Pressure–Strain Correlation Modeling: Towards Achieving Consistency with Rapid Distortion Theory

In turbulence closure modeling, it is widely accepted that the rapid pressure–strain correlation (RPSC) model be consistent with the rapid distortion theory (RDT). It is desirable to achieve this consistency with a closure model that is computationally tractable and satisfies the requisite mathematical constraints of realizability and linearity in the appropriate variables. In this investigation, starting from a detailed modal analysis of two-dimensional mean flows, we identify important flow features to be incorporated into the model. However, the dynamical system analysis shows that the suggested physics cannot be embodied in a model with all desired computational and mathematical attributes. To resolve this conflict, we propose a slight compromise in the physical requirement and ease one of the linearity constraints leading to a “best possible” tractable model. Overall, the present work provides important insight into RPSC closure modeling challenges—arising from the interplay among physical fidelity, computational viability and mathematical constraints—and proposes avenues for future improvement.     

A three-fluid approach in bipolar semiconductors with generation–recombination: constitutive laws and Onsager symmetry

If electrons (e) and holes (h) in metals or semiconductors are heated to the temperatures \(T_{e}\) and \(T_{h}\) greater than the lattice temperature \(T_{p}\), the electron–phonon interaction causes energy relaxation. In the nonuniform case, a momentum relaxation occurs as well. In view of such an application, a new model based on an asymptotic procedure for solving the kinetic equations of carriers and phonons is proposed, with generation–recombination of electrons and holes, which gives naturally the displaced Maxwellian at the leading order. After that, balance equations for the electron number, hole number, energy densities, and momentum densities are constructed, which constitute now a system of eight equations for the chemical potentials (carriers), the temperatures (carriers and phonons), and the drift velocities (carriers and phonons). In the drift-diffusion approximation the constitutive laws are derived and the Onsager relations recovered.     

马氏体相变过程中形核、失稳及多界面微结构的动力学模拟

材料发生相变的过程中会出现失稳、滞后回线及多界面的微结构等复杂现象，而稳定性的丧失使其动力学方程的求解十分困难。对于形状记忆合金中的马氏体相变，相变过程中材料的等效杨氏模量变为负值，使得传统的动力学方程成为病态的，无法直接求解，必须要进行正则化。而相变的滞后回线与微结构的出现也说明经典的弹性理论不再适用，必须要引入新的能量项以能刻画这些现象。本文在非线性弹性理论的框架下，引入应变梯度界面能和位移非均匀能，利用变分原理建立了材料相变的一维动力学模型。高阶项的引入极大地改善了方程的性质，使数值求解成为可能。计算结果表明，该模型确能较有效地描述相变时的失稳与微结构。     

Theoretical modeling of microstructured liquids: a simple thermodynamic approach

A new method is presented for accounting for microstructural features of flowing complex fluids at the level of mesoscopic, or coarse-grained, models by ensuring compatibility with macroscopic and continuum thermodynamics and classical transport phenomena. In this method, the microscopic state of the liquid is described by variables that are local expectation values of microscopic features. The hypothesis of local thermodynamic equilibrium is extended to include information on the microscopic state, i.e., the energy of the liquid is assumed to depend on the entropy, specific volume, and microscopic variables. For compatibility with classical transport phenomena, the microscopic variables are taken to be extensive variables (per unit mass or volume), which obey convection-diffusion-generation equations. Restrictions on the constitutive laws of the diffusive fluxes and generation terms are derived by separating dissipation by transport (caused by gradients in the derivatives of the energy with respect to the state variables) and by relaxation (caused by non-equilibrated microscopic processes like polymer chain stretching and orientation), and by applying isotropy. When applied to unentangled, isothermal, non-diffusing polymer solutions, the equations developed according to the new method recover those developed by the Generalized Bracket [J. Non-Newtonian Fluid Mech. 23 (1987) 271; A.N. Beris, B.J. Edwards, Thermodynamics of Flowing Systems with Internal Microstructure, first ed., Oxford University Press, Oxford, 1994] and by the Matrix Model [J. Rheol. 38 (1994) 769]. Minor differences with published results obtained by the Generalized Bracket are found in the equations describing flow coupled to heat and mass transfer in polymer solutions. The new method is applied to entangled polymer solutions and melts in the general case where the rate of generation of entanglements depends nonlinearly on the rate of strain. A link is drawn between the mesoscopic transport equations of entanglements and conformation and the microscopic equation describing the configurational distribution of polymer segment stretch and orientation. Constraints are derived on the generation terms in the transport equations of entanglements and conformation, and the formula for the elastic stress is generalized to account for reversible formation and destruction of entanglements. A simplified version of the transport equation of conformation is presented which includes many previously published constitutive models, separates flow-induced polymer stretching and orientation, yet is simple enough to be useful for developing large-scale computer codes for modeling coupled fluid flow and transport phenomena in two- and three-dimensional domains with complex shapes and free surfaces.     

空间可展机构非光滑力学模型和动力学研究

空间可展机构广泛应用于展开和支撑柔性太阳能帆板和航天工程领域中的有效载荷, 包括抛物面天线、平面相控阵雷达和合成孔径雷达等. 非光滑特性及其相应的动力学现象在空间可展机构的设计中有着非常重要的作用. 该文系统地综述了空间可展机构非光滑力学建模与非线性动力学的研究进展. 首先详细描述了含间隙铰链的接触碰撞力和摩擦力等非光滑特点;然后系统地介绍了含间隙机构的动力学建模方法、分析方法和参数设计;进一步简单介绍了含间隙铰链空间可展机构的非线性动力学特性, 如谐波共振、周期运动的稳定性和各类分岔等;最后提出了空间可展机构非光滑动力系统动力学、稳定性与控制中亟待解决的若干问题.     

Lower-Dimensional Manifold (Algebraic) Representation of Reynolds Stress Closure Equations

For complex turbulent flows, Reynolds stress closure modeling (RSCM) is the lowest level at which models can be developed with some fidelity to the governing Navier–Stokes equations. Citing computational burden, researchers have long sought to reduce the seven-equation RSCM to the so-called algebraic Reynolds stress model which involves solving only two evolution equations for turbulent kinetic energy and dissipation. In the past, reduction has been accomplished successfully in the weak-equilibrium limit of turbulence. In non-equilibrium turbulence, attempts at reduction have lacked mathematical rigor and have been based on ad hoc hypotheses resulting in less than adequate models.?In this work we undertake a formal (numerical) examination of the dynamical system of equations that constitute the Reynolds stress closure model to investigate the following questions. (i) When does the RSCM equation system formally permit reduced representation? (ii) What is the dimensionality (number of independent variables) of the permitted reduced system? (iii) How can one derive the reduced system (algebraic Reynolds stress model) from the full RSCM equations? Our analysis reveals that a lower-dimensional representation of the RSCM equations is possible not only in the equilibrium limit, but also in the slow-manifold stage of non-equilibrium turbulence. The degree of reduction depends on the type of mean-flow deformation and state of turbulence. We further develop two novel methods for deriving algebraic Reynolds stress models from RSCM equations in non-equilibrium turbulence. The present work is expected to play an important role in bringing much of the sophistication of the RSCM into the realm of two-equation algebraic Reynolds stress models. Another objective of this work is to place the other algebraic stress modeling efforts in the lower-dimensional modeling context. Received 19 November 1999 and accepted 3 August 2000     

On a closure of mass conservation equation and stability analysis in the mathematical theory of vehicular traffic flowSur une fermeture des équations de conservation de la masse et l'analyse de la stabilité dans la théorie mathématique de la circulation véhiculaire

This Note deals with the development of mathematical methods for the closure of the mass conservation equation for macroscopic hydrodynamical models of traffic flow on roads. The closure is obtained by a phenomenological model, relating the local mean velocity to local density earlier in time. An evolution equation is obtained for the flux and a stability analysis is performed; this qualitatively describes some features of congested flow. To cite this article: V. Coscia, C. R. Mecanique 332 (2004).     

On the damping of nonlinear vibrations in some second-order dynamical systems

This paper continues the study of damping of nonlinear vibrations in second-order dynamical systems for the case considered in [1]. The estimates obtained for energy scattering are applied to a system of two bodies connected by a weightless elastic cable. The system rotates in a plane, and the viscous friction forces in the cable are taken into account. The external drag forces are neglected. Such a problem arises, for example, if the motion of a system of elastically connected spacecraft far from attracting centers is considered [2, 3].     

The boundary layers of a fully ionized two-temperature plasma with given component temperatures at an electrode

The flow of a plasma with different component temperatures in the boundary layers at the electrodes of an MHD channel is investigated without any assumptions as to self-similarity. For the calculation of the electron temperature, the full energy equation for an electron gas [1] is solved with allowance for the estimates given in [2]. In contrast to [3, 4], the calculation includes the change in temperature of electrons and ions along the channel caused by the collective transport of energy, the work done by the partial pressure forces, and the Joule heating and the energy exchange between the components. The problem of the boundary layers in the flow of a two-temperature, partially ionized plasma past an electrode is solved in simplified form by the local similarity method in [5–7]. In these papers, either the Kerrebrock equation is used [5, 6] or the collective terms are omitted from the electron energy equation [7].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–10, September–October, 1972.The author thanks V. V. Gogosov and A. E. Yakubenko for interest in this work.     

Internal rotation in the hydrodynamics of weakly conducting dielectric suspensions

Hydrodynamic phenomena in weakly conducting single-phase media due to interphase electric stresses are reviewed in [1]. In the present paper, a model is constructed of a dielectric suspension with body couples due to the field acting on free charges distributed on the surface of the particles of the suspension. Averaging of the microscopic fields yields macroscopic equations for the field and the polarization of the dielectric suspension with allowance for the finite relaxation time of the distribution of the free charge on the phase interface. The developed model is used to consider the occurrence of spontaneous rotation of a dielectric cylinder in a weakly conducting suspension in the presence of an electric field; compared with the case of single-phase media [2], this is characterized by a significant reduction in the threshold intensity of the electric field with increasing concentration of the particles [3]. In the present model of a dielectric suspension, the destabilization of the cylinder is due to the occurrence of rotations of the particles of the suspension due to the interaction between the polarization and the motion of the medium. The relaxation equation for the polarization for the given model is analogous to the corresponding equation for media which can be magnetized [4–6].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 86–93, March–April, 1980.     

Equations of motion and boundary conditions of physical meaning of micropolar theory of thin bodies with two small cuts

Nowadays, microcontinuous mechanics (mechanics of media with microstructure) is being developed very intensively, which is testified by recently published papers [1–14] and by many others, as well as by the symposiumdedicated to the hundredth anniversary of the brothers Cosserat monograph [15], held inParis in 2009. A survey of foreign papers is given in [16], and a special place is occupied by earlier publications of Soviet scientists on micropolar theory of elasticity [17–24]. A brief survey of Cosserat theory of elasticity and an analysis and prospects of such theories in mechanics of rigid deformable bodies is given in [21]. It should be noted that, in a majority of cases, the structure strength calculations are based on the classical theory of elasticity. But there are materials such as animal bones, graphite, several polymers, polyurethane films, porous materials (pumice), various synthetic materials, and materials with inclusions which, under certain conditions, exhibit micropolar properties. There are effects which cannot be prescribed by the classical theory. In statics, nonclassical behavior can be observed in bending of thin films and cantilevers, in torsion of thin and thin-walled rods, and in the case of stress concentration near holes, corner points, cracks, and inclusions. For example, thin specimens are more rigid in bending and torsion as is prescribed by the classical theory [25–27]. The stress concentration near holes decreases, and the concentration factor depends on the radius [28]. The stress concentration near cracks also becomes lower. Conversely, the stress concentration near inclusions is higher than predicted by the classical theory [29–31]. If the material has no center of symmetry of elastic properties, then calculations according to the micropolar theory shows that the specimen is twisted in tension [32]. In dynamical problems, several phenomena also differ from the classical concepts. For example, shear waves propagate with dispersion, microrotation waves arise, and the vibration natural modes differ from the classical ones [2, 7, 11–13, 33]. All these phenomena are used to determine material constants of the micropolar theory of elasticity. There are many methods for determining such constants [2, 34]. Since thin bodies (one-, two-, three-, and multilayer structures) are widely used, it is necessary to create new refined microcontinual theories of thin bodies and advanced methods for their computations. In the present paper, various representations of the system of equations of motion are obtained in the micropolar theory of thin bodies with two small parameters in momenta with respect to a system of Legendre polynomials in the case where an arbitrary line is taken for the base. In this connection, a vector parametric equation of the region of a thin body is given for the parametrization under study, different families of bases (frames) are introduced, and expressions for components of the unit tensor of rank two (UTRT) are obtained. Representations of gradient, tensor divergence, equations of motion, and boundary conditions for the considered parametrization are given. Definitions of (m, n)th-order moment of a variable with respect to an arbitrary system of orthogonal polynomials and a system of Legendre polynomials is given. Expressions for themoments of partial derivatives and several expressions with respect to a system of Legendre polynomials and boundary conditions in moments are obtained.     

Possibility of forming an inverted distribution over the rotational levels of nitrogen in an expanding gas stream

A number of theoretical papers have been devoted to an investigation of the relaxation kinetics of the population of a system of rotational levels of molecules in a stream of gas freely expanding from a sonic nozzle [1–3]. The complexity of the task of constructing models of relaxation and of collisions consistent in accuracy, however, as well as the difficulties in solving the resulting system of kinetic and gas-dynamic equations, lead to the necessity of using substantial approximations. Some disagreement between the experimental data and calculated results [1, 2] requires an evaluation of the accuracy of the various approximations used and further refinement of the theoretical models. In contrast to [1], in order to bring out the possible mutual influence of nonequilibrium energy exchange between the degrees of freedom of nitrogen molecules and the variation of the gas-dynamic parameters, the calculation presented below is based on a numerical solution of a self-consistent system of kinetic and gas-dynamic equations for the populations of rotational states and the temperature, density, and velocity of gas in the stream. Collisional probabilities of rotational transitions, calculated with allowance for the long-range part of the potential of the interaction between molecules [4], are used for this.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 9–16, May–June, 1986.     

Thermodynamically consistent coefficient calibration in nonlinear and anisotropic closure models for turbulence

At this first order level of closure, the scalars, turbulent kinetic energy and its dissipation rate, emerge as natural basic variables for the turbulence modeling in contrast to the considerations in Wang [8] and others. Received September 16, 1999     

Dynamics of biological soft tissue and rubber: internally pressurized spherical membranes surrounded by a fluid

The behavior of a family of dynamical systems representing the elastodynamic response of an internally pressurized, non-linearly elastic spherical membrane lying in an incompressible external fluid is governed primarily by the strain energy function for the membrane, the specific forcing function due to the internal pressure, and the viscosity of the external fluid. It is shown that such systems with an inviscid external fluid and having a constant internal pressure are integrable but not Hamiltonian. Under periodic internal loading, and for a small spherical radius and constitutive relations typical of many biological soft tissues, a periodic orbit in phase space exists near a static equilibrium. A viscous external fluid causes the periodic orbit to be an attractor. The dynamical system is robust under small loading perturbations common in normal biological systems. Rubber models, on the other hand, may admit structural catastrophes. For small initial sphere radii, a jump from one periodic orbit to another is possible for rubber models but not for the classical soft tissue models. It is dangerous, therefore, to model soft biological tissue as a rubber either mathematically or physically in experiments because the predicted instabilities may not exist in tissue.     

Plane incompressible fluid flow past an array of arbitrary profiles vibrating with arbitrary phase shift

We consider the problem of the vibration of an array of arbitrary profiles with arbitrary phase shift. Account is taken of the influence of the vortex wakes. The vibration amplitude is assumed to be small. The problem reduces to a system of two integral Fredholm equations of the second kind, which are solved on a digital computer. An example calculation is made for an array of arbitrary form.A large number of studies have considered unsteady flow past an array of profiles. Most authors either solve the problem for thin and slightly curved profiles or they consider the flow past arrays of thin curvilinear profiles [1].In [2] a study is made of the flow past an array of profiles of arbitrary form oscillating with arbitrary phase shift in the quasi-stationary formulation. The results are reduced to numerical values. Other approaches to the solution of the problem of unsteady flow past an array of profiles of finite thickness are presented in [3–5] (the absence of numerical calculations in [3, 4] makes it impossible to evaluate the effectiveness of these methods, while in [5] the calculation is made for a symmetric profile in the quasi-stationary formulation).     

Dynamical temperature and generalized heat-conduction equation

By means of a dynamical non-equilibrium temperature we derive a generalized heat-conduction equation which accounts for non-local, non-linear, and relaxation effects. The dynamical temperature is also capable to reproduce several enhanced heat equations recently proposed in literature. The heat flux is supposed to be proportional to the gradient of the dynamical temperature, and the material functions are allowed to depend on temperature. It is also pointed out that the heat flux cannot assume arbitrary values, but it is limited from above by a maximum value which ensures that the thermal conductivity remains positive.     

Rheological models with microstructural constraints

Rheological models of complex fluids with a physically restricted microstructure are analyzed to obtain general classes of dynamical evolution equations for these materials. These classes insure that the appropriate mathematical constraints, associated with each type of physical restriction, are consistently incorporated into the corresponding model development. Describing the microstructure of the complex fluid with a second-rank tensor variable, a general class of dynamical evolution equations is derived for three physically meaningful constraints associated with constancy of the invariants of this microstructural tensor. The physical rationale for each of these constraints is discussed, and a corresponding set of constrained dynamical evolution equations is derived in general terms. Abdellatif Ait-Kadi passed away suddenly during the course of this research. The surviving authors express their gratitude to Abdellatif for our many hours of productive work and companionship.     

Continuous symmetries and constitutive laws of dissipative materials within a thermodynamic framework of relaxation: Part I: Formal aspects

An analysis of the continuous symmetries of the constitutive laws of inelastic materials written within a thermodynamical framework of relaxation is performed. This framework relies on the generalization of Gibb’s relationship outside the equilibrium of a uniform system, and the use of the fluctuation theory to model the material dissipation due to its internal microstructure change [Cunat, C., 2001. The DNLR approach and relaxation phenomena. Part I – Historical account and DNLR formalism. Mech. Time-depend. Mater. 5, 39–65]. The approach leads to a viscoelastic like formulation for small deformations, and changes gradually for finite strains towards elastoviscoplasticity (with or without damage) via a dependence of characteristic times with the loading path, in a way similar to the endochronic approach developed by Valanis [Valanis, K.C., 1975. On the fundations of the endochronic theory of viscoplasticity. Arch. Mech. 27, 857–868]. The present thermodynamic framework has been previously applied to elastoviscoplastic materials under cyclic and non-proportional loadings [Dieng, L., Abdul-Latif, A., Haboussi, M., Cunat, C., 2005b. Cyclic plasticity modeling with the distribution of non-linear relaxations approach. Int. J. Plasticity 21, 353–379]. The constitutive laws split into the state laws relating intensive variables (thermodynamics forces) to extensive-like variables, and the complementary evolution laws of the internal variables associated to the dissipative mechanisms. An interpretation of a non-equilibrium thermodynamic approach of irreversible processes in terms of an extremum principle is proposed, associated to a Lagrangian functional. It is shown that one possible choice for the Lagrangian kernel is the material derivative of the internal energy density, augmented by a complementary term that accounts for the evolution laws of the internal variables. Interpreting the material behavior during the non-equilibrium evolution as the Euler–Lagrange equations of the resulting action integral, a differential condition expressing both the local and variational symmetries encapsulated into the Lagrangian formulation is formulated. It is further shown that both symmetry conditions are fully equivalent along the optimal path corresponding to the satisfaction of the constitutive laws. In terms of both practical and methodological aspects, the predictive nature of the symmetry analysis is highlighted, as a systematic tool for the exploitation of the constitutive response. Its performance and utility are exemplified by the construction of a time–temperature equivalence principle for a dry viscous polymer (PA66); the calculated shift factor is shown to well agree with the empirical shift factor given by Williams–Landel–Ferry (WLF) expression. A systematic interpretation of the calculated symmetry groups of the constitutive laws in terms of master curves for various plastic and viscoplastic materials shall be presented in a forthcoming contribution.     

Relaxation behavior and modeling

Load relaxation tests deliver several orders of magnitude of inelastic strain rate data while elastic strains are converted into inelastic strains [see Lemaitre and Chaboche, 1994. (Mechanics of Solid Materials, Oxford University Press, Cambridge p. 264)]. Hart used this test for providing information on the inelastic deformation behavior for modeling purposes. Characteristic relaxation curves were obtained with ductile metals and alloys at room and high temperature showing a scaling relation derived from Hart's theory. Subsequent testing with servo-controlled testing machines and strain measurement on the gage length showed that an increase of prior strain rate also increased the average relaxation rate. For relaxation tests starting in the flow stress region, the relaxation curves can be independent of the stress and strain at the start of the relaxation tests. For the modeling of these newly found relaxation behaviors and other phenomena the viscoplasticity theory based on overstress (VBO) has been introduced. It is shown that VBO admits a long-term (asymptotic) solution that can be used with sufficient accuracy for the flow stress region of the stress–strain diagram. The long-term solution predicts the observed relaxation behaviors and that the relaxation curves coincide when shifted along the stress axis. This behavior is observed for the recently obtained data and is confirmed by two sets of the Hart-type data when they are plotted according to the new method.