Finite inflation of a hyperelastic toroidal membrane over a cylindrical rim

The present paper is devoted to the study of finite inflation of a hyperelastic toroidal membrane on a cylindrical rim under uniform internal pressure. Both compliant and rigid frictionless rims have been considered. The compliant cylindrical rim is modeled as a linear distributed stiffness. The initial cross-section of the torus is assumed to be circular, and the membrane material is assumed to be a homogeneous and isotropic Mooney–Rivlin solid. The problem is formulated as a two point boundary value problem and solved using a shooting method by employing the Nelder–Meads search technique. The optimization function is constructed on a two (three) dimensional search space for the compliant cylinder (rigid cylinder). The effect of the inflation pressure, material properties and elastic properties of the rim on the state of stretch and stress, and on the geometry of the inflated torus have been studied, and some interesting results have been obtained. The stability of the inflated configurations in terms of occurrence of the impending wrinkling state in the membrane has also been studied.     

On the response and stability of an inflated toroidal membrane under radial loading

Two flat annular hyperelastic membranes, stacked and bonded together at both the boundaries (equators), form a closed inflatable structure of toroidal topology. The response and stability of the inflated toroidal structure subjected to a radial line force distribution at the inner boundary are studied. The forcing is considered under constant pressure and constant amount of gas inflation conditions. Two hyperelastic models described by the corresponding relaxed strain energy density functions are considered for the membrane material. The influence of geometry, material and level of inflation on the response and stability of the structure under load has been brought out. The structure exhibits pressure limit points with increasing levels of inflation. The force–deflection (stiffness) behaviour is found to be qualitatively different below and above the pressure limit points. Below the pressure limit point, wrinkling and pull-in under loading are revealed for different inflation conditions, and the stability boundaries are determined. Under certain conditions, a counter-intuitive stretch-softening behaviour is also observed.     

悬臂铁磁板磁弹性耦合作用的力学分析

对有限板宽的悬臂铁磁板在外磁场中的磁弹性相互耦合作用的力学行为，建立了描述板弯曲和稳定性的理论模型及有限元定量分析程序，研究了外磁场倾斜角对板磁弹性失稳的临界磁场值的影响。     

Elastic instabilities for strain-stiffening rubber-like spherical and cylindrical thin shells under inflation

This paper is concerned with investigation of the effects of strain-stiffening on the classical limit point instability that is well-known to occur in the inflation of internally pressurized rubber-like spherical thin shells (balloons) and circular cylindrical thin tubes composed of incompressible isotropic non-linearly elastic materials. For a variety of specific strain-energy densities that give rise to strain-stiffening in the stress-stretch response, the inflation pressure versus stretch relations are given explicitly and the non-monotonic character of the inflation curves is examined. While such results are known for constitutive models that exhibit a gradual stiffening (e.g. exponential and power-law models), our primary focus is on materials that undergo severe strain-stiffening in the stress-stretch response. In particular, we consider two phenomenological constitutive models that reflect limiting chain extensibility at the molecular level. It is shown that for materials with sufficiently low extensibility no limit point instability occurs and so stable inflation is then predicted for such materials. Potential applications of the results to the biomechanics of soft tissues are indicated.     

Magnetoelasticity of highly deformable thin films: Theory and simulation

A non-linear two-dimensional theory is developed for thin magnetoelastic films capable of large deformations. This is derived directly from the three-dimensional theory. Significant simplifications emerge in the descent from three dimensions to two, permitting the self-field generated by the body to be computed a posteriori. The model is specialized to isotropic elastomers and numerical solutions are obtained to equilibrium boundary-value problems in which the membrane is subjected to lateral pressure and an applied magnetic field.     

Two-Dimensional Magnetoelastic Problem for a Multiply Connected Piezomagnetic Body

A general method based on complex variable theory is proposed to determine the magnetic and elastic fields of a piezomagnetic body. This method is used to derive the basic relations for complex potentials in the two-dimensional problem of magnetoelasticity, their general representations for a multiply connected domain, expressions for stresses, displacements, vectors of magnetic field intensity and magnetic flux density, and magnetic field potential. A closed-form solution is obtained for a body with an elliptic (circular) hole or crack subjected at infinity to the action of a constant magnetoelastic field. Numerical results for a piezomagnetic plate with a circular hole are presented __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 64–74, October 2005.     

横向磁场中轴向运动铁磁梁的磁弹性主共振

研究在外激励力与磁场作用下轴向运动铁磁梁的磁弹性非线性主共振问题．基于弹性理论和电磁理论,给出梁的动能和弹性势能表达式,根据哈密顿原理,推导出磁场中轴向运动铁磁梁的磁弹性双向耦合非线性振动方程．通过伽辽金积分法进行离散,得出两端简支边界条件下铁磁梁磁弹性非线性强迫振动方程．应用多尺度法对方程进行求解,得出幅频响应方程．最后通过算例,给出铁磁梁的幅频特性曲线、振幅-磁感应强度和振幅-外激励力曲线并进行分析．结果显示,在幅频响应曲线中铁磁梁的轴向运动速度、外激励力、轴向拉力越大,共振振幅越大;而磁感应强度越大,振幅越小．     

考虑非局部效应的双层纳米板的磁弹性随机振动

研究了四边简支双层纳米板在外加磁场的作用下的磁弹性随机振动问题．基于非局部弹性理论和板壳 磁弹性理论建立了系统的磁弹性随机振动方程．通过模态分析法对其进行位移响应分析,得到了通入平稳随机 电流时双层纳米板位移响应均值、功率谱密度函数等数字特征．在此基础上,分析了非局部参数、磁场强度、 板厚比等对功率谱密度的影响．结果表明,非局部参数、磁场强度、板厚比等因素的变化会影响系统的振动能 量变化及振动响应带宽分布.     

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.     

Inflation of a bonded non-linear viscoelastic toroidal membrane

The inflation of a bonded viscoelastic toroidal membrane under finite deformations is considered. Three new variables, viz. the two principal stretch ratios and the angle between the normal vector of a deformed membrane and the axis of symmetry are introduced as dependent variables. The governing equations are reduced thereafter to a set of three first-order partial differential integral equations. The constitutive equation developed by Pipkin and Rogers for the non-linear response of a viscoelastic material is used. The creep phenomenon for an inflated viscoelastic toroidal membrane under a constant pressure is presented.     

MULTI-FIELD COUPLING BEHAVIOR OF SIMPLY-SUPPORTED CONDUCTIVE PLATE UNDER THE CONDITION OF A TRANSVERSE STRONG IMPULSIVE MAGNETIC FIELD

In order to study the multi-field coupling mechanical behavior of the simply-supported conductive rectangular thin plate under the condition of an externally lateral strong impulsive magnetic field, that is the dynamic buckling phenomenon of the thin plates in the effect of the magnetic volume forces produced by the interaction between the eddy current and the magnetic fields, a FEM analysis program is developed to characterize the phenomena of magnetuelastic buckling and instability of the plates. The critical values of magnetic field for the three different initial vibrating modes are obtained, with a detailed discussion made on the effects of the length-thickness ratio a/h of the plate and the length-width ratio a/b as well as the impulse parameterτon the critical value BOcr of the applied magnetic field.     

软铁磁材料平面裂纹问题的耦合场

由磁弹性问题的线性化理论导出磁场下平面软铁磁体问题的控制方程和复势解。利用复势解和奇异积分方程方法,对面内磁场和远场载荷作用下的含裂纹无限大软铁磁平面问题进行了求解,得到耦合场的解。并对不同磁力模型的结果和磁场与机械载荷共同作用下的裂尖应力强度因子进行了讨论。     

Perturbed magnetic field of an infinite plate with a centered crack

Deforming a cracked magnetoelastic body in a magnetic field induces a perturbed magnetic field around the crack. The quantitative relationship between this perturbed field and the stress around the crack is crucial in developing a new generation of magnetism-based nondestructive testing technologies. In this paper, an analytical expression of the perturbed magnetic field induced by structural deforma- tion of an infinite ferromagnetic elastic plate containing a centered crack in a weak external magnetic field is obtained by using the linearized magnetoelastic theory and Fourier transform methods. The main finding is that the perturbed magnetic field intensity is proportional to the applied tensile stress, and is dominated by the displacement gradient on the boundary of the magnetoelastic solid. The tangential component of the perturbed magnetic-field intensity near the crack exhibits an antisymmetric distribution along the crack that reverses its direction sharply across its two faces, while the normal component shows a symmetric distribution along the crack with singular points at the crack tips.     

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.      

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     

Nonaxisymmetric magnetoelastic coupling natural vibration analysis of annular plates in an induced nonuniform magnetic field

Nonlinear Dynamics - The nonaxisymmetric magnetoelastic nonlinear coupling free vibration study is performed for a conductive thin annular plate in the nonuniform toroidal magnetic field generated...     

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.     

Closed-form solutions for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix

This paper presents two different analytical methods to investigate the magneto-mechanical coupling effect for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix. First, the magnetoelastic solution is expressed in terms of magnetoelastic Green's function that can be decoupled into elastic Green's function and magnetic Green's function. Second, the problem is analyzed by the equivalent inclusion method, and then, the formulation of the inhomogeneity problem can be decoupled into an elastic problem and a magnetic inhomogeneity problem connected by some eigenstrain and eigenmagnetic fields. For the piezomagnetic composites with a non-piezomagnetic matrix, these two solutions are completely equivalent each other though they are obtained by means of two different methods. Moreover, based upon the unified energy method, the effective magnetoelastic moduli of the composites are expressed explicitly in terms of phase properties and volume fractions. Then the dilute and Mori–Tanaka schemes are discussed, respectively. Finally, the calculations are made to predict the effective magnetoelastic moduli and illustrate the performance of each model.     

Equilibrium theory for magnetic elastomers and magnetoelastic membranes

A theory for the equilibrium response of magnetoelastic membranes under pressure and applied magnetic fields is formulated on the basis of three-dimensional magnetoelasticity. A variational principle admitted by the three-dimensional theory is used to generate a model for membranes regarded as thin three-dimensional bodies. Minimum energy considerations in the presence of applied magnetic fields are used to motivate a direct theory of magnetoelastic membranes which does not require information about bulk properties. The theory is applicable to conventional elastomers magnetized through infusion with uniformly dispersed ferrous particles.     

Infinitesimal Stability of the Equilibrium States of?an?Incompressible, Isotropic Elastic Tube Under?Pressure

The infinitesimal stability of the equilibrium states of an arbitrary incompressible, isotropic and homogeneous elastic cylindrical shell in a pure radial expansion under a constant inflation pressure is studied for both thick- and thin-walled shells. The classical criterion of infinitesimal stability yields a general stability theorem relating the frequency and pressure response and reveals that points at which the pressure is stationary define the domain of unstable or neutrally stable states. All results are expressed in terms of a general shear response function, and specific results are provided for the Mooney-Rivlin, Gent and Ogden models, the second having limited extensibility, the last including experimental data. Every static state of a Mooney-Rivlin tube is stable so long as the pressure is less than an asymptotic limit that increases with the thickness. Otherwise, only the Ogden model exhibits static states of instability for all long cylindrical tubes of thickness less than a transitional value above which all static states are infinitesimally stable. A long cylindrical cavity in all three unbounded models is stable for all pressures. All results are illustrated graphically.