Dynamic responses of functionally graded magneto-electro-elastic shells with closed-circuit surface conditions using the method of multiple scales

A three-dimensional (3D) free vibration analysis of simply supported, doubly curved functionally graded (FG) magneto-electro-elastic shells with closed-circuit surface conditions is presented using the method of perturbation. By means of the direct elimination, we firstly reduce the twenty-nine basic equations of 3D magneto-electro-elasticity to ten differential equations in terms of ten primary variables of magnetic, electric and elastic fields. The method of multiple scales is introduced to eliminate the secular terms in various order problems of the present formulation so that the present asymptotic expansion to the primary field variables leads to be uniform and feasible. Through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of governing equations for various order problems. The coupled classical shell theory (CST) is derived as a first-order approximation to the 3D magneto-electro-elasticity. Higher-order modifications can be further determined by considering the solvability and orthonormality conditions in a systematic and consistent way. Some benchmark solutions for the free vibration analysis of FG elastic and piezoelectric plates are used to validate the performance of the present asymptotic formulation. The influence of the material-property gradient index on the natural frequencies and corresponding modal field variables of the FG shells is mainly concerned.     

Exact electroelastic analysis of functionally graded piezoelectric shells

A paper focuses on implementation of the sampling surfaces (SaS) method for the three-dimensional (3D) exact solutions for functionally graded (FG) piezoelectric laminated shells. According to this method, we introduce inside the nth layer I_n not equally spaced SaS parallel to the middle surface of the shell and choose displacements and electric potentials of these surfaces as basic shell variables. Such choice of unknowns yields, first, a very compact form of governing equations of the FG piezoelectric shell formulation and, second, allows the use of strain–displacement equations, which exactly represent rigid-body motions of the shell in any convected curvilinear coordinate system. It is worth noting that the SaS are located inside each layer at Chebyshev polynomial nodes that leads to a uniform convergence of the SaS method. As a result, the SaS method can be applied efficiently to 3D exact solutions of electroelasticity for FG piezoelectric cross-ply and angle-ply shells with a specified accuracy by using a sufficient number of SaS.     

Free vibration of functionally graded truncated conical shells under internal pressure

The influence of internal pressure on the free vibration behavior of functionally graded (FG) truncated conical shells are investigated based on the first-order shear deformation theory (FSDT) of shells. The initial mechanical stresses are obtained by solving the static equilibrium equations. Using Hamilton’s principle and by including the influences of initial stresses, the free vibration equations of motion around this equilibrium state together with the related boundary conditions are derived. The material properties are assumed to be graded in the thickness direction. The differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to discretize the governing equations and the related boundary conditions. The convergence behavior of the method is numerically investigated and its accuracy is demonstrated by comparing the results in the limit cases with existing solutions in literature. Finally, the effects of internal pressure together with the material property graded index, the semi-vertex angle and the other geometrical parameters on the frequency parameters of the FG truncated conical shells subjected to different boundary conditions are studied.     

Three-dimensional general magneto-electro-elastic finite element model for multiphysics nonlinear analysis of layered composites

In this paper, by defining a general potential energy for the multiphase coupled multiferroics and applying the minimum energy principle, the coupled governing equations are derived. This system of equations is then discretized as a general three-dimensional(3D) finite element(FE) model based on the COMSOL software. After validating the formulation, it is then applied to the analysis and design of the common sandwich structure of multiferroics composites. Under the typical static loading, the ef...     

NONLINEAR TRANSIENT RESPONSE OF FUNCTIONALLY GRADED SHALLOW SPHERICAL SHELLS SUBJECTED TO MECHANICAL LOAD AND UNSTEADY TEMPERATURE FIELD

This paper is concerned with the transient deformation of functionally graded （FG） shallow spherical shells subjected to time-dependent thermomechanical load. Based on Timoshenko- Mindlin hypothesis and yon Karman nonlinear theory, a set of nonlinear governing equations of motion for FG shallow spherical shells in regard to transverse shear deformation and all the inertia terms are established using Hamilton＇s principle. The collocation point method and Newmark- beta scheme in conjunction with the finite difference method are adopted to solve the governing equations of motion and the unsteady heat conduction equation numerically. In the numerical examples, the transient deflection and stresses of FG shallow spherical shells with various material properties under different loading conditions are presented.     

Multi-field Response of Anisotropic Magneto-Electro-Elastic Materials at a Rigid Conducting Cylinder

Contact problem of anisotropic magneto-electro-elastic materials indented by a perfectly conducting cylindrical punch is investigated based on a complete coupling theory. The 12 cases of the distinctive eigenvalue distribution of the related governing equations are detailed. For 3 available eigenvalue distribution cases involving semi-infinite anisotropic magneto-electro-elastic materials, real fundamental solutions are provided. A system of singular integral equations is obtained and solved exactly. The explicit expressions for the coupled magneto-electro-elastic fields in the half-plane are presented in the form of elementary functions. Figures are plotted to show the effects of various parameters, such as the volume fraction of the piezoelectric phase, on the contact behaviors. In-depth analyses are given to explain how the various parameters cause the contact properties to change and develop. Connections between the present study and practical application are presented. This article may greatly benefit the experimental and numerical test involving magneto-electro-elastic materials.     

Exact solutions of functionally graded piezoelectric shells under cylindrical bending

Within a framework of the three-dimensional (3D) piezoelectricity, we present asymptotic formulations of functionally graded (FG) piezoelectric cylindrical shells under cylindrical bending type of electromechanical loads using the method of perturbation. Without loss of generality, the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constants in single-layer homogeneous shells and to obey an identical exponent-law in FG shells. The transverse normal load and normal electric displacement (or electric potential) are, respectively, applied on the lateral surfaces of the shells. The cylindrical shells are considered to be fully simple supports at the edges in the circumferential direction and with a large value of length in the axial direction. The present asymptotic formulations are applied to several benchmark problems. The coupled electro-elastic effect on the structural behavior of FG piezoelectric shells is evaluated. The influence of the material property gradient index on the variables of electric and mechanical fields is studied.     

A sampling surfaces method and its application to three-dimensional exact solutions for piezoelectric laminated shells

The application of the sampling surfaces (SaS) method to piezoelectric laminated composite plates is presented in a companion paper (Kulikov, G.M., Plotnikova, S.V., Three-dimensional exact analysis of piezoelectric laminated plates via sampling surfaces method. International Journal of Solids and Structures 50, http://dx.doi.org/10.1016/j.ijsolstr.2013.02.015). In this paper, we extend the SaS method to shells to solve the static problems of three-dimensional (3D) electroelasticity for cylindrical and spherical piezoelectric laminated shells. For this purpose, we introduce inside the nth layer I_n not equally spaced SaS parallel to the middle surface of the shell and choose displacements of these surfaces as basic kinematic variables. Such choice of displacements permits, first, the presentation of governing equations of the proposed piezoelectric shell formulation in a very compact form and, second, gives an opportunity to utilize the strain–displacement equations, which precisely represent all rigid-body shell motions in any convected curvilinear coordinate system. It is shown that the developed piezoelectric shell formulation can be applied efficiently to finding of 3D exact solutions for piezoelectric cross-ply and angle-ply shells with a specified accuracy using a sufficient number of SaS, which are located at Chebyshev polynomial nodes and layer interfaces as well.     

Bending and free vibrational analysis of bi-directional functionally graded beams with circular cross-section

The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified.     

Bending analysis of five-layer curved functionally graded sandwich panel in magnetic field: closed-form solution

In this paper,an exact closed-form solution for a curved sandwich panel with two piezoelectric layers as actuator and sensor that are inserted in the top and bottom facings is presented.The core is made from functionally graded(FG)material that has heterogeneous power-law distribution through the radial coordinate.It is assumed that the core is subjected to a magnetic field whereas the core is covered by two insulated composite layers.To determine the exact solution,first characteristic equations are derived for different material types in a polar coordinate system,namely,magneto-elastic,elastic,and electro-elastic for the FG,orthotropic,and piezoelectric materials,respectively.The displacement-based method is used instead of the stress-based method to derive a set of closed-form real-valued solutions for both real and complex roots.Based on the elasticity theory,exact solutions for the governing equations are determined layer-by-layer that are considerably more accurate than typical simplified theories.The accuracy of the presented method is compared and validated with the available literature and the finite element simulation.The effects of geometrical and material parameters such as FG index,angular span along with external conditions such as magnetic field,mechanical pressure,and electrical difference are investigated in detail through numerical examples.     

Vibration characteristics of piezoelectric functionally graded carbon nanotube-reinforced composite doubly-curved shells

This paper presents an analytical solution for the free vibration behavior of functionally graded carbon nanotube-reinforced composite(FG-CNTRC) doubly curved shallow shells with integrated piezoelectric layers. Here, the linear distribution of electric potential across the thickness of the piezoelectric layer and five different types of carbon nanotube(CNT) distributions through the thickness direction are considered. Based on the four-variable shear deformation refined shell theory, governing equations are obtained by applying Hamilton's principle. Navier's solution for the shell panels with the simply supported boundary condition at all four edges is derived. Several numerical examples validate the accuracy of the presented solution. New parametric studies regarding the effects of different material properties, shell geometric parameters, and electrical boundary conditions on the free vibration responses of the hybrid panels are investigated and discussed in detail.     

Local boundary integral equations for orthotropic shallow shells

A meshless local Petrov–Galerkin (MLPG) formulation is presented for bending problems of shear deformable shallow shells with orthotropic material properties. Shear deformation of shells described by the Reissner theory is considered. Analyses of shells under static and dynamic loads are given here. For transient elastodynamic case the Laplace-transform is used to eliminate the time dependence of the field variables. A weak formulation with a unit test function transforms the set of governing equations into local integral equations on local subdomains in the plane domain of the shell. Nodal points are randomly spread in that domain and each node is surrounded by a circular subdomain to which local integral equations are applied. The meshless approximation based on the moving least-squares (MLS) method is employed for the implementation. Unknown Laplace-transformed quantities are computed from the local boundary integral equations. The time-dependent values are obtained by the Stehfest’s inversion technique.     

Free vibrations of multilayered doubly curved shells based on a mixed variational approach and global piecewise-smooth functions

This paper presents the extension of a two-dimensional model that, recently appeared in literature, deals with freely vibrating laminated plates. The extension takes into account the corresponding theory describing the dynamic of freely vibrating multilayered doubly curved shells. The relevant governing differential equations, associated boundary conditions and constitutive equations are derived from one of Reissner’s mixed variational theorems. Both the governing differential equations and the related boundary conditions are presented in terms of resultant stresses and displacements. In spite of the multi-layer nature of the shell, the theory is developed as if the shell were virtually made of a single layer. This choice does not limit the performances of the model, which are comparable to the corresponding three-dimensional theory. This ability is accomplished by an appropriate global expansion of the relevant kinetic and stress quantities, through the thickness of the multilayered shell. The mentioned expansion is realized by a novel selection of global piecewise-smooth functions. Numerical tests illustrate the performance of the model with respect to several elements subjected to a class of simply supported boundary conditions: plates, circular cylindrical shells, spherical and saddle-shape laminates. The model is first tested by comparing its resulting eigen-parameters, with those few existing of exact or approximate three-dimensional models and, finally, new results are provided for several geometrical and material characteristics for plates and shells.     

旋转薄壳轴对称自由振动的转点问题

在用渐近法求解任意旋转薄壳(圆柱壳和球壳除外)的轴对称自由振动方程时,在一定的频率参数范围内,存在转点问题。其中,对于存在唯一简单转点的情况,至今未获解决。本文解决了这一问题。     

Thermally induced dynamic instability of laminated composite conical shells

Thermally induced dynamic instability of laminated composite conical shells is investigated by means of a perturbation method. The laminated composite conical shells are subjected to static and periodic thermal loads. The linear instability approach is adopted in the present study. A set of initial membrane stresses due to the elevated temperature field is assumed to exist just before the instability occurs. The formulation begins with three-dimensional equations of motion in terms of incremental stresses perturbed from the state of neutral equilibrium. After proper nondimensionalization, asymptotic expansion and successive integration, we obtain recursive sets of differential equations at various levels. The method of multiple scales is used to eliminate the secular terms and make an asymptotic expansion feasible. Using the method of differential quadrature and Bolotin's method, and imposing the orthonormality and solvability conditions on the present asymptotic formulation, we determine the boundary frequencies of dynamic instability regions for various orders in a consistent and hierarchical manner. The principal instability regions of cross-ply conical shells with simply supported–simply supported boundary conditions are studied to demonstrate the performance of the present asymptotic theory.     

Solution of generalized coordinate for warping for naturally curved and twisted beams

A theoretical method for static analysis of naturally curved and twisted beams under complicated loads was presented, with special attention devoted to the solving process of governing equations which take into account the effects of torsion-related warping as well as transverse shear deformations. These governing equations, in special cases, can be readily solved and yield the solutions to the problem. The solutions can be used for the analysis of the beams, including the calculation of various internal forces, stresses, strains and displacements. The present theory will be used to investigate the stresses and displacements of a plane curved beam subjected to the action of horizontal and vertical distributed loads. The numerical results obtained by the present theory are found to be in very good agreement with the results of the FEM results. Besides, the present theory is not limited to the beams with a double symmetric cross-section, it can also be extended to those with arbitrary cross-sectional shape.     

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.     

Nonlinear vibrations of orthotropic shallow shells of revolution

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated.following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.     

3D coupled field in a transversely isotropic magneto-electro-elastic half space punched by an elliptic indenter

The present paper is concerned with three-dimensional (3D) coupled field in a transversely isotropic magneto-electro-elastic half space punched by a rigid flat-ended elliptic indenter. Closed form solutions and corresponding numerical results are presented in this work, in a systematic manner. The material in question is transversely isotropic with the axis of symmetry normal to the surface of the half space. The indenter is assumed to be either electrically and magnetically conducting or insulating. Corresponding boundary integral equations (BIEs), to indenter with different magneto-electric properties, are solved by virtue of the method of generalized potential theory. For all four physical cases, corresponding coupled magneto-electro-elastic fields in the half space are obtained. The present analytical solutions indicate that the indentation forces and stiffness may be written as intrinsic combinations of a physical factor and a geometrical factor. The present study is an extension of the previous work on circular punch, and may find applications in guiding future indentation experiments.