integral, Δ Jcycl, are elucidated The unified formulation is also shown to be derivable from the thermodynamic theory of irreversible processes, had the cracked surface area been taken as an internal state variable. The measured FCP rates for SFC together with two other polymeric materials, PMMA and PVC, are analyzed using the unified formulation and its validity for producing a master diagram is verified when all FCP data are successfully placed in one single line yielding a satisfactory coefficient of correlation of 0.987.

Cyclic J-integral in relation to fatigue crack initiation and propagation

Fatigue crack initiation (threshold) and propagation are associated with an appreciable sensitivity to one basic factor, among others (environment, microstructure, frequency, temperature etc.), namely load ratio R. The crack closure concept has frequently been hypothesized as a possible explanation for this. Nevertheless, existing experimental results with respect to crack closure are somewhat conflicting due to significant variabilities of such factors as specimen geometries and stress intensity factor ranges. This paper intends to describe the R- dependent fatigue threshold as well as fatigue crack growth from a different point of view, i.e. from the global energy equivalence concept. The generalized driving force attached to a fatigue crack tip is in the first instance re-examined with the introduction of a new cyclic J-integral. The original definition of a cyclic J-integral is shown to be inadequate as it fails to account for the total energy flux into the crack tip region from external loading mechanisms. Under small scale yielding conditions, the newly defined cyclic J-integral is equivalent to the range of elastic energy release rate, ΔG, for a linear elastic crack independent of loading processes. It is proposed to use ΔJ as an appropriate criterion for fatigue crack growth so long as the crack does not extend during the unloading portion of one load reversal. ΔJ as such is also interpreted as the source supplier for the energy flow rate dissipated on crack tip cyclic plastic deformations. It is then assumed that, under fixed test conditions, the minimum specific work of fracture required for fatigue crack initiation, ΔJ th , is a material property independent of the load ratio R. The so-predicted values of fatigue threshold, ΔK th , are correlated favorably with experimental measurements. For fatigue crack growth (FCG), a unified formulation is capable of being derived from the thermodynamic theory of irreversible processes, if the cracked surface area is taken as an internal variable and the rate of energy dissipated on FCG depends on ΔJ only. The applicability of the formulation for producing a master FCG diagram is examined for both metals and non-metals, including steel, aluminum alloys, PMMA and PVC, etc.

Spin-orbit band effects on silicon hot-hole transport

High-field transport of silicon holes has been analyzed considering carrier excitation to the spin-orbit band, which is separated by 44 meV from the heavy-hole and light-hole bands. Velocity-field relationships from 6 to 300 K, were interpreted by transport characteristics for holes excited to that band. According to a thermodynamic theory of irreversible processes in hot-carrier transports, the hole temperature at high electric fields was estimated to be lower than the electron temperature at the same electric fields, due to the short momentum and energy relaxation times for holes in the spin-orbit band.

Dissipation in steady states of chemical systems and deviations from minimum entropy production

According to the linear thermodynamic theory of irreversible processes, entropy production is minimized at the steady states of nonequilibrium systems. The principle of minimum entropy production is obtained if the dissipation is expanded in the deviation δ from equilibrium, truncated to lowest order (δ 2), and then differentiated. At this level of apprximation, the derivative of the dissipation is linear in δ and vanishes at the steady state. For a simple chemical reaction mechanism in a well-defined model system, we have derived the corrections to this approximation, by first evaluating the dissipation and its derivative exactly, and then expanding as a series in δ. To leading order in δ, the ratio of the derivative to the dissipation at the steady state is 1 2 (in dimensionless units), and this ratio does not decrease as equilibrium is approached. The steady state coincides with the state of minimum entropy production only at equilibrium. Given sufficient accuracy in measurements of species concentrations, the breakdown of the principle of minimum entropy production can be detected experimentally when the relative standard deviation in measurements of the dissipation is less than δ. Our example shows that the dissipation in a reaching chemical system may increase in time, in the later stages of relaxation toward a near-equilibrium steady state.