含界面裂纹的不可压缩双金属胶接件的解析变分－守恒积分解法

本文根据平面问题的复变函数理论推导了含界面裂纹双金属胶接件满足微分方程、开裂界面边界条件与未开裂界面连续条件的应力与位移本征函数展开式，并建立了不可压缩双金属界面裂纹的复合型守恒积分及其与应力强度因子之关系，进而利用分区广义变分原理满足其余边界条件确定包含应力强度因子在内的展开式系数，得到守恒积分并求出应力强度因子．数值计算表明，沿不同回路的在恒积分具有很好的守恒性而且由这两种方法所得应力强度因子具有很好的一致性．     

含界面裂纹的不可压缩双金属胶接件的解析变分－守恒积分解法

本文根据平面问题的复变函数理论推导了含界面裂纹双金属胶接件满足微分方程、开裂界面边界条件与未开裂界面连续条件的应力与位移本征函数展开式，并建立了不可压缩双金属界面裂纹的复合型守恒积分及其与应力强度因子之关系，进而利用分区广义变分原理满足其余边界条件确定包含应力强度因子在内的展开式系数，得到守恒积分并求出应力强度因子．数值计算表明，沿不同回路的在恒积分具有很好的守恒性而且由这两种方法所得应力强度因子具有很好的一致性．     

关于线性非局部弹性理论的应力边界条件

应力边界条件的提法是线性非局部弹性理论尚未解决的一个理论问题。文中针对这一问题进行了研究，所导出的应力边界条件包含了物体微观结构的长程相互作用，这个结果不仅解释了在裂纹混合边界值问题中非线性局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以自然地得到裂纹尖端的分子内聚力模型。     

应力边界元法解平面弹性问题

本文提出了求解平面弹性问题的应力边界元法。简述了边界积分方程的建立,给出了常单元离散化时求系数的解析式。这种方法适用于应力边界值问题。边界积分方程中的一个边界函数就是边界点法向应力和切向应力之和,因此计算孔边应力非常方便。作为数值算例,计算了有孔无限板的孔边应力。应力边界元法也可应用于平面热弹性问题和平板弯曲问题。     

关于非局部场论的两点注记

研究了非局部场论中尚未完全解决的两个基本问题：其一为局部化体力，力矩残余之间的相关性，由此得到了一个描述两者关系的定理；其二为线性非局部弹性理论的应力边界条件的提法；文中所得到的应力边界条件不仅解释了在裂纹混合边界值问题中线性非局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以给出裂纹尖端的分子内聚力模型。     

线性非局部弹性理论的修正

本文通过考虑局部化残余力的影响对线性非局部弹性理论进行了修正，由修正后的理论所导出的应力边界条件包含了物体微观结构的长程力的作用，这个结果不仅解释了在裂纹混合边界值问题中线性非局部弹性理论方程的解在常应力边界条件下不存在的问题，而且可以自然地得到裂纹尖端的Ｂａｒｅｎｂｌａｔｔ分子内聚力模型。     

双周期平面裂纹问题的特征展开-变分方法

根据弹性力学的变分原理,利用双周期问题位移场的双准周期性质和应力应变场的双周期性质,构造了双周期平面问题的单胞泛函变分表达式. 然后结合针对裂纹问题的复应力函数特征展开式,发展了基于单胞模型的双周期裂纹平面问题的特征展开-变分方法. 由于该方法考虑了最一般的双周期边界条件,因而能够分析一般非对称排列的双周期裂纹问题. 通过结果的收敛性分析说明了该方法具有计算效率和精度都高的优点. 最后,对于裂纹呈平行四边形排列的情况,分析了不同的裂纹排列对应力强度因子的影响.      

横观各向同性圆柱体端部问题的三维弹性理论解

从位移的通解出发，用分离变量法得到横观各向同性圆柱体的位移和应力的特征函数展开式，并把位移势函数的解用付里叶积分的形式表示。利用留数运算，该积分解可以转换成类似于特征函数的展开式。通过混合端部边界问题，得到与特征函数解成双正交关系的另一组函数。利用这种双正交关系，可以处理不同的端部边界问题。     

基于地应力场加卸载的地下洞室稳定模型试验研究

加卸载响应比理论是研究非线性系统稳定的一种重要方法。本文分析了该理论的本质含义,研究了该理论在洞室稳定分析中存在的问题,提出了基于应力边界值即地应力场增减实现加卸载的方法。文中以某矿山巷道稳定研究作为工程背景,建立了相似材料平面应变模型。通过对顶部边界应力进行逐级加卸载,主要研究了拱顶竖向压力、拱顶沉降和声发射的响应特征。结果表明,当应力不太大时,拱顶竖向压力和拱顶沉降的加卸载响应比值接近1;当围岩接近破坏时,比值出现突然的增加,并伴随较大的波动。声发射能量和事件数虽未出现类似特征,但总体来说,随着荷载的增加,声发射呈现增加的趋势。     

反平面剪切下带椭圆孔有限物体的应力集中

文中提出了一种复应力函数展开式,它适用于反平面剪切条件下带椭圆孔口有限物体的应力集中问题。这个展开式恒满足:(a)反平面剪切的全部弹性力学方程组,(b)椭圆孔口即内周界上的自由条件。此时,外周界上的边界条件是待满足的。利用广义变分原理,展开式中的全部待定系数都可以定出。文中计算了两个数值例子,得出了一些应力集中系数值,又从效值上讨论了解的收敛情况。计算表明,本文的方法是有效的。     

Stress intensity factors for interface crack in finite size specimen using a generalized variational approach

A generalized variational approach together with eigenfunction expansion is applied to determine the stress intensity factors for interface crack in finite size specimen. Application is also made of the complex potentials such that a complex stress intensity factor with components corresponding to the Mode I and II stress intensity factors can be identified with one of the leading coefficients in the eigenfunction expansion. Obtained are the numerical values of the stress intensity factors for an interface edge crack in a bimaterial rectangular specimen. The outside boundary is subjected to uniform stress normal and parallel to the crack. Solutions are also obtained for the same crack aand specimen geoinetry is subjected to a pair of equal and opposite concentrated forces along the open end away from the edge crack. The third example pertains to the case of three-point bending where the centre concentrated load is directed along the interface dividing the two materials. Numerical results are obtained for four different combinations of the bimaterial specimen with an interface edge crack.     

基于哈密顿原理的两种材料界面裂纹奇性研究

研究了两种材料组成的弹性体在交界面上含裂纹时的裂纹尖端奇异场。通过变量代换及变分原理，将平面弹性扇形域的方程导向哈密体系，从而可通过分离变量及共轭辛本征函数展开法解析法求解扇形域方程，得到求解双材料界面裂纹尖点奇性的一般表达式，由此为该类问题的求解开辟了一条新途径。     

含界面裂纹Reissner板弯曲问题分析的奇异单元

首先,采用特征函数渐近展开法,推导了Reissner板弯曲界面裂纹尖端附近位移场渐近展开的前两阶显式表达式,并利用所获得的位移场渐近表达式构造了一种可用于Reissner板弯曲界面裂纹分析的奇异单元。然后,将该奇异单元与外部的常规有限单元相结合,开展了含界面裂纹Reissner板弯曲断裂问题的数值分析。奇异单元可以较好地描述裂纹尖端附近的内力场与位移场,其优势是它与常规单元进行连接时不需要使用过渡单元,并且可以直接给出应力强度因子等断裂参数的高精度数值结果。最后,通过两个数值算例验证了本文方法的有效性。     

界面裂纹无摩擦接触裂尖应力场

基于哈密顿原理，通过分离变量及共轭辛本征函数展开法，解析地球解界面裂纹无摩擦接触裂尖的扇形域方程，从另一条途径研究了界面裂尖应力场的特征。     

A new approach to the pseudo-orthogonal properties of eigenfunction expansion form of the crack-tip complex potential function in anisotropic and piezoelectric fracture mechanics

Over the past twenty years, the well-known weight function theory based on the Bueckner work conjugate integral has been widely used to calculate crack tip fracture dominant parameter such as the stress intensity factor, the energy release rate (or the J-integral) and the T-stress in various kinds of cracked materials (e.g. isotropic materials, anisotropic materials and piezoelectric materials). Meanwhile, the pseudo-orthogonal property of the eigenfunction expansion form of the crack tip stress complex potential function has been proved to play a very important role in the theory. In this paper, we provide a new approach to establish the pseudo-orthogonal properties for crack problems in anisotropic and/or piezoelectric materials. In the latter case associated with mechanical-electric coupling, the electrical boundary conditions under both impermeable and permeable crack models are considered. The approach developed is much simpler than the classical complex variable separation technique proposed by previous researchers and hence the cumbersome and lengthy manipulations are avoided. Moreover, it is shown that, unlike previous works, the orthogonal properties of the material characteristic matrices A and B induced by the Stroh theory are no longer necessary in establishing the pseudo-orthogonal properties of eigenfunction expansion form in cracked piezoelectric materials. The approach can be easily extended to treat many other different crack problems concerning the Bueckner integral involving several complex arguments.     

STUDY ON DYNAMIC STRESS INTENSITY FACTORS OF DISK WITH A RADIAL EDGE CRACK SUBJECTED TO EXTERNAL IMPULSIVE PRESSURE

A dynamic weight function method is presented for dynamic stress intensity factors of circular disk with a radial edge crack under external impulsive pressure. The dynamic stresses in a circular disk are solved under abrupt step external pressure using the eigenfunction method. The solution consists of a quasi-static solution satisfying inhomogeneous boundary conditions and a dynamic solution satisfying homogeneous boundary conditions. By making use of Fourier-Bessel series expansion, the history and distribution of dynamic stresses in the circular disk are derived. Furthermore, the equation for stress intensity factors under uniform pressure is used as the reference case, the weight function equation for the circular disk containing an edge crack is worked out, and the dynamic stress intensity factor equation for the circular disk containing a radial edge crack can be given. The results indicate that the stress intensity factors under sudden step external pressure vary periodically with time, and the ratio of the maximum value of dynamic stress intensity factors to the corresponding static value is about 2.0.     

基于哈密尔顿体系的裂纹尖端应力奇性分析及计量

对弹性平面扇形域问题,将径向坐标模拟成时间坐标,通过适当的变换,将扇形域问题导向哈密尔顿体系,利用分离变量法及本征函数向量展开等方法,推导出裂纹尖端的应力奇性解的计算公式,结合变分原理,提出一种解决应力奇性计算的断裂分析元,将此分析元与有限元法相结合,可以进行某些断裂力学或复合材料等应力奇性问题的计算及分析,数值计算结果表明,该方法具有精度高,使用十分方便,灵活等优点,是哈密尔顿体系和辛数学优越性的一次具体体现。     

A Partially Built-in Plate under Uniform Load

A thin plate has the form of the infinite strip ?∞<x<∞, 0≤y≤aand has the edge y=abuilt-in. The edge y=0 has its right half 0<x<∞ built-in while the left half ?∞<x<0 is free. The whole plate is now subjected to a uniform load p ₀applied to its upper surface. What is the resulting deflection of the plate and what are the induced moment and shear resultants? We present a solution to this classical problem based on eigenfunction expansions. In the right and left halves of the strip, the deflection can be expanded as separate eigenfunction expansion series, but these are difficult to match across the line x=0 because of the singularity at (0,0) induced by the boundary conditions. We adopt the novel technique of expanding the field near the centre of the strip in its correct form as a series of Williams polar eigenfunctions, and then linking this expansion to the right and left eigenfunction expansions by using a special form of elastic reciprocity. These right and left reciprocity conditions give two infinite systems of linear equations satisfied by the polar expansion coefficients, and we prove that these equations are sufficient to determine these coefficients. Further applications of reciprocity give closed form expressions for the right and left eigenfunction expansion coefficients so that the whole solution is then determined. The method yields accurate results using small systems of linear equations. We present numerical results for the deflection of the plate and the induced moment and shear resultants.     

Hamiltonian principle based stress singularity analysis near crack corners of multi-material junctions

This paper presents a new method for the stress singularity analysis near the crack corners of a multi-material junctions. The stress singularities near the crack corners of multi-dissimilar isotropic elastic material junctions are studied analytically in terms of the methods developed in Hamiltonian system. The governing equations of plane elasticity in a sectorial domain are derived in Hamiltonian form via variable substitution and variational principle respectively. Both of the methods of global state variable separation and symplectic eigenfunction expansion are used to find the analytical solution of the problem. The relationships among the state vectors in different material spaces are obtained by means of coordinate transformation and consistent conditions between the two adjacent domains. The expression of the original problem is thus changed into a new form where the solutions of symplectic generalized eigenvalues and eigenvectors are needed. The closed form of expressions is established for the stress singularity analysis near the corner with arbitrary vertex angles. Numerical results are presented with several chosen angles and multi-material constants. To show the potential of the new method proposed, a semi-analytical finite element is furthermore developed for the numerical analysis of crack problems.     

Delamination crack originating from transverse cracking in cross-ply composite laminates under extension

Based upon the Stroh formalism for anisotropic elastic materials and upon the method of eigenfunction expansion, the stress redistribution due to delamination cracks originating from transverse cracking is examined from [90/0], and [0/90], laminates under extension. The structure of the solution, in the form of a series expansion, is determined from the eigenvalue equation resulting from appropriate near-field conditions. To complete the solution, use is made of a boundary collocation technique in conjunction with the eigenfunction series that includes a large number of terms, enough to represent the elastic state throughout the appropriate domain concerned. The fracture mechanics parameters, such as stress intensity factors and energy release rates, are calculated and the major characteristics of stress distribution are discussed. The stability of delamination cracks is examined for varying ratios of ply thickness in terms of the energy release rate.