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.     

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.      

Phase mixtures in dynamics of pseudoelasticity

We give a numerical treatment of phase mixtures in pseudoelasticity from a purely mathematical point of view. It is based on a surprising result that the approximate solution may consist of persistent oscillations in strain which resemble the experimentally observed interface patterns. Such a solution is obtained from a sequence of solutions for a rate-type viscoelastic problem with a non-monotone equilibrium stress-strain relation, for which in the limit as the viscosity tends to infinity the viscoelastic problem reduces to the rate-independent elastic problem describing phase transitions. In this manner, it seems to give yet another perspective for the phase mixture from dynamic point of view as the evolution of an unstable state, in contrast to the traditional treatment from stability analysis for phase equilibrium.     

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

On the hysteresis of wire cables in Stockbridge dampers

Stockbridge dampers are used e.g. for reducing wind-excited oscillations due to vortex shedding in conductors of overhead lines. In these dampers, mechanical energy is dissipated in wire cables (“damper cables”). The damping mechanism is due to statical hysteresis resulting from Coulomb (dry) friction between the individual wires of the cable undergoing bending deformation. Systems with statical hysteresis can be modelled by means of Jenkin elements arranged in parallel, consisting of linear springs and Coulomb friction elements. The damper cable is a continuous system and damping takes place throughout the whole length of the cable, so that distributed Jenkin elements are used. The local mechanical properties of the wire cable are identified experimentally in the time domain. In particular, the moment–curvature relation is determined experimentally at every location of the wire cable subjected to dynamic flexural deformations. Using such a model for the damper cables, the equations of motion can be formulated for a Stockbridge damper, and discretization of the damper cable leads to a system of non-linear ordinary differential equations. In order to test this dynamical model of a Stockbridge damper we compute impedance curves and compare them to experimental results.     

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 hysteresis in elasto-plasticity and in ferromagnetism

We define hysteresis as rate-independent memory, illustrate some of its properties, and review some scalar models of elasto-plasticity: the stop, the play, the Prandtl–Ishlinski models. In particular we study the Prager model of linear kinematic hardening, which encompasses stops and plays. We then couple the latter model with the dynamic equation for a one-dimensional system, show existence of a weak solution, and deal with its homogenization. We also discuss the extension to tensors and to three-dimensional systems.

We then deal with ferromagnetic hysteresis. We review the classic Preisach model and a vector extension. Finally, we formulate a model of vector ferromagnetic hysteresis, couple it with the magnetostatic equations, and discuss its homogenization. The latter consists in a two-length-scale model, and corresponds to a variant of the vector Preisach model.     

On the hysteresis of vortex shedding in a periodically-varying flow

Efforts are made to explore the hysteresis characteristics of vortex shedding in a pipe flow, whose velocity varies periodically in time. Results obtained show that during acceleration of the flow, the vortex strength tends to be stronger, whereas during deceleration of the flow, the situation is reversed. As reconstructed from the velocity signals measured at a point in the flow field, the shed vortex arrays appear to possess uneven vortex strengths in response to periodically-varying incoming flows. Furthermore, in the hysteresis range, the streamwise spacings between the vortices appear to be unequal.     

INTERNAL VARIABLES AND THERMODYNAMIC MODELLING OF PSEUDOELASTICITY

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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.     

On the response of rubbers at high strain rates—III. Effect of hysteresis

In this series of papers, we examine the propagation of waves of finite deformation in rubbers through experiments and analysis; in the present paper, Part III, the effect of hysteretic material behavior on the free retraction of prestretched rubber is considered. A rubber strip stretched to many times its initial length is released at one end and the resulting unloading is examined. A high-speed video camera was used to monitor the motion and to determine the evolution of strain and particle velocity in rubber strips. Simple waves as well as shock waves are observed in these unloading experiments. The measurements are modeled using a power-law model of the material behavior. The hysteretic material response and the formation of shocks are characterized.     

Torsional hysteresis in plastic clay

Summary Plastic clay when subjected to symmetrical cyclic torsional strain behaves in a non-linear manner and produces a stress-strain hysteresis loop of characteristic shape. A mathematical model introducing the idea of an energy-controlled breakdown process involving the conversion of elastic elements to viscous elements is proposed and shown to be capable of representing the observed loops.

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Zusammenfassung Wenn man plastische Tone einer symmetrischen zyklischen Torsionsbeanspruchung aussetzt, so weisen sie ein nichtlineares Verhalten auf, wobei sich charakteristische Hystereseschleifen ergeben. Es wird ein mathematisches Modell vorgeschlagen, welches auf der Vorstellung eines energieabhängigen Abbauvorganges beruht, bei dem eine Umwandlung elastischer in viskose Elemente vor sich geht. Es wird gezeigt, daß sich die ermittelten Hystereseschleifen an Hand des vorgeschlagenen Modells erklären lassen.

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Paper presented at the British Society of Rheology Conference, held at Shrivenham, from 9th–12th September, 1968.     

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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On non-monotonic rate dependence of stress hysteresis of superelastic shape memory alloy bars

This paper presents a simple thermo-mechanical model to explain and quantify the observed strain-rate dependence of the stress hysteresis of shape memory alloys (SMAs) bars/strips during stress-induced forward/reverse phase transition with latent heat release/absorption. By solving the convective heat transfer equation and employing the temperature dependence of the SMA’s transformation stresses, we are able to prove that the stress hysteresis depends non-monotonically on the applied strain rate with a peak appearing at an intermediate strain rate. We further showed that such a non-monotonic rate dependence is governed by the competition of phase-transition time (or latent-heat release/absorption time) and the time of heat exchange with the environment, and that the hysteresis peak is achieved when the two time scales become comparable. A bell-shaped scaling law of the rate dependence is derived, agreeing quantitatively well with the results of experiments.