线弹性断裂力学和线弹性动力学中的集中载荷

本文认为线弹性断裂力学（ＬＥＦＭ）中受集中力和集中力偶作用的含裂纹体应力强度因子｀精确解＇的适用范围是值得进一步讨论的问题，并对线弹性动力学中的集中载荷问题进行了讨论     

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。     

压电介质中受拉伸与弯曲联合作用的圆币形裂纹问题

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质非轴对称三维问题的控制微分方程是四个二阶线性偏微分方程相联立的方程组。本文导出了用四个调和函数表示位移及电势的该方程组的势函数通解。作为通解的应用举例，文中求解了压电陶瓷材料中受拉伸与弯曲联合作用的圆币形裂纹问题，得到了裂纹尖端附近应力场及电位移场的解析表达式。结果表明裂尖场以及应力强度因子和电位移强度因子均表现出复杂的机－电耦合行为。     

压电介质平面问题的一般解和基本解

本文从压电介质平面问题基本方程出发，得到了含体积力的基本方程组的一般解。对状态方程进行Ｆｏｕｒｉｅｒ变换，由一般解得到Ｆｏｕｒｉｅｒ变换下状态方程的通解，对于单位集中力和单位点电荷情形，给出了压电介质平面问题各种情况下的有限形式的基本解。     

弹性层内部有集中力的精确解

利用Ｈａｎｋｅｌ变换及矩阵理论，获得了位于水平刚性基础上的弹性层在其内部受垂直于边界的集中力作用了的精确率，推广了已有的结论。Ｋｅｌｖｉｎ解，Ｍｉｎｄｌｉｎ解及弹性层表面受集中的用的解都是其特殊情形下的结论。     

线弹性理论中的集中载荷

集中载荷的概念起源于刚体力学,后来又常用于线弹性理论。但是两者的含义已有不同,须加区别。首先是在线弹性理中有限的集中力将产生无限大的位移和应力,与线性和弹性的基本前提相矛盾。它只能是圣维南含义下的近似解法。在应用叠加原理时遇到的无限小集中力才与线弹性理论严格相容。其次是在线弹性理论中集中力偶的概念完全不能应用,必须代之以集中力对。     

压电陶瓷中圆币形裂纹在横向剪力下的机－电耦合行为

以弹性位移分量和电势函数为基本未知量时，横观各向同性压电介质三维问题的场方程可化为四个联立的二阶线性偏微分方程组，本文导出了用四个调和函数表示位移分量及电势函数的表达式，即得到了该场方程的势函数通解，作为通解的应用举例，文中求解了圆币形裂纹受横向剪切作用的问题，得到了裂尖附近应力场及电位移场的解析表达式，结果表明，在横向剪切载荷下圆币形裂纹的尖端场及应力、电位移强度因子均具有明显的机－电耦合性质，而应力和电位移分量在裂尖仍具有－１／２的奇异性。     

压电陶瓷中圆币形裂纹在横向剪力下的机—电耦合行为

以弹性位移分量和电热函数基本未知量时，横观各向同性压电介质三维问题的场方程可化为四个联立的二阶线性偏微分方程组，本文导出了用四个调和函数表示位移分量及电势函数的表达式，即得到了该场方程的势函数能通解，作为通解的应用举例，文中求解了圆币形裂纹受横向剪切载荷下圆币形裂纹的尖端场及应力、电位移强度因子均具有明显的机－电耦合性质，而应力和电位移分量在裂尖仍具有－１／２的奇异性。     

求弹性半平面问题基本解的一个新方法

本文所提到的弹性半平面问题的基本解是一个满足特殊条件的弹性半平面的应力位移解答。这些条件为:(1)半平面内一点处作用有集中力X,Y或集中力偶M;(2)半平面边界为自由或固定边。利用平面弹性的复变函数方法,文中把弹性半平面基本解的问题归结为下列问题,使一个特定解析函数和另一个解析函数的共轭值在半平面边界上相等。对上述转化后的问题,只要利用复变函数的性质,不难从基本解的第一部分推导出基本解的第二部分。其中,基本解的第一部分是弹性全平面的本基解。从而,半平面问题基本解可以方便地得到。此外,文中还首次给出了:(1)集中力偶作用于半平面内一点时的基本解;(2)当半平面边界固定情况下的基本解。     

含刚性椭圆夹杂的弹性平面奇异解

运用复变函数方法,求解了含刚性椭圆夹杂的无限弹性平面在任意位置作用集中力和集中力偶的问题,导出了界面应力公式,绘出了应力分布曲线。     

The general solution of spherically isotropic magnetoelectroelastic media and its applications

A general solution of the three-dimensional equilibrium problem of spherically isotropic magnetoelectroelastic media is presented. Base on the obtained general solution, exact and compact form solutions are obtained for (1) a spherically isotropic magnetoelectroelastic cone subjected to concentrated force, concentrated couple, a point charge and a point electric current at its apex; (2) a spherically isotropic magnetoelectroelastic space with a concentrated force at the origin; (3) a spherical shell under spherically symmetric deformation; and (4) stress concentration around a spherical cavity subjected to remote uniform tensile force, electric charge and electric current.     

Elasticity solutions for a piezoelectric cone under concentrated loads at its apex

IntroductionTheproblemofaconesubjectedtoconcentratedloadsatitsapexisaclassicalprobleminthetheoryofelasticity.AnumberofscholarshavestUdiedtheproblem.LovereportedthesolutionstotheproblemofanisotropicconeunderconcentfatCdforcesatitsapex['].Lur'estudiedthisclassofproblemssystematicallybymeansofPapkovich-Neubergeneralsolution[2].LekniskiiandHu,byusingtheirrespectivegeneralsolutions,studiedcompressionandbendingproblemsofatransverselyisotropicconesubjectedtoaxialconcentfatedforcesandtfansverseconc…     

General solution for the coupled equations of transversely isotropic magnetoelectroelastic solids

IntroductionMechanicsandphysicsofmediapossessingsimultaneouslypiezoelectric ,piezomagneticandmagnetoelectriceffects ,namely ,magnetoelectroelasticsolids,haveattractedmoreandmoreattentionduetotheirgreatpotentialapplicationsinthetechnologiesofsmartandadaptivematerialsystem[1] .Sometheoreticalinvestigationsappearedintheliteratureinclude :1)Theexistenceproblemofsurfacewavesinsemi_infiniteanisotropicmagnetoelectroelasticmediawithvariousboundaryconditions[2 ,3 ] ;2 )Green’sfunctions[4～ 7] ;3)Inho…     

Crack tip field in piezoelectric/piezomagnetic media

This paper considers the magnetoelectroelastic problem of a crack in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects. Based on the extended Stroh formalism, the general two-dimensional solutions to the magnetoelectroelastic problem are obtained, involving five analytic functions of different variables. The magnetoelectroelastic field around the crack tip is given. It contains five modes of square root singularities. Expressions of the stresses, electric displacements and magnetic inductions in the vicinity of the crack tip are derived and the field intensity factors are provided. The path-independent conservative integral is derived. The energy release rate is written in terms of those field intensity factors. The explicit algebraic results are given for a special case of an anti-plane crack in a magnetoelectroelastic medium.     

SYMPLECTIC DUALITY SYSTEM ON PLANE MAGNETOELECTROELASTIC SOLIDS

By means of the generalized variable principle of magnetoelectroelastic solids, the plane magnetoelectroelastic solids problem was derived to Hamiltonian system. In symplectic geometry space, which consists of original variables, displacements, electric potential and magnetic potential, and their duality variables, lengthways stress, electric displacement and magnetic induction, the effective methods of separation of variables and symplectic eigenfunction expansion were applied to solve the problem. Then all the eigen-solutions and the eigen-solutions in Jordan form on eigenvalue zero can be given, and their specific physical significations were shown clearly. At last, the special solutions were presented with uniform loader, constant electric displacement and constant magnetic induction on two sides of the rectangle domain.     

A mode III crack crossing the magnetoelectroelastic bimaterial interface under concentrated magnetoelectromechanical loads

A mode III crack cutting perpendicularly across the interface between two dissimilar semi-infinite magnetoelectroelastic solid is studied under the combined loads of a line force, a line electric charge and a line magnetic charge at an arbitrary location. The impermeable conditions are implied on the crack faces. The technique developed in literature for the elastic bimaterial with a crack cutting interface is exploited to treat the magnetoelectroelastic bimaterial. The Riemann-Hilbert problem can be formulated and solved based on complex variable method. Analytical solutions can be obtained for the entire plane. The intensity factors around crack tips can be defined for the elastic, electric and magnetic fields. It shows that, no matter where the load position is, the electric displacement intensity factors (EDIFs), as well as the magnetic induction intensity factors (MIIFs), are identical in magnitude but opposite in sign for both crack tips, on condition that a line force is solely applied. Alternatively, if only a line electric charge is considered, then the stress intensity factors (SIFs) and the MIIFs exhibit the behavior. Likewise, if only a line magnetic charge is applied, it turns to the SIFs and the EDIFs instead. In addition, the dependence of the intensity factors is graphically shown with respect to the location of a line force. It is found that the SIF for a crack tip tends to be infinite if the applied force is approaching the tip itself, but the EDIF, with the complete opposite trend, tends to be vanishing. Finally, focusing on the more practical case of piezoelectric/piezomagnetic bimaterial, variation of the SIF along with the moduli as well as the piezo constitutive coefficients is explored. These analyses may provide some guidance for material selection by minimizing the SIF. It is also believed that the results obtained in this paper can serve as the Green’s function for the dissimilar magnetoelectroelastic semi-infinite bimaterial with a crack cutting the interface under general magnetoelectromechanical loads.     

Fracture of piezoelectromagnetic materials

The crack problem in a medium possessing coupled piezoelectric, piezomagnetic and magnetoelectric effects is considered. A conservative integral is derived based on the governing equations for magnetoelectroelastic media. Closed-form solution is obtained for an anti-plane crack in an infinite medium. The conservative integral is used to obtain the path-independent integral near the crack tip. Expressions for stresses, electric displacements and magnetic inductions in the vicinity of a crack tip are derived. It is found that the path-independent integral around the crack tip equals the energy release rate. In the absence of applied mechanical loads, the energy release rate is always negative.     

Analysis of cracked magnetoelectroelastic composites under time-harmonic loading

This paper presents a numerical model for the analysis of cracked magnetoelectroelastic materials subjected to in-plane mechanical, electric and magnetic dynamic time-harmonic loading. A traction boundary integral equation formulation is applied to solve the problem in combination with recently obtained time-harmonic Green’s functions (Rojas-Diaz et al., 2008). The hypersingular boundary integral equations appearing in the formulation are first regularized via a simple change of variables that permits to isolate the singularities. Relevant fracture parameters, namely stress intensity factors, electric displacement intensity factor and magnetic induction intensity factor are directly evaluated as functions of the computed nodal opening displacements and the electric and magnetic potentials jumps across the crack faces. The method is checked by comparing numerical results against existing solutions for piezoelectric solids. Finally, numerical results for scattering of plane waves in a magnetoelectroelastic material by different crack configurations are presented for the first time. The obtained results are analyzed to evaluate the dependence of the fracture parameters on the coupled magnetoelectromechanical load, the crack geometry and the characteristics of the incident wave motion.     

Analysis method of planar cracks of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium

The hyper-singular boundary integral equation method of crack analysis in three-dimensional transversely isotropic magnetoelectroelastic media is proposed. Based on the fundamental solutions or Green’s functions of three-dimensional transversely isotropic magnetoelectroelastic media and the corresponding Somigliana identity, the boundary integral equations for a planar crack of arbitrary shape in the plane of isotropy are obtained in terms of the extended displacement discontinuities across crack faces. The extended displacement discontinuities include the displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, and correspondingly the extended tractions on crack face represent the conventional tractions, the electric displacement and the magnetic induction boundary values. The near crack tip fields and the intensity factors in terms of the extended displacement discontinuities are derived by boundary integral equation approach. A solution method is proposed by use of the analogy between the boundary integral equations of the magnetoelectroelastic media and the purely elastic materials. The influence of different electric and magnetic boundary conditions, i.e., electrically and magnetically impermeable and permeable conditions, electrically impermeable and magnetically permeable condition, and electrically permeable and magnetically impermeable condition, on the solutions is studied. The crack opening model is proposed to consider the real crack opening and the electric and magnetic fields in the crack cavity under combined mechanical-electric-magnetic loadings. An iteration approach is presented for the solution of the non-linear model. The exact solution is obtained for the case of uniformly applied loadings on the crack faces. Numerical results for a square crack under different electric and magnetic boundary conditions are displayed to demonstrate the proposed method.     

磁-电-弹性半空间在轴对称热载荷作用下的三维问题研究

考虑力-电-磁-热等多场耦合作用, 基于线性理论给出了磁-电-弹性半空间在表面轴对称温度载荷作用下的热-磁-电-弹性分析, 并得到了问题的解析解. 利用Hankel 积分变换法求解了磁-电-弹性材料中的热传导及控制方程, 讨论了在磁-电-弹性半空间在边界表面上作用局部热载荷时的混合边值问题, 利用积分变换和积分方程技术, 通过在边界表面上施加应力自由及磁-电开路条件, 推导得到了磁-电-弹性半空间中位移、电势及磁势的积分形式的表达式. 获得了磁-电-弹性半空间中温度场的解析表达式并且给出了应力, 电位移和磁通量的解析解. 数值计算结果表明温度载荷对磁-电-弹性场的分布有显著影响. 当温度载荷作用的圆域半径增大时, 最大正应力发生的位置会远离半无限大体的边界; 反之当温度载荷作用的圆域半径减小时, 最大应力发生的位置会靠近半无限大体的边界. 电场和磁场在温度载荷作用的圆域内在边界表面附近有明显的强化, 而磁-电-弹性场强化区域的强化程度跟温度载荷的大小和作用区域大小相关. 本研究的相关结果对智能材料和结构在热载荷作用下的设计和制造具有指导意义.