线夹杂和线夹杂的相互作用

以短纤维复合材料为工程背景，本文利用线夹杂的工程计算模型以及无限平面中单夹杂的基本解，导出了线夹杂和线夹杂相互作用的平面问题的奇异积分方程。给出了夹杂端点的应力强度因子和夹杂界面应力的表达式，并作了具体的数值计算。     

Interaction between a rigid line inclusion and an elastic circular inclusion

I.IntroductionManypracticalproblemsinengineering,suchascompositematerial,weldedjointorribbedslab,needustostudytheinteractionproblemoflineinclusionandcircularinclusionasshowninFig.1.Sotheproblemwasdiscussedinthispaper.Proceedingfromthestressfieldofplanecon…     

Interaction of harmonic axisymmetric waves with a thin circular absolutely rigid separated inclusion

We study stress concentration near a circular rigid inclusion in an unbounded elastic body (matrix). In the matrix, there are wave motions symmetric with respect to the axis passing through the inclusion center and perpendicular to the inclusion. It is assumed that one of the inclusion sides is completely fixed to the matrix, while the other side is separated and the conditions of smooth contact are realized on that side. The solution method is based on the fact that the displacements caused by waves reflected from the inclusion are represented as a discontinuous solution of the Lamé equations. This permits reducing the original problem to a system of singular integral equations for functions related to the stress and displacement jumps on the inclusion. Its solution is constructed approximately by the collocation method with the use of special quadrature formulas for singular integrals. The approximate solution thus obtained permits numerically studying the stress state in the matrix near the inclusion. Technological defects or constructive elements in the form of thin rigid inclusions contained in machine parts and engineering structure members are stress concentration sources, which may result in structural failure. It is shown that the largest stress concentration is observed near separated inclusions. Static problems for elastic bodies with such inclusions have been studied rather comprehensively [1, 2]. The stress concentration near separated inclusions under dynamic actions on the bodies has been significantly less studied even in the case of harmonic vibrations. The results of these studies can be found in [3, 4], where bodies with a thin separated inclusion were considered, and in [5], where the problem about torsional vibrations of a body with a thin circular separated inclusion was studied. The aim of the present paper is to study stress concentration near such an inclusion in the case of interaction with harmonic waves under axial symmetry conditions.     

Problem of interaction of thin rigid needle-shaped inclusions located between different elastic materials

We study a piecewise-homogeneous elastic plane composed of two half-planes with different elastic parameters and two thin rigid needle-shaped inclusions located between them. One inclusion is rigidly connected with the environment, and the other inclusion is not, while contacting with it like a smooth rigid punch. We consider the plane deformed state generated by stresses given at infinity. The problem is reduced to a combination of a matrix Riemann boundary-value problem from the theory of analytic functions and a matrix Hilbert problem, which can be solved in terms of integrals through the reduction to two separate scalar Riemann boundary-value problems on a twosheeted Riemann surface.We explicitly obtain the complex potentials of the composite elastic plane, the stress intensity factors near the tips of the inclusion, and the rotation angles of the inclusions. We also present numerical examples illustrating how the stresses near the inclusions depend on the elastic and geometric parameters of the problem.     

PLANE ELASTIC PROBLEM ON RIGID LINES ALONG CIRCULAR INCLUSION

The plane elastic problem of circular-arc rigid line inclusions is considered. The model is subjected to remote general loads and concentrated force which is applied at an arbitrary point inside either the matrix or the circular inclusion. Based on complex variable method, the general solutions of the problem were derived. The closed form expressions of the sectionally holomorphic complex potentials and the stress fields were derived for the case of the interface with a single rigid line. The exact expressions of the singular stress fields at the rigid line tips were calculated which show that they possess a pronounced oscillatory character similar to that for the corresponding crack problem under plane loads. The influence of the rigid line geometry, loading conditions and material mismatch on the stress singularity coefficients is evaluated and discussed for the case of remote uniform load.     

A self-consistent statistical mechanics approach for determining effective elastic properties of composites

A self-consistent statistical mechanics approach for determining the effective elastic properties of composites with random structure is developed. The problem is reduced to the model of a single inclusion with a non-homogeneous elastic neighbourhood in a medium with effective elastic properties. The inhomogeneous elastic properties and size of neighbourhood are defined by randomness of the geometry, random size of inclusions and random elastic properties of the inclusions. Numerical results are given for the effective elastic properties of a composite with hollow spherical inclusions.     

双周期圆柱形夹杂纵向剪切问题的精确解

研究无限介质中矩形排列双周期圆柱形夹杂的纵向剪切问题．利用Eshelby等效夹杂理论并结合双周期与双准周期解析函数工具，为这类考虑夹杂相互影响的问题提供了一个严格又实用的分析方法，求得了问题的全场级数解．作为退化情形得到单夹杂问题的经典解答，双周期孔洞、双周期刚性夹杂及单行(列)周期弹性夹杂等问题也可作为特殊情况被解决．数值结果揭示了这类非均匀材料力学性质随微结构参数变化的规律．     

Analysis of the Singularity for the Vertical Contact Problem of Line Crack-Inclusion

ＩｎｔｒｏｄｕｃｔｉｏｎＵｐｔｏｎｏｗ ,ｔｈｅｔｅｃｈｎｉｃａｌｌｉｔｅｒａｔｕｒｅｏｎｓｅｐａｒａｔｅｃｒａｃｋｓ,ｖｏｉｄｓ,ｉｎｃｌｕｓｉｏｎｓａｎｄｔｈｅｉｎｔｅｒａｃｔｉｏｎｓｂｅｔｗｅｅｎｃｒａｃｋｓａｎｄｉｎｃｌｕｓｉｏｎｓｈａｖｅｂｅｅｎｑｕｉｔｅｅｘｔｅｎｓｉｖｅ.Ｈｏｗｅｖｅｒ,ｔｈｅｃｏｎｔａｃｔｐｒｏｂｌｅｍｓｏｆｃｒａｃｋ＿ｉｎｃｌｕｓｉｏｎｄｏｎｏｔｓｅｅｍｔｏｂｅａｓｗｉｄｅｌｙｓｔｕｄｉｅｄ .Ｔｈｉｓｐａｐｅｒｃａｎｂｅｒｅｇａｒｄｅｄａｓｔｈｅｆｕｒｔｈｅｒｒｅｓｅ…     

Interfacial Energies for Incoherent Inclusions

We study a variational problem describing an incoherent interface between a rigid inclusion and a linearly elastic matrix. The elastic material is allowed to slip relative to the inclusion along the interface, and the resulting mismatch is penalized by an interfacial energy term that depends on the surface gradient of the relative displacement. The competition between the elastic and interfacial energies induces a threshold effect when the interfacial energy density is non-smooth: small inclusions are coherent (no mismatch); sufficiently large inclusions are incoherent. We also show that the relaxation of the energy functional can be written as the sum of the bulk elastic energy functional and the tangential quasiconvex envelope of the interfacial energy functional.     

The principle of equivalent eigenstrain for inhomogeneous inclusion problems

In this paper, based on the principle of virtual work, we formulate the equivalent eigenstrain approach for inhomogeneous inclusions. It allows calculating the elastic deformation of an arbitrarily connected and shaped inhomogeneous inclusion, by replacing it with an equivalent homogeneous inclusion problem, whose eigenstrain distribution is determined by an integral equation. The equivalent homogeneous inclusion problem has an explicit solution in terms of a definite integral. The approach allows solving the problems about inclusions of arbitrary shape, multiple inclusion problems, and lends itself to residual stress analysis in non-uniform, heterogeneous media. The fundamental formulation introduced here will find application in the mechanics of composites, inclusions, phase transformation analysis, plasticity, fracture mechanics, etc.     

Geometrical shape of in-plane inclusion characterized by polynomial internal stress field under uniform eigenstrains

The internal stress field of an inhomogeneous or homogeneous inclusion in an infinite elastic plane under uniform stress-free eigenstrains is studied. The study is restricted to the inclusion shapes defined by the polynomial mapping functions mapping the exterior of the inclusion onto the exterior of a unit circle. The inclusion shapes, giving a polynomial internal stress field, are determined for three types of inclusions, i.e., an inhomogeneous inclusion with an elastic modulus different from the surrounding matrix, an inhomogeneous inclusion with the same shear modulus but a different Poisson’s ratio from the surrounding matrix, and a homogeneous inclusion with the same elastic modulus as the surrounding matrix. Examples are presented, and several specific conclusions are achieved for the relation between the degree of the polynomial internal stress field and the degree of the mapping function defining the inclusion shape.     

Independence of the Presence of Cracks or Inclusions of the Net Interaction Force Acting on a Dislocation by a Skewed Line Singularity

Orlov and Indenbom [1] have shown that the net (integrated) interaction force F between two skew dislocations with Burgers vectors separated by a distance h in an infinite anisotropic elastic medium is independent of h. Nix [2] computed numerically the net interaction force F between two skew dislocations that are parallel to the traction-free surface X2=0 of an isotropic elastic half-space. His numerical results showed that F was independent of h; a partial result of what Barnett [3] called Nix"s theorem. The separation-independence portion of Nix"s theorem has been proved to hold for a general anisotropic elastic half-space with a traction-free, rigid, or slippery surface, and for bimaterials [3-5]. In this paper, we show that the net interaction force is independent of the presence of inclusions. We will consider the case in which the line dislocation b is a more general line singularity which can include a coincident line force with strength f per unit length of the line singularity. An inclusion is an infinitely long dissimilar anisotropic elastic cylinder of an arbitrary cross-section whose axis is parallel to the line singularity (f, b). The (skew) line dislocation does not intersect the inclusion. The special cases of an inclusion are a void, crack, or rigid inclusion. There can be more than one inclusion of different cross sections and different materials. The line singularity (f, b) can be outside the inclusions or inside one of the inclusions. The inclusions and the matrix need not have a perfect bonding. One can have a debonding with or without friction. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts

The present paper deals with the problem of load transfer from elastic inclusions to an infinite elastic orthotropic plane with cuts located on one of the principal orthotropy directions. The constitutive system of equations of this problem is derived under the assumption that the inclusions are in a uniaxial stress state. The obtained system consists of a singular integro-differential equation and a singular integral equation for the jumps of the tangential stresses acting on the inclusion shores and for the derivative of the the cut opening function. The behavior of solutions of the system of constitutive equations at the endpoints of the inclusions and cuts is studied, and the solution of this system is constructed by the numerical-analytic discrete singularity method.     

Stress state of a piezoceramic body with a plane crack opened by a rigid inclusion

The paper establishes a relationship between the solutions for cracks located in the isotropy plane of a transversely isotropic piezoceramic medium and opened (without friction) by rigid inclusions and the solutions for cracks in a purely elastic medium. This makes it possible to calculate the stress intensity factor (SIF) for cracks in an electroelastic medium from the SIF for an elastic isotropic material, without the need to solve the electroelastic problem. The use of the approach is exemplified by a penny-shaped crack opened by either a disk-shaped rigid inclusion of constant thickness or a rigid oblate spheroidal inclusion in an electroelastic medium __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 47–60, July 2008.     

The Stress State of Three-Dimensional Bodies with Nonsmooth Inclusions

The stress state of a three-dimensional body with inclusions bounded by surfaces with singular lines (sets of corner points) and a conical point is studied. By determining the asymptotics of displacements and stresses at the singularities of interfaces and using the generalized elastic potentials of single and double layers, the problem posed is reduced to a system of singular integral equations. The results obtained are used to analyze the stress state of a body with a circular conical inclusion     

椭圆类夹杂分析的杂交应力有限元方法

本文给出了一种分析椭圆类夹杂周边应力场的新型杂交应力有限元方法。基于弹性力学中平面问题的Muskhelishvili复势方法,应用保角变换映射技术,以Laurent级数和Faber级数为工具,借助Hellinger-Reissner原理构建一个能够反映椭圆类夹杂周边弹性现象同时包含椭圆夹杂的多边形超级单元。将该超级单元与标准的4节点杂交应力单元耦合在一起即可建立一种分析椭圆类夹杂周边弹性场的新型特殊杂交应力有限元方法。文中考核算例表明：本文方法不但使用简单、有效,而且精度高、单元少。作为本文方法的一个拓展应用,文章最后给出了一个分析含二个椭圆夹杂无限大各向同性板在远场均布载荷作用下椭圆夹杂周边弹性场的算例,并讨论了椭圆夹杂间距和弹性刚度比对应力集中系数的影响。     

Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

The Crack-Inclusion Interaction and the Analysis of Singularity for the Horizontal Contact

ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｉｌｅｗｅｓｔｕｄｙｔｈｅｓｔｒｅｎｇｔｈａｎｄｔｈｅｃｒａｃｋｏｆｐｒａｃｔｉｓｉｎｇｃｏｍｐｏｎｅｎｔｓ,ｔｈｅｍａｔｅｒｉａｌｄｅｆｅｃｔｉｏｎｓｈｏｕｌｄｂｅｃｏｎｓｉｄｅｒｅｄ .ＩｎｔｈｅｏｐｉｎｉｏｎｓｏｆＣｒａｃｋＭｅｃｈａｎｉｃｓ,ｔｈｅｍａｔｅｒｉａｌｄｅｆｅｃｔｉｏｎｃａｎｂｅｒｅｄｕｃｅｄｔｏｐｌａｎａｒｃｒａｃｋｓａｎｄｉｎｃｌｕｓｉｏｎｓ.Ｂｅｓｉｄｅｓ,ｔｈｅｐｒｏｂｌｅｍｏｆｓｈｏｒｔ＿ｆｉｂｅｒｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｓｕｃｈａｓ…     

双周期分布圆形弹性夹杂平面热弹性问题

研究了含双周期分布圆形弹性夹杂的无限弹性平面在均匀拉伸和均匀温变下的弹性响应问题．运用Isida的区域单元法和复势函数的级数展开技术,将问题转化为线性方程组的求解．数值结果表明：相邻夹杂问距过大或过小都不利于减小界面应力,当相邻夹杂中心间距与夹杂半径之比为2．2～2．8时,界面剪切应力与环向应力的极大值最小;比值为2．5～3．5时,界面最大径向应力值最小;并且该比值范围不随两相材料弹性模量之比和热膨胀系数之比而变化．     

Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.