Taking account of the martensite inelasticity in the reverse phase transformation in shape memory alloys

The model of nonlinear deformation of shape memory alloys (SMA) is generalized to the case in which the possible structural transition in the reverse martensitic transformation is taken into account. The statement of active thermomechanical processes of proportional variation in the stress deviatoric components is justified. The problem of buckling on an SMA bar due to the reverse martensitic transformation is solved. It is shown that taking account of the structural transition under buckling in the process of reverse transformation significantly changes the solution of this problem.     

Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation

We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing. In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant. The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [3–11]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [5–7], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the one-phase martensite state in which the elastic moduli are minimal but in the two-phase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [3–9] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account. According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed. The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the “fixed phase atio” concept. The opposite approach can be classified as the “supplementary phase transition” concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only one-sided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive. The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the “fixed phase ratio” concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible. In the framework of the “fixed phase ratio” hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio. In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire cross-section of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load. Here we must distinguish the “fixed load” hypothesis where no perturbations of the external loads are admitted and the “variable load” hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the “supplementary phase transition” concept and the “fixed load” concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach. In the framework of the “variable load” concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the “fixed phase ratio” concept, and the lower bound corresponds to the case where the entire cross-section of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "all-round supplementary phase transition" hypothesis. In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the above-listed statements except for the case of free support in the “fixed load” concept, for which we obtain a numerical solution.     

Modeling of phase and structure transformations occurring in shape memory alloys under nonmonotonically varying stresses

A model of deformation of shape memory alloys (SMA) under nonmonotone loading is proposed. The model takes into account the fact that there is no strain hardening in the process of accumulation of the first phase transformation strains and describes both the usual hardening and the cross-hardening observed in martensite inelasticity experiments. Several examples illustrate the process of solving the model one-dimensional deformation problem with a given law of variation of the stress and the phase composition parameter and the problem on the direct transformation that occurs when cooling an SMA rod subjected to a constant bending moment.     

A CONSTITUTIVE MODEL FOR TRANSFORMATION, REORIENTATION AND PLASTIC DEFORMATION OF SHAPE MEMORY ALLOYS

A constitutive model is developed for the transformation, reorientation and plastic deformation of shape memory alloys (SMAs). It is based on the concept that an SMA is a mixture composed of austenite and martensite, the volume fraction of each phase is transformable with the change of applied thermal-mechanical loading, and the constitutive behavior of the SMA is the combination of the individual behavior of its two phases. The deformation of the martensite is separated into elastic, thermal, reorientation and plastic parts, and that of the austenite is separated into elastic, thermal and plastic parts. Making use of the Tanaka’s transformation rule modified by taking into account the effect of plastic deformation, the constitutive model of the SMA is obtained. The ferroelasticity, pseudoelasticity and shape memory effect of SMA Au-47.5 at.%Cd, and the pseudoelasticity and shape memory effect as well as plastic deformation and its effect of an NiTi SMA, are analyzed and compared with experimental results.     

Solution of The Double-Coupled Problem of Buckling of a Shape Memory Alloy Rod Due to The Direct Thermoelastic Phase Transformation

Problems of the stability of a shape memory alloy rod during the direct martensitic phase transformation under constant compressive loading are solved taking into account the effect of stresses on the phase transition and the effect of the phase transition on the temperature regime using various statements. A comparison of the results of the obtained solutions is made.     

Coupled thermo-mechanical analysis of shape memory alloy circular bars in pure torsion

Pure torsion of shape memory alloy (SMA) bars with circular cross section is studied by considering the effect of temperature gradient in the cross sections as a result of latent heat generation and absorption during forward and reverse phase transformations. The local form of energy balance for SMAs by taking into account the heat flux effect is coupled to a closed-form solution of SMA bars subjected to pure torsion. The resulting coupled thermo-mechanical equations are solved for SMA bars with circular cross sections. Several numerical case studies are presented and the necessity of considering the coupled thermo-mechanical formulation is demonstrated by comparing the results of the proposed model with those obtained by assuming an isothermal process during loading–unloading. Pure torsion of SMA bars in various ambient conditions (free and forced convection of air, and forced convection of water flow) subjected to different loading–unloading rates are studied and it is shown that the isothermal solution is valid only for specific combinations of ambient conditions and loading rates.     

Constitutive modeling of shape memory alloy wire with non-local rate kinetics

A constitutive modeling approach for shape memory alloy (SMA) wire by taking into account the microstructural phase inhomogeneity and the associated solid–solid phase transformation kinetics is reported in this paper. The approach is applicable to general thermomechanical loading. Characterization of various scales in the non-local rate sensitive kinetics is the main focus of this paper. Design of SMA materials and actuators not only involve an optimal exploitation of the hysteresis loops during loading–unloading, but also accounts for fatigue and training cycle identifications. For a successful design of SMA integrated actuator systems, it is essential to include the microstructural inhomogeneity effects and the loading rate dependence of the martensitic evolution, since these factors play predominant role in fatigue. In the proposed formulation, the evolution of new phase is assumed according to Weibull distribution. Fourier transformation and finite difference methods are applied to arrive at the analytical form of two important scaling parameters. The ratio of these scaling parameters is of the order of 10⁶ for stress-free temperature-induced transformation and 10⁴ for stress-induced transformation. These scaling parameters are used in order to study the effect of microstructural variation on the thermo-mechanical force and interface driving force. It is observed that the interface driving force is significant during the evolution. Increase in the slopes of the transformation start and end regions in the stress–strain hysteresis loop is observed for mechanical loading with higher rates.      

An analytical study on the instability phenomena during the phase transitions in a thin strip under uniaxial tension

In the experiments on stress-induced phase transitions in SMA strips, several interesting instability phenomena have been observed, including a necking-type instability (associated with the stress drop), a shear-type instability (associated with the inclination of the transformation front) and an orientation instability (associated with the switch of the inclination angle). In order to shed more lights on these phenomena, in this paper we conduct an analytical study. We consider the problem in a three-dimensional setting, which implies that one needs to study the difficult problem of solution bifurcations of high-dimensional nonlinear partial differential equations. By using the smallness of the maximum strain, the thickness and width of the strip, we use a methodology, which combines series expansions and asymptotic expansions, to derive the asymptotic normal form equations, which can yield the leading-order behavior of the original three-dimensional field equations. An important feature of the second normal form equation is that it contains a turning point for the localization (necking) solution of the first equation. It is the presence of such a turning point which causes the inclination of the phase transformation front. The WKB method is used to construct the asymptotic solutions, which can capture the shear instability and the orientation instability successfully. Our analytical results reveal that the inclination of the transformation front is a phenomenon of localization-induced buckling (or phase-transition-induced buckling as the localization is caused by the phase transition). Due to the similarities between the development of the Luders band in a mild steel and the stress-induced transformations in a SMA, the present results give a strong analytical evidence that the former is also caused by macroscopic effects instead of microscopic effects. Our analytical results also reveal more explicitly the important roles played by the geometrical parameters.     

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.     

A two-level micromechanical theory for a shape-memory alloy reinforced composite

A two-level micromechanical theory is developed to study the influence of the shape and volume concentration of shape-memory alloy (SMA) inclusions on the overall stress–strain behavior of a SMA-reinforced composite. The first level exists on the smaller SMA level, in which, under the action of stress, parent austenite may transform into martensite. The second level is on the larger scale consisting of the metastable SMA inclusions and an inactive polymer matrix. The evolution of martensite microstructure is evaluated from the irreversible thermodynamics, in conjunction with the micromechanics and physics of martensitic transformation. By taking martensite to exist in the form of thin plates on the micro scale and assuming SMA inclusions to be homogeneously aligned spheroids on the macro scale, the overall stress–strain behaviors of a NiTi-reinforced composite are calculated for various SMA shapes and concentrations. The results indicate that, under a tensile axial loading, martensitic transformation is easier to take place when SMA inclusions exist in the form of long fibers, but most difficult to occur when they are in the form of flat discs. In general the levels of the applied stress at which martensite transformation commences, finishes, and austenitic transformation starts, and finishes, are found to decrease with increasing aspect ratio of the SMA inclusions while the damping capacity increases with it; these properties point to the advantage of using fibrous composites for actuators or sensors under a tensile loading.     

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.

     

Multi-modal approach for large natural flexural vibrations of thermally stressed plates

By means of Berger's approximation, suitable for plates with immovable edges, the geometrically nonlinear problem is considerably simplified. Thermally loaded plates with polygonal planform under hard-hinged support conditions are considered, taking into account the effect of shear in transverse isotropy. The class of symmetric vibrations about the flat plate position is represented by a homogeneous and coupled set of Duffing-oscillators as a result of a multi-mode expansion. A unifying non-dimensional closed-form solution for the corresponding nonlinear natural vibration periods is given, which is independent of the special planform. The individual shape of the plate enters the transformation into real time through the linear natural frequencies, or, equivalently, through the linear eigen-values of an effectively prestressed membrane of the same planform.     

SMA短纤维增强复合材料在热载下的相交特性

在形状记忆合金（ＳＭＡ）复合材料研究中,相变特性的研究是一个主要的工作．基于Ｅｓｈｅｌｂｙ的等效夹杂模型和Ｍｏｒｉ和Ｔａｎａｋａ的场平均法,考虑到ＳＭＡ材料的强物理非线性,发展了增量型的等效夹杂模型（ＩｎｃｒｅｍｅｎｔａｌＥｑｕｉｖａｌｅｎｔＩｎｃｌｕｓｉｏｎＭｏｄｅｌ）．考虑在某一温度循环条件下讨论形状记忆合金短纤维增强的铝基复合材料在热载下的相变行为．特别研究了ＳＭＡ短纤维复合材料在变温过程中纤维几何尺寸、体积分数等参数对ＳＭＡ复合材料的相变行为和ＳＭＡ内残余应力等的影响．这些工作对于指导材料设计和了解ＳＭＡ复合材料热机械特性是颇有意义的．     

A finite element model for shape memory alloys considering thermomechanical couplings at large strains

In the present work we propose a new thermomechanically coupled material model for shape memory alloys (SMA) which describes two important phenomena typical for the material behaviour of shape memory alloys: pseudoelasticity as well as the shape memory effect. The constitutive equations are derived in the framework of large strains since the martensitic phase transformation involves inelastic deformations up to 8%, or even up to 20% if the plastic deformation after the phase transformation is taken into account. Therefore, we apply a multiplicative split of the deformation gradient into elastic and inelastic parts, the latter concerning the martensitic phase transformation. An extended phase transformation function has been considered to include the tension–compression asymmetry particularly typical for textured SMA samples. In order to apply the concept in the simulation of complex structures, it is implemented into a finite element code. This implementation is based on an innovative integration scheme for the existing evolution equations and a monolithic solution algorithm for the coupled mechanical and thermal fields. The coupling effect is accurately investigated in several numerical examples including pseudoelasticity as well as the free and the suppressed shape memory effect. Finally, the model is used to simulate the shape memory effect in a medical foot staple which interacts with a bone segment.     

A thermomechanical model for a 1-D shape memory alloy wire with propagating instabilities

A thermomechanical boundary value problem and constitutive model are presented for a shape memory alloy (SMA) wire under uniaxial loading. The intent is to develop a one-dimensional continuum model of an SMA element that includes all the relevant thermomechanical couplings and is suitable for inclusion in finite element analyses. Thermodynamic relations are derived from phenomenological considerations consistent with recent experimental observations and are calibrated to a typical commercially available NiTi wire material. The model includes both temperature-induced and stress-induced transformations that are necessary to exhibit the shape memory effect and pseudoelastic behaviors. The model accommodates possible unstable mechanical behavior during stress-induced transformations by allowing softening transformation paths and including strain gradient effects. This should provide a tool to study propagating transformation fronts and localized latent heat transfer with the surroundings and a variety of interesting future structural applications, such as composites with embedded SMA elements.     

Macro modelling and homogenization for transformation induced plasticity of a low-alloy steel

Metal forming processes are important technologies for the production of engineering structures. In order to optimize the resulting material properties, it becomes necessary to simulate the entire forming process by taking into account physical effects such as phase transformations. In this work, we concentrate on the phase change from austenite to martensite and present a macroscopic material model, which combines the effect of classical plasticity with the effect of transformation induced plasticity (TRIP). An extensive experimental database for a low-alloy steel is used for parameter identification, thus taking into account the effects of uniaxial compressive and tensile stress on the kinetics of phase transformation at different temperatures. For temperatures below the martensite start temperature with simultaneous stresses above the yield limit, it is difficult to obtain experimental data. Consequently, a numerical homogenization technique is employed for this case. In a further part of this paper, an effective integration scheme is provided, which is implemented into a commercial finite element program. In a finite element simulation, the austenite to martensite phase transformation in a shaft subjected to thermal loading is investigated.     

Investigation of the stability of a plane-channel suspension flow with account for finite particle volume fraction

Within the framework of the two-fluid approach, a variant of a heterogeneous-medium model which takes into account a finite volume fraction of the inclusions and a small but finite phase velocity slip is proposed. The interphase momentum exchange is described by the Stokes force with the Brinkman correction for the finite particle volume fraction. The suspension viscosity depends on the particle volume fraction in accordance with the Einstein formula. Within the framework of the model constructed, a formulation of the problem of linear stability of plane-parallel two-phase flows is proposed. As an example, the stability of a channel suspension flow is considered. The system of equations for small disturbances with the boundary conditions is reduced to an eigenvalue problem for a fourth-order ordinary differential equation. Using the orthogonalization method, the dependence of the critical Reynolds number on the governing nondimensional parameters of the problem is studied numerically. It is shown that taking a finite volume fraction of the inclusions into account significantly affects the laminar-turbulent transition limit.     

Analysis of the rate-dependent coupled thermo-mechanical response of shape memory alloy bars and wires in tension

In this paper, the coupled thermo-mechanical response of shape memory alloy (SMA) bars and wires in tension is studied. By using the Gibbs free energy as the thermodynamic potential and choosing appropriate internal state variables, a three-dimensional phenomenological macroscopic constitutive model for polycrystalline SMAs is derived. Taking into account the effect of generated (absorbed) latent heat during the forward (inverse) martensitic phase transformation, the local form of the first law of thermodynamics is used to obtain the energy balance relation. The three-dimensional coupled relations for the energy balance in the presence of the internal heat flux and the constitutive equations are reduced to a one-dimensional problem. An explicit finite difference scheme is used to discretize the governing initial-boundary-value problem of bars and wires with circular cross-sections in tension. Considering several case studies for SMA wires and bars with different diameters, the effect of loading–unloading rate and different boundary conditions imposed by free and forced convections at the surface are studied. It is shown that the accuracy of assuming adiabatic or isothermal conditions in the tensile response of SMA bars strongly depends on the size and the ambient condition in addition to the rate dependency that has been known in the literature. The data of three experimental tests are used for validating the numerical results of the present formulation in predicting the stress–strain and temperature distribution for SMA bars and wires subjected to axial loading–unloading.     

Statistically equilibrium two-dimensional vortices

Equilibrium statistical mechanics is used for describing two-dimensional vortices in an unbounded incompressible ideal fluid. Both the energy and angular momentum integrals and a set of invariants are taken into account. The latter follows from the condition that any vorticity distribution can be obtained from an initial distribution by a differentiable areas-preserving transformation. The equations for the statistically equilibrium vorticity and passive admixture distributions are derived. It is argued that taking subsidiary invariants into account weakens the arbitrariness associated with the choice of a finite-dimensional approximation of the flow. The case in which the vorticity cloud behaves like a thermodynamic system undergoing an ordering phase transition is discussed.Novosibirsk. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 5, pp. 47–55, September–October, 1995.