Nonequilibrium thermodynamics of pseudoelasticity

Solid-solid phase transitions often exhibit hystereses, and a hysteresis indicates energy dissipation. Pseudoelasticity refers to a hysteretic loadingunloading characteristic observed in the stress-induced martensitic transformation of shape memory alloys.This paper describes the thermodynamic model ofideal pseudoelasticity, a largely schematized adaptation of the experimental observations, and it reviews the works of other authors on thermodynamics of pseudoelasticity. Different approaches vary widely and we have chosen to put them into perspective by contrasting their assumptions and predictions against those of ideal pseudoelasticity.Ideal pseudoelasticity receives support from the experimental results of Fu [1] and its thermodynamic properties have been exploited by Huo [2]. The model makes use of an analytical ansatz proposed by Müller [3] in which the hysteresis is assumed to be due to the presence of a coherency energy in solid phase mixtures. This model permits the study of stability of the equilibrium states and the calculation of the energy dissipation or entropy production during the phase transition: The equilibrium states of a phase mixture are found to be unstable in load-controlled processes and the dissipated energy is related to the coherency coefficient.We also discuss some open problems concerning the states inside the hysteresis loop and the formation of interfaces.     

Phase equilibrium in non-fluids and non-fluid mixtures

A criterion of phase equilibrium for mixtures of materials with arbitrary symmetry (e.g. between solid and fluid or two solid mixture phases) is deduced using a rational thermodynamics approach. This criterion, known also as Maxwell relation, is expressed via the difference of chemical potential tensors (Eshelby tensors) on the singular surface dividing the bulk phases.The thermomechanical balance equations, the entropy inequality and the Maxwell relation for phase equilibrium are given first for the case of pure (one-constituent) materials of arbitrary symmetry and then for the case of mixtures (including chemically reacting ones) of arbitrary symmetry.In the special case of fluids it is shown that the chemical potential tensors reduce to the classical scalar chemical potentials and the Maxwell relations to the classical thermodynamic criterion for the phase equilibrium.     

Equilibrium thermodynamics of pseudoelasticity and quasiplasticity

Motivated by recent experimental results by Glasauer [1], a thermodynamic theory of shape memory alloys is proposed, which includes not only the high temperature – pseudoelastic – behavior but also the low temperature range of quasiplasticity. Due to the occurance of three different phases – austenite and two martensitic variants – several cases of two-phase equilibria and a three-phase equilibrium have to be taken into account. Their relevance is determined by minimization of the total free energy and subsequently illustrated by the construction of phase charts. A special point of interest is the influence of interfacial energy effects on these phase charts, resulting in phenomena like, for example, the apparent violation of Gibbs' phase rule. Furthermore, the role of interfacial energies in the hysteretic load-displacement behavior is discussed in the light of the additional quasiplastic case. Received June 12, 1996     

On the size of the hysteresis in pseudoelasticity

Pseudoelasticity is a phenomenon that occurs in alloys with shape memory: In a loading-unloading cycle a body will return to its original configuration, but its path in a load-deformation diagram will run through a hysteresis loop.Pseudoelastic behaviour can be modelled by statistical mechanics which produces a non-monotone load-deformation isotherm. Once such a nonmonotone curve has been calculated there is the possibility of a phase transition that is connected with a hysteresis loop.The paper describes a method for the calculation of the width of the hysteresis. It comes to the conclusion that this width is determined by the interfacial energy associated with domain boundaries between the phases. The bigger that energy is the bigger is the hysteresis loop.     

INTERNAL VARIABLES AND THERMODYNAMIC MODELLING OF PSEUDOELASTICITY

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Study of two-phase bubbly flow dynamics in binary mixtures

The results of an investigation of the bubble dynamics and wave phenomena in two-component vapor-liquid mixtures are presented. These mixtures are widely used in industrial systems as heat transfer media. The effects of various additives on the wave dynamics of vapor-liquid mixtures are of particular interest. A single-velocity two-pressure model was used which takes into account both the liquid radial inertia due to medium volume changes, and the temperature distribution around the bubbles. The claim that mixture composition may have a peculiar effect on the bubble dynamics of a boiling non-ideal solution is substantiated. It is noted that the small free radial oscillation damping ratio for some binary systems lies outside the domain defined by the damping ratio of the constituents as a result of phase change diffusion effects. A criterion is proposed to identify cases of diffusion resistance responsible for the anomalous effect of component concentration on bubble behavior. The phase change delay due to diffusion results in observably higher mixture wave velocities and a smaller damping ratio than for respective single-component systems.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1992 and the AMS fonts, developed by the American Mathematical Society.     

On the spatial behaviour in dynamics of elastic mixtures

This paper studies the spatial behaviour of the dynamic processes in elastic mixtures. With this aim we use the time-weighted surface power function method associated with the linear processes. Consequently, we obtain a precise determination of the domain of influence and the spatial decay estimates with time-independent decay rate inside of the domain of influence. A uniqueness theorem is proved for finite and infinite bodies and we note that it is free of any kind of a priori assumptions on the solutions at infinity.     

A new form of the coherency energy in pseudoelasticity

In some previous papers [1], [2] pseudoelasticity in tensile experiments has been treated thermodynamically under the assumption that the relevant constitutive ingredients are

Continuum characterization of novel pseudoelasticity of ZnO nanowires

A novel pseudoelastic behavior was recently discovered in [0 1 1¯ 0]-oriented ZnO nanowires under uniaxial tensile loading and unloading. This behavior results from a reversible transformation from the parent wurtzite (WZ) structure to a previously unknown graphitic structure (HX) and is associated with recoverable strains up to 16%. In this paper, a micromechanical continuum model is developed to characterize this behavior. Using the first law of thermodynamics, the model decomposes the transformation into an elastic process of structural transitions between WZ and HX through a sequence of thermodynamically reversible phase equilibrium states and a thermodynamically irreversible process of interface propagation. The elastic equilibrium transition process is modeled with strain energy functions of the two constituent phases which are obtained from independent molecular dynamics calculations. The dissipative interface propagation process is modeled phenomenologically with a function which relates dissipation to the interfacial area between the two phases. The model captures major characteristics of the behavior of wires with lateral dimensions between 20 and 40 Å over the temperature range of 100-500 K.     

Thermomechanics of transformation pseudoelasticity and shape memory effect in alloys

Transformation pseudoelasticity and shape memory effect of alloy materials are investigated from the thermomechanical point of view. The thermomechanical constitutive equations and the kinetics of transformation established by the theory are applied to explain the stress-strain-temperature behavior of the material. Numerical illustrations for the uniaxial stress state are given.     

Rate dependence of temperature fields and energy dissipations in non-static pseudoelasticity

The temperature fields and the energy dissipations of shape memory alloys during the stress-induced martensitic transformations are studied theoretically and experimentally. The effect of the loading rate is analyzed. It was found that the temperature field inside a shape memory alloy sample varies strongly in space and time. The increase rate of the temperature is given by the difference between the rate of the latent heat release and the rate of the heat convection and conduction. The notion and the rate dependence of the energy dissipation are discussed in connection with the stress–strain hysteresis, the entropy production, and the Clausius–Duhem inequality.     

Three-dimensional phenomenological thermodynamic model of pseudoelasticity of shape memory alloys at finite strains

The familiar small strain thermodynamic 3D theory of isotropic pseudoelasticity proposed by Raniecki and Lexcellent is generalized to account for geometrical effects. The Mandel concept of mobile isoclinic, natural reference configurations is used in order to accomplish multiplicative decomposition of total deformation gradient into elastic and phase transformation (p.t.) parts, and resulting from it the additive decomposition of Eulerian strain rate tensor. The hypoelastic rate relations of elasticity involving elastic strain rate are derived consistent with hyperelastic relations resulting from free energy potential. It is shown that use of Jaumann corotational rate of stress tensor in rate constitutive equations formulation proves to be convenient. The formal equation for p.t. strain rate , describing p.t. deformation effects is proposed, based on experimental evidence. Phase transformation kinetics relations are presented in objective form. The field, coupled problem of thermomechanics is specified in rate weak form (rate principle of virtual work, and rate principle of heat transport). It is shown how information on the material behavior and motion inseparably enters the rate virtual work principle through the familiar bridging equation involving Eulerian rate of nominal stress tensor.

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Phase transition and dynamics of iron under ramp wave compression

The ramp wave compression experiments of iron with different thicknesses were performed on the magnetically driven ramp loading device CQ-4. Numerical simulations of this process were done with Hayes multi-phase equation of state (H-MEOS) and dynamic equations of phase transition. The calculated results of H-MEOS are in good agreement with those of shock phase transition, but are different from those under ramp wave compression. The reason for this is that the bulk modulus of the material in the Hayes model and the wave velocity are considered constant. Shock compression is a jump from the initial state to the final state, and the sound speed is related to the slope of the Rayleigh line. However, ramp compression is a continuous process, and the bulk modulus is no longer a constant but a function of pressure and temperature. Based on Murnaghan equation of state, the first-order correction of the bulk modulus on pressure in the Hayes model was carried out. The numerical results of the corrected H-MEOS agree well with those of pure iron in both ramp and shock compression phase transition experiments. The calculated results show that the relaxation time of iron is about 30 ns and the phase transition pressure is about 13 GPa. There are obvious differences between the isentropic and adiabatic process in terms of pressure–specific volume and temperature–pressure. The fluctuation of the sound speed after 13 GPa is caused by the phase transition.     

Phase synchronization dynamics of coupled neurons with coupling phase in the electromagnetic field

Nonlinear Dynamics - Based on the law of electromagnetic theory, phase synchronization of coupled extended Hindmarsh–Rose neurons with magnetic and electrical couplings is discussed. It is...     

Boundary-value problems on the dynamics of the theory of elastic mixtures for a disk

We consider the first and second dynamic boundary-value problems in the theory of elastic mixtures. These problems are reduced to the corresponding problems for systems of equations for pseudooscillation by Laplace transformation relative to time. The solutions are represented in terms of four metaharmonic functions. It is proved that the problem of pseudooscillation has a unique solution. Conditions are given for existence of inverse transformations that provide solutions for the initial problem. Translated from Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 86–92, December, 1998.     

Interfacial energy and dissipation in martensitic phase transformations. Part II: Size effects in pseudoelasticity

This paper is a continuation of the Part I (H. Petryk, S. Stupkiewicz, Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory. J. Mech. Phys. Solids, 2010, doi:10.1016/j.jmps.2009.11.003). A fully three-dimensional model of an evolving martensitic microstructure is examined, taking into account size effects due to the interfacial energy and also dissipation related to annihilation of interfaces. The elastic micro-strain energy at microstructured interfaces is determined with the help of finite element computations and is approximated analytically. Three interface levels are examined: of grain boundaries attained by parallel martensite plates, of interfaces between austenite and twinned martensite, and of twin interfaces within the martensite phase. Minimization of the incremental energy supply, being the sum of the increments in the free energy and dissipation of the bulk and interfacial type at all levels, is used as the evolution rule, based on the theory presented in Part I. An example of the formation and evolution of a rank-three laminated microstructure of finite characteristic dimensions in a pseudoelastic CuAlNi shape memory alloy is examined quantitatively.     

A 3-D constitutive model for shape memory alloys incorporating pseudoelasticity and detwinning of self-accommodated martensite

A 3-D constitutive model for polycrystalline shape memory alloys (SMAs), based on a modified phase transformation diagram, is presented. The model takes into account both direct conversion of austenite into detwinned martensite as well as the detwinning of self-accommodated martensite. This model is suitable for performing numerical simulations on SMA materials undergoing complex thermomechanical loading paths in stress–temperature space. The model is based on thermodynamic potentials and utilizes three internal variables to predict the phase transformation and detwinning of martensite in polycrystalline SMAs. Complementing the theoretical developments, experimental data are presented showing that the phase transformation temperatures for the self-accommodated martensite to austenite and detwinned martensite to austenite transformations are different. Determination of some of the SMA material parameters from such experimental data is also discussed. The paper concludes with several numerical examples of boundary value problems with complex thermomechanical loading paths which demonstrate the capabilities of the model.     

DDT in methane-air mixtures

Experimental results from a study on the critical condition for deflagration-to-detonation transition (DDT) in methane-air mixtures are presented. Experiments were carried out at 293 K and 1 atm using methane-air mixtures with methane concentrations ranging from 5.5 to 17% vol. The tests were performed in detonation tubes with inner-diameters of 174 mm and 520 mm. Detonation cell widths l\lambda were determined in the tests for a range of methane concentrations. The results of DDT tests indicate that for a tube cross-sectional area blockage ratio (BR) of 0.3 the critical condition for DDT can be characterized by the d/l = 1d/\lambda = 1 criterion. However, for a BR = 0.6 the critical value d/ld/\lambda was significantly higher. The data also show that the critical condition for DDT can be described by L/l = 7L/\lambda = 7, where L is the characteristic length-scale of the channel volume between orifice plates. This length-scale is defined by a grouping of the orifice plate dimensions (inner and outer-diameter) and plate spacing.