Micromechanical modelling of the effect of plastic deformation on the mechanical behaviour in pseudoelastic shape memory alloys

Except for the recoverable strain induced by phase transformation, NiTi alloys are very ductile even in the martensite phase. The purpose of the present paper is to study the influence of permanent deformation, which results from plastic deformation of martensite, on the mechanical behaviour of pseudoelastic NiTi alloys. Based on phenomenological theory of martensitic transformation and crystal plasticity, a new three dimensional micromechanical model is proposed by coupling both the slip and twinning deformation mechanisms. The present model is implemented as User MATerial subroutine (UMAT) into ABAQUS/Standard to study the influences of plastic deformation on the stress and strain fields, and on the evolution of martensite transformation. Results show that with the increasing of plastic deformation the residual strain increases and the phase transformation stress–strain curves from the martensite to austenite become steeper and less obvious. Both characteristics, stabilisation of martensite and impedance of the reverse transformation, due to plastic deformation are captured.     

Optimization of hysteretic characteristics of damping devices based on pseudoelastic shape memory alloys

In order to develop a fundamental understanding and the feasibility of SMA devices for passive vibration control, an undamped SDOF system with a pseudoelastic SMA restoring force is investigated to find the basic relationship between the shape of the hysteresis loop of SMA elements and their performance as a damping device. The dynamic characteristics of the device are evaluated by the steady-state response at the resonance point in order to focus on the damping effect. Dynamic analysis utilizing the equivalent linearization approach results in two major findings that, to the best of the authors’ knowledge, have not yet been reported in the literature. These results which characterize the unique behavior of the SMA hysteresis include: (a) for a given excitation amplitude, the “scale” of the hysteresis loop, which is a measure of displacement and restoring force, needs to be adjusted so that the response sweeps the maximum loop but does not exceed it; (b) the ratio of the area confined within the hysteresis loop to the area of a corresponding envelope of triangular shape should be as large as possible. The results of this study would be quite useful not only as a guideline for the design of actual SMA devices, but also as a basis for the development of new autoadaptive materials in future.     

On the thermomechanical modeling of shape memory alloys

In this work, a general inelastic framework for the derivation of general three-dimensional thermomechanical constitutive laws for materials undergoing phase transformations is proposed. The proposed framework is based on the generalized plasticity theory and on some basic elements from the theory of continuum damage mechanics. More specifically, a new elaborate formulation of generalized plasticity theory capable of accommodating the multiple and interacting loading mechanisms, which occur during the phase transformations, is developed. Furthermore, the stiffness variations occurring during phase transformations are taken into account by the proposed framework. For this purpose, the free energy is decomposed into elastic and inelastic parts, not in a conventional way, but in one which resembles the elastic-damage cases. Also, a rate-dependent version of the theory is provided. The concepts presented are applied for the derivation of a three-dimensional thermomechanical constitutive model for Shape Memory Alloy materials. Numerical simulations to show qualitatively the ability of the model to capture the behavior of the shape memory alloys are also presented. Furthermore, the model has been fitted to actual experimental results from the literature.     

The influence of surface energy on stress-free microstructures in shape memory alloys

Certain alloys such as In-Tl, Ni-Ti, Ag-Cd or Cu-Al-Ni display remarkable mechanical properties such as the shape memory effect or pseudo-elasticity. This behaviour is related to a solid-solid phase transformation which leads to a complicated microscopic arrangement of different phases. In recent studies such microstructures have been analyzed by the minimization of elastic energy. We discuss the influence of additional surface energy terms on the existence of stress-free microstructures both in the nonlinear and a geometrically linear setting.

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Sommario Certe leghe come quelle di In-Tl, Ni-Ti, Ag-Cd o Cu-Al-Ni mostrano proprietà meccaniche notevoli quali la memoria di forma o la pseudoelasticità. Questo comportamento è determinato da una trasformazione di fase solido-solido che conduce a complicati arrangiamenti a livelo microscopico. In studi recenti tali microstrutture sono state analizzate attraverso la minimizzazione dell'energia elastica. Noi discutiamo l'influenza di termini addizionali di energia superficiale sull'esistenza di microstrutture in uno stato naturale sia in un contesto lineare che non lineare.

     

Crack growth resistance of shape memory alloys by means of a cohesive zone model

Crack growth resistance of shape memory alloys (SMAs) is dominated by the transformation zone in the vicinity of the crack tip. In this study, the transformation toughening behavior of a slowly propagating crack in an SMA under plane strain conditions and mode I deformation is numerically investigated. A small-scale transformation zone is assumed. A cohesive zone model is implemented to simulate crack growth within a finite element scheme. Resistance curves are obtained for a range of parameters that specify the cohesive traction-separation constitutive law. It is found that the choice of the cohesive strength t₀ has a great influence on the toughening behavior of the material. Moreover, the reversibility of the transformation can significantly reduce the toughening of the alloy. The shape of the initial transformation zone, as well as that of a growing crack is determined. The effect of the Young's moduli ratio of the martensite and austenite phases is examined.     

A three-dimensional phenomenological model for martensite reorientation in shape memory alloys

In this work, we propose a macroscopic phenomenological model that is based on the classical framework of thermodynamics of irreversible processes and accounts for the effect of multiaxial stress states and non-proportional loading histories. The model is able to account for the evolution of both twinned and detwinned martensite. Moreover, reorientation of the product phase according to loading direction is specifically accounted for. Towards this purpose the inelastic strain is split into two contributions deriving, respectively, from creation of detwinned martensite and reorientation of previously existing martensite variants. Computational tests demonstrate the ability of the model to simulate the main aspects of the shape memory response in a one-dimensional setting and some of the features that have been experimentally found in the case of multiaxial non-proportional loading histories. Experimental non-proportional loading paths have also been simulated and a good qualitative agreement between numerical and experimental response is observed.     

The step-wise martensite to austenite reversible transformation

The step-wise martensite to austenite reversible transformation (SMART) in shape memory alloys (SMA) is a martensitic thermoelastic transformation where a step-wise kinetics is induced by a partial cycling procedure within the hysteresis cycle (incomplete cycle on heating (ICH) procedure). The ICH procedure has been proved effective in inducing a reversible microstructural modification of the martensitic phase. Results till now obtained both on the SMART behaviour and on the effects of the ICH procedure are reviewed here: the hypotheses advanced till now are discussed to explain experimental evidences.

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Sommario Con l'acronimo di SMART (step-wise martensite to austenite reversible transformation) si indica una trasformazione martensitica termoelastica in cui la cinetica di trasformazione risulta modificata dall'esecuzione di un'appropriata procedura di ciclaggio termico parziale (procedura ICH — Incomplete Cycle on Heating) nell'intervallo di temperatura in cui ha luogo la trasformazione martensitica inversa. Vengono qui presentati i risultati sperimentali relativi sia alla SMART sia agli effetti indotti dalla procedure ICH e responsabili delle modifiche della cinetica di trasformazione. Vengono inoltre discusse le ipotesi sino ad ora avanzate per spiegare i risultati sperimentali.

     

An upper bound to the free energy of n-variant polycrystalline shape memory alloys

In the last two decades, the problem of computing the elastic energy of phase transforming materials has been studied by a variety of research groups. Due to the non-quasiconvexity of the underlying multi-well landscape, different relaxation methods have been used in order to estimate the quasiconvex envelope of the energy density, for which no explicit expression is known at present.This paper combines a recently developed lamination bound for monocrystalline shape memory alloys which relies on martensitic twinned microstructures with the work of Smyshlyaev and Willis [1998a. A ‘non-local’ variational approach to the elastic energy minimization of martensitic polycrystals. Proc. R. Soc. London A 454, 1573–1613]. As a result, a lamination upper bound for n-variant polycrystalline martensitic materials is obtained.The lamination bound is then compared with Reuß- and Taylor-type estimates. While, for given volume fractions, good agreement of lamination upper and convexification lower bounds is obtained, a comparison using energy-minimizing volume fractions computed from the various bounds yields larger differences. Finally, we also investigate the influence of the polycrystal's texture. For a strong ellipsoidal texture, we observe even better agreement of upper and lower bounds than for the case of isotropic statistics.     

A 3-D phenomenological constitutive model for shape memory alloys under multiaxial loadings

This paper presents a new phenomenological constitutive model for shape memory alloys, developed within the framework of irreversible thermodynamics and based on a scalar and a tensorial internal variable. In particular, the model uses a measure of the amount of stress-induced martensite as scalar internal variable and the preferred direction of variants as independent tensorial internal variable. Using this approach, it is possible to account for variant reorientation and for the effects of multiaxial non-proportional loadings in a more accurate form than previously done. In particular, we propose a model that has the property of completely decoupling the pure reorientation mechanism from the pure transformation mechanism. Numerical tests show the ability to reproduce main features of shape memory alloys in proportional loadings and also to improve prediction capabilities under non-proportional loadings, as proven by the comparison with several experimental results available in the literature.     

Martensitic transformation and phonon localization in Ni-Al alloys by atomistic simulations

We discuss an atomistic model for the potential energy of Ni-Al alloys based on the Embedded Atom Method. The potential is applied in a Molecular Dynamics and Quasi Harmonic investigation of the Martensitic Transformation (MT) that occurs in Ni _x Al_1– x for compositions 0.61<x<0.64 at a temperatureT _M ranging from ~ 0 K atx=0.61 to ~ 400 K atx=0.64. We determine the transition temperature as a function of composition and pressure and we show that our potential reliably reproduces the known properties of the alloy.We exploit the model to investigate the microscopic dynamical properties underlying the transition. Our computation shows that in the austenite the compositional disorder induces several bands of localized phonons, that discontinuously de-localize at the MT. We show that specific localized modes associated with Ni-rich clusters identify regions of incipient lattice instability, and provide favourable sites for the nucleation.

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Sommario L'articolo discute un modello Embedded Atom per l'energia potenziale di leghe Ni-Al. Il potenziale viene utilizzato in uno studio di Dinamica Molecolare e nell'approssimazione Quasi Armonica della Transformazione Martensitica (MT) che ha luogo in Ni _x Al_1–x per 0.61<x<0.64 a una temperatutaT _M compresa tra ~ 0 K perx=0.61 e ~ 400 K perx=0.64. La temperature di transizione è determinata in funzione della composizione e della pressione. I risultati dimostrano la validità del potenziale nel riprodurre le proprietà note della lega.Il modello viene utilizzato per studiare la dinamica microscopica alla base della trasformazione. Lo studio mostra che nella austenite il disordine chimico legato alla non-stechiometria induce bande di fononi localizzati che si delocalizzano in modo discontinue allaMT. Viene mostrato che i modi localizzati sono associati a regioni rieche di Ni ed individuano siti di incipiente instabilità reticolare. Tali siti forniscono centri preferenziali per la nucleazione.

     

A non-local implicit gradient-enhanced model for unstable behaviors of pseudoelastic shape memory alloys in tensile loading

In this paper, a gradient-enhanced 3-D phenomenological model for shape memory alloys using the non-local theory is developed based on a 1-D constitutive model. The method utilizes a non-local field variable in its constitutive framework with an implicit gradient formulation in order to achieve results independent of the finite element discretization. An efficient numerical approach to implement the non-local gradient-enhanced model in finite element codes is proposed. The model is used to simulate stress drop at the onset of transformation, and its performance is evaluated using different experimental data. The potential of the presented numerical approach for behavior of shape memory alloys in eliminating mesh-dependent simulations is validated by conducting various localization problems. The numerical results show that the developed model can simulate the observed unstable behaviors such as stress drop and deviation of local strain from global strain during nucleation and propagation of martensitic phase.     

Cyclic thermomechanical behavior of a polycrystalline pseudoelastic shape memory alloy

In this work, 3D finite element modeling is employed to examine the thermomechanical behavior of a polycrystalline Ni-Ti shape memory alloy in the pseudoelastic regime. It is shown that the tension-compression asymmetry during uniaxial cyclic loading is due to a preferred orientation of the crystallographic texture. In pure shear loading, the thermomechanical behavior exhibits symmetry in both senses of shear, due to the fiber texture of the specimen bar stock. It is also shown that the apparent strain rate-dependence is due to thermomechanical coupling with latent heat generation/absorption during phase transformation.     

Recoverable strains in composite shape memory alloys

New upper bounds are proposed for a generic problem of geometric compatibility, which covers the problem of bounding the effective recoverable strains in composite shape memory alloys (SMAs), such as polycrystalline SMAs or rigidly reinforced SMAs. Both the finite deformation and infinitesimal strain frameworks are considered. The methodology employed is a generalization of a homogenization approach introduced by Milton and Serkov [2000. Bounding the current in nonlinear conducting composites. J. Mech. Phys. Solids 48, 1295-1324] for nonlinear composites in infinitesimal strains. Some analytical and numerical examples are given to illustrate the method.     

Molecular-dynamics of a 2D Model of the Shape Memory Effect

This work investigates the thermodynamic properties of a qualitative atomistic model for austenite–martensite transitions. The model, still in 2D, employs Lennard-Jones potentials for the determination of the atomic interactions. By use of two atom species it is possible to identify three stable lattice structures in 2D, interpreted as austenite and two variants of martensite. The model is described in the first part of the work [6] in detail. The present work studies the thermodynamic properties of the model concerning a small, 2-dimensional test assembly consisting of 41 atoms. The phase stability is investigated by exploitation of the condition of minimal free energy. The free energy is calculated from the thermal equation of state, which is measured in numerical tensile tests. In the second part of this work a chain of eleven 41-atom assemblies is investigated. The chain is interpreted as an idealized larger body, where the individual crystallites represent crystallographic layers allowing for the creation of micro structure. By use of tensile tests at various temperature conditions we sketch how such chain may exhibit quasi-plasticity, pseudo-elasticity and the shape memory effect.     

Modeling the dynamic behavior of shape memory alloys

The paper studies the single degree of freedom vibration of a rigid mass suspended by a thin-walled shape memory alloy tube under torsional loading. The behavior is analyzed for the cases of quasiplasticity (low temperatures) and pseudoelasticity (high temperatures) on the basis of an improved version of the Müller–Achenbach model. To illustrate the strong hysteresis-induced damping capacity and the non-linear vibration characteristics, both, free and forced vibrations are considered in the first part of the paper. This is done on the basis of an isothermal version of the model, while the second part of the paper focuses on the effect of non-constant temperature caused by the rate-dependent release and absorption of latent heats.     

Application of a partially relaxed shape memory free energy function to estimate the phase diagram and predict global microstructure evolution

In this work recent results regarding bounds of the relaxed free energy functions of a broad class of shape memory materials of arbitrary symmetry are leveraged to develop a simple and efficient numerical method to analyze several aspects of the thermomechanical response of such materials. This approach is shown to be useful for construction of the austenitic phase diagram used in many early phenomenological models. It is also demonstrated that the resulting implementation is suitable for finite element analysis, has desirable numerical properties, and makes realistic quantitative predictions regarding the evolution of the lattice correspondence variants, transformation stress and strain levels, and orientation dependence.     

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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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 transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

A constitutive relation for pseudo-elastic behaviour in shape memory alloys

A constitutive relation to describe pseudo-elastic deformation in shape memory alloys is presented in this paper. It is capable of describing deformation behaviour of polycrystalline materials under triaxial stress state as well as of monocrystalline materials under one-dimensional condition. Total strain rate is supposed to be composed of elastic strain rate and transformation strain rate. Deformation behaviour of Cu−Zn−Sn alloy and Ti−ni alloy is simulated by use of the proposed constitutive relation. it is shown that simulated results are in a good agreement with experimental data. The project supported by National Natural Science Foundation of China.