On beams made of a phase-transforming material

This paper presents a theory for the mechanics of beams made of single crystals of shape memory alloys. The behavior of such beams can be quite unexpected and complicated due to the presence and propagation of phase boundaries. It is shown that the usual laws of mechanics do not fully determine the propagation of phase boundaries and that there is a need for additional constitutive information in the form of a kinetic relation. A simple experiment to measure this kinetic relation is proposed. Finally, a strategy to use such beams for propulsion at small scales is presented.     

Multiple Equilibrium States of Slightly Extensible Elastic Strings

We consider the equilibrium states of elastic strings loaded transversely by their own weight. We are interested in the relation between the equilibrium states of inextensible strings and those of strings which are nonlinearly elastic and slightly extensible. The inextensible string has a unique concave equilibrium state while the concave equilibrium states of a nonlinearly elastic string come in pairs. In this paper we explore the relation between the two models. This revised version was published online in August 2006 with corrections to the Cover Date.     

A hybrid Eulerian–Lagrangian method to simulate the dispersed phase in turbulent gas-particle flows

This paper presents a methodology to combine stochastic Lagrangian approach and continuum model to simulate the dispersed phase in gas-particle turbulent flows using that both approaches are based on the same Boltzmann-like kinetic equation governing the joint fluid-particle probability density function (pdf). The proposed hybrid method is based on the separate application of each approach in two adjacent domains and their coupling at the interface via flux boundary conditions. Validation of the method is carried out for non-colliding solid particles suspended in homogeneous turbulent shear flow without two-way coupling.     

Modeling of dynamics, heat transfer, and combustion in two-phase turbulent flows: 1. Isothermal flows

This paper presents a review of authors' collective works in the field of two-phase flow modeling done in the past few decades. The paper is aimed at the construction of mathematical models for simulation of particle-laden turbulent flows. A kinetic equation was obtained for the probability density function (PDF) of the particle velocity distribution in turbulent flows. The proposed kinetic equation describes both the interaction of particles with turbulent eddies of the carrier phase and particle-particle collisions. This PDF equation is used for the derivation of different schemes describing turbulent momentum transfer in the dispersed particle phase. The turbulent characteristics of the gaseous phase are calculated on the basis of the k - turbulence model with a modulation effect of particles on the turbulence.

The constructed models have been applied to the calculation of various two-phase gas-particle turbulent flows in jets and channels as well as particle deposition in tubes and separators. For validating the theoretical and numerical results, a wide range of comparisons with experimental data from Russian and foreign sources has been done.     

The concept of material forces in phase transition problems within the level-set framework

The dynamics of a phase transition front in solids using the level set method is examined in this paper. Introducing an implicit representation of singular surfaces, a regularized version of the sharp interface model arises. The interface transforms into a thin transition layer of nonzero thickness where all quantities take inhomogeneous expressions within the body. It is proved that the existence of an inhomogeneous energy of the material predicts inhomogeneity forces that drive the singularity. The driving force is a material force entering the canonical momentum equation (pseudo-momentum) in a natural way. The evolution problem requires a kinetic relation that determines the velocity of the phase transition as a function of the driving force. Here, the kinetic relation is produced by invoking relations that can be considered as the regularized versions of the Rankine–Hugoniot jump conditions. The effectiveness of the method is illustrated in a shape memory alloy bar.     

Viscosity of Strain Gradient Effects on the Kinetics of Propagating Phase Boundaries in Solids

The theory of thermoelastic materials undergoing solid-solid phase transformations requires constitutive information that governs the evolution of a phase boundary. This is known as a kinetic relation which relates a driving traction to the speed of propagation of a phase boundary. The kinetic relation is prescribed in the theory from the onset. Here, though, a special kinetic relation is derived from an augmented theory that includes viscous, strain gradient and heat conduction effects. Based on a special class of solutions, namely travelling waves, the kinetic relation is inherited from the augmented theory as the viscosity, strain gradient and heat conductivity are removed by a suitable limit process.     

Sudden tensile loading of a rubberlike bar

In uniaxial tension, the stress–strain curve for rubber changes curvature from concave to convex as the strain increases. For sudden tensile loading of a bar, a one-dimensional model that reflects this behavior leads to an under-determined problem reminiscent of that arising in materials capable of undergoing phase transitions. In the latter setting, adding the kinetic relation underlying the phase change to the conventional statement of the problem removes the indeterminacy; the same is true when such a relation is used in a formal way in the problem for rubber. This presents a physical question: What is the evolutionary process at the microscale whose kinetics are needed in the dynamics of rubber?     

Non-isothermal kinetics of a moving phase boundary

In this paper we derive an explicit formula for a kinetic relation governing the motion of a phase boundary in a bilinear thermoelastic material capable of undergoing solid-solid phase transitions. To obtain the relation, we study traveling wave solutions of a regularized problem that includes viscosity, heat conduction and convective heat exchange with an ambient medium. Both inertia and latent heat of transformation are taken into account. We investigate the effect of material parameters on the kinetic relation and show that in a certain range of parameters the driving force becomes a non-monotone function of the interface velocity. The model also predicts a nonzero resistance to phase boundary motion, part of which is caused by the thermal trapping. Received: November 15, 2001 / Published online September 4, 2002 RID="*" ID="*" e-mail: annav@math.pitt.edu Communicated by Lev Truskinovsky, Minneapolis     

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.     

Dynamics of propagating phase boundaries: Thermoelastic solids with heat conduction

This paper is concerned with the incorporation of thermal effects into the continuum modeling of dynamic solid-solid phase transitions. The medium is modeled as a one-dimensional thermoelastic solid characterized by a specific Helmholtz free-energy potential and a specific kinetic relation. Heat conduction and inertia are taken into account. An initial-value problem that gives rise to both shock waves and a propagating phase boundary is analyzed on the basis of this model.     

An atomistic investigation of the kinetics of detwinning

This paper presents a “first principles” atomistic study of the dynamics of detwinning in a shape-memory alloy. In order to describe the macroscopic motion of twin boundaries, the continuum theory of twinning must be provided with a “kinetic relation”, i.e. a relation between the driving force and the propagation speed. This kinetic relation is a macroscopic characterization of the underlying atomistic processes. The goal of the present atomistic study is to provide the continuum theory with this kinetic relation by extracting the essential macroscopic features of the dynamics of the atoms. It also aims to elucidate the mechanism underlying the process of detwinning.The material studied is stoichiometric nickel-manganese, and interatomic interactions are described using three physically motivated Lennard-Jones potentials. The effect of temperature and shear stress on detwinning — specifically on the rate of transformation from one variant of martensite to the other — is examined using molecular dynamics. An explicit formula for this (kinetic) relation is obtained by fitting an analytic expression to the simulation results. The numerical experiments also verify that transverse ledge propagation is the mechanism underlying twin-boundary motion. All calculations are carried out in a two-dimensional setting.     

On kinetics of hysteresis

An initiation criterion and a kinetic relation describing hysteresis in shape memory alloys consistent with non-equilibrium thermodynamics are proposed. The initiation of phase transition begins always at zero driving force, defined as the negative derivative of free energy with respect to the volume fraction of high strain phase. The kinetic relation is obtained from the dissipation potential which is proportional to the magnitude of the volume fraction rate times the newly formed volume fraction. The proposed theory turns out to be rate-independent, but history dependent, and can describe all features of hysteresis loops observed in experiments.      

Propagating phase boundaries: Formulation of the problem and existence via the Glimm method

This paper treats the hyperbolic-elliptic system of two conservation laws which describes the dynamics of an elastic material having a non-monotone strain-stress function. FollowingAbeyaratne &Knowles, we propose a notion of admissible weak solution for this system in the class of functions of bounded variation. The formulation includes an entropy inequality, a kinetic relation (imposed along any subsonic phase boundary) and an initiation criterion (for the appearance of new phase boundaries). We prove theL ¹-continuous dependence of the solution to the Riemann problem. Our main result yields the existence and the stability of propagating phase boundaries. The proofs are based onGlimm's scheme and in particular on the techniques ofGlimm andLax. In order to deal with the kinetic relation, we prove a result of pointwise convergence of the phase boundary.     

Shocks versus kinks in a discrete model of displacive phase transitions

We consider dynamics of phase boundaries in a bistable one-dimensional lattice with harmonic long-range interactions. Using Fourier transform and Wiener–Hopf technique, we construct traveling wave solutions that represent both subsonic phase boundaries (kinks) and intersonic ones (shocks). We derive the kinetic relation for kinks that provides a needed closure for the continuum theory. We show that the different structure of the roots of the dispersion relation in the case of shocks introduces an additional free parameter in these solutions, which thus do not require a kinetic relation on the macroscopic level. The case of ferromagnetic second-neighbor interactions is analyzed in detail. We show that the model parameters have a significant effect on the existence, structure, and stability of the traveling waves, as well as their behavior near the sonic limit.     

Thermo-elastic aspects of dynamic nucleation

In spite of recent progress in our understanding of the absolute stability of elastic phases under loads, the generic presence of metastable configurations and the possibility of their dynamic breakdown remains a major problem in the mechanical theory of phase transitions in solids. In this paper, by considering the simplest one-dimensional model, we study the interplay between inertial and thermal effects associated with nucleation of a new phase, and address the crucial question concerning the size of a perturbation breaking metastability. We begin by reformulating the nucleation problem as a degenerate Riemann problem. By choosing a specific kinetic relation, originating from thermo-visco-capillary (TVC) regularization, we solve a self-similar problem analytically and demonstrate the existence of two types of solutions: with nucleation and without it. We then show that in the presence of a non-zero latent heat, solution with nucleation may by itself be non-unique. To understand the domain of attraction of different self-similar solutions with and without nucleation, we regularize the model and study numerically the full scale initial value problem with locally perturbed data. Through numerical experiments we present evidence that the TVC regularization is successful in removing deficiencies of the classical thermo-elastic model and is sufficient in specifying the limits of metastability.     

Void fraction and pressure drop during external upward two-phase cross flow in tube bundles – part II: Predictive methods

The present paper is the Part II of a broad study concerning void fraction and pressure drop for air-water upward external flow across tube bundles. In the Part I, the experimental facility and the data regression procedures were described and the experimental results are presented and discussed. Initially, Part II presents a literature review concerning void fraction and pressure drop predictive methods available in the open literature for two-phase upward flow across tube bundles. Next, the methods from literature are compared among them and with the database presented in paper Part I. Significant discrepancies are observed among the predictive methods, and deviations as high as two orders of magnitude are verified among the predicted values of pressure drop. Then, a new void fraction predictive method is proposed based on the experimental results and on the minimum kinetic energy principle. This method provides satisfactory predictions of the results described in paper Part I and also of independent data from the literature. A new predictive method for frictional pressure drop during two-phase flow based on two-phase multiplier is also proposed. This method predicted 94% of the experimental data obtained in the present study within an error margin of ± 30%, and also provides accurate predictions of independent results for triangular tube bundles gathered in the open literature.     

一般动力学系统的精确不变量和绝热不变量

在增广相空间中研究一般动力学系统的精确不变量与绝热不变量，建立了该空间中力学系统的对称性与不变量之间的关系，提出了一般动力学系统的高阶绝热不变量的概念及其构造方法，揭示了高阶绝热不变量与无穷小对称变换之间的正反关系，讨论了ＶａｎｄｅｒＰｏｌ方程和Ｄｕｆｆｉｎｇ－ＶａｎｄｅｒＰｏｌ方程的一阶绝热不变量与相应的无穷小对称变换．     

Energy losses of ice slurry in pipe sudden contractions

The paper presents the results of the experimental research on the ice slurry loss coefficient during its flow through sudden contractions. Experimental studies were conducted using a few of the most common contractions of copper pipes. Six contraction ratios were covered: 0.500, 0.615, 0.650, 0.769, 0.800 and 0.813. In the experimental research, the mass fraction of solid particles in the slurry ranged from 20 to 30 (20%, 25%, 30%). Research results allow for determining the theoretical correlation for the ice slurry, in order to calculate the loss coefficient in contractions during laminar flow. The results of experimental investigations also confirmed that the loss coefficients in contractions in turbulent (transitional) flow of the ice slurry are the same as in the case of Newtonian liquids. The paper also presents original theoretical correlation for calculating the kinetic energy correction factor in the laminar range flow of liquid in the Bingham model. This correlation is used in this work for calculation of the ice slurry flow through contractions.     

Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.