Abstract.

Linearized Burnett and super-Burnett equations are considered for steady state Couette flow. It is shown that the linear super-Burnett equations lead to periodic velocity and temperature curves, i.e. unphysical solutions. The problem is discussed as well for the so-called augmented Burnett equations by Zhong et al. (AIAA Journal 31, 1036-1043 (1993)), and for the recently introduced regularized 13 moment equations (R13) of Struchtrup and Torrilhon (Phys. Fluids 15(9), 2668-2680 (2003) ). It is shown that both theories exhibit proper Knudsen boundary layers for velocity and temperature. However, the heat flux parallel to the wall has different signs for the Burnett and the R13 equations, and a comparison with DSMC results shows that only the R13 equations predict the proper sign.

Elastic structural members are considered with plastic hinges at which damage may occur. Equations of balance of linear momentum, angular momentum and energy are obtained expressing the fact that energy may be absorbed by damage at a plastic hinge. It is shown that all the work done by the moments in bending of a plastic hinge that does not appear as heat must be absorbed by damage. The fraction of the bending work which is transformed into damage is thecoefficient of damage. In general, this coefficient is smaller than unity, which implies that the plastic hinge is a heat source. The heat source manifests itself by producing a jump discontinuity in axial heat flux and thus, in virtue of Fourier's law, a jump discontinuity in temperature gradient ensues. The general thermodynamics of the system is formulated: It is shown that agreement with the second law of thermodynamics prevails if the coefficient of damage does not exceed unity. Using the condition that a plastic hinge must break after absorbing a critical amount of damage energy, we show that a structure composed of such structural members with a finite number of plastic hinges must fail before it absorbs an infinite amount of energy through a cyclic loading and unloading program. Although the failure criterion for a single plastic hinge may be simple, the complex distribution of damage among the plastic hinges of an entire structure can make the failure criteria for a structure defy simple rules. It is also argued that the introduction of a distributed damage energy rate does not seem feasible in view of the second law of thermodynamics. This suggests that damage in a microstructure is localized rather than uniformly distributed.

Extended Thermodynamics is an elegant and powerful material theory which yields, when applied, a symmetric hyperbolic system of evolution equations for the independent field variables. This system is obtained by imposing general physical principles, such as that of material objectivity and the entropy principle. With respect to thermodynamic equilibrium, the latter is carried through to second order. However, the following question arises: If this principle is imposed to even higher order terms, could it then yield restrictions also on the lower order terms, beyond those previously obtained in the literature? In this paper the entropy principle is applied to fourth order so that constitutive functions up to this order are obtained. In the process of computations many complicated equations involving only the lower order terms are deduced, which must be satisfied as identities. These equations, after long and tedious calculations, turn indeed out to be identically satisfied. This fact cannot be casual, thus confirming that this theory is trustworthy.

The paper addresses the fundamental problem of strained epitaxial surface growth on the basis of a front capturing approach. The particular problem arising in this context is the correct handling of (1) asymmetric attachment kinetics as well as (2) proper inclusion of energetic terms to model the elastic effects. The latter has not been included in most previous work concerned with the modeling of epitaxial surface growth. In the case of front capturing (1) and (2) pose fundamental challenges. In contrast to this, front tracking offers the possibility of a direct translation of the kinetics and the dynamics under consideration into a coupled set of free boundary equations to simulate. However, their numerical realization is troublesome. A numerical procedure is presented to overcome the technical problems arising in the case of front tracking as a benchmark for the front capturing model without elastic contributions. Developing the front tracking model further to include elastic effects remains an open challenge. In contrast front capturing allows for a straightforward extension to elastic driving forces based on variational principles of irreversible thermodynamics. The extended front capturing model is successfully applied to experiments by Dorsch et al. [J. Cryst. Growth 183, 305 (1998)]. The respective numerical study elucidates the question why these experiments contradict previous experimental findings with respect to final surface morphologies.

Abstract.

Dear CMT readers, it is my pleasure to introduce you to this topical issue dealing with a new research field of great interest, nonextensive statistical mechanics. This theory was initiated by Constantino Tsallis’ work in 1998, as a possible generalization of Boltzmann-Gibbs thermostatistics. It is based on a nonadditive entropy, nowadays referred to as the Tsallis entropy.

Nonextensive statistical mechanics is expected to be a consistent and unified theoretical framework for describing the macroscopic properties of complex systems that are anomalous in view of ordinary thermostatistics. In such systems, the long-standing problem regarding the relationship between statistical and dynamical laws becomes highlighted, since ergodicity and mixing may not be well realized in situations such as the edge of chaos. The phase space appears to self-organize in a structure that is not simply Euclidean but (multi)fractal. Due to this nontrivial structure, the concept of homogeneity of the system, which is the basic premise in ordinary thermodynamics, is violated and accordingly the additivity postulate for the thermodynamic quantities such as the internal energy and entropy may not be justified, in general. (Physically, nonadditivity is deeply relevant to nonextensivity of a system, in which the thermodynamic quantities do not scale with size in a simple way. Typical examples are systems with long-range interactions like self-gravitating systems as well as nonneutral charged ones.) A point of crucial importance here is that, phenomenologically, such an exotic phase-space structure has a fairly long lifetime. Therefore, this state, referred to as a metaequilibrium state or a nonequilibrium stationary state, appears to be described by a generalized entropic principle different from the traditional Boltzmann-Gibbs form, even though it may eventually approach the Boltzmann-Gibbs equilibrium state. The limits $t\to \infty $ and $N\to \infty $ do not commute, where t and N are time and the number of particles, respectively.

The present topical issue is devoted to summarizing the current status of nonextensive statistical mechanics from various perspectives. It is my hope that this issue can inform the reader of one of the foremost research areas in thermostatistics.

This issue consists of eight articles. The first one by Tsallis and Brigatti presents a general introduction and an overview of nonextensive statistical mechanics. At first glance, generalization of the ordinary Boltzmann-Gibbs-Shannon entropy might be completely arbitrary. But Abe’s article explains how Tsallis’ generalization of the statistical entropy can uniquely be characterized by both physical and mathematical principles. Then, the article by Pluchino, Latora, and Rapisarda presents a strong evidence that nonextensive statistical mechanics is in fact relevant to nonextensive systems with long-range interactions. The articles by Rajagopal, by Wada, and by Plastino, Miller, and Plastino are concerned with the macroscopic thermodynamic properties of nonextensive statistical mechanics. Rajagopal discusses the first and second laws of thermodynamics. Wada develops a discussion about the condition under which the nonextensive statistical-mechanical formalism is thermodynamically stable. The work of Plastino, Miller, and Plastino addresses the thermodynamic Legendre-transform structure and its robustness for generalizations of entropy. After these fundamental investigations, Sakagami and Taruya examine the theory for self-gravitating systems. Finally, Beck presents a novel idea of the so-called superstatistics, which provides nonextensive statistical mechanics with a physical interpretation based on nonequilibrium concepts including temperature fluctuations. Its applications to hydrodynamic turbulence and pattern formation in thermal convection states are also discussed.

Nonextensive statistical mechanics is already a well-studied field, and a number of works are available in the literature. It is recommended that the interested reader visit the URL http: //tsallis.cat.cbpf.br/TEMUCO.pdf. There, one can find a comprehensive list of references to more than one thousand papers including important results that, due to lack of space, have not been mentioned in the present issue. Though there are so many published works, nonextensive statistical mechanics is still a developing field. This can naturally be understood, since the program that has been undertaken is an extremely ambitious one that makes a serious attempt to enlarge the horizons of the realm of statistical mechanics.

The possible influence of nonextensive statistical mechanics on continuum mechanics and thermodynamics seems to be wide and deep. I will therefore be happy if this issue contributes to attracting the interest of researchers and stimulates research activities not only in the very field of nonextensive statistical mechanics but also in the field of continuum mechanics and thermodynamics in a wider context.

As the editor of the present topical issue, I would like to express my sincere thanks to all those who joined up to make this issue. I cordially thank Professor S. Abe for advising me on the editorial policy. Without his help, the present topical issue would never have been brought out.

This paper presents an analysis of volume average relations in connection with internal energy, net power and the balance of energy in continuum mechanics. The role of the so-called Hill and Hill–Mandel conditions in connection with the averaged energy balance equation has been elucidated. The significance of volume averages of stress moments for the calculation of the average net power is demonstrated.

We study delamination of two elastic bodies glued together by an adhesive that can undergo a unidirectional inelastic rate-independent process. The quasistatic delamination process is thus activated by time-dependent external loadings, realized through body forces and displacements prescribed on parts of the boundary. The novelty of this work consists in considering the glue as infinitesimally thin and ideally rigid in the sense that a crack in the glue cannot be seen before, speaking “microscopically”, all macromolecular links of the adhesive are fully debonded. The concept of energetic solution is applied and existence of such solutions is proved by showing Γ-convergence of a suitable approximation that, in addition, allows for a direct computer implementation, unlike the original problem.

The most usual formulation of the Laws of Thermodynamics turns out to be suitable for local or simple materials, while for non-local systems there are two different ways: either modify this usual formulation by introducing suitable extra fluxes or express the Laws of Thermodynamics in terms of internal powers directly, as we propose in this paper. The first choice is subject to the criticism that the vector fluxes must be introduced a posteriori in order to obtain the compatibility with the Laws of Thermodynamics. On the contrary, the formulation in terms of internal powers is more general, because it is a priori defined on the basis of the constitutive equations. Besides it allows to highlight, without ambiguity, the contribution of the internal powers in the variation of the thermodynamic potentials. Finally, in this paper, we consider some examples of non-local materials and derive the proper expressions of their internal powers from the power balance laws.

Kinetic Monte-Carlo (KMC) methods are used as an approach to simulate precipitation in Cu-alloyed bcc Fe. In order to characterize the process, transformed fractions, that is, the precipitated atoms, are related to Johnson-Mehl-Avrami-Kolmogorov theory. However, simulated data often deviate from corresponding fit curves and so does the resulting growth exponent when compared to theoretical expectations. Furthermore, some data may suggest the development of a metastable phase. In our study, we show that the characteristics of the transformed fraction and, as a consequence, the derived growth exponents sensitively depend on the number of atoms that are considered to form a particle and hence contribute to the transformed fraction. With a temperature dependence of the critical cluster size and additionally accounting for severe impingement of the particles, we obtain growth exponents which lie close to the expected range between n = 1.5 and n = 2.5 for pre-existing nuclei or continuous nucleation, respectively. From these, we obtain activation energies for nucleation and growth of precipitates. In this way, atomistic KMC simulations yield thermodynamical quantities, which can be valuable input parameters for larger length scale simulation methods, for example, for Phase Field Method simulations.

Abstract.

Acceleration waves propagating in isotropic solids at finite temperatures are studied by applying the method of singular surfaces to a new continuum model derived statistical-mechanically from a three-dimensional lattice model. The continuum model explicitly takes into account the microscopic thermal vibrations of the constituent atoms as one of the field variables. The propagation speeds and the ratios of mechanical and thermal amplitudes for both longitudinal and transverse waves are consistently determined. The differential equations that govern the time variation of the amplitudes of the waves are also derived. The analytical results, which are valid over a wide temperature range that includes the melting point, are evaluated numerically for several materials, and their physical implications are discussed. One of the findings to be emphasized is that of the singularities of the characteristic quantities at the melting point.