Reissner板弯曲的辛求解体系

基于Reissner板弯曲问题的Hellinger-Reissner变分原理,通过引入对偶变量,导出Reissner板弯曲的Hamilton对偶方程组．从而将该问题导入到哈密顿体系,实现从欧几里德空间向辛几何空间,拉格朗日体系向哈密顿体系的过渡．于是在由原变量及其对偶变量组成的辛几何空间内,许多有效的数学物理方法如分离变量法和本征函数向量展开法等均可直接应用于Reissner板弯曲问题的求解．这里详细求解出Hamilton算子矩阵零本征值的所有本征解及其约当型本征解,给出其具体的物理意义．形成了零本征值本征向量之间的共轭辛正交关系．可以看到,这些零本征值的本征解是Saint-Venant问题所有的基本解,这些解可以张成一个完备的零本征值辛子空间．而非零本征值的本征解是圣维南原理所覆盖的部分．新方法突破了传统半逆解法的限制,有广阔的应用前景．     

二维矩形域内Stokes流问题的辛解析和数值方法

给出了一种新的解析求解二维矩形域中的Stokes流动问题的方法——辛体系方法（Hamilton体系方法）．在辛体系下,基本问题归结为本征值和本征解的问题．由于辛本征解之间存在辛正交共轭关系,问题的解和边界条件均可以由本征解描述和表示．利用辛本征解空间的完备性,建立一套封闭的求解问题方法．研究结果表明零本征值本征解描述了基本流动,而非零本征值本征解则表示问题的局部效应．数值结果给出了几种有代表性的流动情况,显示了该求解方法对求解许多问题的有效性．同时,这种方法也为研究其他问题提供了一条思路．     

两种材料组成的弹性楔的佯谬解

从Hellinger-Reissner变分原理出发,通过引入适当的变换可以将两种材料组成的弹性楔问题导入极坐标哈密顿体系,从而可以在由原变量和其对偶变量组成的辛几何空间,利用分离变量法和辛本征向量展开法求解该问题的解。在极坐标哈密顿体系下的所有辛本征值中,本征值-1是一个特殊的本征值。一般情况下本征值-1为单本征值,求解其对应的基本本征函数向量就直接给出了顶端受有集中力偶的经典弹性力学解。但当两种材料的顶角和弹性模量满足特殊关系时,本征值-1成为重本征值,同时经典弹性力学解的应力分量变成无穷大,即出现佯谬。此时重本征值-1存在约当型本征解,通过对该特殊约当型本征解的直接求解就给出了两种材料组成的弹性楔顶端受有集中力偶的佯谬问题的解。结果进一步表明经典弹性力学中弹性楔的佯谬解对应的就是极坐标哈密顿体系的约当型解。     

横观各向同性电磁弹性介质中裂纹对SH波的散射

研究横观各向同性电磁弹性介质中裂纹和反平面剪切波之间的相互作用．根据电磁弹性介质的平衡运动微分方程、电位移和磁感应强度微分方程,得到SH波传播的控制场方程．引入线性变换,将控制场方程简化为Helmholtz方程和两个Laplace方程A·D2通过Fourier变换,并采用非电磁渗透型裂面边界条件,得到了柯西奇异积分方程组．利用Chebyshev多项式求解积分方程,得到应力场、电场和磁场以及动应力强度因子的表达,并给出了数值算例．     

含椭圆形夹杂的压电体平面应变问题

利用复变函数理论,对在无限远处均匀应力和电位移载荷作用下的含有椭圆形弹性夹杂的横观各向同性压电材料,作了力电分析．在该文有限元结果和前人相关理论解的基础上,提了出一个可接受的认为弹性夹杂体内的应力场为常应力场的假设．在采用了不导通电边界条件之后,获得了以复势形式表示的压电基体的和弹性夹杂体内部的应力场解．     

双功能梯度纳米梁系统振动分析的辛方法

在辛力学与非局部Timoshenko(铁木辛柯)梁理论的基础上,针对黏弹性介质中的双功能梯度纳米梁系统的自由振动问题,提出了一种全新的解析求解方法.在Hamilton(哈密顿)体系下,位移与广义剪力、转角与广义弯矩互为对偶变量.以对偶变量为基本未知量,Lagrange(拉格朗日)体系下的高阶偏微分控制方程简化为一系列常微分方程.该纳米梁系统的振动问题归结为辛空间下的本征问题,解析频率方程和振动模态可以通过辛本征解和边界条件直接获得.数值结果验证了该方法的正确性与有效性,并针对纳米梁系统的小尺度效应、纳米梁间的相互作用以及黏弹性地基的影响进行了系统的参数分析.     

含椭圆夹杂二维各向异性电磁弹性介质的耦合问题

研究了无限大二维各向异性电磁弹性固体的椭圆夹杂问题.基于广义Stroh方法、保角变换和扰动概念,得到了基体和夹杂内电场、磁场和弹性场的封闭解.本文结果可用于研究电磁弹性复合材料的有效性能.     

压电材料中两个非对称平行裂纹的基本解

采用Schmidt方法分析压电材料中非对称平行的双可导通裂纹的断裂性能．利用Fourier变换使问题的求解转换为求解两对以裂纹面位移之差为未知变量的对偶积分方程．为了求解对偶积分方程,直接把裂纹面位移差函数展开成Jacobi多项式形式．最终得到了裂纹的应力强度因子与电位移强度因子之间的关系．数值结果表明,应力强度因子和电位移强度因子与裂纹间的距离、裂纹的几何尺寸有关;与不可导通裂纹有关结果相比,可导通裂纹的电位移强度因子远小于相应问题不可导通裂纹的电位移强度因子．同时可以发现裂纹间的“屏蔽”效应也在压电材料中出现．     

辛求解体系下功能梯度材料平面梁的完整解析解

采用辛弹性力学解法,求取弹性模量沿轴向指数变化,而Poisson比保持不变的功能梯度材料平面梁的完整解析解．通过求解被Saint-Venant原理覆盖的一般本征解,建立起完整的解析分析过程,进而给出平面梁位移和应力的精确分布规律．传统的弹性力学分析方法常常忽略被Saint-Venant原理覆盖的解,但这些衰减的本征解对材料的局部效应起着较大的影响作用,可能导致材料或结构的突然失效．采用辛求解方法,充分利用本征向量之间的辛共轭正交关系,得到了功能梯度材料梁的完整解析解．两个数值算例分别将功能梯度材料平面梁的位移和应力分布与相应均匀材料情形的结果进行比较,研究了材料非均匀性对位移和应力解的影响．     

横观各向同性电磁弹性固体耦合方程的一般解

横观各向同性电磁弹性固体的耦合特征由5个关于弹性位移、电位和磁位的二阶偏微分方程控制.基于势函数理论,耦合的方程组被简化为5个非耦合的关于势函数的广义Laplace方程.弹性场和电磁场由势函数表示,这构成了横观各向同性电磁弹性固体的一般解.     

An interface crack moving between magnetoelectroelastic and functionally graded elastic layers

A constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading is investigated by the integral transform method. Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The singular stress, electric displacement and magnetic induction near the crack tip are obtained asymptotically and the corresponding field intensity factors are defined. Numerical results show that the stress intensity factors are influenced by the crack moving velocity, the material properties, the functionally graded parameter and the geometric size ratios. The propagation of the moving crack may bring about crack kinking, depending on the crack moving velocity and the material properties across the interface.     

Magneto–electro interaction of two offset indenters in frictionless contact with magnetoelectroelastic materials

Within the theory of linear full-field magneto–electro–elasticity, magneto–electro interaction of two electrically-conducting and magnetically-conducting indenters acting over the surface of magnetoelectroelastic materials widely used in practical industries is examined. The operation theory, Fourier transform technique and integral equation technique are employed to address the two-dimensional, mixed boundary-value problem explicitly. The surface stresses, electric displacement and magnetic induction and their respective intensity factors are obtained in closed forms for two perfectly conducting semi-cylindrical indenters. Degradation from two perfectly conducting semi-cylindrical indenters to one single perfectly conducting cylindrical indenter is discussed. Numerical analyses are detailed to reveal the effects of the interactions between two semi-cylindrical indenters on contact behaviors subjected to multi-field loadings.     

Closed-form solutions for two collinear dielectric cracks in a magnetoelectroelastic solid

The problem of two collinear electromagnetically dielectric cracks in a magnetoelectroelastic material is investigated under in-plane electro-magneto-mechanical loadings. The semi-permeable crack-face boundary conditions are adopted to simulate the case of two collinear cracks full of a dielectric interior. Utilizing the Fourier transform technique, the boundary-value problem is reduced to solving singular integral equations with Cauchy kernel, which then are solved explicitly. The intensity factors of stress, electric displacement, magnetic induction, crack opening displacement (COD) and the energy release rates near the inner and outer crack tips are determined in closed forms for two cases of possible far-field electro-magneto-mechanical loadings respectively. Numerical results for a BaTiO₃–CoFe₂O₄ composite are carried out to show the effects of applied mechanical loadings on the crack-face electric displacement and magnetic induction, the stress intensity factor and the COD intensity factor, respectively. The obtained results reveal that when the applied mechanical loading is stress, applied electromagnetic loadings have no influences on the stress intensity factor. When the applied mechanical loadings is train, the applied positive electromagnetic loadings decrease the intensity factors of stress and COD, and the applied negative ones increase the intensity factors of stress and COD. The variations of energy release rates are also given to show the effects of the geometry of two collinear dielectric cracks.     

Interfacial fracture of layered magnetoelectroelastic materials

A problem for an interface crack located in a layered magnetoelectroelastic material strip of semi-infinite length is solved. A closed-form solution is obtained for anti-plane mechanical and in-plane electric and magnetic fields. Explicit expressions for stresses and electric and magnetic fields, together with their intensity factors and the energy release rate, are obtained. The extreme cases of impermeable and permeable cracks are discussed. Using the basic solution for a single crack, solutions for two collinear interface cracks in an infinitely long layered magnetoelectroelastic medium, an interface crack in an infinitely long layered magnetoelectroelastic medium, and an edge crack at the interface of a semi-infinitely long layered magnetoelectroelastic medium are also obtained. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 145–164, March–April, 2008.     

基于辛对偶体系的层合板自由边缘效应的分析解

在原变量——位移和其对偶变量——应力组成的辛几何空间,建立了Pipes-Pagano模型的复合材料层合板问题的辛对偶求解体系．与传统的单类变量不同,辛对偶变量有利于同时描述层间位移连续性条件和应力平衡条件．进入辛对偶体系以后,就可以应用辛对偶体系的统一解析求解方法,如分离变量和辛本征展开的方法对层合板问题进行解析分析和求解．对层合板自由边缘效应的分析求解,验证了辛对偶体系的方法对层合板问题的分析求解是十分有效的．     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

Magnetoelectroelastodynamic interaction of multiple arbitrarily oriented planar cracks

The problem of multiple arbitrarily oriented planar cracks in an infinite magnetoelectroelastic space under dynamic loadings is considered. An explicit solution to the problem is given in the Laplace transform domain in terms of suitable exponential Fourier integral representations. The unknown functions in the Fourier integrals are directly related to the Laplace transform of the jumps in the displacements, electric potential and magnetic potential across opposite crack faces and are to be determined by solving a system of hypersingular integral equations. Once the hypersingular integral equations are solved, the displacements, electric potential, magnetic potential and other quantities of interest such as the crack tip intensity factors may be easily computed in the Laplace transform domain and recovered in the physical space with the help of a suitable algorithm for inverting Laplace transforms.     

Hamilton体系辛半解析法及在非均匀电磁波导中的应用

力学中的Hamilton体系需用对偶变量来描述,而电磁场正好有电场和磁场这一对对偶变量．尝试将力学中的Hamilton体系理论应用于电磁波导的分析,以横向电场和磁场作为对偶变量,将电磁波导的基本方程导向辛几何的形式．基于Hamilton变分原理, 导出横向离散的半解析系统方程, 保持体系的辛结构．以非均匀波导为例, 求解了方程的辛本征值问题, 计算结果与解析解相当吻合．