Elementary properties of the enstrophy and strain fields in confined two-dimensional flows

The evolution of decaying two-dimensional turbulent flows in a bounded domain is considered. It is shown that the global enstrophy is always equal to the total squared strain field, when integrated over a domain with no-slip boundaries, despite the complex evolution of the flow and strong vortex-wall interactions. This property is also valid for a square domain with stress-free walls, and in general for any polygonal boundary. In contrast, the enstrophy is always greater than the squared strain field in a stress-free circular domain, and in general for any closed domain with negatively-signed curvature at all points of the boundary.     

Utilizing mechanical loads and flexoelectricity to induce and control complicated evolution of domain patterns in ferroelectric nanofilms

We have conducted a systematical investigation to reveal the stability and evolution path of various ferroelectric domain patterns in nanofilms subjected to mechanical loads and related flexoelectric field. Within a rigorous framework of flexoelectricity, a phase-field approach has been established for simulating the domain structure of ferroelectric nanofilms. The electromechanical fields of the nanofilms are numerically solved by a fast Fourier transform technique (FFT) based on the combination of Khachaturyan's microscopic elastic theory and Stroh's formalism of anisotropic elasticity. Using this approach, we simulate eight types of domain patterns that can be stabilized in the nanofilms. It is further demonstrated that these domain patterns can be significantly affected by the mechanical loads and related flexoelectric field and exhibit fruitful evolution paths. To adapt the applied mechanical strain and strain gradient, the domain pattern may remain stable, evolve into another polydomain pattern, or become a monodomain state (an effect of domain erasing). The domain fraction, detailed domain morphology, average stresses in the nanofilms, average polarization and temporal evolution characteristics of the domain patterns under various mechanical loads and sources of flexoelectric field have been analyzed. This investigation should provide instructive information for the practical application of ferroelectric nanofilms under complex and changeable mechanical conditions.     

Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials

Elastomeric materials experience stretch-induced softening as evidenced by a pre-stretched material exhibiting a significantly more compliant response than that of the virgin material. In this paper, we propose a fully three-dimensional constitutive model for the observed softening of the stress-strain behavior. The model adopts the Mullins and Tobin concept of an evolution in the underlying hard and soft domain microstructure whereby the effective volume fraction of the soft domain increases with stretch. The concept of amplified strain is then utilized in a mapping of the macroscopic deformation to the deformation experienced by the soft domain. The strain energy density function of the material is then determined from the strain energy of the soft domain and thus evolves as the volume fraction of soft domain evolves with deformation. Comparisons of model results for cyclic simple extension with the experimental data of Mullins and Tobin show the efficacy of the model and suggest that an evolution in the underlying soft/hard domain microstructure of the elastomer captures the fundamental features of stretch-induced softening. Model simulations of the cyclic stress-strain behavior and corresponding evolution in structure with strain for uniaxial tension, biaxial tension and plane strain tension are also presented and demonstrate three-dimensional features of the constitutive model.     

A real-space phase field model for the domain evolution of ferromagnetic materials

A real-space phase field model based on the time-dependent Ginzburg–Landau (TDGL) equation is developed to predict the domain evolution of ferromagnetic materials. The phase field model stems from a thermodynamic theory of ferromagnetic materials which employs the strain and magnetization as independent variables. The phase field equations are shown to reduce to the common micromagnetic model when the magnetostriction is absent and the magnitude of magnetization is constant. The strain and magnetization in the equilibrium state are obtained simultaneously by solving the phase field equations via a nonlinear finite element method. The finite-element based phase field model is applicable for the domain evolution of ferromagnetic materials with arbitrary geometries and boundary conditions. The evolution of magnetization domains in ferromagnetic thin film subjected to external stresses and magnetic fields are simulated and the magnetoelastic coupling behavior is investigated. Phase field simulations show that the magnetization vectors form a single magnetic vortex in ferromagnetic disks and rings. The configuration and size of the simulated magnetization vortex are in agreement with the experimental observation, suggesting that the phase field model is a powerful tool for the domain evolution of ferromagnetic materials.     

Experimental investigation on macroscopic domain formation and evolution in polycrystalline NiTi microtubing under mechanical force

This paper reports the experimental results on macroscopic deformation instability and domain morphology evolution during stress-induced austenite → martensite (A→M) phase transformation in superelastic NiTi polycrystalline shape memory alloy microtubes. High-speed data and image acquisition techniques were used to investigate the dynamic and quasi-static events which took place in a displacement-controlled quasi-static tensile loading/unloading process of the tube. These events include dynamic formation, self-merging, topology transition, convoluted front motion and front instability of a macroscopic deformation domain. The reported phenomena brought up several fundamental issues regarding the roles of macroscopic domain wall energy and kinetics as well as their interplay with the bulk strain energy of the tube in the observed morphology instability and pattern evolution under a mechanical force. These issues are believed to be essential elements in the theoretical modeling of macroscopic deformation patterns in polycrystals and need systematic examination in the future.     

基于虚拟激励法的大跨度桥梁非平稳随机振动分析闭合解

针对大跨度结构考虑地面空间运动的非一致地震响应分析问题,结合虚拟激励法,应用傅里叶分析建立了结构非平稳随机振动响应演变功率谱分析的频域方法。建立的方法完全基于频域执行,给出了响应演变功率谱的闭合解表达式。由于实现了确定性调制过程与随机过程的有效分离,应用离散傅里叶变换进行计算不需要较高的采样分析频率就可以获得较好精度的数值结果。数值算例研究了某斜拉桥的考虑地面运动空间效应非平稳随机地震响应,与通常时频分析方法进行对比,验证了本文频域方法的正确性和有效性。     

A continuum thermodynamics formulation for micro-magneto-mechanics with applications to ferromagnetic shape memory alloys

A continuum thermodynamics formulation for micromagnetics coupled with mechanics is devised to model the evolution of magnetic domain and martensite twin structures in ferromagnetic shape memory alloys. The theory falls into the class of phase-field or diffuse-interface modeling approaches. In addition to the standard mechanical and magnetic balance laws, two sets of micro-forces and their associated balance laws are postulated; one set for the magnetization order parameter and one set for the martensite order parameter. Next, the second law of thermodynamics is analyzed to identify the appropriate material constitutive relationships. The proposed formulation does not constrain the magnitude of the magnetization to be constant, allowing for spontaneous magnetization changes associated with strain and temperature. The equations governing the evolution of the magnetization are shown to reduce to the commonly accepted Landau-Lifshitz-Gilbert equations for the case where the magnetization magnitude is constant. Furthermore, the analysis demonstrates that under certain limiting conditions, the equations governing the evolution of the martensite-free strain are shown to be equivalent to a hyperelastic strain gradient theory. Finally, numerical solutions are presented to investigate the fundamental interactions between the magnetic domain wall and the martensite twin boundary in ferromagnetic shape memory alloys. These calculations determine under what conditions the magnetic domain wall and the martensite twin boundary can be dissociated, resulting in a limit to the actuating strength of the material.     

Boltzmann superposition principle for a time-dependent soft material: assessment under creep flow field

Variety of time-dependent soft materials that undergo evolution of microstructure are known to follow the Boltzmann superposition principle when appropriately transformed from the real to the effective time domain. This behavior is attributed to obliteration of time dependency in the effective time domain by normalizing the real time by the time-dependent relaxation time. This work is aimed at assessing validity of the Boltzmann superposition principle in the effective time domain under application of step changes and ramps in stress for a time-dependent soft glassy material: an aqueous suspension of Laponite, whose rheological properties are known to show strong time and stress dependency. Interestingly, creep experiments started at different aging times and for different constant stresses indeed lead to a comprehensive time–aging time–stress superposition resulting in time and stress dependence of relaxation time and modulus. Subsequently, we analyze compliance response of the material to different kinds of stress step change and ramp combinations. We observe that except few, most of the compliance data plotted in the effective time domain does not overlap on the comprehensive superposition. We analyze this behavior and attribute the observed mismatch to the history dependence of the time evolution of modulus.     

铁电陶瓷微观电畴演化的成核率模型

大量的实验已经证实电畴翻转是铁电材料非线性和迟滞性本构曲线的根本原因.根据这些描述电畴运动的物理实验结果,提出了成核率模型来模拟铁电材料的非线性本构行为.进一步由模型建立了基于微观电畴运动的电畴翻转体积分数演化方程.应用该模型得到的理论计算结果与实验数据的比较表明,模型能够描述铁电材料的非线性本构行为.同时模型所揭示的微观电畴成核的演化行为可进一步的指导宏观唯象模型的建立与改进.     

A SIMPLIFIED MODEL FOR DOMAIN SWITCHING OF FERROELECTRIC CRYSTAL

I. INTRODUCTION Domain switching is the main source of nonlinear characteristics of ferroelectric materials. The trans-formation performance of domain is the basis for ferroelectric constitutive research. In many literatures[1??10], complete switching models were adopted, in which ferroelectric materials are considered tobe consist of several basic sorts of domains, which are independent of each other. Under the actionof su?ciently strong electric ?elds or mechanical stress, the orienta…     

Evolution of the boundary of a viscous fluid occupying a half-space type domain in a Hele-Shaw slot

New exact solutions to an idealized single-phase unsteady Hele-Shaw problem of the evolution of the air-viscous fluid interface are constructed for the case when the fluid occupies a half-space type domain in an unbounded slot. It is shown that in this problem the qualitative features of the interface evolution do not differ from the previously considered case of the Hele-Shaw flow in a channel-type cell.     

Continuum thermodynamics of ferroelectric domain evolution: Theory, finite element implementation, and application to domain wall pinning

A continuum thermodynamics framework is devised to model the evolution of ferroelectric domain structures. The theory falls into the class of phase-field or diffuse-interface modeling approaches. Here a set of micro-forces and governing balance laws are postulated and applied within the second law of thermodynamics to identify the appropriate material constitutive relationships. The approach is shown to yield the commonly accepted Ginzburg-Landau equation for the evolution of the polarization order parameter. Within the theory a form for the free energy is postulated that can be applied to fit the general elastic, piezoelectric and dielectric properties of a ferroelectric material near its spontaneously polarized state. Thereafter, a principle of virtual work is specified for the theory and is implemented to devise a finite element formulation. The theory and numerical methods are used to investigate the fields near straight 180° and 90° domain walls and to determine the electromechanical pinning strength of an array of line charges on 180° and 90° domain walls.     

Elastic-plastic fields near crack tip growing step-by-step in a perfectly plastic solid

In this paper, based on the three-dimensional flow theory of plasticity, the fundametal equations for plane strain problem of elastic-perfectly plastic solids are presented. By using these equations the elastic-plastic fields near the crack tip growing step-by-step in an elastic incompressible-perfectly plastic solid are analysed.The first order asymptotic solutions for the stress field and velocity fields near the crack tip are obtained. The solutions show the evolution process of elastic unloading domain and the development process of central fan domain and reveal the possibility of the presence of the secondary plastic domain. The second order asymptotic solution for stress field is also presented.     

A SIMPLE CONSTITUTIVE MODEL FOR FERROELECTRIC CERAMICS UNDER ELECTRICAL/MECHANICAL LOADING

A simple phenomenological model is developed for describing the macroscopic constitutive response of ferroelectric materials based on consideration of the fact that domain switching is a progressive evolution process with loading. The volume fraction of domain switching is taken as an internal variable, which is derived from the domain nucleation theory. The proposed theory can simulate the dielectric hysteresis, reversed butterfly hysteresis, nonlinear strain-stress hysteresis, as well as electric displacement-stress relation of ferroelectric materials. Its comparison with experimental results and two other theoretical models reveals that the model presented can well predict the nonlinear hysteresis of ferroelectrics under electrical or mechanical loading.     

Spanwise domain effects on the evolution of the plane turbulent mixing layer

Large Eddy Simulation is used to simulate a series of plane mixing layers. The influence of the spanwise domain on the development of the mixing layer, and the evolution of the coherent structures, are considered. The mixing layers originate from laminar conditions, and an idealised inflow condition is found to produce accurate flow predictions when the spanwise computational domain extent is sufficient to avoid confinement effects. Spanwise domain confinement of the flow occurs when the ratio of spanwise domain extent to local momentum thickness reaches a value of ten. Flow confinement results in changes to both the growth mechanism of the turbulent coherent structures, and the nature of the interactions that occur between them. The results demonstrate that simulations of the two-dimensional mixing layer flow requires a three-dimensional computational domain in order that the flow will evolve in a manner that is free from restraints imposed by the spanwise domain.     

Time dependence of effective properties of polycrystalline ferroelectric ceramics

In this paper, based on Merz[7] experimental results and classical nucleation theory, a micromechanics statistical model is proposed to describe the relation between the special microstructure-level evolution phenomena-domain switching and macro-response. The polycrystalline ferroelectric ceramics treated as a composition of switched domain and unswitched domain, the approaches of Eshelby's equivalent inclusion and Mori-Tanaka's mean field theory are used to analyze and predict its effective electroelastic properties. The model can incorporate the effects of time dependence of domain switching and shape of individual crystalline. To the BaTiO₃ polycrystalline ceramics, the analytical results are in good agreement with the experimental results.     

Energy principle of ferroelectric ceramics and single domain mechanical model

Many physical experiments have shown that the domain switching in a ferroelectric material is a complicated evolution process of the domain wall with the variation of stress and electric field. According to this mechanism, the volume fraction of the domain switching is introduced in the constitutive law of ferroelectric ceramic and used to study the nonlinear constitutive behavior of ferroelectric body in this paper. The principle of stationary total energy is put forward in which the basic unknown quantities are the displacement u _i , electric displacement D _i and volume fraction ρ _I of the domain switching for the variant I. Mechanical field equation and a new domain switching criterion are obtained from the principle of stationary total energy. The domain switching criterion proposed in this paper is an expansion and development of the energy criterion. On the basis of the domain switching criterion, a set of linear algebraic equations for the volume fraction ρ _I of domain switching is obtained, in which the coefficients of the linear algebraic equations only contain the unknown strain and electric fields. Then a single domain mechanical model is proposed in this paper. The poled ferroelectric specimen is considered as a transversely isotropic single domain. By using the partial experimental results, the hardening relation between the driving force of domain switching and the volume fraction of domain switching can be calibrated. Then the electromechanical response can be calculated on the basis of the calibrated hardening relation. The results involve the electric butterfly shaped curves of axial strain versus axial electric field, the hysteresis loops of electric displacement versus electric filed and the evolution process of the domain switching in the ferroelectric specimens under uniaxial coupled stress and electric field loading. The present theoretic prediction agrees reasonably with the experimental results given by Lynch. The project supported by the National Natural Science Foundation of China (10572138).     

铁电陶瓷微观电畴的成核率模型

大量的实验已经证实电畴翻转是铁电材料非线性和迟滞性本构曲线的根本原因。研究者已经对铁电陶瓷的微观电畴翻转行为进行了大量详细的研究。针对描述电畴成核的物理实验结果和经典的成核率实验数据,为了建立电畴翻转体积分数的演化方程提出了反应微观电畴翻转的成核率模型。针对铁电试样电畴随机分布的情况,应用该模型对铁电陶瓷的非线性本构行为进行了研究。理论结果与实验数据的比较表明,模型能较好的描述铁电材料的非线性本构行为。同时模型所揭示的微观反转的物理本质可进一步的指导宏观唯象模型的改进。     

On equilibrium domains in superelastic NiTi tubes — helix versus cylinder

Macroscopic cylinder- and helix-shaped deformation domains were observed in NiTi polycrystalline shape memory alloy tubes during the phase transition under uniaxial quasi-static isothermal stretching. Further experiments showed that the occurrence of cylinder or helix domain and its subsequent isothermal evolution strongly depend on the domain volume, tube wall-thickness and loading history. This paper studies the energetics of the cylindrical and helical domains using an elastic inclusion model. It is demonstrated that the total misfit strain energy of an equilibrium domain in tube essentially depends on two nondimensional length scales: the normalized effective domain length and the normalized wall-thickness. Based on such understanding, we quantify the length scale dependence in the energy of the equilibrium cylindrical and helical domains. The energetic preference of each type of domain is predicted and the critical condition for the helix → cylinder domain transition is established.     

可压缩理想塑性体中逐步扩展裂纹顶端的弹塑性场

本文利用理想塑性固体平面应变问题的基本方程,分析了可压缩理想塑性体中逐步扩展裂纹顶端的弹塑性场,得到了关于应力的渐近场,分析了弹性卸载区的演变过程和修正的中心扇形区的发展过程,预示了出现二次塑性区的可能性,弹性可压缩性的影响明显表现在经典的中心扇形区必需加以修正,垂直于板面方向的应力偏量不再为零,而且随着新裂纹面的形成,裂纹前方的均匀应力场和紧连着的修正的中心扇形区的应力偏量将发生变化,这种变化是由于垂直于板面方向的应力偏量发生变化造成的。