The contact problem of a rigid stamp with friction on a functionally graded magneto-electro-elastic half-plane

We consider in this study the frictional sliding contact problem between a functionally graded magneto-electro-elastic material and a perfectly conducting rigid punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the unknowns are the normal contact stress, the electric displacement and the magnetic induction. These integral equations are then solved numerically to obtain the distributions of the normal contact stress, electric displacement and magnetic induction at the surface of the graded medium. The main objective of this paper is to study the effect of the non-homogeneity parameter, the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat and circular punch profiles.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

Sensitivity of Dynamic Response of a Simply Supported Functionally Graded Magneto-electro-elastic Plate to its Elastic Parameters

Dynamic response sensitivity of a simply supported functionally graded magneto-electro-elastic plates have been studied by combining analytical method with finite element method. The functionally graded material parameters are assumed to obey exponential law in the thickness direction. A series solution of double trigonometric function agreed with the simply supported boundary condition is adopted in the plane of the plate and finite element method is used across the thickness of the plate. The finite element model is established based on energy variational principle. The coupled electromagnetic dynamic characteristics of a simply supported functionally graded magneto- electro-elastic plate are decided by its dynamics differential equation into which displacement components, electric potential and magnetic potential as nodal degree of freedom are incorporated. Dynamic response sensitivity is defined as a partial differential of dynamic response with respect to material parameter. Sensitivity of dynamic response of a simply supported functionally graded magneto-electro-elastic plate to its elastic parameters has been studied. The influence of the different exponential factor on dynamic response sensitivity has also been investigated.     

功能梯度曲梁弯曲问题的解析解

该文采用弹性力学逆解法,求得了功能梯度曲梁在端部受弯矩作用的解析解。假设弹性模量E=E₀rn沿径向呈幂函数的梯度分布。根据弹性力学平面问题的基本方程,在极坐标系下,引入应力函数,得到了弯曲问题的解析解。进而将功能梯度曲梁问题进行扩展,求得了整环或厚壁圆筒以及向错问题的解析解。将所得到的解退化到均匀弹性情况,与经典的理论解一致。最后对梯度函数按幂函数变化的算例进行了分析,结果显示梯度因子n对应力及位移的分布产生了巨大的影响。该文所得到的结论可以作为功能梯度曲梁构件优化设计的理论基础。     

Multiphysical time-dependent creep response of FGMEE hollow cylinder in thermal and humid environment

In this paper, we investigate the history of radial displacement, stresses, electric potential, and magnetic potential of a functionally graded magneto-electro-elastic (FGMEE) hollow cylinder subjected to an axisymmetric hygro-thermo-magneto-electro-mechanical loading for the plane strain condition. The material properties are taken as a power-law function of radius. Using stress-displacement relations, equations of equilibrium, electrostatic and magnetostatic equations, we find a differential equation including creep strains. Initially, eliminating creep strains, we obtain an analytical solution for the primitive stresses and electric and magnetic potential. In the next step, considering creep strains, we find the creep stress rates by applying the Norton law and Prandtl–Reuss equations for steady-state hygrothermal boundary condition. Finally, using an iterative method, we find the time-dependent creep stresses, radial displacement, and magnetic and potential field redistributions at any time. In numerical section, are comprehensively investigate the effects of grading index, hygrothermal environmental conditions, rotating speed, and temperature- and moisture-dependency of elastic constant of FGMEE.

     

On approximate solutions for the deformation of plane anisotropic beams

Plane deformation of anisotropic beams with narrow rectangular cross sections exhibits coupling of stretching, bending and transverse shearing. For anisotropic cantilever beams with a stiff end-cap under end forces and an end couple, assessments were made for approximate solutions by comparing these with numerically exact finite element (FE) solutions. Specific attention is given to point-wise or approximate satisfaction of the end-fixity conditions. As approximate methodologies, (i) the elementary polynomial form of Airy's stress function for the plane stress problem in a rectangular region, (ii) a Timoshenko-type beam theory, and (iii) the Bernoulli-Euler beam theory were selected. Among these, only the polynomial form of Airy's stress function violates the point-wise end-fixity conditions. Both the polynomial Airy stress function and the Timoshenko-type beam theory successfully model the effects of transverse shear deformation and the coupling of stretching and transverse deflection. Analytical solutions demonstrate that the normal shear coupling effect increases linearly with the thickness-to-span ratios in axial normal stress and axial displacement, while the coupling manifests quadratically in transverse displacement. The comparison of end displacements with the numerically exact FE solutions indicates that the polynomial form of Airy's stress function is no better than the Timoshenko-type beam theory. Similar conclusions were reached for the problem of uniformly loaded cantilever beams. It has been found that the accurate prediction of the deformation of thick anisotropic beams with significant normal-shear coupling requires the use of higher order theories.     

Surface effects on the bending,buckling and free vibration analysis of magneto-electro-elastic beams

Surface effects are responsible for the size dependence and should be taken into account for dielectric structures at nanoscale dimensions. By incorporating the effects of surface stress, surface piezoelectricity, surface elasticity and surface piezomagneticity, this paper investigates the bending, buckling and free vibration of magneto-electro-elastic (MEE) beams based on the Euler–Bernoulli beam theory. The governing differential equation and its corresponding boundary conditions are derived by Hamilton’s principle. The analytical solutions for the magneto-electro-elastic bending deflection, buckling magnetic potentials and frequency equations of MEE beams are obtained. In contrast to the previously published works, the positive surface stress is found to stiffen the MEE beams, as evidenced by the decrease in the deflections, the increase in the buckling magnet potentials and the increase in the resonant frequencies. Numerical studies show the importance of the surface effects, the electric and magnetic potentials and boundary conditions on the static and dynamic behavior of MEE beams. This work may be of special interest in the design and application of smart composite MEE beams.     

An exact Peano Series solution for bending analysis of imperfect layered functionally graded neutral magneto-electro-elastic plates resting on elastic foundations

In this article, a three-dimensional solution is presented for the bending analysis of functionally graded and layered neutral magneto-electro-elastic plates resting on two-parameter elastic foundations, considering imperfect interfacial bonding. The equations of motion, Gauss's equations for electrostatics and magnetostatics, and boundary and interface conditions are satisfied exactly regardless of the number of layers. No assumptions on deformations, stresses, and magnetic and electric fields along the thickness direction are introduced. The interfacial imperfection is modeled using a generalized spring layer. The state-space method is employed for solving the governing partial differential equations. Effects of a two-parameter elastic foundation, gradient index, bonding imperfection, and applied mechanical and electrical loads on the response of the functionally graded magneto-electro-elastic plate are discussed. The obtained exact solution can serve as a benchmark for assessing the accuracy of layered functionally graded magneto-electro-elastic plate theories.     

Analytical solution of a cantilever functionally graded beam

The problem of a cantilever functionally graded beam subjected to different loads is studied. In terms of Airy stress function a general two-dimensional solution is presented for a cantilever functionally graded beam, assuming that all the elastic moduli of the material have the same variations along the beam-thickness direction. Explicit expressions of analytical solutions to some specific examples under different boundary conditions are obtained to demonstrate the usefulness of the proposed general solution technique. This solution will be useful in analyzing functionally graded beam with arbitrary variations of material properties and it can serve as a basis for establishing simplified functionally graded beam theories.     

Static behavior of functionally graded magneto-electro-elastic shells under electric displacement and magnetic flux

Three-dimensional (3D) static behavior of doubly curved functionally graded (FG) magneto-electro-elastic shells under the mechanical load, electric displacement and magnetic flux is studied by an asymptotic approach. Without loss of generality, the material properties of the shells are regarded as heterogeneous through the thickness coordinate. The edge boundary conditions are considered as the full simple supports. The basic equations of 3D magneto-electro-elasticity are firstly reduced to the state-space vector equations by means of direct elimination. By introducing a geometric perturbation parameter and following the regular asymptotic expansion, we finally decompose the 3D problem as recursive sets of two-dimensional (2D) problems with governing equations of the coupled classical shell theory (CCST). It is shown that the 3D solutions can be obtained by repeatedly solving the CCST-type equations order-by-order in view of the recursive property among various order problems. Parametric studies for both the coupling magneto-electro-elastic effect and the influence of the gradient index of material properties on the structural behavior of various FG shells under magneto-electro-mechanical loads are presented.     

压电梯度薄壳的高阶理论解

压电功能梯度执行器能产生较大的位移、降低应力峰值并避免了粘结层带来的问题,压电梯度超声换能器能拓展频带宽度。本文作者提出了一个简单而有效的求解压电梯度薄壳力、电行为特性的高阶理论。设定位移分量为壳厚的线性函数,而电势沿厚度方向为二次分布。考虑了压电作动元的驱动信号不同时所具有的不同形式的电荷平衡方程。应用Fourier级数法得到压电系数沿厚度坐标变化的梯度壳的力电耦合的解析解。所得结果可退化至梁、板等多种特殊情况。利用所得方程分析了一非均匀简支压电层合板,并与三维精确结果作了对比,两者吻合得很好,表明该理论的正确性。最后具体求解了压电梯度圆柱壳的力、电特性,给出了位移、应力、电势沿厚度方向的变化规律。     

Stroh formalism based boundary integral equations for 2D magnetoelectroelasticity

This paper presents a novel approach for obtaining boundary integral equations of 2D anisotropic magnetoelectroelasticity. This approach is based on the holomorphy theorems and the Stroh formalism and allows developing of the integral equations for the aperiodic, singly and doubly periodic problems of magnetoelectroelasticity. Obtained equations contain the unknown discontinuities of displacement, electric and magnetic potentials and also traction, electric displacement and magnetic induction that allow adopting the existing boundary element procedures for their solution. Analytical solutions for systems of collinear permeable or impermeable cracks are obtained. Numerical boundary element solutions are obtained for the singly and doubly periodic sets of permeable and impermeable cracks in the magnetoelectroelastic medium and a half-plane. Comparison with analytical solutions and other available results validate the present formulations and numerical computation.     

Non-local theory solution for the anti-plane shear of two collinear permeable cracks in functionally graded piezoelectric materials

In this paper, the non-local theory of elasticity is firstly applied to obtain the behavior of two collinear cracks in functionally graded piezoelectric materials under anti-plane shear loading for permeable electric boundary conditions. To make the analysis tractable, it is assumed that the material properties vary exponentially with coordinate vertical to the crack. By means of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations that the unknown variable is the jump of the displacement across the crack surfaces. These equations are solved by use of the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present near the crack tips. The non-local elastic solutions yield finite stresses at the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses at the crack tips depend on the distance between two collinear cracks, the functionally graded parameter and the lattice parameter of the materials, respectively.     

Inelastic bending of beam including transverse shear

An incremental force–displacement relation is derived using the finite element matrix method to study the inelastic bending of beams including transverse shear. Solutions are obtained for the case of a cantilever beam with a point load and a simply supported beam with distributed loading considering and also ignoring the effect of transverse shear. They show good agreement for those cases where solutions are available. Also, the effect of transverse shear is found to be quite significant.     

湿-热-机-弹耦合FGM梁的稳定性及振动特性

研究了初始轴向机械载荷作用下Winkler-Pasternak弹性地基上功能梯度材料（FGM）梁在湿-热环境中的稳定性及振动特性。假设温度和湿度沿梁厚度方向稳态分布,材料的物性依赖于温度且按Voigt混合幂律模型连续分布。首先,基于一种扩展的n阶广义梁理论,应用Hamilton原理,统一建立了以轴向位移、弯曲变形项挠度及剪切变形项挠度为基本未知函数FGM梁的屈曲及自由振动方程,采用Navier解法获得了FGM简支梁静动态响应的精确解。其次,通过算例验证并给出了该广义梁理论阶次n的理想取值,丰富梁理论的同时,可供验证或改进其他各种剪切变形梁理论。最后,着重探讨了3种湿-热分布下湿度与温度增加、初始轴向机械载荷、跨厚比、地基刚度、梯度指标等诸多参数对FGM梁稳定性和振动特性的影响。     

Analysis of functionally graded plates by meshless method: A purely analytical formulation

Functionally graded plates under static and dynamic loads are investigated by the local integral equation method (LIEM) in this paper. Plate bending problem is described by the Reissner moderate thick plate theory. The governing equations for the functionally graded material with respect to the neutral plane are presented in the Laplace transform domain and therefore the in-plane and bending problems are uncoupled. Both isotropic and orthotropic material properties are considered. The local integral equation method is developed with the locally supported radial basis function (RBF) interpolation. As the closed forms of the local boundary integrals are obtained, there are no domain or boundary integrals to be calculated numerically in this approach. The solutions of the nodal values for the entire plate are obtained by solving a set of linear algebraic equation system with certain boundary conditions. Details of numerical procedures are presented and the accuracy and convergence characteristics of the method are examined. Several examples are presented for the functionally graded plates under static and dynamic loads and the accuracy for proposed method has been observed compared with 3D analytical solutions.     

Non-Local Theory Solution for an Anti-Plane Shear Permeable Crack in Functionally Graded Piezoelectric Materials

In this paper, the non-local theory solution of a Griffith crack in functionally graded piezoelectric materials under the anti-plane shear loading is obtained for the permeable electric boundary conditions, in which the material properties vary exponentially with coordinate parallel to the crack. The present problem can be solved by using the Fourier transform and the technique of dual integral equation, in which the unknown variable is the jump of displacement across the crack surfaces, not the dislocation density function. To solve the dual integral equations, the jump of the displacement across the crack surfaces is directly expanded in a series of Jacobi polynomials. From the solution of the present paper, it is found that no stress and electric displacement singularities are present near the crack tips. The stress fields are finite near the crack tips, thus allows us to use the maximum stress as a fracture criterion. The finite stresses and the electric displacements at the crack tips depend on the crack length, the functionally graded parameter and the lattice parameter of the materials, respectively. On the other hand, the angular variations of the strain energy density function are examined to associate their stationary value with locations of possible fracture initiation.     

A complex variable approach for asymptotic Mode-III crack-tip fields in an anisotropic functionally graded material

This paper addresses asymptotic full crack-tip fields for an anti-plane (Mode-III) stationary crack in an anisotropic functionally graded material. A monoclinic material that has a material symmetry plane is considered. The complex variable approach and the asymptotic analysis are used to solve a perturbed Laplace equation resulting from material anisotropy and gradation. The out-of-plane displacement and stress solutions are provided for a crack in exponentially and linearly graded anisotropic materials by considering material gradation either parallel or perpendicular to the crack. The characteristics of the asymptotic solutions in an anisotropic functionally graded material are compared with those for anisotropic homogeneous and isotropic graded materials. Finally, engineering significance of the present work is discussed.     

Performance of functionally graded plates under localised transverse loading

This paper presents a study of the bending of an isotropic functionally graded plate under localised transverse load through a combination of analytical and computational means. The analytical modelling is based on the recently developed three-dimensional elasticity solution, expanded to cover different loading types, whilst the Finite Element model uses graded isoparametric elements. The plate under consideration is assumed to be simply supported, with Young’s and shear moduli varying exponentially through the thickness and the Poisson’s ratio constant. Comparative analysis of stress and displacement fields in functionally graded and homogeneous plates subjected to uniformly distributed and patch loadings is carried out.     

一阶剪切理论下功能梯度梁与均匀梁静态解之间的相似关系

基于一阶剪切理论,研究了功能梯度材料Timoshenko 梁的静态弯曲解与对应的均匀材料梁的解的线性转换关系。通过比较功能梯度材料梁和均匀材料梁的无量纲控制方程,发现了它们弯曲解的线性相关性。在给定材料弹性模量沿横向非均匀变化规律后,可将功能梯度材料Timoshenko 梁在静载荷作用下的弯曲变形解用相同尺寸、相同载荷以及相同边界条件下的均匀材料Timoshenko 梁的弯曲变形解线性表示。这样,可将非均匀Timoshenko 梁弯曲问题的求解转化为对应的均匀材料Timoshenko 梁弯曲问题的求解和转换系数的计算,从而使得求解过程得以简化。