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

Interfacial energy and dissipation in martensitic phase transformations. Part I: Theory

A model of evolving martensitic microstructures is formulated that incorporates the interfacial energy and dissipation on three different scales corresponding to the grain boundaries attained by martensite plates, the interfaces between austenite and martensite plates, and the twin interfaces within martensite plates. Three different time scales are also considered in order to clarify the meaning of rate-independent dissipation related to instabilities at more refined temporal and spatial scales. On the slowest time scale, the process of deformation and martensitic phase transformation is investigated as being composed of segments of smooth quasi-static evolution separated by sudden jumps associated with creation or annihilation of interfaces. A general evolution rule is used in the form of minimization of the incremental energy supply to the whole compound thermodynamic system, including the rate-independent dissipation. Close relationship is shown between the evolution rule and the thermodynamic condition for stability of equilibrium, with no a priori assumption on convexity of the dissipation function. It is demonstrated that the extension of the minimum principle from the first-order rates to small but finite increments requires a separate symmetry restriction imposed on the state derivative of the dissipation function. Formulae for the dissipation associated with annihilation of interfaces are proposed that exhibit limited path-independence and satisfy that symmetry requirement. By exploiting the incremental energy minimization rule with the help of the transport theorems, local propagation conditions are derived for both planar and curved phase transformation fronts. The theory serves as a basis for the algorithm for calculation of the stress-induced evolution of martensitic microstructures along with their characteristic dimensions and related hysteresis loops in shape memory alloys; the examples are given in Part II of the paper.     

Phase transitions induced by extension in a slender SMA cylinder: Analytical solutions for the hysteresis loop based on a quasi-3D continuum model

In this paper, we propose a quasi-3D continuum model to study the rate-independent hysteresis phenomenon in phase transitions of a slender shape memory alloy (SMA) cylinder subject to the uniaxial tension. Based on the three-dimensional field equations and the traction-free boundary conditions, by using a coupled series-asymptotic expansion method, we manage to express the total elastic potential energy of the cylinder in terms of the leading order term of the axial strain. We further consider the rate-independent dissipation effect in a purely one-dimensional setting. The mechanical dissipation functions are also expressed in terms of the axial strain. The equilibrium configuration of the cylinder is then determined by using the principle of maximizing the total energy dissipation. An illustrative example with some special chosen material constants is further considered. Free end boundary conditions are proposed at the two ends of the cylinder. By conducting a phase plane analysis and through some calculations, we obtain the analytical solutions of the equilibrium equation. We find that the engineering stress–strain curves corresponding to the obtained solutions can capture some important features of the experimental results. It appears that the analytical results obtained in this paper reveal the multiple solutions nature of the problem and shed certain light on the instability phenomena during the phase transition process.     

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.     

Thermomechanical modeling of nonlinear internal hysteresis due to incomplete phase transformation in pseudoelastic shape memory alloys

This paper presents a thermomechanical model for pseudoelastic shape memory alloys (SMAs) accounting for internal hysteresis effect due to incomplete phase transformation. The model is developed within the finite-strain framework, wherein the deformation gradient is multiplicatively decomposed into thermal dilation, rigid body rotation, elastic and transformation parts. Helmholtz free energy density comprises three components: the reversible thermodynamic process , the irreversible thermodynamic process and the physical constraints of both. In order to capture the multiple internal hysteresis loops in SMA, two internal variables representing the transition points of the forward and reverse phase transformation, \(\phi _s^f\) and \(\phi _s^r\), are introduced to describe the incomplete phase transformation process. Evolution equations of the internal variables are derived and linked to the phase transformation. Numerical implementation of the model features an Euler discretization and a cutting-plane algorithm. After validation of the model against the experimental data, numerical examples are presented, involving a SMA-based vibration system and a crack SMA specimen subjected to partial loading–unloading case. Simulation results well demonstrate the internal hysteresis and free vibration behavior of SMA.

     

Young‐Measure Solutions for a Viscoelastically Damped Wave Equation with Nonmonotone Stress‐Strain Relation

We study the viscoelastically damped wave equation with a nonmonotone stress‐strain relation σ. This system describes the dynamics of phase transitions, which is closely related to the creation of microstructures. In order to analyze the dynamic behavior of microstructures we consider highly oscillatory initial states. Two questions are addressed in this work: How do oscillations propagate in space and time? What can be said about the long‐time behavior? An appropriate tool to deal with oscillations are Young measures. They describe the local distribution or one‐point statistics of a sequence of fast fluctuating functions. We demonstrate that highly oscillatory initial states generate in a unique fashion an evolution in the space of Young measures and we derive the determining equations. Further on we prove a generalized dissipation identity for Young‐measure solutions. As a consequence, it is shown that every low‐energy solution converges to a Young‐measure equilibrium as t→∞. This is a generalization of G. Friesecke's & J. B. McLeod's [FM96] convergence result for classical solutions to the case of Young‐measure solutions. (Accepted November 12, 1997)     

Dynamics of Phase Transitions and Hysteresis in a Viscoelastic Ericksen's Bar on an Elastic Foundation

This work is a follow-up on the study [32] of interface dynamics and hysteresis in materials undergoing solid-solid phase transitions. We consider the dynamics of a viscoelastic bar with a nonmonotone stress-strain relation and viscous stress linearly proportional to the strain rate. The bar is placed on an elastic foundation with stiffness β mimicking the interaction of phases in higher dimensions. Time-dependent displacement-controlled loading of the bar results in a tilted and serrated hysteresis loop, in qualitative agreement with some experimental observations in shape-memory alloys. The model exhibits three phase transition processes: phase nucleation, interface slip and phase annihilation. Between these dynamic processes the system gets stuck in local minimizers of the potential energy. As β increases from zero, a slip-dominated solution behavior transforms to the one where slip and annihilation events are preceded by a step-by-step nucleation process. We show that this transition is caused by an interplay between the slip-favoring inertia term and the nucleation-favoring elastic foundation terms. This revised version was published online in August 2006 with corrections to the Cover Date.     

A micromechanical model for pretextured polycrystalline shape-memory alloys including elastic anisotropy

We present a micromechanical model for polycrystalline shape-memory alloys which is capable of reproducing important aspects of the material behavior such as pseudoelasticity, pseudoplasticity, tension–compression asymmetry and the influence of texture inhomogeneities which may occur from the production process of components or specimens. Our model is based on the optimization of the material’s free energy density and uses a dissipation ansatz which is homogeneous of first order. Considering the full anisotropic material properties of both the austenite and the martensite phase, we compute the evolution of the orientation distributions of austenite and martensite as internal variables of our model.      

考虑损伤的内变量黏弹-黏塑性本构方程

基于Rice 不可逆内变量热力学框架,在约束构型空间中讨论材料的蠕变损伤问题. 通过给定具体的余能密度函数和内变量演化方程推导出考虑损伤的内变量黏弹-黏塑性本构方程. 通过模型相似材料单轴蠕变加卸载试验对一维情况下的本构方程进行参数辨识和模型验证,本构方程能很好地描述黏弹性变形和各蠕变阶段.不同的蠕变阶段具有不同的能量耗散特点. 受应力扰动后,不考虑损伤的材料系统能自发趋于热力学平衡态或稳定态. 在考虑损伤的整个蠕变过程中,材料系统先趋于平衡态再背离平衡态发展. 能量耗散率可作为材料系统热力学状态偏离平衡态的测度;能量耗散率的时间导数可用于表征系统的演化趋势;两者的域内积分值可作为结构长期稳定性的评价指标.      

On the Three Phase Mixtures in Martensitic Transformations of Shape Memory Alloys: Thermodynamical Modeling and Characteristic Temperatures

We study here the thermally-induced martensitic transformation process of shape memory alloys. Taking the internal energy of phase mixtures as the potential function and introducing coherency energies between martensite and austenite and between different variants of the martensitic phase, we are able to use thermodynamical arguments to obtain hysteresis diagrams which could be measured experimentally. The characteristic temperatures for martensitic transformation, (martensite start and finish) and (austenite start and finish) can be identified explicitly and are closely related to the coherency parameters of the coherency energies. Received September 12, 1997     

Hydrodynamical problems of first-order phase transitions

The two-phase liquid-vapor system in a state of thermodynamic equilibrium is considered. If a shock wave propagates in this medium, during its passage the material undergoes shock compression and transforms into a new equilibrium state. Not only the initial velocity changes in this case, but so does the quantitative composition of the phases. Due to the complication of the process, analytic results have practically not been available so far. Calculations of parameters behind the shock discontinuity were carried out approximately by using various tables and nomograms, restricted basically to only one two-phase system, H₂O. Thus, condensation jumps were treated in [1–4] in two-phase supersonic flows within the single-velocity model and a low content of the liquid phase in the mixture. Using the assumptions mentioned, the various parameters were found at the front of the shock wave by numerical solution of the conservation equations of mass, momentum, and energy at the discontinuity. The thermodynamic parameters are usually given in tabulated form as a function of pressure or temperature for equilibrium conditions of the phases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 81–87, September–October, 1977.     

Phase transitions in the shape memory alloy CuAlNi

Shape memory alloys exhibit a complex load-deformation temperature behaviour. In CuAlNi different maximal recoverable deformations may be observed in tensile experiments. We have found five phases and their corresponding phase transitions, two of them are reversible and the others exhibit hysteresis. We use a thermodynamic theory to calculate the energy landscape that describes the behaviour of the CuAlNi specimen.Received: 8 March 2004, Accepted: 9 March 2004, Published online: 12 May 2004 Correspondence to: A. Musolff     

Uniaxial Modeling of Multivariant Shape-Memory Materials with Internal Sublooping using Dissipation Functions

Models for the macroscopic behavior of Shape Memory Materials can be conveniently constructed within the Ziegler–Green–Naghdi framework where all the constitutive information is encoded in two ingredients: the free energy and the dissipation function. In a previous work, we have proposed various expressions for the basic functions suitable to model pseudoelasticity with complete transformations cycles. In this work we consider additional effects due to Martensite reorientation and to transformation reversal prior to transformation completion. The new constitutive model allows for the modeling of a variety of effects including: shape memory associated with thermally induced transformation, internal pseudoelastic subloops and the determination of limit cycles associated with repetitive stress cycling.     

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     

Effects of stress dependent electrochemical reaction on voltage hysteresis of lithium ion batteries

Intercalation of lithium ions into the electrodes of lithium ion batteries is affected by the stress of active materials, leading to energy dissipation and stress dependent voltage hysteresis. A reaction-diffusion-stress coupling model is established to investigate the stress effects under galvanostatic and potentiostatic operations. It is found from simulations that the stress hysteresis contributes to the voltage hysteresis and leads to the energy dissipation. In addition, the stress induced voltage hysteresis is small in low rate galvanostatic operations but extraordinarily significant in high rate cases. In potentiostatic operations, the stresses and stress induced overpotentials increase to a peak value very soon after the operation commences and decays all the left time. Therefore, a combined charge-discharge operation is suggested, i.e., first the galvanostatic one and then the potentiostatic one. This combined operation can not only avoid the extreme stress during operations so as to prevent electrodes from failure but also reduce the voltage hysteresis and energy dissipation due to stress effects.     

On consistent micromechanical estimation of macroscopic elastic energy,coherence energy and phase transformation strains for SMA materials

An apparatus of micromechanics is used to isolate the key ingredients entering macroscopic Gibbs free energy function of a shape memory alloy (SMA) material. A new self-equilibrated eigenstrains influence moduli (SEIM) method is developed for consistent estimation of effective (macroscopic) thermostatic properties of solid materials, which in microscale can be regarded as amalgams of n-phase linear thermoelastic component materials with eigenstrains. The SEIM satisfy the self-consistency conditions, following from elastic reciprocity (Betti) theorem. The method allowed expressing macroscopic coherency energy and elastic complementary energy terms present in the general form of macroscopic Gibbs free energy of SMA materials in the form of semilinear and semiquadratic functions of the phase composition. Consistent SEIM estimates of elastic complementary energy, coherency energy and phase transformation strains corresponding to classical Reuss and Voigt conjectures are explicitly specified. The Voigt explicit relations served as inspiration for working out an original engineering practice-oriented semiexperimental SEIM estimates. They are especially conveniently applicable for an isotropic aggregate (composite) composed of a mixture of n isotropic phases. Using experimental data for NiTi alloy and adopting conjecture that it can be treated as an isotropic aggregate of two isotropic phases, it is shown that the NiTi coherency energy and macroscopic phase strain are practically not influenced by the difference in values of austenite and martensite elastic constants. It is shown that existence of nonzero fluctuating part of phase microeigenstrains field is responsible for building up of so-called stored energy of coherency, which is accumulated in pure martensitic phase after full completion of phase transition. Experimental data for NiTi alloy show that the stored coherency energy cannot be neglected as it considerably influences the characteristic phase transition temperatures of SMA material.     

Thermo-mechanically coupled model of diffusionless phase transformation in austenitic steel

A constitutive model of thermo-mechanically coupled finite strain plasticity considering martensitic phase transformation is presented. The model is formulated within a thermodynamic framework, giving a physically sound format where the thermodynamic mechanical and chemical forces that drive the phase transformation are conveniently identifiable. The phase fraction is treated through an internal variable approach and the first law of thermodynamics allows a consistent treatment of the internal heat generation due to dissipation of inelastic work. The model is calibrated against experimental data on a Ni–Cr steel of AISI304-type, allowing illustrative simulations to be performed. It becomes clear that the thermal effects considered in the present formulation have a significant impact on the material behavior. This is seen, not least, in the effects found on forming limit diagrams, also considered in the present paper.