ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL—II:FINITE ELEMENT MODELING AND APPLICATION

Based on the generalized variational principle of magneto-thermo-elasticity of a ferromagnetic thin shell established(see,Analyses on nonlinear coupling of magneto-thermo-elasticity of ferromagnetic thin shell—I),the present paper developed a finite element modeling for the mechanical-magneto-thermal multi-field coupling of a ferromagnetic thin shell.The numerical modeling composes of finite element equations for three sub-systems of magnetic,thermal and deformation fields,as well as iterative methods for n...     

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.     

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.     

THEORY OF THICK-WALLED SHELLS AND ITS APPLICATION IN CYLINDRICAL SHELL

In this paper, a theory of thick-walled shells is established by means of Hellinger-Reissner's variational principle, with displacement and stress assumptions. The displacements are expanded into power series of the thickness coordinate. Only the first four and the first three terms are used for the displacements parallel and normal to the middle surface respectively. The normal extruding and transverse shear stresses are assumed to bt, cubic polynomials and to satLyfy the boundary stress conditions on the outer and inner surfaces of the shell. The governing equations and boundary conditions are derived by means of variational principle. As an example, a thick-walled cylindrical.shell is disscussed with the theory proposed. Furthermore, a photoelastic experiment has been carried out, and the results are in fair agreement with the computations.     

VIBRATION AND ACOUSTIC RADIATION FROM SUBMERGED STIFFENED SPHERICAL SHELL WITH DECK-TYPE INTERNAL PLATE

Based on the motion differential equations of vibration and acoustic coupling system for thin elastic spherical shell with an elastic plate attached to its internal surface, in which Dirac-δ functions are employed to introduce the moments and forces applied by the attachment on the surface of shell, by means of expanding field quantities as Legendre series, a semi-analytic solution is derived for the vibration and acoustic radiation from a submerged stiffened spherical shell with a deck-type internal plate, which has a satisfactory computational effectiveness and precision for an arbitrary frequency range. It is easy to analyze the effect of the internal plate on the acoustic radiation field by using the formulas obtained by the method proposed. It is concluded that the internal plate can significantly change the mechanical and acoustic characteristics of shell, and give the coupling system a very rich resonance frequency spectrum. Moreover, the method can be used to study the acoustic radiation mechanism in similar structures as the one studied here.     

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.     

Corotational nonlinear analyses of laminated shell structures using a 4-node quadrilateral flat shell element with drilling stiffness

A new 4-node quadrilateral flat shell element is developed for geometrically nonlinear analyses of thin and moderately thick laminated shell structures. The fiat shell element is constructed by combining a quadrilateral area co- ordinate method （QAC） based membrane element AGQ6- II, and a Timoshenko beam function （TBF） method based shear deformable plate bending element ARS-Q12. In order to model folded plates and connect with beam elements, the drilling stiffness is added to the element stiffness matrix based on the mixed variational principle. The transverse shear rigidity matrix, based on the first-order shear deformation theory （FSDT）, for the laminated composite plate is evaluated using the transverse equilibrium conditions, while the shear correction factors are not needed. The conventional TBF methods are also modified to efficiently calculate the element stiffness for laminate. The new shell element is extended to large deflection and post-buckling analyses of isotropic and laminated composite shells based on the element independent corotational formulation. Numerical re- sults show that the present shell element has an excellent numerical performance for the test examples, and is applicable to stiffened plates.     

THE EXACT SOLUTION FOR THE GENERAL BENDING PROBLEMS OF CONICAL SHELLS ON THE ELASTIC FOUNDATION

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell, the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived. Under the arbitrary load, and boundary conditions, the general bending problem of the conical shell on the elas     

BUCKLING AND POSTBUCKLING OF MODERATELY THICK PLATES

This paper gives the basic differential equations for finite deflections of elastic platesaccording to Reissner’s approximate stress distributions.The buckling and postbucklingproblems of elastic rectangular plates.including the effect of transverse shear deformation.are solved and discussed by using perturbation method suggested in ref.[8].Thepostbuckling equilibrium paths of perfect and imperfect moderately thick rectangularplates are presented and compared with the results based on thin plate theory.     

Nonlinear analysis on dynamic buckling of eccentrically stiffened functionally graded material toroidal shell segment surrounded by elastic foundations in thermal environment and under time-dependent torsional loads

The nonlinear analysis with an analytical approach on dynamic torsional buckling of stiffened functionally graded thin toroidal shell segments is investigated. The shell is reinforced by inside stiffeners and surrounded by elastic foundations in a thermal environment and under a time-dependent torsional load. The governing equations are derived based on the Donnell shell theory with the von K′arm′an geometrical nonlinearity,the Stein and McE lman assumption, the smeared stiffeners technique, and the Galerkin method. A deflection function with three terms is chosen. The thermal parameters of the uniform temperature rise and nonlinear temperature conduction law are found in an explicit form. A closed-form expression for determining the static critical torsional load is obtained. A critical dynamic torsional load is found by the fourth-order Runge-Kutta method and the Budiansky-Roth criterion. The effects of stiffeners, foundations, material,and dimensional parameters on dynamic responses of shells are considered.     

Vibrations of FGM thin cylindrical shells with exponential volume fraction law

In this paper, the influence of an exponential volume fraction law on the vibration frequencies of thin functionally graded cylindrical shells is studied. Material properties in the shell thickness direction are graded in accordance with the exponential law. Expressions for the strain-displacement and curvature-displacement relationships are taken from Love＇s thin shell theory. The Rayleigh-Ritz approach is used to derive the shell eigenfrequency equation. Axial modal dependence is assumed in the characteristic beam functions. Natural frequencies of the shells are observed to be dependent on the constituent volume fractions. The results are compared with those available in the literature for the validity of the present methodology.     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material(FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-ofvariables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.     

STRESS SINGULARITY ANALYSIS AT CRACK TIP ON BI-MATERIAL INTERFACES BASED ON HAMILTONIAN PRINCIPLE

In this paper, the stress singularity analysis at the crack tip on elastic bi-material inter-faces is considered. The governing equations of plane elasticity in sectorial domain are derived to be inHamiltonian form via variable substitution and variational principle. The methods of separation ofvariables and conjugate symplectic eigen-function expansion are developed to solve the equations insectorial domain. The general formulae for the solution of stress singularities at the crack tip on bi-ma-terial interfaces are put forward, and a new solution technique for fracture problems is presented.     

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.     

Nonlinear stability of double-deck reticulated circular shallow spherical shell

Based on the variational equation of the nonlinear bending theory of doubledeck reticulated shallow shells, equations of large deflection and boundary conditions for a double-deck reticulated circular shallow spherical shell under a uniformly distributed pressure are derived by using coordinate transformation means and the principle of stationary complementary energy. The characteristic relationship and critical buckling pressure for the shell with two types of boundary conditions are obtained by taking the modified iteration method. Effects of geometrical parameters on the buckling behavior are also discussed.     

Analytical model of sound transmission through laminated composite cylindrical shells considering transverse shear deformation

Composite structures are often used in the aerospace industry due to the advantages offered by a high strength to weight ratio. Sound transmission through an infinite laminated composite cylindrical shell is studied in the context of the transmission of airborne sound into the aircraft interior. The shell is immersed in an external fluid medium and contains internal fluid. Airflow in the external fluid medium moves with a constant velocity. An exact solution is obtained by simultaneously solving the first-order shear deformation theory （FSDT） of a laminated composite shell and the acoustic wave equations. Transmission losses （TL） obtained from numerical solutions are compared with those of other authors. The effects of structural properties and flight conditions on TL are studied for a range of values, especially, the Mach number, stack sequences, and the angle of warp. Additionally, comparisons of the transmission losses are made between the classical thin shell theory （CST） and FSDT for laminated composite and isotropic cylindrical shells.     

THE METHOD OF MIXED BOUNDARY CONDITION FOR A KION OF LINEAR AND NONLINEAR COMPOSITE STRUCTURE

This paper deals with the axisymmetrical deformation of shallow shells in largedeflection,which are in conjunction with linear elastic structures at the boundary.A methodof mixed boundary condition for this problem is introduced.then the problem of a compositestructure is transformed into a problem of a single structure and the integral equations aregiven.The perturbation method is used to obtain the solutions and an example of compositestructure consisting of a shallow spherical and a cylindrical shell is presented.     

GURTIN VARIATIONAL PRINCIPLE AND FINITE ELEMENT SIMULATION FOR DYNAMICAL PROBLEMS OF FLUID-SATURATED POROUS MEDIA

Based on the theory of porous media,a general Gurtin variational principle for theinitial boundary value problem of dynamical response of fluid-saturated elastic porous media isdeveloped by assuming infinitesimal deformation and incompressible constituents of the solid andfluid phase.The finite element formulation based on this variational principle is also derived.Asthe functional of the variational principle is a spatial integral of the convolution formulation,thegeneral finite element discretization in space results in symmetrical differential-integral equationsin the time domain.In some situations,the differential-integral equations can be reduced to sym-metrical differential equations and,as a numerical example,it is employed to analyze the reflectionof one-dimensional longitudinal wave in a fluid-saturated porous solid.The numerical results canprovide further understanding of the wave propagation in porous media.     

WAVELET METHOD FOR CALCULATING THE DEFECT STATES OF TWO-DIMENSIONAL PHONONIC CRYSTALS

Based on the variational theory, a wavelet-based numerical method is developed to calculate the defect states of acoustic waves in two-dimensional phononic crystals with point and line defects. The supercell technique is applied. By expanding the displacement field and the material constants （mass density and elastic stiffness） in periodic wavelets, the explicit formulations of an eigenvalue problem for the plane harmonic bulk waves in such a phononic structure are derived. The point and line defect states in solid-liquid and solid-solid systems are calculated. Comparisons of the present results with those measured experimentally or those from the plane wave expansion method show that the present method can yield accurate results with faster convergence and less computing time.