Singularities in auxetic elastic bimaterials

In this paper we study the effects of negative Poisson's ratios on elastic problems containing singularities. Materials with a negative Poisson's ratio are termed auxetic. We present a brief review of such materials. The elasticity problem of a bimateral wedge is presented, then two particular cases of this problem are investigated: the free-edge problem and the interface crack problem. We study the effect on the stress singularity due to one portion of the bimaterial becoming auxetic. We find that the auxetic material has a significant effect on the singularity order, even causing the singularity to vanish for certain values of the elastic constants.     

Singularity under a concentrated force in elasticity

We first discuss singularity problem of a sort of partial differential equation involving δ function. Using this result we, then have the answer to various singularity problems in elasticity due to the presentation of a concentrated force. Lastly corresponding conclusions in vibration problem are drawn. Project of research foundation supported by National Education Committee     

Existence of Traveling-Wave Solutions for Hyperbolic Systems of Balance Laws

This paper is concerned with traveling-wave solutions for hyperbolic systems of balance laws satisfying a stability condition and a Kawashima-like condition. We are interested in the case where the traveling-wave equations have a singularity, which is absent for 2 × 2 systems satisfying the two conditions. To deal with the singularity, we reduce the problem to a parametrized one without singularity by using the center manifold theorem. For the parametrized problem, we prove the existence of solutions by modifying an existing argument in the literature. In this way, we show the existence of traveling-wave solutions.     

计算动力学中的伪弧长方法研究

动力学问题通常采用微分方程来描绘,但由于工程实际问题的复杂性,微分方程模型常伴随着解的不连续性、刚性或激波间断奇异性特点,传统方法很难求解,奇异性问题是计算动力学难点,同时也是国内外学者研究的热点.伪弧长数值算法是针对计算动力学中的奇异性问题所提出的,其基本思想为通过在解曲线上引入伪弧长参数,并增加一个约束方程,在伪弧长参数作用下,使得原始离散单元发生扭曲形变,从而达到消除或减弱奇异性的目的.本文首先介绍伪弧长方法求解定常对流-扩散方程的奇异性问题,并提出针对双曲守恒定律的局部伪弧长算法,其思想在于首先通过间断解的梯度变换来确定强间断所处位置,进而通过局部网格点重构以及数值修正来达到强间断处奇异性消除与降低的目的.针对高维问题,提出全局伪弧长方法,通过对整个计算区域内的网格点进行重构,使得所有网格点向奇异间断点处移动,从而降低间断点的影响域,达到降低奇异性的目的.重点讨论了三维全局伪弧长算法问题的计算难点,即三维空间网格扭曲大变形导致的数值算法不收敛,并提出在算法设计过程中采用分块重构与整体计算相结合的策略,实现了三维空间中的伪弧长数值算法,最后通过数值实验来验证伪弧长算法对于奇异性问题的有效性.     

Interaction of an anti-plane singularity with interfacial anti-cracks in cylindrically anisotropic composites

Anti-plane problem for a singularity interacting with interfacial anti-cracks (rigid lines) under uniform shear stress at infinity in cylindrically anisotropic composites is investigated by utilizing a complex potential technique in this paper. After obtaining the general solution for this problem, the closed solution for the interface containing one anti-crack is presented analytically. In addition, the complex potentials for a screw dislocation dipole inside matrix are obtained by the superimposing method. Expressions of stress singularities around the anti-crack tips, image forces and torques acting on the dislocation or the center of dipole are given explicitly. The results indicate that the anisotropy properties of materials may weaken the stress singularity near the anti-crack tip for the singularity being a concentrated force but enhance the one for the singularity being a screw dislocation and change the equilibrium position of screw dislocation. The presented solutions are valid for anisotropic, orthotropic or isotropic composites and can be reduced to some new or previously known results.     

三维切口应力奇性指数计算

将三维切口根部的位移渐近展开式引入线弹性力学平衡方程,导得关于切口应力奇性指数的特征微分方程组。再采用插值矩阵法,一次性地计算出三维切口的各阶应力奇性指数,它们具有同阶精度,并可同时获取相应的特征角函数。算例显示该法是分析三维切口应力奇异指数的一个有效的路径。计算结果表明,若直接用平面应变理论预测三维切口应力奇性指数将导致部分重要的奇性指数丢失。     

三维弹塑性有限变形问题边界元法中奇异积分的直接分析去奇解法

作为本文作者研究工作的继续,本文提出了处理三维弹塑性有限变形问题边界元法中二次元区域弱奇及Cauchy 主值奇异积分的二次极坐标变换—分析去奇法.该方法先通过适当的二次极坐标变换降低奇异积分的奇异性,然后利用Causs 散度定理去除Cauchy主值积分的奇异性.通过三维弹塑性及三维有限变形问题数值算例说明该方法具有良好的精度及数值稳定性,并且实施较方便.本文方法可直接推广应用于二阶以上高阶元离散模型奇异积分处理.     

Boundary Singularity for Thermal Transpiration Problem of the Linearized Boltzmann Equation

We study the boundary singularity of the fluid velocity for the thermal transpiration problem in the kinetic theory. Logarithmic singularity has been demonstrated through the asymptotic and computational analysis. The goal of this paper is to confirm this logarithmic singularity through exact analysis. We use an iterated scheme, with the “gain” part of the collision operator as a source. The iterated scheme is appropriate for large Knudsen numbers considered here and yields an explicit leading term.     

Zero-Hopf Polynomial Centers of Third-Order Differential Equations

We study the 3-dimensional center problem at the zero-Hopf singularity in some families of polynomial vector fields arising from third-order polynomial differential equations. After proving some general properties we check that the quadratic family has no 3-dimensional centers. Later we characterize all the 3-dimensional centers in the cubic homogeneous family. Finally we give a partial classification of the 3-dimensional centers at just one singularity of the full cubic family and propose one open problem to close this classification.     

The singular behaviour in the vicinity of intersection between the body and free surface

The singular behaviour in the vicinity of intersection between the body and free surface is presented.It is shown that in the linear regime the singularity of velocity potential for transient problem is in d~2|nd.The singular behaviour for harmonic problem is the same as the result for the transient problem.In particular,the singularity for the harmonic problem with infinite frequency is in d~2 lnd for velocity potential(d is the distance between field point and intersection).     

Interface end stress field of antiplane of orthotropic bimaterials

The orthotropic bimaterial antiplane interface end of a flat lap is studied by constructing new stress functions and using the composite complex function method of material fracture. The expressions of stress fields, displacements fields and the stress intensity factor around the flat lap interface end are derived by solving a class of generalized bi-harmonic equations. The result shows that this type of problem has one singularity, the stress field has no singularity when two materials have constant ratio F 〉 0, the stress field has power singularity, and the singularity index has a trend to -1/2 as F increases. The derived equation is verified with FEM analysis.     

插值矩阵法分析双材料平面V形切口奇异阶

对二维V形切口问题提出奇异阶分析的一个新方法.首先,以V形切口尖端附近位移场沿其径向渐近展开为基础,将其线弹性理论控制方程转换成切口尖端附近关于周向变量的常微分方程组特征值问题,然后将数值求解两点边值问题的插值矩阵法进一步拓展为求解一般常微分方程组特征值问题,插值矩阵法是在离散节点上采用微分方程中待求函数的最高阶导数作为基本未知量.由此,V形切口的应力奇性阶问题通过插值矩阵法获得,同时相应的切口附近位移场和应力场特征向量一并求出.     

复合材料切口应力奇性指数计算

提出一种计算广义平面应交状态下复合材料切口应力奇性指数的新方法.在切口尖端的位移幂级数渐近展开式被引入正交各向异性材料的物理方程后,将用位移表示的应力分量代入切口端部柱状邻域的线弹性理论控制方程,切口应力奇性指数的计算被转化为常微分方程组特征值的求解.采用插值矩阵法求解该常微分方程组,可一次性地获取切口尖端多阶应力奇性指数.本法适合平面和反平面应力场耦合或解耦的情形,并可退化计算裂纹或各向同性材料切口的应力奇性指数.算例表明,所提方法对分析复合材料切口应力奇性指数是一种准确有效的手段.     

On the solution singularity in the plane elasticity problem for a cantilever strip

The plane elasticity problem of bending of a cantilever strip whose material is assumed to be incompressible in the transverse direction is solved. It is shown that, in the classical statement of of the boundary condition for the fixed edge of the strip, the solution has a singularity at the corner points of the edge. Several cases of the strip fixation and loading characterized by the presence or absence of the solution singularity are considered. The strength of glass beams of three types, for which the theory of elasticity predicts whether the normal stress has a singularity, is studied experimentally. It is shown that the limit stresses for the beams of the types under study are practically the same, which testifies that the solution singularity does not have any physical nature.     

An Interface Inclusion between Two Dissimilar Piezoelectric Materials

The generalized two-dimensional problem of a dielectric rigid line inclusion, at the interface between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity, is studied by means of the Stroh theory. The problem was reduced to a Hilbert problem, and then closed-form expressions were obtained, respectively, for the complex potentials in piezoelectric media, the electric field inside the inclusion and the tip fields near the inclusion. It is shown that in the media, all field variables near the inclusion-tip show square root singularity and oscillatory singularity, the intensity of which is dependent on the material constants and the strains at infinity. In addition, it is found that the electric field inside the inclusion is singular and oscillatory too, when approaching the inclusion-tips from inside the inclusion.     

考虑尺寸效应的双材料轴对称界面端应力奇异性

提出了一种分析双材料轴对称界面端的应力奇异行为的特征值法.基于弹性力学空间轴对称问题的基本方程和一阶近似假设,利用分离变量形式的位移函数和无网格算法,导出了关于应力奇异性指数的离散形式的奇异性特征方程.由奇异性特征方程的特征值和特征向量,即可确定应力奇异性指数、位移角函数和应力角函数.数值求解了纤维/基体轴对称界面端模型的奇异性特征方程, 结果表明:尺寸效应参数δ(奇异点与轴对称轴的距离和应力奇异性支配区域大小的比值)影响着应力奇异性的强弱与阶次, 准一阶近似解析解只是δ>>1时的一个特例.      

多重应力奇异性及其强度系数的数值分析方法

以具有两个应力奇异性次数的平面问题为例,提出了一种利用普通的数值分析结果确定奇异点附近多重应力奇异性的各阶次数以及相应的应力强度系数的数值分析方法,计算实例表明,本方法可以精确地求得各阶应力奇异性的次数,并且可以很方便地应用外插法确定出对应的应力强度系数。     

Stress singularities at crack corners

In this paper the stress and displacement fields near an embedded crack corner in a linear elastic medium are analytically computed. The conical-spherical coordinate system is introduced to solve this problem. It is observed that the strength of the stress singularity depends on the angle of the crack corner. The singularity becomes weaker, varying from r ^-1 to r ⁰, as the angle of the crack corner varies from 360° to 0°. Both symmetric and skew-symmetric loadings give the same variation of the behavior of the stress singularity. It is also found that the order of the singularity is independent of the Poisson's ratio, unlike the corner cracks at a free surface where Poisson's ratio affects the results.     

Analytical removal of singularity and direct evaluation of singular integrals in 3-D elastoplastic finite deformation analysis by BEM

As a further development of the present authors' research work [1,2], in this paper a method of the so-called quadratic pentahedron polar co-ordinate transformation and analytical removal of singularity of Cauchy principal value singular integrals is proposed to evaluate the strongly singular integrals in the sense of Cauchy principal values and the weakly singular integrals over quadratic internal cells in 3-D elastoplastic finite deformation analysis by BEM. First, a quadratic pentahedron polar co-ordinate transformation technique is used to reduce the order of singularity of the singular integrals. Then, a form of Gauss' theorem is introduced to remove the singularity in the Cauchy principal value singular integrals analytically. Therefore, the evaluation of all those strongly and weakly singular integrals can be carried out by standard Gaussian quadrature accurately and efficiently. Numerical examples of the 3-D elastoplastic problem and 3-D finite deformation problem are given to demonstrate that the method possesses good accuracy and numerical stability, and is convenient to implement. The method in this paper can be applied extensively to evaluating the singular integrals over cubic and higher order elements.