Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat 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 main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of 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 stamp profile.     

磁电弹复合材料和刚性导电导磁圆柱压头的二维微动接触分析

本文求解平面应变状态下磁电弹复合材料半平面和刚性导电导磁圆柱压头的二维微动接触问题。假设压头具有良好的导电导磁性,且表面电势和磁势是常数。微动接触依赖载荷的加载历史,所以首先求解单独的法向加载问题,然后在法向加载问题的基础上求解循环变化的切向加载问题。整个接触区可以分为内部的中心粘着区和两个外部的滑移区,其中滑移区满足Coulomb摩擦法则。利用Fourier积分变换,磁电弹半平面的微动接触问题将简化为耦合的Cauchy奇异积分方程组,然后数值离散为线性代数方程组,利用迭代法求解未知的粘着/滑移区尺寸、电荷分布、磁感应强度、法向接触压力和切向接触力。数值算例给出了摩擦系数、总电荷和总磁感应强度对各加载阶段的表面接触应力、电位移和磁感应强度的影响。     

Electro-mechanical frictionless contact behavior of a functionally graded piezoelectric layered half-plane under a rigid punch

The frictionless contact problem of a functionally graded piezoelectric layered half-plane in-plane strain state under the action of a rigid flat or cylindrical punch is investigated in this paper. It is assumed that the punch is a perfect electrical conductor with a constant potential. The electro-elastic properties of the functionally graded piezoelectric materials (FGPMs) vary exponentially along the thickness direction. The problem is reduced to a pair of coupled Cauchy singular integral equations by using the Fourier integral transform technique and then is numerically solved to determine the contact pressure, surface electric charge distribution, normal stress and electric displacement fields. For a flat punch, the normal stress intensity factor and electric displacement intensity factor are also given to quantitatively characterize the singularity behavior at the punch ends. Numerical results show that both material property gradient of the FGPM layer and punch geometry have a significant influence on the contact performance of the FGPM layered half-plane.     

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.     

Static analysis of simply supported functionally graded and layered magneto-electro-elastic plates

In this article, static analysis of functionally graded, anisotropic and linear magneto-electro-elastic plates have been carried out by semi-analytical finite element method. A series solution is assumed in the plane of the plate and finite element procedure is adopted across the thickness of the plate such a way that the three-dimensional character of the solution is preserved. The finite element model is derived based on constitutive equation of piezomagnetic material accounting for coupling between elasticity, electric and magnetic effect. The present finite element is modeled with displacement components, electric potential and magnetic potential as nodal degree of freedom. The other fields are calculated by post-computation through constitutive equation. The functionally graded material is assumed to be exponential in the thickness direction. The numerical results obtained by the present model are in good agreement with available functionally graded three-dimensional exact benchmark solutions given by Pan and Han [Pan, E., Han, F., in press. Green’s function for transversely isotropic piezoelectric functionally graded multilayered half spaces. Int. J. Solids Struct.]. Numerical study includes the influence of the different exponential factor, magneto-electro-elastic properties and effect of mechanical and electric type of loading on induced magneto-electro-elastic fields. In addition further study has been carried out on non-homogeneous transversely isotropic FGM magneto-electro-elastic plate available in the literature [Chen, W.Q., Lee, K.Y., Ding, H.J., 2005. On free vibration of non-homogeneous transversely isotropic magneto-electro-elastic plates].     

Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch

This paper presents the fundamental contact solutions of a magneto-electro-elastic half-space indented by a smooth and rigid half-infinite punch. The material is assumed to be transversely isotropic with the symmetric axis perpendicular to the surface of the half-space. Based on the general solutions, the generalized method of potential theory is adopted to solve the boundary value problems. The involved potentials are properly assumed and the corresponding boundary integral equations are solved by using the results in literature. Complete and exact fundamental solutions are derived case by case, in terms of elementary functions for the first time. The obtained solutions are of significance to boundary element analysis, and an important role in determining the physical properties of materials by indentation technique can be expected to play.     

Inclined circular flat punch on a transversely isotropic piezoelectric half-space

Summary Utilizing the general solution of transversely isotropic piezoelectricity, the paper analyzes the problem of an inclined rigid circular flat punch indenting a transversely isotropic piezoelectric half-space. The potential theory method is employed and generalized to take into account the effect of the electric field in piezoelectric materials. Assuming that the punch is maintained at a constant electric potential, exact expressions for the elastoelectric field are derived in terms of elementary functions. It is noted that the solution corresponding to a flat circular punch centrally loaded by a concentrated force can be obtained as a special case. Received 15 December 1998; accepted for publication 9 March 1999     

Exact solution of a flat smooth punch on a piezoelectric half plane

This paper derives an exact solution for a flat smooth punch applied on a piezoelectric half plane. The piezoelectric solid occupies the lower half plane, and a flat rigid punch is applied on it. As the permittivity of the air (environment) is far less than that of the piezoelectric material, the electric induction of air may be neglected. The permittivity of the punch is also far less than that of the piezoelectric material and consequently the normal component of the electric displacement vanishes at the contact boundary. The exact solution is obtained by eigen-function representation and analytic continuation. The distribution of pressure under the punch has been found. The electric field along the surface of the lower half plane is extracted in a closed form.     

Dynamic analysis of a penny-shaped dielectric crack in a magnetoelectroelastic solid under impacts

The transient response of a magneto-electro-elastic material with a penny-shaped dielectric crack subjected to in-plane magneto-electro-mechanical impacts is made. To simulate an opening crack with a dielectric interior, the crack-face electromagnetic boundary conditions are supposed to depend on the crack opening displacement and the jumps of electric and magnetic potentials across the crack. Four ideal crack-face electromagnetic boundary conditions involving a combination of electrically permeable or impermeable and magnetically permeable or impermeable assumptions can be reduced. The Laplace and Hankel transform techniques are further utilized to solve the mixed initial-boundary-value problem. Three coupling Fredholm integral equations are obtained and solved by the composite Simpson's rule. Dynamic field intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD), electric potential and magnetic potential are given in the Laplace transform domain. By means of a numerical inversion of the Laplace transform, numerical results are calculated to show the variations of the physical parameters of concern versus the normalized time in graphics. The effects of applied electric and magnetic loads on the dynamic intensity factors of stress and COD, and the dynamic energy release rate for a BaTiO₃-CoFe₂O₄ composite with a penny-shaped vacuum crack are discussed in detail.     

Closed-form solutions for two anti-plane collinear cracks in a magnetoelectroelastic layer

In this paper we develop closed form solutions for anti-plane mechanical and in-plane electric and magnetic fields for two collinear cracks in magneto-electro-elastic layer of finite thickness under the conditions of permeable crack faces using integral transform method. The anti-plane mechanical shear or displacement and in-plane electrical and magnetic loading are applied to the top and bottom surfaces of the layer for the two cases considered. Expressions for shear stresses, electric displacements and magnetic inductions in the vicinity of the cracks are derived as well as intensity factors for two cracks in magneto-electro-elastic layer. Numerical results for stress intensity factors and energy release rate are shown graphically.     

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.     

刚性圆形压头作用下热电材料半平面的无摩擦接触问题

热电材料可以将热能转化为电能,反之亦然,这一优良的性质将有助于研发更具成本效益的设备和器件。本文研究了刚性圆形压头作用在热电材料半平面的无摩擦接触问题。假定压头为电导体、热导体,且压头压入深度及与材料的接触区域宽度未知。首先求解电场和温度场,利用傅里叶变换得到了电势函数、温度、电流密度和能量通量的解析表达式。然后求解弹性场,利用积分变换和边界条件,将该热弹性接触问题转化为第一类奇异积分方程并数值求解。数值结果讨论了压头半径和热电载荷对法向接触应力、电流强度因子和能量通量强度因子的影响。结果表明,对于圆压头,热电材料的法向电流密度、法向能量通量在接触边缘表现出奇异性,而表面法向接触应力在接触边缘为零。本文建立的研究模型有助于更深层次的了解热电材料的接触行为。     

Frictional moving contact over the surface between a rigid punch and piezomagnetic materials – Terfenol-D as example

A general theory of the frictional moving contact of piezomagnetic materials indented by a flat or cylindrical punch is set up. The rigid punch moves at a constant speed and the Coulomb friction law applies inside the contact region. Terfenol-D with high magnetostriction and coupling is chosen. Employing the Galilean transformation and Fourier transform, Cauchy integral equations of the second kind are obtained and solved exactly. Closed-form expressions of physical quantities on the surface in terms of elementary functions are given. Numerical analyses are conducted to reveal the effects of the friction coefficient and moving speed of the punch on various surface stresses and magnetic induction. The singularity, discontinuity and spike of the surface magnetic induction may be important factors to explain why surface damage occurs for piezomagnetic materials.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

Crack growth of an interface crack between two dissimilar magneto-electro-elastic materials under anti-plane mechanical and in-plane electric magnetic impact

This paper examines the dynamic response of an interface crack between two dissimilar magneto-electro-elastic materials subjected to the mechanical and electric magnetic impacts. The magneto-electric impermeable boundary conditions are adopted. Laplace and Fourier transforms and dislocation density functions are employed to reduce the mixed boundary value problem to Cauchy singular integral equations in Laplace transform domain, which are solved numerically. Lots of numerical results are given graphically in time domain. The effects of electric impact loading and magnetic impact loading on dynamic energy density factors are discussed. Crack growth and propagation is predicted. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

均布荷载作用下悬臂磁电弹性梁的解析解

对磁电弹性平面问题进行了研究，给出了用拟调和位移函数表达的通解，进而以试凑法按平面应力问题推导出了均布荷载作用下悬臂磁电弹性粱的解析解，所得解有易于理解、便于校对、形式统一简洁的特点。本文还将计算结果与压电材料和弹性材料相应结果进行了分析、比较，为验证各种数值计算方法提供了参考依据。     

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric tor-sional contact problem of a half-space coated with func-tionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thick-ness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral trans-form technique and reduce the problem to a singular integral equation, and then numerically determine the unknown con-tact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.     

The axisymmetric viscoelastic contact problem with rotational friction

The boundary value problem that arises when a mechanically rough rigid punch of arbitrary axisymmetric profile is pressed against the surface of a linear aging viscoelastic half space and is also made to rotate about its axis, so that there is total slip between the contacting surfaces is analysed and solved. The moment required to make the punch rotate, on the assumption that the coefficient of friction obeys a power law or is a constant, and the total normal pressure acting on the punch may each be evaluated in terms of the history of the radius of the contact area. Application is made to the special cases where the punch has the form of (i) a cone, (ii) a paraboloid of revolution and (iii) a flat ended cylinder. Apart from case (iii) where the contract area is constant we can only find an explicit expression for the moment in terms of total pressure so long as the contact area is increasing. The case of constant total pressure and Maxwell viscoelastic material is examined in more detail.     

Wave propagation analysis of a functionally graded magneto-electro-elastic nanobeam rest on Visco-Pasternak foundation

Wave propagation analysis of a nanobeam made of functionally graded magneto-electro-elastic materials with rectangular cross section rest on Visco-Pasternak foundation is studied in this paper. For modeling the axial, rotation and transverse deformations, Timoshenko beam model is used. Fundamental magneto-electro-elastic equations of the model are derived for a general functionally graded beam excited to electric and magnetic potentials. Surface elasticity is employed for more confident modeling the behavior of nanobeam. Using Hamilton principle and calculation of kinetic and strain energies, the equations of motion are derived. Considering the harmonic wave propagation of infinite domain yields characteristic equation of the system in terms of different parameters of model. The effects of different parameters such as non-homogeneous index, wave number and residual surface stress are investigated on the different phase velocities corresponding to modes of deformation. One can find that increasing the non-homogeneous index and wave number leads to decrease in wave propagation phase velocities.     

Uniform motion of a plane punch on the boundary of an elastic half-plane

The dynamic contact problem of a plane punch motion on the boundary of an elastic half-plane is considered. The punch velocity is constant and does not exceed the Rayleigh wave velocity. The moving punch deforms the elastic half-plane penetrating into it so that the punch base remains parallel to itself at all times. The contact problem is reduced to solving a two-dimensional integral equation for the contact stresses whose two-dimensional kernel depends on the difference of arguments in each variable. A special approximation to the kernel is used to obtain effective solutions of the integral equation. All basic characteristics of the problem including the force of the punch elastic action on the elastic half-plane and the moment stabilizing the punch in the horizontal position in the process of penetration are obtained. A similar problem was considered in [1] and earlier in the “mode of steady-state motions” in [2, 3] and in other publications.