Hysteresis in magnetic shape memory composites: Modeling and simulation

Magnetic shape memory alloys are characterized by the coupling between the reorientation of structural variants and the rearrangement of magnetic domains. This permits to control the shape change via an external magnetic field, at least in single crystals. Composite materials with single-crystalline particles embedded in a softer matrix have been proposed as a way to overcome the blocking of the reorientation at grain boundaries.We investigate hysteresis phenomena for small NiMnGa single crystals embedded in a polymer matrix for slowly varying magnetic fields. The evolution of the microstructure is studied within the rate-independent variational framework proposed by Mielke and Theil (1999). The underlying variational model incorporates linearized elasticity, micromagnetism, stray field and a dissipation term proportional to the volume swept by the twin boundary. The time discretization is based on an incremental minimization of the sum of energy and dissipation. A backtracking approach is employed to approximately ensure the global minimality condition.We illustrate and discuss the influence of the particle geometry (volume fraction, shape, arrangement) and the polymer elastic parameters on the observed hysteresis and compare with recent experimental results.     

Hysteresis modeling of two-way shape memory effect in NiTi alloys

In the present study the two-way shape memory effect (TWSME) of a Ni-51 at.% Ti alloy was investigated and a numerical model, able to simulate its hysteretic behaviour in the strain-temperature response, is proposed. In particular, the TWSME was induced through a proper thermo-mechanical training, carried out at increasing number of training cycles and for two values of training deformation, and the thermal hysteretic behaviour, between M _f (Martensite finish temperature) and A _f (Austenite finish temperature), was recorded. The experimental measurements were used to develop a phenomenological model, based on the Prandtl-Ishlinksii hysteresis operator, which was implemented in a Matlab^® function and a Simulink^® model. A systematic comparison between experimental results and numerical predictions is illustrated and a satisfactory accuracy and efficiency has been observed, therefore the method looks suitable for real-time control of NiTi based actuators.     

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.     

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.     

Hysteresis in transverse galloping: The role of the inflection points

Transverse galloping is here considered as a one-degree-of-freedom oscillator subjected to aerodynamic forces, which are described by using the quasi-steady hypothesis. The hysteresis of transverse galloping is also analyzed. Approximate solutions of the model are obtained by assuming that the aerodynamic and damping forces are much smaller than the inertial and stiffness ones. The analysis of the approximate solution, which is obtained by means of the method of Krylov–Bogoliubov, reveals the existing link between the hysteresis phenomenon and the number of inflection points at the aerodynamic force coefficient curve, C_y(α); C_y and α being, respectively, the force coefficient normal to the incident flow and the angle of attack. The influence of the position of these inflection points on the range of flow velocities in which hysteresis takes place is also analyzed.     

An incremental variational approach to coupled thermo-mechanical problems in anelastic solids. Application to shape-memory alloys

We study the coupled thermo-mechanical problem that is obtained by combining generalized standard materials with Fourier’s law for heat conduction. The analysis is conducted in the framework of non-smooth mechanics in order to account for possible constraints on the state variables. This allows models of damage and phase-transformation to be included in the analysis. In view of performing numerical simulations, an incremental thermo-mechanical problem and corresponding variational principles are introduced. Conditions for existence of solutions to the incremental problem are discussed and compared with the isothermal case. The numerical implementation of the proposed approach is studied in detail. In particular, it is shown that the incremental thermo-mechanical problem can be recast as a concave maximization problem and ultimately amounts to solve a sequence of linear thermal problems and purely mechanical (i.e. at a prescribed temperature field) problems. Therefore, using the proposed approach, thermo-mechanical coupling can be implemented with low additional complexity compared to the isothermal case, while still relying on a sound mathematical framework. As an application, thermo-mechanical coupling in shape memory alloys is studied. The influence of the loading strain-rate on the phase transformation and on the overall stress–strain response is investigated, as well as the influence of the thermal boundary conditions. The numerical results obtained by the proposed approach are compared with numerical and experimental results from the literature.     

An isotropic-plasticity-based constitutive model for martensitic reorientation and shape-memory effect in shape-memory alloys

In this work, we develop an isotropic-plasticity-based constitutive model for initially martensitic shape-memory alloys (SMA) which exhibit martensitic reorientation and the shape-memory effect. The constitutive model is then implemented in the [Abaqus reference manuals. 2006. Providence, R.I.] finite-element program by writing a user-material subroutine. The results from the constitutive model and numerical procedure are then compared to representative physical experiments conducted on polycrystalline rod and sheet Ti–Ni. The constitutive model and the numerical simulations are able to reproduce the stress–strain responses from these physical experiments to good accuracy. Finally, two different boundary value problems utilizing the one-way shape-memory effect are studied: (a) the deformation of an arterial stent, and (b) a micro-clamper. We show that our constitutive model can be used to model the response of the aforementioned boundary value examples.     

Thermomechanical coupling in shape memory alloys under cyclic loadings: Experimental analysis and constitutive modeling

In this paper, we examine the influence of thermomechanical coupling on the behavior of superelastic shape memory alloys subjected to cyclic loading at different loading rates. Special focus is given to the determination of the area of the stress-strain hysteresis loop once the material has achieved a stabilized state. It is found that this area does not evolve monotonically with the loading rate for either transient or asymptotic states. In order to reproduce this observation analytically, a new model is developed based on the ZM model for shape memory alloys which was modified to account for thermomechanical coupling. The model is shown to predict the non-monotonic variation in hysteresis area to good accord. Experimentally observed variations in the temperature of SMA test samples are also correctly reproduced for lower strain rates.     

A constitutive model for shape memory alloys accounting for thermomechanical coupling

This paper presents a generalized Zaki-Moumni (ZM) model for shape memory alloys (SMAs) [cf. Zaki, W., Moumni, Z., 2007a. A three-dimensional model of the thermomechanical behavior of shape memory alloys. J. Mech. Phys. Solids 55, 2455-2490 accounting for thermomechanical coupling. To this end, the expression of the Helmholtz free energy is modified in order to derive the heat equation in accordance with the principles of thermodynamics. An algorithm is proposed to implement the coupled ZM model into a finite element code, which is then used to solve a thermomechanical boundary value problem involving a superelastic SMA structure. The model is validated against experimental data available in the literature. Strain rate dependence of the mechanical pseudoelastic response is taken into account with good qualitative as well as quantitative accuracy in the case of moderate strain rates and for mechanical results in the case of high strain rates. However, only qualitative agreement is achieved for thermal results at high strain rates. It is shown that this discrepancy is mainly due to localization effects which are note taken into account in our model. Analyzing the influence of the heat sources on the material response shows that the mechanical hysteresis is mainly due to intrinsic dissipation, whereas the thermal response is governed by latent heat. In addition, the variation of the area of the hysteresis loop with respect to the strain rate is discussed. It is found that this variation is not monotonic and reaches a maximum value for a certain value of strain rate.     

A finite-deformation-based phenomenological theory for shape-memory alloys

In this work we develop a finite-deformation-based, thermo-mechanically-coupled and non-local phenomenological theory for polycrystalline shape-memory alloys (SMAs) capable of undergoing austenite ↔$\leftrightarrow$ martensite phase transformations. The constitutive model is developed in the isotropic plasticity setting using standard balance laws, thermodynamic laws and the theory of micro-force balance (Fried and Gurtin, 1994). The constitutive model is then implemented in the ABAQUS/Explicit (2009) finite-element program by writing a user-material subroutine. Material parameters in the constitutive model were fitted to a set of superelastic experiments conducted by Thamburaja and Anand (2001) on a polycrystalline rod Ti–Ni. With the material parameters calibrated, we show that the experimental stress-biased strain–temperature-cycling and shape-memory effect responses are qualitatively well-reproduced by the constitutive model and the numerical simulations. We also show the capability of our constitutive mode in studying the response of SMAs undergoing coupled thermo-mechanical loading and also multi-axial loading conditions by studying the deformation behavior of a stent unit cell.     

Hysteresis modelling and chaos prediction in one- and two-DOF hysteretic models

In the present work hysteresis is simulated by means of internal variables. The analytical models of different types of hysteresis loops allow the reproduction of major and minor loops and provide a high degree of correspondence with experimental data. In models of this type adding an external periodic excitation or increasing the number of dimensions can lead to the occurrence of chaotic behaviour. Using an effective algorithm based on numerical analysis of the wandering trajectories [1–7], the evolution of the chaotic behaviour regions of oscillators with hysteresis is presented in various parametric planes. The substantial influence of a hysteretic dissipation value on the form and location of these regions, as well as the restraining and generating effects of hysteretic dissipation on the occurrence of chaos, are ascertained. Conditions for pinched hysteresis are defined. Furthermore, autonomous coupled hysteretic oscillators under sliding friction are investigated. Conditions for the occurrence of chaotic behaviour in a two-degree-of-freedom (two-DOF) hysteretic system are found in the plane of maximal static friction forces of both oscillators versus belt velocity.     

Hysteresis losses in oscillatory systems

Hysteresis losses restrict the quality factors of oscillatory systems. We have considered the energy dissipation mechanism using as an example the contact spots of the ball supports of a pendulum tribometer with plane samples. The bending of a contact spot leads to a storage of the system׳s elastic energy. Measuring the quality factor of the pendulum at oscillations, owing to the inclination of the contact spot, makes it possible to determine the coefficient of hysteresis losses in materials and thin coatings without destroying their structures.     

A phenomenological model for microstructure-dependent inelasticity in shape-memory alloys

Degradation in shape-memory alloy response is a crucial concern for a variety of innovative applications. Under cyclic loadings, these materials generally experience permanent inelastic deformations. The onset of plasticization is known to be very sensitive to the microstructure of the polycrystalline specimen. Moving from recent experimental findings (Malard et al. in Funct Mater Lett 2:45–54, 2009; Acta Mater 59:1542–1556, 2011), we present a phenomenological model for permanent inelastic effects in shape-memory alloys taking into account the polycrystalline microstructure. In particular, the mechanical response under cyclic loadings is investigated in connection with the mean crystal grain size. Formulated within the variational frame of generalized standard materials, the model consists in an extension of the model in Auricchio et al. (Int J Plast 23:207–226, 2007) to the case of microstructure-dependent parameters. The mathematical setting is discussed and numerical simulations showing the capability of the model to reproduce experiments are presented.     

Approximation in multiscale modelling of microstructure evolution in shape-memory alloys

Various models of microstructure in deformation gradient and its evolution arising in martensitic mechanically-induced isothermal phase transformation are surveyed and scrutinized, focusing on over-bridging of various scales of the problem and its numerical approximation. In particular, numerically efficient model of a relaxed problem is shown to be approximated by conventional but computationally less efficient model based on standard partial differential inequalities.     

Hysteresis and imperfection sensitivity in small ferromagnetic particles

The classical results of Stoner and Wohlfarth for the prediction of hysteresis loops in small ferromagnetic particles are extended to specimens of non-ellipsoidal shape, and shown to be a consequence of micromagnetics. The insensitivity to surface roughness is proposed as a possible explanation of the high coercivity behavior of small particles.

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Sommario Si deducono i classici risultati di Stoner e Wohlfarth dalla teoria del micromagnetismo, e si dimostra la loro validità per la previsione di cicli di isteresi magnetica anche per particelle di forma non ellissoidale. Viene proposta, quale possibile spiegazione della notevole ampiezza dei cicli di isteresi caratteristici di particelle di piccole dimensioni, l'indifferenza alla presenza di rugosità superficiale.

     

Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys

A crystal-inelasticity-based constitutive model for martensitic reorientation and detwinning in shape-memory alloys (SMAs) has been developed from basic thermodynamics principles. The model has been implemented in a finite-element program by writing a user-material subroutine. We perform two sets of finite-element simulations to model the behavior of polycrystalline SMAs: (1) The full finite-element model where each finite element represents a collection of martensitic microstructures which originated from within an austenite single crystal, chosen from a set of crystal orientations that approximates the initial austentic crystallographic texture. The macroscopic stress-strain responses are calculated as volume averages over the entire aggregate: (2) The Taylor model (J. Inst. Metals 62 (1938) 32) where an integration point in a finite element represents a material point which consist of sets of martensitic microstructures which originated from within respective austenite single-crystals. Here the macroscopic stress-strain responses are calculated through a homogenization scheme.Experiments in tension and compression were conducted on textured polycrystalline Ti-Ni rod initially in the martensitic phase by Xie et al (Acta Mater. 46 (1998) 1989). The material parameters for the constitutive model were calibrated by fitting the tensile stress-strain response from a full finite-element calculation of a polycrystalline aggregate to the simple tension experiment. With the material parameters calibrated the predicted stress-strain curve for simple compression is in very good accord with the corresponding experiment. By comparing the simulated stress-strain response in simple tension and simple compression it is shown that the constitutive model is able to predict the observed tension-compression asymmetry exhibited by polycrystalline Ti-Ni to good accuracy. Furthermore, our calculations also show that the macroscopic stress-strain response depends strongly on the initial martensitic microstructure and crystallographic texture of the material.We also show that the Taylor model predicts the macroscopic stress-strain curves in simple tension and simple compression reasonably well. Therefore, it may be used as a relatively inexpensive computational tool for the design of components made from shape-memory materials.     

Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes

Results are presented for the numerical simulation of vortex-induced vibrations (VIVs) of a cylinder at low Reynolds numbers (Re). A stabilized space–time finite-element formulation is utilized to solve the incompressible flow equations in primitive variables. The cylinder, of low nondimensional mass (m^*=10), is free to vibrate in, both, the transverse and in-line directions. To investigate the effect of Re and reduced natural frequency, F_n, two sets of computations are carried out. In the first set of computations the Reynolds number is fixed (=100) and the reduced velocity (U^*=1/F_n) is varied. Hysteresis, in the response of the cylinder, is observed at the low- as well as high-end of the range of reduced velocity for synchronization/lock-in. In the second set of computations, the effect of Reynolds number (50Re500) is investigated for a fixed reduced velocity (U^*=4.92). The effect of the Reynolds number is found to be very significant for VIVs. While the vortex-shedding mode at low Re is 2S (two single vortices shed per cycle), at Re300 and larger, the P+S mode of vortex shedding (a single vortex and one pair of counter-rotating vortices are released in each cycle of shedding) is observed. This is the first time that the P+S mode has been observed for a cylinder undergoing free vibrations. This change of vortex-shedding mode is hysteretic in nature and results in a very large increase in the amplitude of in-line oscillations. Since the flow ceases to remain two-dimensional beyond Re200, it remains to be seen whether the P+S mode of shedding can actually be observed in reality for free vibrations.     

Internal loops in superelastic shape memory alloy wires under torsion – Experiments and simulations/predictions

Understanding torsional responses of shape memory alloy (SMA) specimens under partial or fully transformed cases with internal loops is of particular importance as the entire response might not be always utilized and only a portion of the entire response (internal loop) might be of significance to designers. In this work, we present experimental results of large complex loading and unloading torsional cycles which were conducted on superelastic SMA wires, under isothermal conditions with the purpose of elucidating the torsional internal loop response during loading and unloading. Such data hereto has not been available in open literature. Utilizing this data, we model the torsional response of superelastic SMA wires subjected to various loading and unloading situations that can result in different extents of transformation.A thermodynamically consistent Preisach model (Rao and Srinivasa, 2013) captures such complex internal loops with a high degree of precision by modeling driving force for phase transformation vs. volume fraction of martensite relationships. This approach is different from capturing purely phenomenological stress–strain or stress–temperature Preisach models. The thermodynamic approach utilized here has broader predictive capability. The model predictions indicate good agreement with the internal loop structures even though only the outer loop information was used for model calibration. The addition of a single inner loop information for model calibration greatly improves the predictions.     

Multi-axial behavior of shape-memory alloys undergoing martensitic reorientation and detwinning

Recently, a rate-independent, finite-deformation-based crystal mechanics constitutive model for martensitic reorientation and detwinning in shape-memory alloys has been developed by Thamburaja [Thamburaja, P., 2005. Constitutive equations for martensitic reorientation and detwinning in shape-memory alloys. Journal of the Mechanics and Physics of Solids 53, 825–856] and implemented in the ABAQUS/Explicit [Abaqus reference manuals. 2005. Providence, RI] finite-element program. In this work, we show that the aforementioned model is able to quantitatively predict the experimental response of an initially textured and martensitic polycrystalline Ti–Ni rod under a variety of uniaxial and multi-axial stress states. By fitting the material parameters in the model to the stress–strain response in simple tension, the constitutive model predicts the stress–strain curves for experiments conducted under simple compression, torsion, proportional-loading tension–torsion, and path-change tension–torsion loading conditions to good accord. Furthermore the constitutive model also reproduces the force–displacement response for an indentation experiment to reasonable accuracy.