Non-linear constitutive relations for magnetostrictive materials

In this paper, non-linear deformation behavior of magnetostrictive materials is studied and three magnetoelastic coupling constitutive models are developed. The standard square (SS) constitutive model is developed by means of truncating the polynomial expansion of the Gibbs free energy. The hyperbolic tangent (HT) constitutive equations, which involve a hyperbolic tangent magnetic-field dependence, are proposed to model the magnetic-field-induced strain saturation of magnetostrictive materials in the region of intense magnetic fields. A new model based on density of domain switching (DDS) is established in terms of the basic truth that magnetic domain switching underlies magnetostrictive deformation. In this model, it is assumed that the relation between density of domain switching, defined by the quantity of magnetic domains switched by per unit magnetic field and magnetic field can be described by a density function with normal distribution. The moduli in these constitutive models can be determined by a material function that is proposed to describe the dependence of the peak piezomagnetic coefficient on the compressive pre-stress for one-dimensional cases based on the experimental results published. The accuracy of the non-linear constitutive relations is evaluated by comparing the theoretical values with experimental results of a Terfenol-D rod operated under both compressive pre-stress and bias magnetic field. Results indicate that the SS constitutive equations can accurately predict the experimental results under a low or moderate magnetic field while the HT model can, to some extent, reflect the trend of saturation of magnetostrictive strain under a high magnetic field. The model based on DDS, which is more effective in simulating the experimental curves, can capture the main characteristics of the mechanism of magnetoelastic coupling deformation of a Terfenol-D rod, such as the notable dependence of magnetoelastic response on external stress and the saturation of magnetostrictive strain under intense magnetic fields. In addition, the SS constitutive relation for a general three-dimensional problem is discussed and an approach to characterize the modulus tensors is proposed.     

Experimental and theoretical study of the nonlinear response of agiant magnetostrictive rod

In this paper, a magnetomechanical coupling constitutive relation of the giant magnetostrictive material was investigated experimentally and theoretically. A grain-oriented magnetostrictive rod of iron and rare earth was tested under a combined magnetomechanical loading. Two types of experimental curves were obtained, i.e., the magnetostrictive curve of the extensional strain vs the magnetic field, and the curve of the magnetic polarization intensity vs the pre-stress. A new theoretical constitutive model, based on the density of domain switching, is developed. Comparison of the theoretical predictions with the experimental results indicates that this model can capture the main characteristics of the magnetoelastic coupling deformation of a giant magnetostrictive rod. The project supported by the National Natural Science Foundation of China (10025209, 10132010, 10102007)     

磁致伸缩材料的非线性本构关系

给出了磁致伸缩材料的两个非线性本构关系,即标准平方型和双曲正切型。在确定一维问题的本构系数时,基于已有的实验结果,引进一个材料函数,用来描述磁致伸缩材料的压磁系数随预应力变化的关系。将非 线性本构关系的理论模型计算结果与实验曲线对比,结果表明标准平方型本构关系在中低磁场下能精确地模拟实验曲线,而双曲正切型本构关系在高磁场时能反映材料的磁致应变饱和现象。讨论了在标准平方型本构的一般三维情形,给出了确定本构系数的方法。     

EXPERIMENTAL RESEARCHES ON MAGNETO-THERMO-MECHANICAL CHARACTERIZATION OF TERFENOL-D

The coupling effects of axial pre-stress,temperature and magnetic field on magne- tostrictive strain and magnetization as well as Young's modulus of a Terfenol-D (Tb_(0.3)Dy_(0.7)Fe_(1.93)) rod are tested to give a good understanding of magneto-thermal-mecha-nical characteristics of giant magnetostrictive materials.Results show that magneto-thermo-mechanical coupling of gi- ant magnetostrictive materials is very strong;and the influences of pre-stress and temperature on magnetostrictive strain and Young's modulus vary with the intensity of magnetic field.     

Numerical simulation on coupling behavior of Terfenol-D rods

Terfenol-D rods, as a kind of giant magnetostrictive materials, are often used as active elements of device for anti-vibration application due to its superior material properties. Their magneto-mechanical responses exhibited in many experiments are nonlinear and coupled. In order to have a good understanding on their coupling characters for accurate control, the numerical simulation on dynamic behavior of a Terfenol-D rod is conducted based on a nonlinear and coupling constitutive model proposed in this paper. The results show that the constitutive model can effectively describe some intrinsic coupling phenomena observed by experiments involving the maximum magnetostrictive strain of a Terfenol-D rod changing with pre-stresses and the corresponding dynamic responses show that the frequency and the amplification of the Terfenol-D rod change with magnetic bias field and pre-stresses, which are also consistent with experimental data and cannot be captured by previous constitutive model.     

Surface effect on band structure of magneto-elastic phononic crystal nanoplates subject to magnetic and stress loadings

This paper presents a theoretical model for the size-dependent band structure of magneto-elastic phononic crystal(PC) nanoplates according to the Kirchhoff plate theory and Gurtin-Murdoch theory, in which the surface effect and magneto-elastic coupling are considered. By introducing the nonlinear coupling constitutive relation of magnetostrictive materials, Terfenol-D/epoxy PC nanoplates are carried out as an example to investigate the dependence of the band structure on the surface effect, magn...     

Magnetostrictive composites in the dilute limit

We calculate the effective properties of a magnetostrictive composite in the dilute limit. The composite consists of well separated identical ellipsoidal particles of magnetostrictive material, surrounded by an elastic matrix. The free energy of the magnetostrictive particles is computed using the constrained theory of DeSimone and James [2002. A constrained theory of magnetoelasticity with applications to magnetic shape memory materials. J. Mech. Phys. Solids 50, 283-320], where application of an external field causes rearrangement of variants rather than rotation of the magnetization or elastic strain in a variant. The free energy of the composite has an elastic energy term associated with the deformation of the surrounding matrix and demagnetization terms. By using results from the constrained theory and from the Eshelby inclusion problem in linear elasticity, we show that the energy minimization problem for the composite can be cast as a quadratic programming problem. The solution of the quadratic programming problem yields the effective properties of Ni₂MnGa and Terfenol-D composite systems. Numerical results show that the average strain of the composite depends strongly on the particle shape, the applied stress, and the elastic modulus of the matrix.     

VIBRATION SUPPRESSION OF CANTILEVER LAMINATED COMPOSITE PLATE WITH NONLINEAR GIANT MAGNETOSTRICTIVE MATERIAL LAYERS

In this paper, a nonlinear and coupled constitutive model for giant magnetostrictive materials(GMM) is employed to predict the active vibration suppression process of cantilever laminated composite plate with GMM layers. The nonlinear and coupled constitutive model has great advantages in demonstrating the inherent and complicated nonlinearities of GMM in response to applied magnetic field under variable bias conditions(pre-stress and bias magnetic field).The Hamilton principle is used to derive the nonlinear and coupled governing differential equation for a cantilever laminated composite plate with GMM layers. The derived equation is handled by the finite element method(FEM) in space domain, and solved with Newmark method and an iteration process in time domain. The numerical simulation results indicate that the proposed active control system by embedding GMM layers in cantilever laminated composite plate can efficiently suppress vibrations under variable bias conditions. The effects of embedded placement of GMM layers and control gain on vibration suppression are discussed respectively in detail.     

考虑膜厚影响薄膜磁致伸缩系数、泊松比和杨氏模量的同时测量

基于经典层合板理论,建立了一个能同时测量薄膜-基底系统中薄膜的磁致伸缩系数、杨氏模量和泊松比的板模型.以前的研究计算薄膜磁致伸缩系数时,大多假设薄膜的弹性属性与相应的块材一致,由此导致的磁致伸缩系数计算是不准确的.在目前大多数方法中仅仅在使用一个单一的弹性各向同性基底中能够避免这个问题.该文模型在各向异性基底下同样适用,并且不要求薄膜的厚度远远小于基底厚度,因此也能够用来计算磁致伸缩应变和设计微电机械系统和生物微电机械系统.对已有的铁基非晶薄膜的实验数据,在不同磁场强度下,磁致伸缩系数的计算结果与已有模型进行了比较,它们之间的差异得到了解释.同时,还可以得到薄膜的弹性常数.     

A constitutive model for magnetostrictive and piezoelectric materials

This paper is concerned with a macroscopic nonlinear constitutive law for magnetostrictive alloys and ferroelectric ceramics. It accounts for the hysteresis effects which occur in the considered class of materials. The uniaxial model is thermodynamically motivated and based on the definition of a specific free energy function and a switching criterion. Furthermore, an additive split of the strains and the magnetic or electric field strength into a reversible and an irreversible part is suggested. Analog to plasticity, the irreversible quantities serve as internal variables. A one-to-one-relation between the two internal variables provides conservation of volume for the irreversible strains. The material model is able to approximate the ferromagnetic or ferroelectric hysteresis curves and the related butterfly hysteresis curves. Furthermore, an extended approach for ferrimagnetic behavior which occurs in magnetostrictive materials is presented. A main aspect of the constitutive model is its numerical treatment. The finite element method is employed to solve the coupled field problem. Here the usage of the irreversible field strength permits the application of algorithms of computational inelasticity. The algorithmic consistent tangent moduli are developed in closed form. Hence, quadratic convergence in the iterative solution scheme of governing balance equations is obtained.     

Some variational principles in elastic dielectric and elastic magnetic materials

The electroelastic and magnetoelastic analyses in the dielectric and magnetic materials should be studied together with their environments. From the first thermodynamic law a general variational theory is proposed. Some nonlinear variational principles with electric or magnetic Gibbs free energy and the complete governing equations, derived from these principles are given in this paper. The governing equations are complete and useful in engineering. It is noted that the virtual displacements do not only produce the variation of the strain, but also produce the variation of electric potential or magnetic potential. It is also noted that even in a small deformation the effect of the volume change on the electromagnetic field cannot be neglected. Just due to these two reasons the Maxwell stress can be naturally derived from the variational equation and it is unique. In these derivations new elctrostrictive and magnetostrictive coefficients are introduced and the anti-symmetric part of the stress can be naturally got. These principles are also useful for computational mechanics.     

超磁致伸缩材料的擦除特性和同余特性实验研究

超磁致伸缩材料在力磁耦合作用下具有复杂的迟滞响应。Preisach模型可有效描述物理过程中的滞后现象,它具有两个重要特性,即擦除特性和同余特性。擦除特性是指输入局部极大值擦除了加载过程中小于该值的所有极大值,或输入局部极小值擦除了加载过程中大于该值的所有极小值,同时,与这些历史极值相应的加载历史也被擦除,不再影响之后的输出。同余特性是指输入极大值与极小值相同的所有闭合曲线一致。本文通过实验系统地研究了超磁致伸缩材料在多轴力磁耦合场作用下的磁致伸缩曲线、磁滞回线和应力应变的迟滞行为,发现其在力磁耦合下的非线性滞后行为具有擦除特性和同余特性。这满足了Preisach模型描述滞后现象的两个基本要求,验证了利用Preisach模型描述超磁致伸缩材料迟滞行为的可行性,为超磁致伸缩材料的非线性理论和器件设计提供了依据。     

Empirical plasticity models applied for paper sheets having different anisotropy and dry solids content levels

The rheological nature of paper or board is usually treated either as elasto-plastic or as viscoelastic depending on the studied paper making process or behavior in converting and end use. In this paper we study several stress–strain curve models and the determination of material parameters from an elasto-plastic point of view. Finally, a suitable approach for all stress–strain curves measured from 180 strips is constructed using a linear function for an elastic region and a nonlinear function for a strain hardening region. This model determines a proportional limit (elastic limit) and gives fairly elegant dependencies between material/fitting parameters and two important factors of mechanical properties of paper: dry solids content and anisotropy. In this paper the dependency of a plastic strain on dry solids content and anisotropy is estimated using the introduced stress–strain curve model. Correspondingly, the model can be used to estimate many other mechanical behaviors, for example, the tension differences arising from non-uniform moisture content of the paper web profile. However, the main target of this study is to produce competent parameters based on modeled stress–strain curves for further construction of a material model. This elasto-plastic material model will be utilized in out-of-plane deformation and fracture models.     

Magnetoelastic fracture of soft ferromagnetic materials

Effects of magnetic field on fracture toughness of soft ferromagnetic materials were studied using experimental techniques and theoretical models. The manganese–zinc ferrite with a single-edge-notch-beam (SENB) were chosen to be the specimen and the Vickers’ indentation specimen subjected to a magnetic field were chosen to be the specimens. Results indicate that there is no significant variations of the measured fracture toughness of the manganese–zinc ferrite ceramic in the presence of the magnetic field. The theoretical model involves an anti-plane shear crack with finite length in an infinite magnetostrictive body where an in-plane magnetic field prevails at infinity. Magnetoelasticity is used. The crack-tip elastic field is different from that of the classical mode III fracture problem. Furthermore, the magnetoelastic fracture of the soft ferromagnetic material was studied by solving the stress field for a soft ferromagnetic plane with a center-through elliptical crack. The stress field at the tip of a slender elliptical crack is obtained for which only external magnetic field normal to the major axis of the ellipse is applied at infinity. The results indicate that the near field stresses are governed by the magnetostriction and permeability of the soft ferromagnetic material. The induction magnetostrictive modulus is a key parameter for finding whether magnetostriction or magnetic-force-induced deformation is dominant near the front an elliptically-shaped crack. The influence of the magnetic field on the apparent toughness of a soft ferromagnetic material with a crack-like flaw can be regarded approximately in two ways: one possesses a large induction magnetostrictive modulus and the other has a small modulus. Finally, a small-scale magnetic-yielding model was developed on the basis of linear magnetization to interpret the experimental results related to the fracture of the manganese–zinc ferrite ceramics under magnetic field. Studied also is the fracture test of the soft ferromagnetic steel with compact tension specimens published in the existing literature.     

超磁致伸缩换能器系统动力学特性分析

根据第一压磁理论建立超磁致伸缩换能器系统含有平方、立方项非线性动力学模型,利用非线性振动近似解析方法对换能器系统进行分析,揭示了换能器系统非线性特性。利用基于MAT-LAB/Simulink的系统仿真理论,建立换能器系统动态仿真模型,发现换能器系统在一定参数下存在着混沌现象。通过数值模拟的方法对换能器系统做深入的理论分析,得出系统在不同参数下存在着倍周期、倒倍周期分岔等复杂非线性现象。本文的分析结果表明,超磁致伸缩材料的弹性模量、压磁系数、相对磁导率和系统阻尼系数等参数是超磁致伸缩换能器系统动力学特性的主要影响因素。     

On stress-based piecewise elasticity for limited strain extensibility materials

A one-dimensional stress-based elasticity model with limited strain extensibility is developed in this paper, based on thermodynamics arguments. Such nonlinear elastic models can be used to model certain rubber-like and biological materials with limiting chain extensibility. The derived constitutive function is a non-smooth piecewise expression, which can be regularized for numerical or physical considerations. This non-smooth constitutive expression is derived from a Gibbs potential. A three-dimensional extension of this stress-based model is also proposed in the paper. Some simple structural examples are investigated for a bar composed of this non-smooth elastic body. A homogeneous bar composed of this new class of nonlinear elastic material that is loaded is studied for different tension states, namely for concentrated or distributed axial loading. It is shown that the displacement limit extensibility can be observed at the structural scale, with finite or infinite axial load parameters.     

A modulus gradient model for an axially loaded inhomogeneous elastic rod

A new gradient elasticity formulation is proposed for a one-dimensional linear elastic inhomogeneous rod. In the new formulation, similar to the differential relation between the local strain and the gradient enhanced strain in the classical models of gradient elasticity, a differential relation is proposed for the Young’s modulus. Analytical and finite element solutions of the proposed formulation are derived. Results of the proposed model are compared with a classical model of gradient elasticity for a model problem of carbon nanotube reinforced polymer composite.     

Wave propagation in a piezoelectric rod of 6 mm symmetry

The wave propagation in a piezoelectric rod of 6 mm symmetry is investigated by applying a 3-D piezoelectric elastic model. A self-adjoint method is introduced to solve this problem, this method avoids calculating the generalized eigenvalue equation, it completely draws the dispersion curves in the forms of Quasi-P wave, Quasi-SV wave and Quasi-SH wave under the self-adjoint boundary condition, and it can evaluate the dispersion curves of all kinds of boundary conditions. As an example, the dispersion curves of PLT-5H are completely drawn, we also found the Quasi-SV wave has standing wave phenomenon in the PLT-5H rod. In addition the relation of dispersion curves among different boundary conditions is discussed, and an experiment method is introduce to decide the dispersion curves for another boundary conditions.     

Nonlinear guided bulk waves in heterogeneous elastic structural elements

Guided nonlinear bulk strain waves are considered in heterogeneous composite materials, containing macro- or micro(nano)inclusions as the main sources of nonuniformity. Theory is based on the combination of the finite deformation and the 5-constant (Murnaghan) nonlinear elasticity, that leads under several assumptions to the only nonlinear doubly dispersive equation with variable coefficients (DDE) for a component of longitudinal strain in rod and in shell. Numerical simulations performed and experimental data are discussed, and lead to conclusions concerning dramatic influence of heterogeneity in dynamic behaviour of strain solitons in solids. We have shown how different smooth variations in elasticity may lead to either amplification or decay of strain solitons in both polymer rod and shell. These date can be useful in the NDT problems, and in problems of the solids integrity under intense elastic pulse loading.     

MODIFICATION OF ONE-DIMENSION COUPLED HYSTERESIS MODEL FOR GMM WITH THE DOMAIN FLEXING FUNCTION

This paper focuses on eliminating the unphysical negative susceptibility which ap- pears when magnetic field is at unsaturated excitation level and reduces from extremity of the hysteresis loop in one-dimension coupled hysteresis model. The domain flexing function c （H） is used to replace the domain flexing constant c in one-dimension coupled hysteresis model. The fea- sibility and rationality of proposed modification are convinced by comparing the magnetization and magnetostriction curves with experimental data and another typical modification results. The effects of pre-stress and temperature on magnetic-elastic-thermal coupling property and hysteresis behavior are investigated.