Changes in the natural frequency of a ferromagnetic rod in a magnetic field due to magnetoelastic interaction

Based on the magnetoelastic generalized variational principle and Hamilton＇s principle, a dynamic theoretical model characterizing the magnetoelastic interaction of a soft ferromagnetic medium in an applied magnetic field is developed in this paper. From the variational manipulation of magnetic scale potential and elastic displacement, all the fundamental equations for the magnetic field and mechanical deformation, as well as the magnetic body force and magnetic traction for describing magnetoelastic interaction are derived. The theoretical model is applied to a ferromagnetic rod vibrating in an applied magnetic field using a perturbation technique and the Galerkin method. The results show that the magnetic field will change the natural frequencies of the ferromagnetic rod by causing a decrease with the bending motion along the applied magnetic field where the magnetoelastic buckling will take place, and by causing an increase when the bending motion of the rod is perpendicular to the field. The prediction by the mode presented in this paper qualitatively agrees with the natural frequency changes of the ferromagnetic rod observed in the experiment.     

ANALYSIS ON THE MAGNETO-ELASTIC-PLASTIC BUCKLING/SNAPPING OF CANTILEVER RECTANGULAR FERROMAGNETIC PLATES

An analysis of buckling/snapping and bending behaviors of magneto-elastic-plastic interaction and coupling for cantilever rectangular soft ferromagnetic plates is presented.Based on the expression of magnetic force from the variational principle of ferromagnetic plates,the buckling and bending theory of thin plates,the Mises yield criterion and the increment theory for plastic deformation,we establish a numerical code to quantitatively simulate the behaviors of the nonlinearly multi-fields coupling problems by the finite element method.Along with the phenom- ena of buckling/snapping and bending,or the characteristic curve of deflection versus magnitude of applied magnetic fields being numerically displayed,the critical loads of buckling/snapping, and the influences of plastic deformation and the width of plate on these critical loads,the plastic regions expanding with the magnitude of applied magnetic field,as well as the evolvement of deflection configuration of the plate are numerically obtained in a case study.     

Combined heat and mass transfer by mixed convection MHD flow along a porous plate with chemical reaction in presence of heat source

An exact and a numerical solutions to the problem of a steady mixed convective MHD flow of an incompressible viscous electrically conducting fluid past an infinite vertical porous plate with combined heat and mass transfer are presented.A uniform magnetic field is assumed to be applied transversely to the direction of the flow with the consideration of the induced magnetic field with viscous and magnetic dissipations of energy.The porous plate is subjected to a constant suction velocity as well as a uniform mixed stream velocity.The governing equations are solved by the perturbation technique and a numerical method.The analytical expressions for the velocity field,the temperature field,the induced magnetic field,the skin-friction,and the rate of heat transfer at the plate are obtained.The numerical results are demonstrated graphically for various values of the parameters involved in the problem.The effects of the Hartmann number,the chemical reaction parameter,the magnetic Prandtl number,and the other parameters involved in the velocity field,the temperature field,the concentration field,and the induced magnetic field from the plate to the fluid are discussed.An increase in the heat source/sink or the Eckert number is found to strongly enhance the fluid velocity values.The induced magnetic field along the x-direction increases with the increase in the Hartmann number,the magnetic Prandtl number,the heat source/sink,and the viscous dissipation.It is found that the flow velocity,the fluid temperature,and the induced magnetic field decrease with the increase in the destructive chemical reaction.Applications of the study arise in the thermal plasma reactor modelling,the electromagnetic induction,the magnetohydrodynamic transport phenomena in chromatographic systems,and the magnetic field control of materials processing.     

PERTURBATION ANALYSIS FOR MAGNETO-PLASTIC INSTABILITY OF FERROMAGNETIC BEAM-PLATES WITH GEOMETRIC IMPERFECTION

The magneto-plastic instability of a ferromagnetic beam-type plate with simple supports and small initial imperfection is analytically investigated in this paper for that the plastic deformation of the plate with a linear-strain hardening relation is co…     

Magnetoelastic combined resonance and stability analysis of a ferromagnetic circular plate in alternating magnetic field

The nonlinear combined resonance problem of a ferromagnetic circular plate in a transverse alternating magnetic field is investigated. On the basis of the deformation potential energy, the strain potential energy, and the kinetic energy of the circular plate, the Hamilton principle is used to induce the magnetoelastic coupling transverse vibration dynamical equation of the ferromagnetic circular plate. Based on the basic electromagnetic theory, the expressions of the magnet force and the Lorenz force of the circular plate are presented. A displacement function satisfying clamped-edge combined with the Galerkin method is used to derive the Duffing vibration differential equation of the circular plate. The amplitude-frequency response equations of the system under various combined resonance forms are obtained by means of the multi-scale method, and the stability of the steady-state solutions is analyzed according to the Lyapunov theory.Through examples, the amplitude-frequency characteristic curves with different parameters, the amplitude of resonance varying with magnetic field intensity and excitation force,and the time-course response diagram, phase diagram, Poincar′e diagram of the system vibration are plotted, respectively. The effects of different parameters on the amplitude and stability of the system are discussed. The results show that the electromagnetic parameters have a significant effect on the multi-valued attribute and stability of the resonance solutions, and the system may exhibit complex nonlinear dynamical behavior including multi-period and quasi-periodic motion.     

A VARIATIONAL PRINCIPLE ON MAGNETOELASTIC INTERACTION OF FERROMAGNETIC THIN PLATES

The key to revealing the behaviors of magnetoelastic interaction is how to express themagnetic forces applying on a ferronqagnetic elastic body.In this paper,a functional for a ferromag-netic thin plate in magnetic fields is proposed by taking the summation of the magnetic energy of themagnetic system and the strain energy of the elastic plates.We present a variational principle for theproblem by choosing the variations of magnetic potential and deflection as independent variates eachother.Based on the principle,not only are the simultancous governing equations for magnetic fieldsand deformation of structures deduced,but also a general expression of magnetic force acting on theplates is gained,which makes it possible to commonly simulate the distinct two experiments of magne-toelastic interaction in a theoretical model.Thus,it can be used to theoretical prediction of the magne-toelastic interaction of ferromagnetic plates in a complex environment of applied magnetic fields.     

Non-linear characteristics of Rayleigh–Taylor instable perturbations

The direct numerical simulation method is adopted to study the non-linear characteristics of Rayleigh-Taylor instable perturbations at the ablation front of a 200 μm planar CH ablation target. In the simulation, the classical electrical thermal conductivity is included, and NND difference scheme is used. The linear growth rates obtained from the simulation agree with the Takabe formula. The ampli- tude distribution of the density perturbation at the ablation front is obtained for the linear growth case. The non-linear characteristics of Rayleigh-Taylor instable perturbations are analyzed and the numerical results show that the amplitude distributions of the compulsive harmonics are very different from that of the fundamental perturbation. The characteristics of the amplitude distributions of the harmonics and their fast growth explain why spikes occur at the ablation front. The numerical results also show that non-linear effects have relations with the phase differences of double mode initial perturbations, and different phase differences lead to varied spikes.     

Effects of magnetohydrodynamic flow past a vertical plate with variable surface temperature

An analysis is performed to study the magnetohydrodynamic flow of an electrically conducting, viscous incompressible fluid past a semi-infinite vertical plate with variable surface temperature under the action of transversely applied magnetic field. The heat due to viscous dissipation and the induced magnetic field are assumed to be negligible. The dimensionless governing equations are unsteady, two-dimensional, coupled and non-linear governing equations. It is found that the magnetic field parameter has a retarding effect on the velocities of air and water.     

ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL I： GENERALIZED VARIATIONAL THEORETICAL MODELING

Based on the generalized variational principle of magneto-thermo-elasticity of the ferromagnetic elastic medium, a nonlinear coupling theoretical modeling for a ferromagnetic thin shell is developed. All governing equations and boundary conditions for the ferromagnetic shell are obtained from the variational manipulations on the magnetic scalar potential, temperature and the elastic displacement related to the total energy functional. The multi-field couplings and geometrical nonlinearity of the ferromagnetic thin shell are taken into account in the modeling. The general modeling can be further deduced to existing models of the magneto-elasticity and the thermo-elasticity of a ferromagnetic shell and magneto-thermo-elasticity of a ferromagnetic plate, which are coincident with the ones in literature.     

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.     

Stability of plane-parallel flow of magnetic fluids under external magnetic fields

In this work,we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields.The fluids are assumed to be incompressible,and their magnetization is coupled to the flow through a simple phenomenological equation.Dimensionless parameters are defined,and the equations are perturbed around the base state.The eigenvalues of the linearized system are computed using a finite difference scheme and studied with respect to the dimensionless parameters of the problem.We examine the cases of both the horizontal and vertical magnetic fields.The obtained results indicate that the flow is destabilized in the horizontally applied magnetic field,but stabilized in the vertically applied field.We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters.Furthermore,we show that the superparamagnetic limit,in which the magnetization of the fluids decouples from hydrodynamics,recovers the same purely hydrodynamic critical Reynolds number,regardless of the applied field direction and of the values of the other dimensionless magnetic parameters.     

Stability of plane-parallel flow of magnetic fluids under external magnetic fields

In this work,we present a theoretical study on the stability of a two-dimensional plane Poiseuille flow of magnetic fluids in the presence of externally applied magnetic fields.The fluids are assumed to be incompressible,and their magnetization is coupled to the flow through a simple phenomenological equation.Dimensionless parameters are defined,and the equations are perturbed around the base state.The eigenvalues of the linearized system are computed using a finite difference scheme and studied with respect to the dimensionless parameters of the problem.We examine the cases of both the horizontal and vertical magnetic fields.The obtained results indicate that the flow is destabilized in the horizontally applied magnetic field,but stabilized in the vertically applied field.We characterize the stability of the flow by computing the stability diagrams in terms of the dimensionless parameters and determine the variation in the critical Reynolds number in terms of the magnetic parameters.Furthermore,we show that the superparamagnetic limit,in which the magnetization of the fluids decouples from hydrodynamics,recovers the same purely hydrodynamic critical Reynolds number,regardless of the applied field direction and of the values of the other dimensionless magnetic parameters.     

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.     

Approximate solutions of the Alekseevskii–Tate model of long-rod penetration

The Alekseevskii–Tate model is the most successful semi-hydrodynamic model applied to long-rod penetration into semi-infinite targets. However, due to the nonlinear nature of the equations, the rod(tail) velocity, penetration velocity, rod length, and penetration depth were obtained implicitly as a function of time and solved numerically By employing a linear approximation to the logarithmic relative rod length, we obtain two sets of explicit approximate algebraic solutions based on the implicit theoretica solution deduced from primitive equations. It is very convenient in the theoretical prediction of the Alekseevskii–Tate model to apply these simple algebraic solutions. In particular, approximate solution 1 shows good agreement with the theoretical(exact) solution, and the first-order perturbation solution obtained by Walters et al.(Int. J. Impac Eng. 33:837–846, 2006) can be deemed as a special form of approximate solution 1 in high-speed penetration. Meanwhile, with constant tail velocity and penetration velocity approximate solution 2 has very simple expressions, which is applicable for the qualitative analysis of long-rod penetration. Differences among these two approximate solutions and the theoretical(exact) solution and their respective scopes of application have been discussed, and the inferences with clear physical basis have been drawn. In addition, these two solutions and the first-order perturbation solution are applied to two cases with different initial impact velocity and different penetrator/target combinations to compare with the theoretical(exact) solution. Approximate solution 1 is much closer to the theoretical solution of the Alekseevskii–Tate model than the first-order perturbation solution in both cases, whilst approximate solution 2 brings us a more intuitive understanding of quasi-steady-state penetration.     

磁性靶向药物递送中铁磁流体的动力学建模

Among the proposed techniques for delivering drugs to specific locations within human body, magnetic drug targeting prevails due to its non-invasive character and its high targeting efficiency. Magnetic targeting drug delivery is a method of carrying drug-loaded magnetic nanoparticles to a target tissue target under the applied magnetic field. This method increases the drug concentration in the target while reducing the adverse side-effects. Although there have been some theoretical analyses for magnetic drug targeting, very few researchers have addressed the hydrodynamic models of magnetic fluids in the blood vessel. A mathematical model is presented to describe the hydrodynamics of ferrofiuids as drug carriers flowing in a blood vessel under the applied magnetic field. In this model, magnetic force and asymmetrical force are added, and an angular momentum equation of magnetic nanoparticles in the applied magnetic field is modeled. Engineering approximations are achieved by retaining the physically most significant items in the model due to the mathematical complexity of the motion equations. Numerical simulations are performed to obtain better insight into the theoretical model with computational fluid dynamics. Simulation results demonstrate the important parameters leading to adequate drug delivery to the target site depending on the magnetic field intensity, which coincident with those of animal experiments. Results of the analysis provide important information and suggest strategies for improving delivery in clinical application.     

COUPLING EFFECTS OF VOID SHAPE AND VOID SIZE ON THE GROWTH OF AN ELLIPTIC VOID IN A FIBER-REINFORCED HYPER-ELASTIC THIN PLATE

The growth of a prolate or oblate elliptic micro-void in a fiber reinforced anisotropic incompressible hyper-elastic rectangular thin plate subjected to uniaxial extensions is studied within the framework of finite elasticity. Coupling effects of void shape and void size on the growth of the void are paid special attention to. The deformation function of the plate with an isolated elliptic void is given, which is expressed by two parameters to solve the differential equation. The solution is approximately obtained from the minimum potential energy principle. Deformation curves for the void with a wide range of void aspect ratios and the stress distributions on the surface of the void have been obtained by numerical computation. The growth behavior of the void and the characteristics of stress distributions on the surface of the void are captured. The combined effects of void size and void shape on the growth of the void in the thin plate are discussed. The maximum stresses for the void with different sizes and different void aspect ratios are compared.     

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.     

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.     

AN ANALYSIS OF THE EFFECTS OF SYMMETRICAL TILT GRAIN BOUNDARIES ON THE PLASTIC DEFORMATION

A numerical investigation on the simple polycrystals containing threesymmetrical tilt grain boundaries(GBs)is carried out within the framework of crystalplasticity which precisely considers the finite deformation and finite lattice rotation aswell as elastic anisotropy.The calculated results show that the slip geometry and theredistribution of stresses arising from the anisotropy and boundary constraint play animportant role in the plastic deformation in the simple polycrystals.The stress levelalong GB is sensitive to the load level and misorientation,and the stresses along GB aredistributed nonuniformly.The GB may exhibit a softening or strengthening feature,which depends on the misorientation angle.The localized deformation bands usuallydevelop accompanying the GB plastic deformation,the impingement of the localizedband on the GB may result in another localized deformation band.The yield stresseswith different misorientation angles are favorably compared with the experimentalresults.     

Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials

The present paper is exposed theoretically to the influence on the dynamic stress intensity factor （DSIF） in the piezoelectric bi-materials model with two symmet- rically permeable interracial cracks near the edges of a circular cavity, subjected to the dynamic incident anti-plane shearing wave （SH-wave）. An available theoretical method to dynamic analysis in the related research field is provided. The formulations are based on Green＇s function method. The DSIFs at the inner and outer tips of the left crack are obtained by solving the boundary value problems with the conjunction and crack- simulation technique. The numerical results are obtained by the FORTRAN language program and plotted to show the influence of the variations of the physical parameters, the structural geometry, and the wave frequencies of incident wave on the dimensionless DSIFs. Comparisons with previous work and between the inner and outer tips are con- cluded.