A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

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.     

A physically based model for stress sensing using magnetostrictive composites

Magnetostrictive composites are of considerable interest for real-time remote force sensing and structural health monitoring. In this paper, we introduce a new procedure for modeling the magnetic field induced by an external load applied on an epoxy-based composite material filled with Terfenol-D particles. This model is based on an assumed sequence of physical processes that occur at the microscopic scale, and it includes both domain switching and magnetization rotation. The modeling procedure is demonstrated on a problem relevant for load sensing applications in which the magnetostrictive composite is subjected to a uniaxial compression. Comparison of the calculated and experimental results strengthens the validity of the assumed sequence of physical processes and provides valuable insights important for application developments.     

Interfacial stiffness dependence of the effective magnetostriction of particulate magnetostrictive composites

Terfenol-D composites attract much attention recently due to their large magnetostriction, small eddy energy loss and large operation frequency bandwidth. Binder layer in the composite usually mechanically weakens the composite and reduces the effective properties. A typical kind of magnetostictive composite is composed of Rare Earth metallic compound powder, matrix material and resin binder. The binder, which is usually flexible and forms mechanically weak interface in the composite, inevitably influences the overall magnetostriction of composites. In this paper, a theoretical model was developed to treat a simple deformation case of this kind of mechanically weak interface, in which the flexible layer has low stiffness to withstand deformation but no de-bonding or cracking. An infinite magnetostrictive plane with a circular inclusion was considered, where the matrix and inclusion are all general magnetostrictive materials which can be modeled by the standard square constitutive relation of magnetostriction. The binder layer of a certain thickness was modeled as a set of springs with no thickness but with an equivalent stiffness. The mathematical formulation was brought into the complex variable framework. The magnetoelastic field was solved and the effective magnetostriction was explicitly obtained. Comparisons with experimental results were also presented. In terms of this analysis, the interfacial stiffness has significant influences on the overall magnetostriction of composite. Increasing the interfacial stiffness can lead to large magnetostriction of composites. The measure for improving the interfacial stiffness includes increasing the binder modulus and reducing its thickness.     

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.     

Non-linear response of laminated composite plates under thermomechanical loading

The non-linear response of laminated composite plates under thermomechanical loading is studied using the third-order shear deformation theory (TSDT) that includes classical and first-order shear deformation theories (CLPT and FSDT) as special cases. Geometric non-linearity in the von Kármán sense is considered. The temperature field is assumed to be uniform in the plate. Layers of magnetostrictive material, Terfenol-D, are used to actively control the center deflection. The negative velocity feedback control is used with the constant gain value. The effects of lamination scheme, magnitude of loading, layer material properties, and boundary conditions are studied under thermomechanical loading.     

A small-scale magnetic-yielding model for an infinite magnetostrictive plane with a crack-like flaw

The effect of an external magnetic field on the fracture toughness of magnetostrictive materials has been investigated by determining the local stress fields around the tip of a very slender elliptical flaw embedded in an infinite magnetostrictive plane subjected to magnetic loading, based on the assumption of linear magnetization. In this paper, the above-mentioned analytical approach is extended to develop a small-scale magnetic-yielding model. The magnetic saturation zone is constructed and the distributions of magnetic field and magnetization are obtained around the tip of a slender elliptical crack. Based on the complex potential theory, the stress field is obtained in the vicinity of the tip of the slender elliptical crack by implementing the continuity conditions of displacement and resultant force at the interface between the magnetic saturation and magnetoelastic zones. The stress fields near the tip of the slender elliptical crack are obtained for two kinds of soft ferromagnetic materials each with a small induction magnetostrictive modulus. The theoretical results obtained show that the stresses in the neighborhood of a crack-tip are finite even when the elliptical crack reduces to a sharp crack, and are much smaller than the yield stress or the nominal fracture stress of the material. This suggests that, generally, the magnetic field has no obvious effects on the apparent fracture toughness of soft ferromagnetic materials, which is in agreement with the existing experimental results published in the existing literature. In addition, the theoretical analysis illustrates that no crack is magnetically impermeable, and the corresponding boundary conditions are inappropriate for fracture analysis of soft ferromagnetic materials.     

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.     

Experiments and modeling of iron-particle-filled magnetorheological elastomers

Magnetorheological elastomers (MREs) are ferromagnetic particle impregnated rubbers whose mechanical properties are altered by the application of external magnetic fields. Due to their coupled magnetoelastic response, MREs are finding an increasing number of engineering applications. In this work, we present a combined experimental and theoretical study of the macroscopic response of a particular MRE consisting of a rubber matrix phase with spherical carbonyl iron particles. The MRE specimens used in this work are cured in the presence of strong magnetic fields leading to the formation of particle chain structures and thus to an overall transversely isotropic composite. The MRE samples are tested experimentally under uniaxial stresses as well as under simple shear in the absence or in the presence of magnetic fields and for different initial orientations of their particle chains with respect to the mechanical and magnetic loading direction.Using the theoretical framework for finitely strained MREs introduced by Kankanala and Triantafyllidis (2004), we propose a transversely isotropic energy density function that is able to reproduce the experimentally measured magnetization, magnetostriction and simple shear curves under different prestresses, initial particle chain orientations and magnetic fields. Microscopic mechanisms are also proposed to explain (i) the counterintuitive effect of dilation under zero or compressive applied mechanical loads for the magnetostriction experiments and (ii) the importance of a finite strain constitutive formulation even at small magnetostrictive strains. The model gives an excellent agreement with experiments for relatively moderate magnetic fields but has also been satisfactorily extended to include magnetic fields near saturation.     

铁电陶瓷PZT53复杂力电耦合行为的实验研究

通过实验研究了平行和垂直于极化方向的正应力对铁电陶瓷锆钛酸铅(PZT53)的电滞回线(E3-P3)和电致应变曲线(E-ε)的影响. 实验发现平行于极化方向的压应力对PZT53陶瓷的电滞回线、电致应变曲线形状以及矫顽场大小都有明显的影响,但是垂直于极化方向的拉、压应力只对PZT53陶瓷的电致应变曲线形状有明显的影响,但对电滞回线形状和矫顽场大小都没有显著影响. 采用畴翻转的模型详细解释了观察到的实验现象,所得结果为建立铁电陶瓷的多轴力、电耦合本构模型,提供了物理基础.     

软铁磁薄板磁弹性屈曲的理论模型

铁磁弹性薄板的磁弹性屈曲问题一直作为电磁——弹性力学相互作用的一个基本模型进行研究，而作用在其磁介质上的磁力计算则是定量理论预测准确与否的关键．到目前为止，文献上已有的理论模型对悬臂铁磁梁式悬臂板在横向磁场中磁弹性屈曲的理论预测值始终高于实验值，有的甚至相差１００％左右．本文基于电磁力计算的微观安培电流模型，严格给出了软铁磁薄板等效横向磁力的宏观计算表达式．在此基础上，建立了电磁——力学相互耦合作用的非线性理论模型．该模型能描述铁磁薄板结构在非均匀横向磁场环境中的磁弹性失稳（或屈曲）．其定量分析采用了有限元法和有限差分法相结合．数值结果显示：本模型给出的磁弹性屈曲的临界磁场值与实验值符合良好．与此同时，文中还对文献中认为较成功的Ｍｏｏｎ－Ｐａｏ模型的基本假设进行了分析．定量结果发现：Ｍｏｏｎ－Ｐａｏ理论模型的基本假设仅在梁式板的长厚比Ｌ／ｈ比较大时（约在２００左右），是可以接受的，而当Ｌ／ｈ较小时，该假设将导致理论值与实验值的较大误差．Ｌ／ｈ比值越小，理论值与实验值的误差越大     

A theoretical model of magnetoelastic bucking for soft ferromagnetic thin plates

As an essential model of magnetoelastic interaction between magnetic field and mechanical deformation, the study on magnetoelastic buckling phenomenon of soft ferromagnetic plates in a magnetic environment has been conducted. One of the key steps for the theoretical prediction of the critical magnetic field is how to formulate magnetic force exerted on the magnetized medium. Till today, the theoretical predictions, from theoretical models in publications, of the magnetoelastic buckling of ferromagnetic cantilevered beam-plate in transverse magnetic field are all higher than their experimental data. Sometimes, the discrepancy between them is as high as 100%. In this paper, the macroscope formulation of the magnetic forces is strictly obtained from the microscope Amperion current model. After that, a new theoretical model is established to describe the magnetoelastic buckling phenomenon of ferromagnetic thin plates with geometrically nonlinear deformation in a nonuniform transverse magnetic field. The numerical method for quantitative analysis is employed by combining the finite elemental method for magnetic fields and the finite difference method for deformation of plates. The numerical results obtained from this new theoretical model show that the theoretical predictions of critical values of the buckling magnetic field for the ferromagnetic cantilevered beam-plate are in excellent agreement with their experimental data. By the way, the region of applicability to the Moon-Pao's model, or the couple model, is checked by quantitative results. This project was supported in part by the National Natural Science Foundation of China and the Foundation of the SEdC of China for Returned Chinese Scholars from Abroad.     

Magnetoelastic problem for a bodywith periodic elastic inclusions

A general approach based on complex variable theory is proposed to determine the magnetoelastic state of a body with an infinite row of elliptic inclusions under the action of magnetic and elastic fields. Numerical solutions to a two-dimensional problem for a body made of Terfenol-D magnetostrictive material and piezomagnetic ceramic material and having circular, elliptic, and rectilinear inclusions made of a different material are presented depending on the geometry of the inclusions, their material characteristics, the spacing between them, and the type of applied load __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 9, pp. 32–40, September 2006.     

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.     

Incremental Magnetoelastic Deformations,with Application to Surface Instability

In this paper the equations governing the deformations of infinitesimal (incremental) disturbances superimposed on finite static deformation fields involving magnetic and elastic interactions are presented. The coupling between the equations of mechanical equilibrium and Maxwell’s equations complicates the incremental formulation and particular attention is therefore paid to the derivation of the incremental equations, of the tensors of magnetoelastic moduli and of the incremental boundary conditions at a magnetoelastic/vacuum interface. The problem of surface stability for a solid half-space under plane strain with a magnetic field normal to its surface is used to illustrate the general results. The analysis involved leads to the simultaneous resolution of a bicubic and vanishing of a 7×7 determinant. In order to provide specific demonstration of the effect of the magnetic field, the material model is specialized to that of a “magnetoelastic Mooney–Rivlin solid”. Depending on the magnitudes of the magnetic field and the magnetoelastic coupling parameters, this shows that the half-space may become either more stable or less stable than in the absence of a magnetic field.