The Somigliana ring dislocation revisited

Solutions for Somigliana ring dislocations in an elastic half space and in two perfectly bonded dissimilar half spaces, with the dislocation ring parallel to the interface, are obtained from the Papkovich potential solutions by using Aderogba's Theorem. Interaction between a dislocation and a free surface or a perfectly bonded interface is considered. Implications of the results for the solution of crack problems are discussed.     

Inplane deformation of a circular inhomogeneity with imperfect interface

The plane elastic problem of a circular inhomogeneity with an imperfect interface of spring-constant-type is reduced to the solution of a Somigliana dislocation problem, when the solution for the corresponding problem with a perfect interface is known. The Burger's vector of the Somigliana dislocation is determined so that its components satisfy two interfacial conditions involving the traction components of the corresponding problem with a perfect interface. Employing complex variables, a two-phase potential solution to the Somigliana dislocation inhomogeneity problem is developed for a general form of the Burger's vector. Detailed results are reported for a uniform eigenstrain in the inhomogeneity, and for a remote uniform heat flow in the matrix. In the latter case, the inhomogeneity behaves as a void, when it begins to slide.     

应用边界积分法求圆形夹杂问题的解析解

边界元方法作为一种数值方法, 在各种科学工程问题中得到了广泛的应用.本文参考了边界元法的求解思路, 从Somigliana等式出发, 利用格林函数性质,得到了一种边界积分法, 使之可以用来寻求弹性问题的解析解.此边界积分法也可以从Betti互易定理得到. 应用此新方法, 求解了圆形夹杂问题.首先设定夹杂与基体之间完美连接, 将界面处的位移与应力按照傅里叶级数展开,根据问题的对称性与三角函数的正交性来简化假设, 减少待定系数的个数.其次选择合适的试函数(试函数满足位移单值条件以及无体力的线弹性力学问题的控制方程),应用边界积分法, 求得界面处的位移与应力的值. 然后再求解域内位移与应力.得到了问题的精确解析解, 当夹杂弹性模量为零或趋向于无穷大时,退化为圆孔或刚性夹杂问题的解析解. 求解过程表明,若问题的求解区域包含无穷远处时, 所取的试函数应满足无穷远处的边界条件.若求解区域包含坐标原点, 试函数在原点处位移与应力应是有限的.结果表明了此方法的有效性.      

Effect of rotation in magneto-thermoelastic transversely isotropic hollow cylinder with three-phase-lag model

This article deals with the various heat source responses in a transversely isotropic hollow cylinder under the purview of three-phase-lag (TPL) generalized thermoelasticity theory. In presence of magnetic field and due to the rotating behavior of the cylinder, the governing equations are redefined for generalized thermoelasticity with thermal time delay. In order to obtain the stress, displacement and temperature field, the field functions are expressed in terms of modified Bessel functions in Laplace transformed domain. When the outer radius of hollow cylinder tends to infinity, the corresponding results are discussed. Finally an appropriate Laplace transform inversion technique is adopted.     

Effect of interface stresses on the image force and stability of an edge dislocation inside a nanoscale cylindrical inclusion

Dislocation mobility and stability in inclusions can affect the mechanical behaviors of the composites. In this paper, the problem of an edge dislocation located within a nanoscale cylindrical inclusion incorporating interface stress is first considered. The explicit expression for the image force acting on the edge dislocation is obtained by means of a complex variable method. The influence of the interface effects and the size of the inclusion on the image force is evaluated. The results indicate that the impact of interface stress on the image force and the equilibrium positions of the edge dislocation inside the inclusion becomes remarkable when the radius of the inclusion is reduced to nanometer scale. The force acting on the edge dislocation produced by the interface stress will increase with the decrease of the radius of the inclusion and depends on the inclusion size which differs from the classical solution. The stability of the dislocation inside a nanoscale inclusion is also analyzed. The condition of the dislocation stability and the critical radius of the inclusion are revised for considering interface stresses.     

Generalization of the Prandtl problem for a creep model

An approximate solution to the problem of compression of an infinite layer of material between rough parallel plates is constructed with the creep equations being fulfilled. Constitutive relations in accordance with which the equivalent stress tends to a finite value as the equivalent strain rate tends to infinity are used. The behavior of the solution in the neighborhood of the maximum friction surface is studied. It is shown that the existence of the solution depends on one of the parameters included in the constitutive equations. If the solution exists, the equivalent strain rate tends to infinity in the neighborhood of the maximum friction surface, and the asymptotic behavior of the solution depends on the same parameter. It is established that there is a range of this parameter in which the nature of the change in the equivalent strain rate near the maximum friction surface is the same as in the solutions for rigid plastic materials.     

On the Elastic Field of a Shpherical Inhomogeneity with an Imperfectly Bonded Interface

This study is devoted to the development of a unified and explicit elastic solution to the problem of a spherical inhomogeneity with an imperfectly bonded interface. Both tangential and normal displacement discontinuities at the interface are considered and a linear interfacial condition, which assumes that the tangential and the normal displacement jumps are proportional to the associated tractions, is adopted. The elastic disturbance due to the presence of an imperfectly bonded inhomogeneity is decomposed into two parts: the first is formulated in terms of an equivalent nonuniform eigenstrain distributed over a perfectly bonded spherical inclusion, while the second is formulated in terms of an imaginary Somigliana dislocation field which models the interfacial sliding and normal separation. The exact form of the equivalent nonuniform eigenstrain and the imaginary Somigliana dislocation are fully determined in this paper.     

共晶基陶瓷复合材料的强度模型

根据细观结构内界面的强约束特性,通过纤维-基体内界面切应力确定了共晶陶瓷棒体的细观应力场.然后分析了两相界面处位错塞积产生的应力集中,获得基体内的最大应力,当最大拉应力等于基体理论断裂强度时,得到共晶棒体的断裂强度的解析表达式.考虑共晶陶瓷棒体长度和方位的随机性,根据概率理论得到共晶陶瓷基复合材料的宏观强度的理论模型.结果表明复合材料的宏观强度与亚微米纤维的直径和长度、以及亚微米纤维、基体、共晶陶瓷棒体的弹性常数有关.理论与实验结果十分接近,说明文中理论模型是合理的,同时证明了共晶界面对陶瓷复合材料的重要影响.     

Radial Capillary Transport from an Infinite Reservoir

Radial capillary transport occurs, for example, when wine spreads in the tablecloth ink in paper, rain drops in textiles, or dye into yarn. It is of technical relevance for propellant and other liquid transport in space. We present a theoretical and experimental study on the more basic situation when liquid spreads radially from an infinite reservoir. Our theoretical model predicts both outward and inward radial transport in a porous screen. While the outward wicking is fed by a circular wick in the center, the inward wicking is fed by a ring-like wick from the outside. For both cases, an analytical solution is obtained in terms of time versus radius as well as radius versus time aided by the Lambert W function. In the experiments, we use four different filter papers combined with three cylindrical wicks for outward wicking and one ring wick for inward wicking, respectively. The wicking process is recorded by a digital camera. Afterward, the resulting image series are evaluated with Matlab routines to detect the wicking front line. From the wetted area, we derive the mean radius versus time. Beside radially outward and inward wicking, we consider also experimental reference data from horizontal and vertical wicking in a strip.     

Interaction between dislocation and two circular cylindrical inclusions

In this paper, the elastic field of the infinite homogeneous medium with two circular cylindrical inclusions under the action of a screw dislocation is investigated and the corresponding analytical solution is obtained. Here, the conformal mapping and the theorem of analytical continuation are used. From the results obtained, it can be seen that the elastic field depends on the shear moduli of individual phases, the geometric parameters of the system, and the position and relative slip of the screw dislocation. In addition, the corresponding specific cases are also considered in this paper when two circular cylindrical inclusions are tangent to each other and they are holes and/or rigid inclusions. Finally, numerical results are illustrated to show the interaction between the screw dislocation and two circular cylindrical inclusions.     

含有非圆形双孔的无限平板中应力的解析解研究

无限平板中含有任意形状单个孔的问题可以使用复变函数方法获得其应力解析解.对于无限平板中含有两个圆孔或两个椭圆孔的双连通域问题,也可以利用多种方法进行求解,比如双极坐标法、应力函数法、复变函数法以及施瓦茨交替法等.其中复变函数中的保角变换方法是获得应力解析解的一个重要方法.但目前尚未见到用此方法求解无限板中含有一个正方形孔和一个椭圆孔的问题.当板在无穷远处受有均布载荷和孔边作用垂直均布压力时,利用保角变换方法可以求解板中含有两个特定形状孔的问题.该方法将所讨论的区域映射成象平面里的一个圆环,其中最关键的一步是找出相应的映射函数.基于黎曼映射定理,提出了该映射函数一般形式,并利用最优化方法,找到了该问题的具体映射函数,然后通过孔边应力边界条件建立了求解两个解析函数的基本方程,获得了该问题的应力解析解.运用ANSYS有限单元法与结果进行了对比.研究了孔距、椭圆形孔大小和两孔布置方位对边界切向应力的影响,以及不同载荷下两孔中心线上应力分布规律.      

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.     

弱界面异质球形夹杂本征应变问题的解析解

本文研究了具有线弹簧弱界面的异质球形夹杂的本征应变问题，所采用的线弹簧界面模型既能界面的切线方向滑动，又能考虑界面的法线方向张开，根据叠加原理、原问题的弹性场可分成三部分；二部分由真实均匀本征应变所引起，另一部分由等效的非均匀本征应变所引起，后一部分则由虚拟的Ｓｏｍｉｇｌｉａｎａ位错场所产生。本文求得了等效非均匀本征应变和虚拟位错场的Ｂｕｒｇｅｒ矢量的解析表达式，进而确定的问题的弹性场。     

一种特殊梯度分布的功能梯度圆柱壳的二维分析

功能梯度材料是一种新型材料,其结构分析已成为当今力学研究的热点。本文对一种特殊梯度分布的功能梯度材料圆柱壳进行了二维精确分析。从弹性力学平面应变问题的基本方程出发,引入应力函数,导出功能梯度材料圆柱壳受静载作用下的控制微分方程。假设材料的杨氏模量沿半径方向呈幂函数分布,泊松比为常数,利用分离变量法,导出了简支边界情况下功能梯度圆柱壳的精确解。通过算例分析了不同梯度变化时,功能梯度圆柱壳内的应力和位移变化规律。计算结果表明不同梯度分布的圆柱壳结构中的应力、位移沿厚度方向的变化规律是不同的,有时甚至差别很大。因此对于材料性质梯度变化的功能梯度材料圆柱壳,必须针对其自身特点,建立相应的理论分析模型。     

Stress field due to elastic mass growth on spherical and cylindrical substrates

Mass growth on cylindrical and spherical substrates is a phenomenon which can be related to the biochemical creation of an elastic actin gel shell by polymerization and cross linking of actin filaments either in vivo on bacteria cylindrical surfaces as a means for their motility or in vitro on spherical beads as a means for experimentally studying the previous in vivo case. Such mass growth is strongly effected by the developed stress field. The objective of this paper is to accurately determine this stress field assuming elasticity of the growing mass and symmetrical growth. Based on the special kinematics of mass growing on spherical and cylindrical substrates, inwards or outwards from them, and various isotropic constitutive laws for both small and finite elastic strains, it is possible to obtain the solution for the stress field in closed analytical form for all cases considered. This expands very significantly recent findings [Dafalias, Y.F., Pitouras, Z., 2008. Stress field in actin gel growing on spherical substrate. Biomechanics and Modeling in Mechanobiology, doi:10.1007/s10237-007-0113-y.] for some constitutive laws and outwards growth on spherical substrates only. The effect of biomass compressibility is shown to be of cardinal importance for the developed stress field, questioning the validity of the simplifying assumption of a zero value Poisson ratio usually made in the relevant biomechanics literature for simplicity. Few selected graphs of stress variation along the radial direction illustrate the analytical findings. The obtained closed form analytical expressions for stress can be a standard reference tool in this important area of stress-modulated soft tissue growth.     

Shielding effect and emission condition of a screw dislocation near a blunt crack in elliptical inhomogeneity

The shielding effect and emission condition of a screw dislocation near a blunt crack in elastic elliptical inhomogeneity is dealt with. Utilizing the Muskhelishvili complex variable method, the explicit series form solutions of the complex potentials in the matrix and the inclusion regions are derived. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The influences of the orientation of the dislocation and morphology of the blunt crack as well as the material elastic dissimilarity upon the shielding effect and emission criterion are discussed in detail. As a result, numerical analysis and discussion show that the positive screw dislocation can reduce the stress intensity factor of the crack tip (shielding effect) only when it is located in the certain region. The shielding effect increases with the increase of the shear modulus of the matrix and the curvature radius of the blunt crack tip, but decreases with the increase of dislocation azimuth angle. The critical loads at infinity for dislocation emission increases with the increment of the emission angle and the curvature radius of the blunt crack tip, and the most probable angle for screw dislocation emission is zero. The present solutions contain previous results as the special cases.     

一维六方准晶压电材料含运动螺型位错弹性问题分析

借助复变函数方法,研究了点群6mm 一维六方准晶压电材料中运动螺型位错,得到位错芯附近的应力、位移、能量的解析表达式.分析发现,在靠近运动螺型位错芯处,声子场和相位子场应力的分量表现出(x2+y2)-1的奇异性.当不考虑相位子场的影响时,文中得到的结果可以退化为含运动螺型位错的压电材料中的结果;当不考虑电场的影响时,退...     

Tangential interaction of elastic spherical particles in contact

The elastic interaction of spherical particles is studied. The distribution of the stress, normal to the contact plane, is determined by the rod model suggested recently, which is applicable in the more wide range of deformations as compared with the classical Hertz law. In the rod model context an inner part of compressed particles is regarded as an elastic cylindrical rod, which radius is equal to the contact radius. The rod reaction is added to the normal particle interaction corresponding with the Hertz solution. The resulting normal force passes into the Hertz solution for infinitesimal deformations and gives stronger particle repulsion for finite deformations. Here we solve the Mindlin problem for the rod model, i.e., derive the tangential interaction of initially compressed particles when a relative displacement takes place. The analytical expressions, which determine the total displacement of the sphere’s centers and the corresponding tangential force, are derived. So, the generalization of the classical Mindlin law is obtained for the rod model.