稀薄气体流动非线性耦合本构关系研究进展

临近空间高超声速飞行器流场蕴含着复杂的非线性流动机理与丰富的热化学非平衡流动现象, 基于Newton摩擦定律和Fourier热传导定律的Navier-Stokes(N-S)方程不足以描述高超声速飞行器从连续流到稀薄流的多尺度非平衡现象。非线性耦合本构关系(nonlinear coupled constitutive relations, NCCR)作为一种全新的本构方程体系, 在严格满足热力学熵条件的基础上, 巧妙地构建了应力与热流的非线性表达形式。然而, NCCR方程的强非线性耦合特性是求解过程的一大难题。为了克服这一技术瓶颈, 提出了混合迭代算法, 为实现NCCR方程的高效稳定求解提供了坚实的理论基础。在该理论研究的基础上, 考虑到原始NCCR方程对热通量演化方程的简化处理, 降低了方程的计算精度, 提出了改进的NCCR+方程。该方程在强激波压缩区域和膨胀区域表现出比传统NCCR方程更高的计算精度与更强的非平衡流动模拟能力。同时, 为了解决临近空间高超声速空气动力学的多尺度与多物理效应耦合难题, 提出了NCCR与转动非平衡的耦合计算模型, 拓展了NCCR方程在双原子气体中的模拟能力。为了揭示稀薄气体效应与真实气体效应的耦合作用机理, 进一步建立了NCCR与热化学反应的耦合计算方法。大量研究结果表明, 考虑多物理效应的NCCR方程在低Kn下能够恢复到与N-S方程一致的解。随着Kn的增加, 流场的非平衡程度逐渐增强, 其结果与N-S方程差异显著, 而与DSMC方法计算结果和实验数据具有更好的一致性。      

On the invariance of constitutive equations according to the kinetic theory of gases

Iterative techniques for solving the Boltzmann equation in the kinetic theory of gases yield expressions for the stress tensor and heat flux vector that are analogous to constitutive equations in continuum mechanics. However, these expressions are not generally invariant under the Euclidean group of transformations, whereas constitutive equations in continuum mechanics are usually required to be by the principle of material frame indifference. This disparity in invariance properties has led some previous investigators to argue that Euclidean invariance should be discarded as a contraint on constitutive equations. It is proven mathematically in this paper that the results of the Chapman-Enskog iterative procedure have no direct bearing on this issue. In order to settle this question, it is necessary to examine mathematically the effect of superimposed rigid body rotations on solutions of the Boltzmann equation. A preliminary investigation along these lines is presented which suggests that the kinetic theory is consistent with material frame indifference in at least a strong approximate sense provided that the disparity in the time scales of the microscopic and macroscopic motions is extremely large—a condition which is usually a prerequisite for the existence of constitutive equations.On leave from Stevens Institute of Technology, Hoboken, New Jersey 07030.     

炸药爆轰的连续介质本构模型和数值计算方法

假定炸药和爆轰产物处于局部热力学平衡状态, 即它们的压力和温度相同, 利用热力学基本关系建立炸药爆轰过程的连续介质本构模型的一般理论框架. 在此框架下, 炸药爆轰本构模型由一组常微分方程构成, 包括炸药和爆轰产物的状态方程、简单混合法则、化学反应速率方程和能量守恒方程, 易于由成熟的计算方法如梯形法等进行求解. 一组广义Maxwell型非线性固体本构形式的微分方程描述了压力和温度随时间的演化速率与应变率和化学反应速率的关系, 借助简单混合物理论, 其中的系数由炸药和爆轰产物的材料参数确定. 未反应的炸药和爆轰产物采用JWL状态方程, 化学反应率方程采用Lee-Tarver点火-燃烧二项式模型, 模拟PBX-9404炸药的一维冲击波起爆过程和爆轰波传播过程. 计算结果表明了本文给出的本构模型和相应计算方法的有效性. 关键词： 炸药爆轰 本构模型 化学反应率方程 数值模拟     

Jerky flow model as a basis for research in deformation instabilities

A phenomenological jerky flow model was developed in which macroscale plastic strain rates are defined by dislocation kinetics. The model takes into account destructive processes governed by shear and bulk defect accumulation. At the heart of the model lie equations of solid mechanics and relaxation-type constitutive equations. A loaded elastoplastic solid is treated as a nonlinear dynamic system whose evolution, according to synergetic laws, is much contributed by negative and positive feedbacks expressed, respectively, through constitutive equations of the first group (relaxation equations) and constitutive equations of the second group (kinetic equations for deformation defect and damage accumulation rates). The negative feedback stabilizes deformation by relaxation, bringing the process to some local dynamic equilibrium. The positive feedback destabilizes deformation, driving the system to a critical state. Numerical experiment was performed in 2D and 3D statements. Statistical analysis of stress fluctuations about the average trend shows that the jerky flow model of an elastoplastic medium demonstrates evolution characteristic of nonlinear dynamic systems: through states of dynamic chaos and self-organized criticality to a global catastrophe.     

Hybrid Particle–Continuum Simulations of Polymer Brushes in Shear Flow

A hybrid particle–continuum method is used to study the shear flow confined between two opposing walls, one of which is coated with polymer chains. Molecular dynamics (MD) is used in the particle region near the brush and Navier–Stokes (NS) equations are applied in the remaining region where the continuum assumption holds. The information exchange from the continuum region to the particle region is implemented using the constrained particle dynamics. Both Couette shear flow and oscillatory flow are considered in the present work. The effect of the shear flow on the conformational characteristics of polymer brushes is analyzed. In the overlap region, the velocities obtained from MD simulations are smoothly connected with those from NS equations. Our investigations demonstrate that the hybrid particle–continuum model is valid in exploring the shear behavior of polymer brushes.     

On the kinetic theory of vehicular traffic flow: Chapman–Enskog expansion versus Grad’s moment method

Based on a Boltzmann-like traffic equation for aggressive drivers we construct in this paper a second-order continuum traffic model which is similar to the Navier–Stokes equations for viscous fluids by applying two well-known methods of gas-kinetic theory, namely the Chapman–Enskog method and the method of moments of Grad. The viscosity coefficient appearing in our macroscopic traffic model is not introduced in an ad hoc way–as in other second-order traffic flow models–but comes into play through the derivation of a first-order constitutive relation for the traffic pressure. Numerical simulation shows that our Navier–Stokes-like traffic model satisfies the anisotropy condition and produces numerical results which are consistent with our daily experiences in real traffic.     

Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity

Stress-strain relation in Eringen's nonlocal elasticity theory was originally formulated within the framework of an integral model. Due to difficulty of working with that integral model, the differential model of nonlocal constitutive equation is widely used for nanostructures. However, paradoxical results may be obtained by the differential model for some boundary and loading conditions. Presented in this article is a finite element analysis of Timoshenko nano-beams based on the integral model of nonlocal continuum theory without employing any simplification in the model. The entire procedure of deriving equations of motion is carried out in the matrix form of representation, and hence, they can be easily used in the finite element analysis. For comparison purpose, the differential counterparts of equations are also derived. To study the outcome of analysis based on the integral and differential models, some case studies are presented in which the influences of boundary conditions, nonlocal length scale parameter and loading factor are analyzed. It is concluded that, in contrast to the differential model, there is no paradox in the numerical results of developed integral model of nonlocal continuum theory for different situations of problem characteristics. So, resolving the mentioned paradoxes by means of a purely numerical approach based on the original integral form of nonlocal elasticity theory is the major contribution of present study.     

A Comparison and Unification of Ellipsoidal Statistical and Shakhov BGK Models

The Ellipsoidal Statistical model (ES-model) and the Shakhov model (S-model) were constructed to correct the Prandtl number of the original BGK model through the modification of stress and heat flux. With the introduction of a new parameter to combine the ES-model and S-model, a generalized kinetic model can be developed. This new model can give the correct Navier-Stokes equations in the continuum flow regime. Through the adjustment of the new parameter, it provides abundant dynamic effect beyond the ES-model and S-model. Changing the free parameter, the physical performance of the new model has been tested numerically. The unified gas kinetic scheme (UGKS) is employed for the study of the new model. In transition flow regime, many physical problems, i.e., the shock structure and micro-flows, have been studied using the generalized model. With a careful choice of the free parameter, good results can be achieved for most test cases. Due to the property of the Boltzmann collision integral, the new parameter in the generalized kinetic model cannot be fully determined. It depends on the specific problem. Generally speaking, the S-model predicts more accurate numerical solutions in most test cases presented in this paper than the ES-model, while ES-model performs better in the cases where the flow is mostly driven by temperature gradient, such as a channel flow with large boundary temperature variation at high Knudsen number.     

A Brief Review of Elasticity and Viscoelasticity for Solids

There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.     

Research on nonlinear constitutive relationship of permanent deformation in asphalt pavements

To predict correctly the rut depths in asphalt pavements, a new nonlinear viscoelastic-elastoplastic constitutive model of permanent deformation in asphalt pavements is presented. The model combines a generalized Maxwell model with an elastoplastic one. Then from the creep theory, the linear and nonlinear constitutive equations of the generalized Maxwell model are obtained. From the nonlinear finite element method for the rutting of the asphalt pavement, the rut depths of 4 asphalt-aggregate mixtures are obtained. And the results are compared with the ones from the finite element method by SHRP and the experiments by SWK/UN. The results in this paper are better than the ones by SHRP, and agree with the ones of the experiment by SWK/UN. This shows that the nonlinear viscoelastic-elastoplastic constitutive model, which is presented in this paper for the rutting of the asphalt pavement, is effective. The properties, such as nonlinear elasticity, plasticity, viscoelasticity and nonlinear viscoelasticity, which affect the rutting of an asphalt pavement, can be shown in the model. And the characteristics of the permanent deformation of the asphalt pavement can be presented entirely in the model.     

Stabilization of the Eulerian model for incompressible multiphase flow by artificial diffusion

The commonly used Eulerian or continuum model for incompressible multiphase flow is known to be unstable to perturbations for all wavenumbers, even if viscosity terms are used in the momentum equations. In the present work the model is stabilized by adding explicit artificial diffusion to the mass equations. The artificial diffusion terms lead to improved stability properties: uniform flow becomes linearly stable for large wavenumbers, and above an analytically derived threshold for the artificial diffusivity, stability for all wavenumbers is achieved. The artificial diffusivity reappears in the momentum equations, in such a way that fundamental properties of the standard equations remain valid: Galilean invariance is maintained, total mass and momentum are conserved, decay of total kinetic energy is ensured in the absence of external forces, and a flow initially at rest at hydrostatic pressure remains unchanged, even if the spatial distribution of volume fractions is nonuniform. A staggered finite volume pressure correction method using central differencing (leading to energy conserving discretization of convective and pressure terms) is presented. Application of the method to one-dimensional two-phase flow of falling particles particles confirms that the equations are stable with and unstable without artificial diffusion in the volume fraction equation.     

A multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks

We present a multiscale model for numerical simulations of dynamics of crystalline solids. The method combines the continuum nonlinear elasto-dynamics model, which models the stress waves and physical loading conditions, and molecular dynamics model, which provides the nonlinear constitutive relation and resolves the atomic structures near local defects. The coupling of the two models is achieved based on a general framework for multiscale modeling – the heterogeneous multiscale method (HMM). We derive an explicit coupling condition at the atomistic/continuum interface. Application to the dynamics of brittle cracks under various loading conditions is presented as test examples.     

Nonlinear mechanical response of supercooled melts under applied forces

We review recent progress on a microscopic theoretical approach to describe the nonlinear response of glass-forming colloidal dispersions under strong external forcing leading to homogeneous and inhomogeneous flow. Using mode-coupling theory (MCT), constitutive equations for the rheology of viscoelastic shear-thinning fluids are obtained. These are, in suitably simplified form, employed in continuum fluid dynamics, solved by a hybrid-Lattice Boltzmann (LB) algorithm that was developed to deal with long-lasting memory effects. The combined microscopic theoretical and mesoscopic numerical approach captures a number of phenomena far from equilibrium, including the yielding of metastable states, process-dependent mechanical properties, and inhomogeneous pressure-driven channel flow.     

Analytical description of ultrasonic hardening and softening

In the presented work, a modeling of ultrasonic hardening and softening has been carried out. The analytical model is constructed by the generalization of the synthetic theory of plastic deformation. A new term, ultrasonic defect intensity, is being introduced so that the phenomenon of both hardening and softening is described by the uniform system of constitutive equations.     

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang {\it et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg--de-Vries (KdV) equation, and the soliton solution is derived.     

On variational symmetry of defect potentials and multiscale configurational force

In this work, we study invariant properties of defect potentials that are capable of describing defect motions in a continuum. By formulating two canonical defect theories, a generalized Nye theory and the Kröner–de Wit theory, we have found three defect potentials that are variational, i.e. their associated Euler–Lagrange equations are differential compatibility conditions of the continuum and defects. Consequently, symmetry properties of these variational functionals render several classes of new conservation laws and invariant integrals that are related with continuum compatibility conditions, which are independent of the constitutive relations of the continuum. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by such an integral is specifically defect-free. The invariant integral is applied to study macroscopically brittle fracture, and a multiscale Griffith criterion is proposed, which leads to a rigorous justification of the well-known Griffith–Irwin theory.     

Hamaker微观连续介质理论的数字密度和Hamaker常数分析

针对Hamaker微观连续介质理论在微观接触力计算中存在的问题,根据Hamaker假设,用连续介质法计算2个原子之间的相互作用力,发现作用力同经典的Lennard-Jones势所反映的作用力不一致.通过分析数字密度,发现数字密度并非是如Hamaker所认为的常数,而是随间距变化的;并得到Hamaker微观连续介质理论仅在间距大于7倍的原子半径时才成立的结论.通过分析Hamaker常数,发现Hamaker常数也随间距变化.     

The density wave in a new anisotropic continuum model

In this paper the new continuum traffic flow model proposed by Jiang et al is developed based on an improved car-following model, in which the speed gradient term replaces the density gradient term in the equation of motion. It overcomes the wrong-way travel which exists in many high-order continuum models. Based on the continuum version of car-following model, the condition for stable traffic flow is derived. Nonlinear analysis shows that the density fluctuation in traffic flow induces a variety of density waves. Near the onset of instability, a small disturbance could lead to solitons determined by the Korteweg-de-Vries （KdV） equation, and the soliton solution is derived.