Clausius-Duhem Inequality Research Articles

Thermal and hydrodynamic effects in variable geometry nanofluid transport with Thermo-Responsive liquid Composites: Dynamics of entropy generation

In this communication, the authors focused on energy loss during the flow of hybrid nanofluid in expanding/contracting channels. The goal is to determine how, in real-world applications, hybrid nanofluids can maximize energy efficiency and reduce entropy production, hence assisting in the development of more effective and sustainable thermal management systems. It is well documented that heat transfer is enhanced due to the presence of nanoparticles in a base fluid and hence decreases energy loss in the system. The second law of thermodynamics defines that heat transfer is not ideal; hence, entropy in a system always rises. Suitable measures can be taken to minimize energy loss. One method is to use a nanofluid, if multiple nanosized particles are used; it is named a hybrid nanofluid. In this communication, a comparative analysis of simple and hybrid nanofluids is provided. A mathematical model outlining the dynamics of the flow is provided. The upper plate acts as a porous plate that is contracting/expanding and through which coolant hybrid nanofluids enter the channel. The lower plate remains stationary and is heated externally. The equations governing the flow are converted into ordinary differential equations by employing the similarity transformation. These ODEs are then solved analytically to get the series solution by using HAM along with the given boundary conditions. The entropy equation is modelled using the Clausius Duhem inequality. Copper and silver nanoparticles are combined with water. The impacts of different dimensionless parameters on stream function, velocity profile, Nusselt number, entropy, and Bejan number are discussed, and their results are shown graphically.

Thermodynamically Consistent Evolution Equations in Continuum Mechanics

This paper addresses the modelling of material behaviour in terms of differential (or rate) equations. To comply with the objectivity principle, recourse is made to invariant fields in the Lagrangian description or to objective time derivatives in the Eulerian description. The thermodynamic consistency is investigated in terms of the Clausius–Duhem inequality with two unusual features. Firstly, the (non-negative) entropy production is viewed as a constitutive function per se. Secondly, the inequality is viewed as a constraint on the pertinent fields and it is solved by using a representation formula, which allows for the the admissibility of a class of models. For definiteness, models of heat conduction are established, within Lagrangian descriptions, while models of the Navier–Stokes–Voigt fluid are investigated within Eulerian descriptions. In connection with thermo-viscous fluids, evolution equations are investigated within the Eulerian description. It is shown that the thermodynamic consistency is compatible with both objective and non-objective evolution equations.

Investigation of the strain rate and stretch level dependent behavior of elastomeric nanocomposites in complex uniaxial tests under finite strains

The mechanical behavior of elastomeric nanocomposites in experiments with nested stress-strain cycles and cyclic tests with increasing strain amplitude were considered. In our proposed testing procedures, long time stops at each stage of the change in loading direction were of great importance. This revealed two significant features of the behavior of elastomeric nanocomposites that have received little attention. It was shown that material softening (the Mullins effect) should be considered not only in the elastic part of the Cauchy stress tensor, but also in its dissipative part. The second peculiarity was the difference between the characteristic relaxation time at loading and the characteristic relaxation time at unloading observed in the experiments.This paper focuses on the behavior of highly-filled elastomeric materials based on different matrices (styrene-butadiene rubber (SBR) and nitrile-butadiene rubber (NBR)) and with different concentrations of carbon black (CB) or a combination of two fillers (CB and purified multi-walled carbon nanotubes (MWCNTs)).A mathematical model of the viscoelastic behavior of elastomeric nanocomposites under finite strains was proposed. It takes into account the peculiarities of the behavior of highly filled elastomers observed in the experiments. The specificity of the model consists in a new variant of the form of the free energy potential. It was shown that the new model satisfies the thermodynamic inequality, which is a consequence of the first law of thermodynamics and the second law in the form of the Clausius-Duhem inequality. A good agreement between theoretical calculations and experimental data was obtained.

Visco-Hyperelastic Constitutive Modeling for High-Damping Rubber Materials During Combined Quasi-Static Compression–Cyclic Shear Deformation Process

High-damping rubber materials utilized in high-damping rubber isolation bearings are frequently subjected to multiple deformations during the occurrence of earthquakes. Typically, large combined compression–shear deformations of the material could potentially cause compressive shear damage to rubber bearings. During this process, visco-hyperelastic properties of rubber materials will greatly change, which would significantly impact the seismic performance of rubber bearings. Thus, to give out a deep insight into their variations, it is necessary and urgent to develop a high-performance numerical method to investigate this process. This paper proposed a visco-hyperelastic constitutive modeling approach for high-damping rubber materials based on the experimental assessment of combined quasi-static compression–cyclic shear deformation process. Within the thermodynamic framework, the Clausius–Duhem inequality associated with the intrinsic dissipation of the material was firstly derived in accordance with the Lagrangian formulism. Then, stress–strain relations were obtained upon considering the occurrence of entropy production due to viscous dissipation. In the model, Stumpf–Marczak strain energy density function, which satisfies the Baker–Ericksen (B–E) inequality, was harnessed to describe the hyperelasticity of the material. By introducing higher orders of strain and strain rates and taking their couplings into account, a generalized viscous dissipation potential was proposed to capture nonlinear strain and strain rate-sensitivity effects of the material. To identify constitutive parameters, the deformation gradient was particularized for the combined quasi-static compression–cyclic shear deformation process. And, an inverse identification procedure was carried out at different levels of compression stress. The prediction results revealed that the proposed model exhibits remarkable prediction ability and adaptivity for different rubber materials during this process. Several new insights were highlighted on the variations of visco-hyperelastic characteristics of high-damping rubber materials with respect to the compression stress. The accuracy of the model was further validated by design parameters including initial shear modulus, secant shear modulus and equivalent viscous damping factor. This work could provide a fundamental guideline for the optimization and reliability analysis of high-damping rubber isolation bearings used in the field of seismic engineering.

Conducting viscous bodies with phase transitions: Deriving Gurtin’s postulate from the second law structure invariance

We prove that the basic structures of phase-field models for phase-transitions in non-isothermal setting can be derived all together from a unique principle requiring structure invariance of the second law of thermodynamics, written as a Clausius-Duhem inequality, under orientation-preserving diffeomorphism-based changes of observer and standard regularity conditions. We thus prove that the independent scalar balance of actions driving phase transitions does not require to be postulated, as it was in Gurtin’s 1996 proposal, rather it is a natural consequence of the invariance requirement adopted and maintains independence from constitutive choices. The approach discussed here is also a model-building framework for the dynamics of complex bodies and allows one to show that the microstructural behavior can be responsible of effects leading to wave-type heat propagation. The results indicate another role for the second law of thermodynamics, in addition to being a source of constitutive restrictions and compatibility or stability conditions.

Finite strain modelling of Shape Memory Alloys in the logarithmic strain space: A comparative study with other finite strain approaches

Finite strain models for Shape Memory Alloys (SMAs) offer significant advantages over small strain models in that they excel in their ability to characterize SMA behaviour under large strain and finite rotations while eliminating the inherent assumptions associated with the small strain framework. The paper at hand presents a numerical model for SMAs at finite strain, employing the logarithmic strain measure to obtain an additive split of the total strain into its elastic and inelastic (transformation) parts. Notably, the model maintains the thermodynamic consistency validated using the Clausius–Duhem inequality. Additionally, we compare this model with two other finite strain SMA models employing logarithmic strain measures: a hypoelastic model, which splits the rate of deformation tensor additively, and a hyperelastic model, which decomposes the deformation gradient multiplicatively. The numerical formulation for the three finite strain models is specifically tailored to address superelasticity in SMAs. It is derived from the established small strain model and seamlessly extended into the domain of finite strain framework. Several numerical examples are carried out to showcase the logarithmic strain space approach applied to SMAs and to compare its response with that of the other aforementioned finite strain models in SMAs. The meticulous examination reveals that the logarithmic strain space approach aligns well with the existing finite strain models, demonstrating its effectiveness and compatibility.

A Multiphysics Thermoelastoviscoplastic Damage Internal State Variable Constitutive Model including Magnetism.

We present a macroscale constitutive model that couples magnetism with thermal, elastic, plastic, and damage effects in an Internal State Variable (ISV) theory. Previous constitutive models did not include an interdependence between the internal magnetic (magnetostriction and magnetic flux) and mechanical fields. Although constitutive models explaining the mechanisms behind mechanical deformations caused by magnetization changes have been presented in the literature, they mainly focus on nanoscale structure-property relations. A fully coupled multiphysics macroscale ISV model presented herein admits lower length scale information from the nanoscale and microscale descriptions of the multiphysics behavior, thus capturing the effects of magnetic field forces with isotropic and anisotropic magnetization terms and moments under thermomechanical deformations. For the first time, this ISV modeling framework internally coheres to the kinematic, thermodynamic, and kinetic relationships of deformation using the evolving ISV histories. For the kinematics, a multiplicative decomposition of deformation gradient is employed including a magnetization term; hence, the Jacobian represents the conservation of mass and conservation of momentum including magnetism. The first and second laws of thermodynamics are used to constrain the appropriate constitutive relations through the Clausius-Duhem inequality. The kinetic framework employs a stress-strain relationship with a flow rule that couples the thermal, mechanical, and magnetic terms. Experimental data from the literature for three different materials (iron, nickel, and cobalt) are used to compare with the model's results showing good correlations.

Entropy and Thermodynamic Temperature in Nonequilibrium Classical Thermodynamics as Immediate Consequences of the Hahn–Banach Theorem: II Properties

In a companion article it was shown in a certain precise sense that, for any thermodynamical theory that respects the Kelvin–Planck second law, the Hahn–Banach theorem immediately ensures the existence of a pair of continuous functions of the local material state—a specific entropy (entropy per mass) and a thermodynamic temperature—that together satisfy the Clausius–Duhem inequality for every process. There was no requirement that the local states considered be states of equilibrium. This article addresses questions about properties of the entropy and thermodynamic temperature functions so obtained: To what extent do such temperature functions provide a faithful reflection of “hotness”? In precisely which Kelvin–Planck theories is such a temperature function essentially unique, and, among those theories, for which is the entropy function also essentially unique? What is a thermometer for a Kelvin–Planck theory, and, for the theory, what properties does the existence of a thermometer confer? In all of these questions, the Hahn–Banach Theorem again plays a crucial role.

On the entropy inequality and its exploitation in continuum physics

AbstractThe paper deals with the statement and the application of the entropy principle, through the Clausius–Duhem inequality, in continuum physics. The conceptual role is taken from the Coleman–Noll paper of 1963 thus leading to the physical admissibility of constitutive equations. The statement is generalized by letting the rate of entropy production be a constitutive property per se. This generalization proves essential in connection with the modelling of hysteretic phenomena. As to the application, the view of the Coleman–Noll procedure is maintained but a representation formula is shown to generalize the consequences of the entropy principle; as an example the modelling of heat conduction is investigated. Furthermore, while applying the entropy principle to magnetic materials, it is shown an interesting connection between the balance of angular momentum and the thermodynamic restrictions. Also, the modelling through rate-type equations shows the need of Lagrangian fields to obey objectivity and that of the entropy production as a constitutive function to account for the difference between loading and unloading processes.

On the development of a constitutive model for steel subjected to fire and explosion

The behavior of steel under thermal and mechanical loading including high transient temperatures and strain rates (e.g., fire, creep or explosion) is extremely complex. To study this behavior, we propose a constitutive model accordance with the laws of thermodynamic, the principle of virtual power and experimental results. The thermodynamic framework considers the Clausius-Duhem inequality and maximum dissipation rate principle, as well as the theory of finite deformations, linear isotropic viscoelasticity, nonlinear isotropic and kinematic hardening, local and non-local viscoplasticity, local and non-local anisotropic viscodamage and parameters associated to material behavior dependent on loadings. The components of the thermodynamic conjugate forces are defined from the Helmholtz free energy function and the rate of energy dissipation, which are postulated considering the creep phenomenon, thermal-mechanical transient phenomenon, cyclic phenomenon and plastic wave propagation phenomenon. The balance of microforces is defined based on the principle of virtual power. A novel rule of non-associated thermo-viscoplastic flow of the Perzyna type and a novel rate dependent local and non-local damage evolution law are introduced from constitutive model. The verification of the model was carried out in two benchmark studies composed of fire tests. The predictions of the model have shown in according with the results of the experiments. A sensitivity study was also performed with a view to analysis the influence of the parameters in achieved results. From this study a simplified constitutive model is presented. Benchmark studies composed of explosion tests are part of future work.

A Perspective on Plasticity, Dissipation and the Second Law of Thermodynamics

Abstract The requirement of a non-negative dissipation rate for all possible deformation histories is generally imposed on plastic constitutive relations. This is a constraint analogous to the Coleman–Noll [Coleman, B. D., and Noll, W., 1964, “The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity,” Arch. Ration. Mech. Anal., 13, pp. 167–178. 10.1007/BF01262690] postulate that the Clausius–Duhem inequality needs to be satisfied for all possible deformation histories. The physical basis for the Clausius–Duhem inequality is as a statistical limit for a large number of discrete events for a long time and is not a fundamental physical requirement for small systems for a short time. The relation between the requirement of a non-negative dissipation rate and the Clausius–Duhem inequality is considered. The consequences of imposing a non-negative dissipation rate for all possible deformation histories are illustrated for: (i) a single crystal plasticity framework that accounts for elastic lattice curvature changes as well as elastic lattice straining and (ii) for discrete defect theories of plasticity, with attention specifically on discrete dislocation plasticity for crystalline solids and discrete shear transformation zone (STZ) plasticity for amorphous solids. Possible less restrictive conditions on the evolution of dissipation in plasticity formulations are considered as are implications for stability. The focus is on open questions and issues.

A review of the second law of thermodynamics and its application

Over thousands of years, many great physicists, such as Carnot, Boltzmann, Plank, Clausius, to name only a few, have put great endeavor into unwinding the mysteries of thermodynamic studies. In recent decades, many innovations have been made by groundbreaking modern technologies, such as refrigerators and air conditioning, and both recent findings and formerly discovered concepts of thermodynamics have been explained deeply by modern scientists. The paper briefly introduces the definition of the second law of thermodynamics and the relative concepts and their application. This paper delves into the history and definition of the Carnot Theorem, the Irreversible Carnot engine and refrigerator principle, as well as the Coefficient of Performance (COP). After that, this paper will introduce a few relative concepts, including entropy, exergy, and the Clausius-Duhem inequality. In summary, this paper covers a basic overall explanation of the aforementioned concepts, as well as drawing conclusions about the experimental achievements of the early explorers of thermodynamics.

Differential geometry-based thermodynamics derivation of isotropic and anisotropic eikonal non-local gradient (ENLG) damage models using a micromorphic media framework

The Eikonal non-local damage (ENL) approach models damage as a space-deforming phenomenon that affects the interaction distance between material points. As damage increases, interactions between points decrease, ultimately resulting in no interaction. In the integral version of such an approach, non-local interaction distances between material points are computed by solving a stationary Hamilton–Jacobi equation with a damage-dependent Riemannian metric. In the implicit gradient version of ENL models, the Riemannian metric figures in the Helmholtz equation to be solved for computing the non-local field controlling damage evolution. However, one of the main criticisms of such formulation is the lack of thermodynamics basis in its derivation. This paper presents a thermodynamics derivation of the Eikonal implicit gradient formulation based on differential geometry concepts to overcome this issue. A free-energy potential is defined considering the non-local strain as a morphological descriptor belonging to the abstract differentiable manifold (where the Riemannian metric is defined). Following a micromorphic media framework, the balance equations of the model are obtained. It is shown that the stress tensor and thermodynamic force associated with damage are the sum of standard and additional non-local contributions. It is also shown that the resulting energy dissipation is always positive, thus verifying the Clausius–Duhem inequality. After presenting all development considering second-order anisotropic continuum damage, the isotropic formulation is obtained as a particular case. Finally, a two-dimensional numerical implementation of an isotropic implicit Eikonal non-local gradient damage model is illustrated. Test cases are simulated to show the relocalization features of the considered formulation and its natural capability of naturally representing damage-to-fracture transition for high damage levels.

Restrictions in a distributed complex fractional order linear constitutive equations of viscoelasticity

We analyze a viscoelastic body in a linear stress state by the use of the distributed complex order fractional derivative in the constitutive equation. A model is formulated so that it takes into account all complex order derivatives of the form t0Dtα+iβ(α), α∈[α1,α2]⊂(0,1), β∈C1([α1,α2]). Using a new procedure, we derive restrictions on a model that follows from the Clausius–Duhem inequality under isothermal conditions. We apply our analysis to the creep test, as well as to generalized wave equation on R, with a specific constitutive equation of Kelvin–Voigt type.

A fourth-order degradation tensor for an anisotropic damage phase-field model

This work proposes a thermodynamically consistent phase-field model for anisotropic brittle material under the hypotheses of plane stress, small deformation and constant temperature. The model is derived from the principle of virtual power, the first and second laws of thermodynamics in the form of the Clausius-Duhem inequality. The degradation effect on the material behavior is given by a fourth-order degradation tensor introduced as an internal variable that evolves according to the current strain state rather than the conventional scalar degradation function of phase-field models. Therefore, local anisotropy can be induced, changing the material mechanical behavior differently in all directions organically. The proposed degradation tensor is defined in the global coordinate system and therefore is sensitive to any change in the principal directions of the strain and stress states. To demonstrate the model’s capability of representing damage in isotropic and transversely isotropic materials, some benchmark examples were carried out and the evolution of the damage components was analyzed.

Tension-compression asymmetric functional degeneration of super-elastic NiTi shape memory alloy: Experimental observation and multiscale constitutive model

The functional degeneration of super-elastic NiTi shape memory alloy (SMA) rods is investigated in the cyclic tension-unloading and compression-unloading experiments. The results show that the functional degeneration exhibits a strong tension–compression asymmetry, which originates from the initial 〈111〉 and 〈110〉 mixed textures along the axial direction of the rods, the polarity of martensite transformation, and the complex interaction between martensite transformation and dislocation slipping. Moreover, the dependence of functional degeneration on the applied stress level is discussed. To understand and predict the effect of initial texture on the cyclic deformation of polycrystalline NiTi SMA, a multiscale constitutive model is constructed at finite deformation. At the mesoscopic scale, two major inelastic deformation mechanisms, i.e., the transformation between austenite and martensite phases and the transformation-induced plasticity caused by the strain mismatch at the austenite–martensite interface, are considered. The total deformation gradient is multiplicatively decomposed into the martensite transformation, transformation-induced plasticity and elastic parts. The deformation gradient parts at the mesoscopic scale are obtained by upscaling these physical quantities at a smaller spatial scale (i.e., the microscopic scale). The proposed model is formulated within the framework of irreversible thermodynamics. The driving forces for martensite transformation and transformation-induced plasticity are derived according to the established Helmholtz free energy and Clausius-Duhem inequality. The complex interaction between martensite transformation and dislocation slipping, and the forward/reverse inheritance of dislocations caused by the moving austenite–martensite interfaces are considered. To calculate the overall cyclic responses of polycrystalline NiTi SMA at the macroscopic scale, the developed constitutive model is implemented into the finite element software ABAQUS/Explicit using a user-defined material subroutine. A polycrystalline representative volume element is constructed by introducing the initial 〈111〉 and 〈110〉 mixed textures observed in the experiment. By comparing the simulated results with the experimental data, the capability of the proposed model to predict the texture-induced tension–compression asymmetric functional degeneration is verified, and the influence of texture type on the cyclic deformation is discussed.

Special Cosserat rods with rate-dependent evolving natural configurations

We present a nonlinear, geometrically exact, and thermodynamically consistent framework for modeling special Cosserat rods with evolving natural configurations. In contrast to the common usage of the point-wise Clausius–Duhem inequality to embody the Second Law of Thermodynamics, we enforce the strictly weaker form that the rate of total entropy production is non-decreasing. The constitutive relations between the state variables and applied forces needed to close the governing field equations are derived via prescribing frame indifferent forms of the Helmholtz energy and the total dissipation rate and requiring that the state variables evolve in a way that maximizes the rate of total entropy production. Due to the flexibility afforded by enforcing a global form of the Second Law, there are two models obtained from this procedure: one satisfying the stronger form of the Clausius–Duhem inequality and one satisfying the weaker global form of the Clausius–Duhem inequality. Finally, we show that in contrast to other viscoelastic Cosserat rod models introduced in the past, certain quadratic strain energies in our model yield both solid-like stress relaxation and creep.

Techniques for the Thermodynamic Consistency of Constitutive Equations

The paper investigates the techniques associated with the exploitation of the second law of thermodynamics as a restriction on the physically admissible processes. Though the exploitation consists of the use of the arbitrariness occurring in the Clausius–Duhem inequality, the approach emphasizes two uncommon features within the thermodynamic analysis: the representation formula, of vectors and tensors, and the entropy production. The representation is shown to be fruitful whenever more terms of the Clausius–Duhem inequality are not independent. Among the examples developed to show this feature, the paper yields the constitutive equation for hypo-elastic solids and for Maxwell–Cattaneo-like equations of heat conduction. The entropy production is assumed to be given by a constitutive function per se and not merely the expression inherited by the other constitutive functions. This feature results in more general expressions of the representation formulae and is crucial for the compact description of hysteretic phenomena.

Constitutive Modeling with Single and Dual Internal Variables.

Phenomenological constitutive models with internal variables have been applied for a wide range of material behavior. The developed models can be classified as related to the single internal variable formalism based on the thermodynamic approach by Coleman and Gurtin. The extension of this theory to so-called dual internal variables opens up new avenues for the constitutive modeling of macroscopic material behavior. This paper reveals the distinction between constitutive modeling with single and dual internal variables using examples of heat conduction in rigid solids, linear thermoelasticity, and viscous fluids. A thermodynamically consistent framework for treating internal variables with as little a priori knowledge as possible is presented. This framework is based on the exploitation of the Clausius-Duhem inequality. Since the considered internal variables are "observable but not controllable", only the Onsagerian procedure with the use of the extra entropy flux is appropriate for the derivation of evolution equations for internal variables. The key distinctions between single and dual internal variables are that the evolution equations are parabolic in the case of a single internal variable and hyperbolic if dual internal variables are employed.

Discrete defect plasticity and implications for dissipation

Inelastic deformation of solids is almost always, if not always, associated with the evolution of discrete defects such as dislocations, vacancies, twins and shear transformation zones. The focus here is on discrete defects that can be modeled appropriately by continuum mechanics, but where the discreteness of the carriers of plastic deformation plays a significant role. The formulations are restricted to small deformation kinematics and the defects considered, dislocations and discrete shear transformation zones (STZs), are described by their linear elastic fields. In discrete defect plasticity both the stress–strain response and the partitioning between defect energy storage and defect dissipation are outcomes of an initial/boundary value problem solution. For such defects an explicit expression for the dissipation rate is presented and, because there is a length scale associated with the discrete defects, the stress–strain response and the evolution of the dissipation rate can be size dependent. Discrete dislocation plasticity modeling results are reviewed that illustrate the implications of defect dissipation evolution for friction, fracture, fatigue crack growth and thermal softening. Examples are also given of the consequences in constrained shear of three modes of the evolution of discrete defects for the size dependence of the stress–strain response and the associated dissipation. For a purely mechanical formulation the Clausius–Duhem inequality specializes to the requirement that the dissipation rate is non-negative. Requiring a non-negative dissipation rate for all points of a body and for all time imposes restrictions on kinetic relations for the evolution of discrete defects. In statistical mechanics, the Clausius–Duhem inequality can be violated for sufficiently small regions for a sufficiently short time. Explicit kinetic relations for discrete dislocation plasticity dissipation and for discrete STZ plasticity dissipation identify conditions that can lead to a negative dissipation rate. In continuum mechanics, at least in some circumstances, satisfaction of the Clausius–Duhem inequality can be regarded as a stability requirement. Nevertheless, simple one-dimensional continuum calculations illustrate that there can be a negative dissipation rate over a small region and for a short time period with overall stability maintained. Implications for discrete defect plasticity modeling are briefly discussed.