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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.

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 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.

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.

The dynamics of multi-layer transducer systems are modelled and simulated by use of a multi-domain spectral method. The model accounts for a nonlinear constitutive relation between the electric displacement and the electric field. Simulation of resonance and wave propagation in the transducer is implemented using a domain-decomposition method and governing equations in each layer are discretized using a Chebyshev collocation method. The model is applied to a reciprocal transducer system, as used in many engineering applications including ultrasonic flowmeter applications and blood-velocity measurement setups, where the transducer consists of a PZT5 piezoceramic layer and a matching layer.

Reversible large electric-field-induced strain caused by reversible orientation switchings in BaTiO3 is modeled using the Landau’s theory of phase transition. A triple well free energy function is constructed. Each of its minima is associated with one of the polarization orientations involved. Nonlinear constitutive laws accounting for reversible orientation switchings and electrostriction effects are obtained by using thermodynamic equilibrium conditions. Hysteretic dynamics of one-dimensional structures is described by coupled nonlinear differential equations. Double hysteretic loops in the electric and mechanic fields are both successfully modeled. Giant reversible electrostriction is modeled as a consequence of reversible orientation switchings via electro-mechanical couplings. Comparisons with experimental results reported in literatures are presented.

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.

It is demonstrated how the formation of macroscopic martensite variant combinations in a two-dimensional shape memory alloy structure can be modified by mechanical loadings during thermally induced phase transformations. The dynamics involving square to rectangular transformations are modelled using a modified Ginzburg–Landau theory. The macroscale dynamics of the structure is described by a set of coupled partial differential equations of nonlinear dynamic thermoelasticity. The switching and modification of martensite variant combinations during thermally induced transformations are simulated using the developed dynamic model.

A numerical model is constructed for modelling macroscale damping effects induced by the first-order martensite phase transformations in a shape memory alloy rod. The model is constructed on the basis of the modified Landau–Ginzburg theory that couples nonlinear mechanical and thermal fields. The free energy function for the model is constructed as a double well function at low temperature, such that the external energy can be absorbed during the phase transformation and converted into thermal form. The Chebyshev spectral methods are employed together with backward differentiation for the numerical analysis of the problem. Computational experiments performed for different vibration energies demonstrate the importance of taking into account damping effects induced by phase transformations.

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.

A mathematical model is constructed for the modelling of two dimensional thermo-mechanical behavior of shape memory alloy patches. The model is constructed on the basis of a modified Landau-Ginzburg theory and includes the coupling effect between thermal and mechanical fields. The free energy functional for the model is exemplified for the square to rectangular transformations. The model, based on nonlinear coupled partial differential equations, is reduced to a system of differential-algebraic equations and the backward differentiation methodology is used for its numerical analysis. Computational experiments with representative distributed mechanical loadings are carried out for patches of different sizes to analyze thermo-mechanical waves, coupling effects, and 2D phase transformations.

Different from the traditional piezoelectric vibration energy harvesting, a new strategy based on stress-induced polarization switching has been proposed in the current paper. Two related prototypes are presented and the associated advantages and drawbacks have been discussed in detail. It has been demonstrated that, with the assistance of a bias electric field, the robustness of the energy harvesters is improved. Furthermore, the real-space phase-field model has been employed to study the nonlinear hysteretic behavior involved in the proposed energy harvesting process. A substantially larger electric current associated with the stress-induced polarization switching has been demonstrated when compared with that with piezoelectric effect. In addition, the effects of bias electric potential, bias resistance, mechanical boundary conditions, charge leakage and electrodes arrangements have also been investigated by the phase-field simulation, which provides a guidance for future real implementations.

Nonlinear kinetic parameter estimation plays an essential role in kinetic study in reaction engineering. In the present study, the feasibility and reliability of the simultaneous parameter estimation problem is investigated for a multi-component photocatalytic process. The kinetic model is given by the L-H equation, and the estimation problem is solved by a hybrid genetic-simplex optimization method. Here, the genetic algorithm is applied to find out, roughly, the location of the global optimal point, and the simplex algorithm is subsequently adopted for accurate convergence. In applying this technique to a real system and analyzing its reliability, it is shown that this approach results in a reliable estimation for a rather wide range of parameter value, and that all parameters can be estimated simultaneously. Using this approach, one can estimate kinetic parameters for all components from data measured in only one time experiment.

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, 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

Model reduction using the proper orthogonal decomposition (POD) method is applied to the dynamics of ferroelastic patches to study the first order square to rectangular phase transformations. Governing equations for the system dynamics are constructed by using the Landau-Ginzburg theory and are solved numerically. By using the POD method, a set of empirical orthogonal basis functions is first constructed, then the system is projected onto the subspace spanned by a small set of basis functions determined by the associated singular values. The performance of the low dimensional model is verified by simulating nonlinear thermo-mechanical waves and square to rectangular transformations in a ferroelastic patch. Comparison between numerical results obtained from the original PDE model and the low dimensional one is carried out., Comment: Keywords: Phase transformation, ferroelastic patch, model reduction, proper orthogonal decomposition, Galerkin projection

Nonlinearities and hysteresis effects in a re- ciprocal ultrasound system is examined by a dynamical mathematical model on the basis of phase-transition theory. In particular, we consider the perovskite piezoelectric ceramic where the polarization process in the material can be modeled by minimizing the Helmholtz free energy function, i.e., each polarization state is associated with a minimum of the free energy function. Based on the free-energy function, nonlinear constitutive laws can be obtained accounting for effects such as hysteresis. The elastic field is assumed to be coupled linearly with other fields while the electric field is nonlinearly coupled to the electric displacement. We present step-response results for the transmitter and receiver of a reciprocal-transducer system and identify the influence of nonlinearities on the system dynamics. components) is modelled with nonlinear constitutive laws capable of capturing the hysteresis features in the E-D relations. The proposed model is based on polarization switching theory employing the Landau - Ginzburg theory so as to characterize different polarization directions. Ther- modynamical equilibrium conditions allow the constitutive laws to be obtained. We finally present the dynamical behaviour of a one-dimensional reciprocal system using the proposed model.

In this paper we analyse the performance of a low dimensional model for the nonlinear thermo-mechanical waves. The model has been obtained by using proper orthogonal decomposition methods combined with a Galerkin projection. First, we analyse the original PDE model in order to obtain the system states at many time instances. Then, by using an empirical orthogonal basis extracted from our numerical results, we construct an empirical low dimensional model. Finally, we compare the results obtained with the original PDE model and those obtained with our low-dimensional model. These comparisons are carried out for mechanically induced phase transformations in a shape-memory alloy rod.

Starting from a two-dimensional model approximating the dynamics of cubic-to-tetragonal and tetragonal-to-orthorhombic phase transformations in shape memory materials, it is shown that the Falk model in the one dimensional case is a special case of the formulated model. Computational experiments based on a conservative difference scheme are carried out to analyse thermomechanical wave interactions in a rod with shape memory effect.

In this paper, an approach based on the nonlinear theory of elastic dynamics is proposed to model microstructure evolution in a shape memory alloy patch. The hysteretic dynamics of the patch at the macro-scale is also modeled by employing Hamilton's principle. The elastic potential energy function is constructed as non-convex on the deviatoric strain, and the governing equations are formulated as two coupled wave equations in terms of the two displacement components. Square to rectangle transformations and rectangle variant re-orientations are modeled by simulating the transitions of system states from one stable branch of the non-convex constitutive relation to another. The proposed model allows a robust implementation of conventional finite element analysis with all typical boundary conditions, but with strong nonlinearity. Numerical results for microstructure evolution caused by boundary stress loadings are presented, laminated structures and needle-like microstructures, which are often observed in experimental investigations, are also obtained. Hysteretic dynamics at the macro-scale is presented together with microstructure evolution.

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.