Kinetic and potential energy in the first law of thermodynamics

Kinetic and potential energy are included in the first law of thermodynamics in quite a contradictory way. Whereas in thermodynamics the total energy is understood as the sum of internal, kinetic and potential energy, the total energy in continuum mechanics incorporates only internal and kinetic energy, the potential energy being part of the work. The Gibbs ' fundamental equation is also occasionally extendend to contain a term for the potential energy. Some serious contradictions may result from this. As is first shown, kinetic and potential energy do not have any influence on the internal energy as long as relativistic effects are excluded. The Gibbs' fundamental equation therefore describes exchange processes between the “internal variables? of a system and its surroundings. Proceeding from this result one obtains a general definition of heat in open systems, including electromagnetic reactions, surfaceeffects and variable mole numbers. Exchange processes between the “external variables? of a system and its surroundings and hence also the influence of kinetic and potential energy are described by another independent equation, i.e. the energy equation of mechanics. Addition of both equations leads to the heat definition which is usually but under some further neglects given in textbooks. This definition has considerable disadvantages compared to the one derived before. In particular it is no longer possible to realize how mechanical and thermal energy are transformed into each other, which may give rise to errors.     

Anti-plane shear Lamb''s problem treated by gradient elasticity with surface energy

The consideration of higher-order gradient effects in a classical elastodynamic problem is explored in this paper. The problem is the anti-plane shear analogue of the well-known Lamb's problem. It involves the time-harmonic loading of a half-space by a single concentrated anti-plane shear line force applied on the half-space surface. The classical solution of this problem based on standard linear elasticity was first given by J.D. Achenbach and predicts a logarithmically unbounded displacement at the point of application of the load. The latter formulation involves a Helmholtz equation for the out-of-plane displacement subjected to a traction boundary condition. Here, the generalized continuum theory of gradient elasticity with surface energy leads to a fourth-order PDE under traction and double-traction boundary conditions. This theory assumes a form of the strain-energy density containing, in addition to the standard linear-elasticity terms, strain-gradient and surface-energy terms. The present solution, in some contrast with the classical one, predicts bounded displacements everywhere. This may have important implications for more general contact problems and the Boundary-Integral-Equation Method.     

Formulation of thermoelastic dissipative material behavior using GENERIC

We show that the coupled balance equations for a large class of dissipative materials can be cast in the form of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). In dissipative solids (generalized standard materials), the state of a material point is described by dissipative internal variables in addition to the elastic deformation and the temperature. The framework GENERIC allows for an efficient derivation of thermodynamically consistent coupled field equations, while revealing additional underlying physical structures, like the role of the free energy as the driving potential for reversible effects and the role of the free entropy (Massieu potential) as the driving potential for dissipative effects. Applications to large and small-strain thermoplasticity are given. Moreover, for the quasistatic case, where the deformation can be statically eliminated, we derive a generalized gradient structure for the internal variable and the temperature with a reduced entropy as driving functional.     

Generalized heat-transport equations: parabolic and hyperbolic models

We derive two different generalized heat-transport equations: the most general one, of the first order in time and second order in space, encompasses some well-known heat equations and describes the hyperbolic regime in the absence of nonlocal effects. Another, less general, of the second order in time and fourth order in space, is able to describe hyperbolic heat conduction also in the presence of nonlocal effects. We investigate the thermodynamic compatibility of both models by applying some generalizations of the classical Liu and Coleman–Noll procedures. In both cases, constitutive equations for the entropy and for the entropy flux are obtained. For the second model, we consider a heat-transport equation which includes nonlocal terms and study the resulting set of balance laws, proving that the corresponding thermal perturbations propagate with finite speed.     

On canonical equations of continuum thermomechanics

It is shown that the canonical balance of momentum of continuum mechanics can be formulated in a general way, but not independently of the usual balance of linear momentum, even in the absence of specified constitutive equations. A parallel construct is made of necessity for the accompanying time-like canonical energy equation. On specifying the energy, previous particular cases can be deduced including pure elasticity, inhomogeneous thermoelasticity of conductors, and the case of dissipative solid-like materials described by means of a diffusive internal variable (such as in damage or weakly non-local plasticity). A redefinition of the entropy flux is necessarily accompanied by a redefinition of the Eshelby stress tensor.     

A multi-field incremental variational framework for gradient-extended standard dissipative solids

The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time-space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.     

A thermodynamic model for magnetorheological fluids

An analysis of the dynamic behavior of a magnetorheological (MR) fluid is given in terms of a vectorial internal variable describing the change of the macroscopic average of the relative position vector of suspensions. Under the restriction of the second law, the constitutive equations of the MR fluid for stress, heat flux, magnetization and internal variable can be derived. The related issue of dissipative and energy transfer mechanisms is treated at some length. Studies on the steady shear flow indicate the direction of the internal variable is independent of shear rate. The Bingham-type constitutive equation for shear stress is obtained and endowed with a new meaning. The pressure-driven flow, another significant flow type for the design of MR devices, is also analyzed to study the plug flow region and the relationship between yield stress and flow rate. In addition, a criterion of flow initiated by the applied shear force is proposed based on the saturation of the internal variable and the condition of the equilibrium of forces in the fluid and solid regions. Received April 08, 1997     

Evolution Equations for Non-Simple Viscoelastic Solids

Objectivity and compatibility with thermodynamics of evolution equations are examined in connection with the modelling of viscoelastic solids. The purpose of the paper is to show that the evolution equation for the stress is eventually obtained by means of a tensorial internal variable within the framework of the reference configuration. The non-simple character is realized by gradients of the internal variable. The thermodynamic analysis is developed by investigating the entropy inequality in the reference configuration and allowing for a non-zero extra-entropy flux. It follows that the evolution for the Cauchy stress tensor involves the Oldroyd derivative, irrespective of the form of the non-local terms.     

Theory for Atomic Diffusion on Fixed and Deformable Crystal Lattices

We develop a theoretical framework, for the diffusion of a single unconstrained species of atoms on a crystal lattice, that provides a generalization of the classical theories of atomic diffusion and diffusion-induced phase separation to account for constitutive nonlinearities, external forces, and the deformation of the lattice. In this framework, we regard atomic diffusion as a microscopic process described by two independent kinematic variables: (i) the atomic flux, which reckons the local motion of atoms relative to the motion of the underlying lattice, and (ii) the time-rate of the atomic density, which encompasses nonlocal interactions between migrating atoms and characterizes the kinematics of phase separation. We introduce generalized forces power-conjugate to each of these rates and require that these forces satisfy ancillary microbalances distinct from the conventional balance involving the forces that expend power over the rate at which the lattice deforms. A mechanical version of the second law, which takes the form of an energy imbalance accounting for all power expenditures (including those due to atomic diffusion and phase separation), is used to derive restrictions on constitutive equations. With these restrictions, the microbalance involving the forces conjugate to the atomic flux provides a generalization of the usual constitutive relation between the atomic flux and the gradient of the diffusion potential, a relation that in conjunction with the atomic balance yields a generalized Cahn–Hilliard equation.     

Second gradient viscoelastic fluids: dissipation principle and free energies

We consider a generalization of the constitutive equation for an incompressible second order fluid, by including thermal and viscoelastic effects in the expression for the stress tensor. The presence of the histories of the strain rate tensor and its gradient yields a non-simple material, for which the laws of thermodynamics assume a appropriate modified form. These laws are expressed in terms of the internal mechanical power which is evaluated, using the dynamical equation for the fluid. Generalized thermodynamic constraints on the constitutive equation are presented. The required properties of free energy functionals are discussed. In particular, it is shown that they differ from the standard Graffi conditions. Various free energy functionals, which are well-known in relation to simple materials, are generalized so that they apply to this fluid. In particular, expressions for the minimum free energy and a more recently introduced explicit functional of the minimal state are proposed. Derivations of various formulae are abbreviated if closely analogous proofs already exist in the literature.     

Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates

The governing equation of motion of gradient elastic flexural Kirchhoff plates, including the effect of in-plane constant forces on bending, is explicitly derived. This is accomplished by appropriately combining the equations of flexural motion in terms of moments, shear and in-plane forces, the moment–stress relations and the stress–strain equations of a simple strain gradient elastic theory with just one constant (the internal length squared), in addition to the two classical elastic moduli. The resulting partial differential equation in terms of the lateral deflection of the plate is of the sixth order instead of the fourth, which is the case for the classical elastic case. Three boundary value problems dealing with static, stability and dynamic analysis of a rectangular simply supported all-around gradient elastic flexural plate are solved analytically. Non-classical boundary conditions, in additional to the classical ones, have to be utilized. An assessment of the effect of the gradient coefficient on the static or dynamic response of the plate, its buckling load and natural frequencies is also made by comparing the gradient type of solutions against the classical ones.     

The equations of continuum mechanics and the laws of thermodynamics

With appropriate constitutive assumptions on the stress tensor, the heat flux vector, and the frictional heating associated to a process, we derive for a fluid media the existence of internal energy and entropy as well as the classical energy balance equation and the Clausius-Duhem inequality.

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Sommario Con riferimento ad un continuo fluido, si dimostra-sotto appropriate ipotesi costitutive sul tensore delle tensioni, il vettore del flusso termico e il riscaldamento per altrito associato a un processo-l'esistenza della energia interna e dell'entropia di un continuo fluido. Si derivano inoltre la classica equazione di bilancio dell'energia e la diseguaglianza di Clausius-Duhem.

     

Study of a collocated Lagrange‐remap scheme for multi‐material flows adapted to HPC

This paper describes a collocated numerical scheme for multi‐material compressible Euler equations, which attempts to suit to parallel computing constraints. Its main features are conservativity of mass, momentum, total energy and entropy production, and second order in time and space. In the context of a Eulerian Lagrange‐remap scheme on planar geometry and for rectangular meshes, we propose and compare remapping schemes using a finite volume framework. We consider directional splitting or fully multi‐dimensional remaps, and we focus on a definition of the so‐called corner fluxes. We also address the issue of the internal energy behavior when using a conservative total energy remap. It can be perturbed by the duality between kinetic energy obtained through the conservative momentum remap or implicitly through the total energy remap. Therefore, we propose a kinetic energy flux that improves the internal energy remap results in this context. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of plate in second strain gradient elasticity

The bending analysis of a thin rectangular plate is carried out in the framework of the second gradient elasticity. In contrast to the classical plate theory, the gradient elasticity can capture the size effects by introducing internal length. In second gradient elasticity model, two internal lengths are present, and the potential energy function is assumed to be quadratic function in terms of strain, first- and second-order gradient strain. Second gradient theory captures the size effects of a structure with high strain gradients more effectively rather than first strain gradient elasticity. Adopting the Kirchhoff’s theory of plate, the plane stress dimension reduction is applied to the stress field, and the governing equation and possible boundary conditions are derived in a variational approach. The governing partial differential equation can be simplified to the first gradient or classical elasticity by setting first or both internal lengths equal to zero, respectively. The clamped and simply supported boundary conditions are derived from the variational equations. As an example, static, stability and free vibration analyses of a simply supported rectangular plate are presented analytically.     

Crack Tip Equation of Motion in Dynamic Gradient Damage Models

We propose in this contribution to investigate the link between the dynamic gradient damage model and the classical Griffith’s theory of dynamic fracture during the crack propagation phase. To achieve this main objective, we first rigorously reformulate two-dimensional linear elastic dynamic fracture problems using variational methods and shape derivative techniques. The classical equation of motion governing a smoothly propagating crack tip follows by considering variations of a space-time action integral. We then give a variationally consistent framework of the dynamic gradient damage model. Owing to the analogies between the variational ingredients of these two models and under some basic assumptions concerning the damage band structuration, one obtains a generalized Griffith criterion which governs the crack tip evolution within the non-local damage model. Assuming further that the internal length is small compared to the dimension of the body, the previous criterion leads to the classical Griffith’s law through a separation of scales between the outer linear elastic domain and the inner damage process zone.     

关于可变区域问题的变分法以及广义数学规划问题

本文具体研究了定义在可变区域上或可变边界表面上的泛函的一阶、二阶变分问题，得到了与经典变分法相对应的关于可变区域问题的变分法。并用该变分法讨论了具有可变区域的弹性系统的势能原理。另一方面，与传统的数学规划相对应，研究了可变区域上泛函的约束极值问题——广义数学规划问题，给出了相应的广义Ｋｕｈｎ－Ｔｕｃｋｅｒ条件。     

Stretched Gaussian Asymptotic Behavior for Fractional Giona–Roman Equation on Biased Heterogeneous Fractal Structure in External Force Fields

We introduce a biased heterogeneous fractional Giona-Roman equation (BHFGRE) on biased heterogeneous fractal structure media describing systems involving external force fields. The BHFGRE is shown to obey generalized Einstein relation, and its stationary solution is the generalized Boltzmann distribution. It is proved that the asymptotic shape of its solution is a stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for the case of constant potentials and generic potentials.     

Investigation on thermomechanical fracture in the framework of configurational forces

A configurational force approach is developed for providing a fresh look onto classical aspects of thermomechanical fracture. The theoretical framework is based on the finite deformation and makes no restrictions on the material response. The integral form of configurational force balance at the crack tip is constructed, and the concentrated configurational body force is decomposed into the inertial and internal parts. The energy release rate is evaluated through the generalized second law of thermodynamics applicable to configurational force system. The theoretical investigation shows that the negative of the projection of the internal configurational force concentrated at the crack tip along the direction of crack propagation plays the role of the energy release rate and acts directly in response to crack propagation. This finding enables us to deal with the thermomechanical fracture problems in material space.