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In the current study, a 1-D phenomenological model is constructed to capture the temperature-induced hysteretic response in polycrystalline shape memory alloys (SMAs). The martensitic and austenitic transformations are regarded as the first-order transitions. A differential single-crystal model is formulated on the basis of Landau theory. It is assumed that the transformation temperatures follow the normal distribution among the grains due to the anisotropic stress field developed in the material. The polycrystalline hysteretic response is expressed as the integration of single-crystal responses. Besides, the prediction strategy for incomplete transitions is presented, and the first-order reversal curves are obtained via density reassignment. The proposed model is numerically implemented for validation. Comparisons between the modeling results and the experimental ones demonstrate the capability of the proposed model in addressing the hysteresis in thermally-induced phase transformations.

Magnetic shape memory alloys (MSMAs) are materials with strong nonlinearity, which will show hysteresis when it works. In the current paper, a differential model is proposed to describe the hysteresis in MSMAs caused by magnetic field induced martensite reorientation based on Landau theory of phase transitions. First, the three-dimensional model is simplified into a one-dimensional case, and the hysteresis in MSMAs is described by three martensite variant orientations. Then, a traditional Landau free energy is introduced, and the traditional Landau model is obtained by using the Euler–Lagrange equation. However, it is found that the prediction accuracy of the model is not ideal compared with the experimental data. Thus, an improved model is proposed, and the traditional Landau free energy is replaced by a function that can be determined by the easily measured physical quantities in the experimental curve. Numerical experiments show that the prediction effect of the improved model is much better than the traditional Landau model, and the stress dependence is also demonstrated. Moreover, the proposed improved model has the advantages for dynamic analysis and control with its differential form. Therefore, the frequency dependence of the improved model is demonstrated and a feedback linearization control methodology is proposed to make the system develops in the desired trajectory.

The Preisach model, which is formulated as a weighted superposition of hysteresis kernels, has been widely used for hysteresis modeling, especially in smart-material-based actuators. However, in the classical Preisach model, a trade-off is always required between the model accuracy and the number of the hysteresis kernels. To deal with this problem, a model order reduction technique based on the discrete empirical interpolation method (DEIM) has recently been proposed. The method can largely reduce the number of the hysteresis kernels while barely losing the model accuracy. It is noted that the kernel weight in the reduced DEIM-based model can be both positive and negative, which means that the monotonicity of the Preisach model could be lost. The monotonicity is a very important property especially when constructing the inverse Preisach model. Furthermore, the loss of the monotonicity can also deteriorate the model predictability. In the current paper, a modification strategy is proposed. In the modified reduced Preisach model, the DEIM is only employed to select the dominant hysteresis kernels while the corresponding weights of the selected hysteresis kernels are re-identified by solving a constraint optimization problem. Systematic simulation studies and experimental validation are carried out to demonstrate the effectiveness of the proposed strategy.

In the current paper, a macroscopic model is proposed to simulate the hysteretic dynamics of ferroelectric ceramics with creep phenomenon incorporated. The creep phenomenon in the hysteretic dynamics is attributed to the rate-dependent characteristic of the polarisation switching processes induced in the materials. A non-convex Helmholtz free energy based on Landau theory is proposed to model the switching dynamics. The governing equation of single-crystal model is formulated by applying the Euler–Lagrange equation. The polycrystalline model is obtained by combining the single crystal dynamics with a density function which is constructed to model the weighted contributions of different grains with different principle axis orientations. In addition, numerical simulations of hysteretic dynamics with creep phenomenon are presented. Comparison of the numerical results and their experimental counterparts is also presented. It is shown that the creep phenomenon is captured precisely, validating the capa...

In the current paper, the coupled hysteretic behaviors of ferroelectric ceramics subjected to combined electromechanical stimulations are modeled. For a single grain, the polarization orientation switching process is modeled by employing the Euler–Lagrange equations and formulated as a coupled differential equation system. For ferroelectric ceramics, the principle axis orientations of the individual grains are assumed to be distributed in a certain profile, the behaviors of the ceramics are modeled as a weighted combination of the response of each grain. The influence of intergranular interactions is carefully discussed. Numerical results for the minor hysteresis loops in strain and polarization are demonstrated. Comparisons between these results and their experimental counterparts are presented to illustrate the attributes of the proposed model.

In the current paper, a nonlinear differential algebraic approach is proposed for the modeling of hysteretic dynamics of polycrystalline ferromagnetic materials. The model is constructed by employing a phenomenological theory to the magnetization orientation switching. For the modeling of hysteresis in polycrystalline ferromagnetic materials, the single crystal model is applied to each magnetic domain along its own principal axis. The overall dynamics of the polycrystalline materials is obtained by taking a weighted combination of the dynamics of all magnetic domains. The weight function for the combination is taken as the distribution function of the principal axes. Numerical simulations are performed and comparisons with its experimental counterparts are presented. The hysteretic dynamics caused by orientation switching processes is accurately captured by the proposed model. Minor hysteresis loops associated with partial-amplitude loadings are also captured. Rate dependence of the hysteresis loops are inherently incorporated into the model due to its differential nature.

In the current paper, the coupled fatigue hysteretic behaviors in ferroelectric ceramics are modeled by a blending a phenomenological hysteresis model with point defects. The time-dependent phenomenological model is employed to capture the hysteresis loops associated with phase transitions. The fatigue hysteresis is modeled by combining the single-crystal model with time functions. The mechanism of fatigue in ferroelectric ceramics is interpreted as an accumulation of residual polarization in the proposed model. Numerical results of static and dynamic hysteretic behaviors are presented and compared to their experimental counterparts. The capability of the proposed model for capturing the fatigue phenomenon is validated.

In the current paper, the classical Preisach model is modified for modeling the rate-dependent hysteresis loops by employing a series of elementary differential Preisach operators. The elementary hysteresis loop is well characterized by a differential operator and the rate-dependence feature is inherently embedded in the operator. By introducing a specific density function, the overall output of the model can be formulated as a weighted sum of the dynamic responses of all the differential operators. Comparisons between the numerical results and the experimental counterparts are presented. A perfect fit of the model results with the experimental measurements is obtained with all input frequencies considered.

This paper is an extended study on the model of the hysteretic dynamics of magnetorheological dampers based on a phenomenological phase transition theory (Wang and Kamath in Smart Mater. Struct. 15(6):1725–1733, 2006). It is demonstrated that, by appropriately choosing model parameters, the frequency dependence of the hysteretic dynamics can be captured very well by the model based on phase transition theory. Whilst by introducing an appropriate rescaling coefficient to account for the strength of the magnetized particle chains with various magnetic field strengths, the field strength dependence of the hysteretic dynamics can also be captured very well by the same differential equation with the same set of model parameters. There are in total eight model parameters introduced for capturing the hysteretic dynamics, including its dependence on the loading rate and field strength.

In the current paper, the rate dependent hysteretic dynamics of a magnetostrictive transducer is investigated by using a coupled nonlinear macroscopic differential model. The transducer is modeled as a one-dimensional magnetostrictive structure based on the Landau phenomenological theory of phase transition. The hysteresis loops and butterfly-shaped behaviors in the magnetic and mechanical fields are both successfully modeled with estimated model parameter values. The capability of the proposed model for capturing the driving rate dependency is illustrated by numerical experiments.

In the most of its engineering applications, ferroelectric materials are often subjected to combined loadings in the electric and mechanical fields. To simulate the influence of the biased stresses on the hysteretic dynamics of the materials, a macroscopic differential model is proposed to model the hysteresis loops and butterfly-shaped behaviors caused by the polarization orientation switching with biased stresses. A group of numerical simulations are presented, and the comparison of theoretical results with its experimental counterparts is also presented.

The loading rate dependency of the hysteretic dynamics of Magnetorheological (MR) dampers is investigated in the current paper. The model is constructed on the basis of a phenomenological phase-transition theory. The hysteretic dynamics is treated as the responses of a nonlinear system upon external loadings. With appropriately chosen coefficients, the proposed model is able to capture the loading-rate dependency feature of the hysteretic dynamics. Comparisons between the numerical and experimental results are presented, and perfect agreements are obtained.

In the current paper, the hysteretic dynamics of magnetorheological dampers is modeled by a differential model. The differential model is constructed on the basis of a phenomenological phase transition theory. The model is expressed as a second order nonlinear ordinary differential equation with bifurcations embedded in. Due to the differential nature of the model, the hysteretic dynamics of the MR dampers can be linearized and controlled by introducing a feedback linearization strategy.

In this paper, a macroscopic differential model for the nonlinear dynamics of the electric field in ferroelectric ceramics is developed on the basis of polarization switching theory. In a one-dimensional description, dynamics with hysteresis caused by polarization switching is modelled by using the Landau theory of phase transitions for single-crystal cases. For ferroelectric ceramics, the orientation of the principal axis of grains is assumed to have a certain distribution. The overall dynamics is determined by making a weighted combination of the dynamics of each grain. The weight function for the combination is taken phenomenologically based on experimental observations. It is shown that experimental hysteresis can be reproduced by the macroscopic differential model precisely.

In the current paper, a macroscopic differential model for the hysteretic dynamics in shape memory alloy actuators is constructed by using the modified Landau theory of the first order phase transformation. An intrinsic thermo-mechanical coupling is achieved by constructing the free energy as a function depends on both mechanical deformation and the material temperature. Both shape memory and pseudoelastic effects are modeled. The hysteretic dynamics is linearized by introducing another hysteresis loop via nonlinear feedback strategy, which cancels the original one.

Udgivelsesdato: 2007-Jan Nonlinearities and hysteresis effects in a reciprocal PZT transducer are examined by use of a dynamical mathematical model on the basis of phase-transition theory. In particular, we consider the perovskite piezoelectric ceramic in which the polarization process in the material can be modeled by Landau theory for the first-order phase transformation, in which each polarization state is associated with a minimum of the Landau free-energy function. Nonlinear constitutive laws are obtained by using thermodynamical equilibrium conditions, and hysteretic behavior of the material can be modeled intrinsically. The time-dependent Ginzburg-Landau theory is used in the parameter identification involving hysteresis effects. We use the Chebyshev collocation method in the numerical simulations. The elastic field is assumed to be coupled linearly with other fields, and the nonlinearity is in the E-D coupling. We present numerical results for the reciprocal-transducer system and identify the influence of nonlinearities on the system dynamics at high and low frequency as well as electrical impedance effects due to tuning by a series inductance. It is found that nonlinear effects are not important at high frequencies (1 MHz) subject to high-input voltages, but they become important under high-voltage and off-resonance conditions.

In this paper, the finite volume method is developed to analyze coupled dynamic problems of nonlinear thermoelasticity. The major focus is given to the description of martensitic phase transformations essential in the modeling of shape memory alloys (SMA). Computational experiments are carried out to study the thermo-mechanical wave interactions in a SMA rod, and a patch. Both mechanically and thermally induced phase transformations, as well as hysteresis effects, in a 1D structure are successfully simulated with the developed methodology. In the 2D case, the main focus is given to square-to-rectangular transformations and examples of martensitic combinations under different mechanical loadings are provided.

Shape memory alloy (SMA) is one of the novel advanced functional materials that has an increasing range of current and potential applications, including smart materials and structures, bio-medical and nanotechnologies. This range includes also applications of SMA for control and vibration tuning of various structures, seismic response mitigation, and others. In vibration tuning in many of these applications, it is often necessary to apply supplementary oscillators to absorb the vibration energy input into the primary system. Moreover, when supplementary oscillators are used in these applications, we often have to deal with a situation where the primary vibration frequency is not known a priori. In such cases, we have to design a robust supplementary oscillator such that it is able to operate in a rather wide range of frequencies. A SMA-based oscillator is an ideal candidate for these purposes. In the present paper, we propose a dynamic nonlinear model and its numerical realization for using SMA oscillators as vibration absorbers. The system under consideration consists of a SMA rod and an end-mass. We demonstrate that due to the thermo-mechanical coupling, the vibration characteristics of the supplementary oscillator can be tuned by changing its temperature. The dynamic nonlinear model of the SMA oscillator is simplified for the vibration analysis and an efficient numerical methodology is proposed to evaluate the performance of the oscillator. It is demonstrated that the vibration of the primary system can be tuned within a rather wide frequency range by using the SMA oscillator. It is also shown that at high temperatures the performance of the oscillator is close to that of a linear oscillator, while at low temperatures, the SMA oscillator behaves as a regular damper by using its dissipation due to mechanically-induced phase transformations.

The dynamic responses of ferroelectric materials upon external mechanical and electrical stimulations are inherently nonlinear and coupled. In the current paper, a macroscopic differential model is constructed for the coupled hysteretic dynamics via modeling the orientation switching induced in the materials. A non-convex potential energy is constructed with both mechanic and electric field contributions. The governing equations are formulated as nonlinear ordinary differential equations by employing the Euler-Lagrange equation, and can be easily recast into a state space form. Hysteresis loops associated with stress induced polarization switching and butterfly-shaped behavior in ferroelectric materials are also successfully captured. The effects of mechanical loadings on the electrically induced switching are numerically investigated, as well as the mechanically-induced switching with various bias electric fields.

To describe the field-dependent hysteretic responses of dielectric materials when the applied electrical loadings beyond small field ranges, the Rayleigh's law and the complex dielectric constant notation have been introduced with some success. In the current paper, we investigate the field-dependent hysteretic responses on a dynamical point of view. It is shown that the hysteretic responses can be modeled by the Duffing equation, and the field-dependency feature can be naturally captured by the model. The complex dielectric coefficients and the Rayleigh's law can be regarded as special cases of the dynamical system approach. Numerical comparisons among the three models illustrate that the proposed nonlinear differential model is able to capture more accurately the hysteretic behavior of dielectric materials. It is also much easier to model the dielectric losses that presented in all dielectric materials. In the proposed dynamical approach, the loss tangent and the hysteretic behavior are both influenced by the frequency of the applied external loadings.

In this paper, a low dimensional model is constructed to approximate the nonlinear ferroelastic dynamics involving mechanically and thermally-induced martensite transformations. The dynamics of the first order martensite transformation is first modeled by a set of nonlinear coupled partial differential equations (PDEs), which is obtained by using the modified Ginzburg–Landau theory. The Chebyshev collocation method is employed for the numerical analysis of the PDE model. An extended proper orthogonal decomposition is then carried out to construct a set of empirical orthogonal eigenmodes of the dynamics, with which system characteristics can be optimally approximated (in a specified sense) within a range of different temperatures and under various mechanical and thermal loadings. The performance of the low dimensional model is analyzed numerically. Results on the dynamics involving mechanically and thermally-induced phase transformations and the hysteresis effects induced by such transformations are presented.

In the current paper, a macroscopic differential model is constructed for the modeling of two-way shape memory effects in one-dimensional shape memory alloy (SMA) structures. The model is based on the phenomenological theory of thermoelastic phase transformations in SMAs. Hysteresis loops in both mechanical and thermal fields are treated as macroscopic illustrations of martensite transformations and martensite variant re-orientations. A non-convex free energy function is constructed such that each of its local equilibriums can be used to characterize one of the phases involved in the transformations. System states (strain) can be transformed upon external loadings (mechanical or thermal) from one stable equilibrium to another, thus the dynamics of phase transformations can be modeled by investigating the system state transformations. Governing equations for the transformation dynamics are formulated by employing the Lagrange's equation, and are expressed as nonlinear differential equations. Numerical examples of thermal and mechanical hysteresis loops associated with the transformations caused by thermal and mechanical loadings are presented. Two way shape memory effects and pseudo-elastic effects are successfully modeled.

In the current paper, the hysteretic dynamics of ferroelectric materials under combined electro-mechanical loadings is investigated using a macroscopic differential model. The model is constructed on the basis of the Landau theory of the first order phase transformations. Hysteresis loops in the electric field and the butterfly-shaped behaviors in the electro-mechanical coupling are modeled as a consequence of polarizations and orientation switchings, together with nonlinear electro-mechanical coupling effects. The effects of bias stress on the orientation switchings are investigated numerically. Comparison of the model results with its experimental counterparts is presented, capability of the model is approved.Copyright © 2008 by ASME

In the current paper, the hysteretic dynamics and butterfly-shaped behavior of ferroelectric materials are modeled and controlled with a macroscopic differential model inspired by the Landau theory of first-order phase transformations. Hysteretic dynamic behavior of the materials is analyzed as a consequence of orientation switching and the governing equations of the dynamics are formulated as coupled differential equations describing system states switching from one equilibrium state to another. The rate dependence of hysteresis is included in the analysis. A nonlinear feedback strategy is introduced for the linearization and a simple and efficient control strategy is proposed. Comparison of the results obtained with the developed model and their experimental counterparts is presented.