含功能梯度材料加强环的任意几何形状孔附近应力集中分析

基于复变函数理论，结合保角变换技术研究含功能梯度材料（FGM）加强环的任意几何形状孔附近应力集中。采用分层均匀化方法，给出了远场均布载荷作用下材料参数沿孔周法线方向任意变化的FGM加强环内的复势函数解。通过数值算例，详细讨论了加强环内杨氏模量不同变化规律对三角形、正方形、矩形等各种几何形状孔附近应力分布的影响。结果表明：通过在孔周衬入FGM加强环并合理选择加强环内材料参数的递变规律，可以有效缓解各种几何形状孔附近的应力集中。同时通过一些特例与已有文献比对验证了本文结果的正确性。     

非理想界面对三相复合材料应力场的影响

对非理想界面的三相复合材料,提出了计算弹性应力场的微观力学模型,在适当的简化假设下,对带界相的颗粒增强和纤维增强复合材料,得到了应力场的计算公式。以剪切载荷为例给出了数值例子。给出的数值结果表明非理想界面对三相复合材料应力场的影响。     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

An inclusion problem

An inclusion is a special region in a material, and this region experiences a transformation of the following nature. If the inclusion were free, then it would acquire a certain deformation with no stress arising in it; but since the inclusion is “pasted” into the material, this prevents free deformations and causes stresses arising in the inclusion itself and in the environment. Three systems of equations describing the problem are derived. For a space with a homogeneous isotropic matrix, an equivalent system of integral equations is obtained whose solution, for a homogeneous anisotropic ellipsoidal inclusion, is reduced to a system of linear algebraic equations. For the case where the moduli of elasticity in the inclusion and the homogeneous matrix coincide, an explicit solution for an inclusion of arbitrary shape is obtained.     

压电螺位错与含界面裂纹圆形涂层夹杂的干涉

研究位于基体或夹杂中任意点的压电螺型位错与含界面裂纹圆形涂层夹杂的电弹耦合干 涉问题. 运用复变函数方法，获得了基体，涂层和夹杂中复势函数的一般解答. 典型例 子给出了界面含有一条裂纹时，复势函数的精确级数形式解. 基于已获得的复势函数和广 义Peach-Koehler公式，计算了作用在位错上的像力. 讨论了裂纹几何条件，涂层厚度和材 料特性对位错平衡位置的影响规律. 结果表明，界面裂纹对涂层夹杂附近的位错运动有很大 的影响效应，含界面裂纹涂层夹杂对位错的捕获能力强于完整粘结情况；并发现界面裂纹长 度和涂层材料常数达到某一个临界值时可以改变像力的方向. 解答的特殊情形包含了以 往文献的几个结果.     

The scattering of P-waves by a piezoelectric particle with FGPM interfacial layers in a polymer matrix

Propagation of P-wave in an unbounded elastic polymer medium which contains a set of nested concentric spherical piezoelectric inhomogeneities is formulated. The polymer matrix is made of Epoxy and is isotropic; each phase of the inhomogeneity is made of a different piezoelectric material and is radially polarized and has spherical isotropy. Note that the individual phases are homogeneous, and all interfaces are perfectly bonded. The scattered displacement and electric potentials in the matrix are expressed in terms of spherical wave vector functions and Legendre functions, respectively. The transmitted displacement and electric potentials within each phase of the piezoelectric particle are expressed in terms of Legendre functions. The equations of motion and electrostatics in each phase of the piezoelectric inhomogeneity lead to a system of coupled second order differential equations, which is solved using the generalized Frobenius series. The present theory is extended to the case where the core of the inhomogeneity is made of PZT-4 and its coating is made of functionally graded piezoelectric material (FGPM) whose microstructural composition varies smoothly from PZT-4 at the core–coating interface to Epoxy at the coating–matrix interface. The effects of different types of variation in the electro-mechanical properties of FGPM on scattering cross-section and other electro-mechanical fields are addressed. The present theory is valid for arbitrary coating thickness, and arbitrary frequencies.     

A coated rigid elliptical inclusion loaded by a couple in the presence of uniform interfacial and hoop stresses

We consider a confocally coated rigid elliptical inclusion, loaded by a couple and introduced into a remote uniform stress field. We show that uniform interfacial and hoop stresses along the inclusion–coating interface can be achieved when the two remote normal stresses and the remote shear stress each satisfy certain conditions. Our analysis indicates that: (i) the uniform interfacial tangential stress depends only on the area of the inclusion and the moment of the couple; (ii) the rigid-body rotation of the rigid inclusion depends only on the area of the inclusion, the coating thickness, the shear moduli of the composite and the moment of the couple; (iii) for given remote normal stresses and material parameters, the coating thickness and the aspect ratio of the inclusion are required to satisfy a particular relationship; (iv) for prescribed remote shear stress, moment and given material parameters, the coating thickness, the size and aspect ratio of the inclusion are also related. Finally, a harmonic rigid inclusion emerges as a special case if the coating and the matrix have identical elastic properties.     

Harmonic elastic inclusions in the presence of point moment

We employ conformal mapping techniques to design harmonic elastic inclusions when the surrounding matrix is simultaneously subjected to remote uniform stresses and a point moment located at an arbitrary position in the matrix. Our analysis indicates that the uniform and hydrostatic stress field inside the inclusion as well as the constant hoop stress along the entire inclusion–matrix interface (on the matrix side) are independent of the action of the point moment. In contrast, the non-elliptical shape of the harmonic inclusion depends on both the remote uniform stresses and the point moment.     

Complex variable function method for the scattering of plane waves by an arbitrary hole in a porous medium

Based on the complex variable function method, a new approach for solving the scattering of plane elastic waves by a hole with an arbitrary configuration embedded in an infinite poroelastic medium is developed in the paper. The poroelastic medium is described by Biot's theory. By introducing three potentials, the governing equations for Biot's theory are reduced to three Helmholtz equations for the three potentials. The series solutions of the Helmholtz equations are obtained by the wave function expansion method. Through the conformal mapping method, the arbitrary hole in the physical plane is mapped into a unit circle in the image plane. Integration of the boundary conditions along the unit circle in the image plane yields the algebraic equations for the coefficients of the series solutions. Numerical solution of the resulting algebraic equations yields the displacements, the stresses and the pore pressure for the porous medium. In order to demonstrate the proposed approach, some numerical results are given in the paper.     

Contact stress analysis of an elastic half-plane containing multiple inclusions

This paper presents a two-dimensional contact stress analysis to investigate the effects of multiple inclusions on the contact pressure and subsurface stresses in an elastic half-plane. The boundary element method is used to analyze the contact problem where a set of integral equations is derived on the contact region and the matrix–inclusion interfaces. As the contact region is unknown a priori, an iterative procedure is implemented to determine the actual contact region and the contact pressure, and the tractions and displacements on the matrix–inclusion interfaces are obtained by solving the integral equations numerically. Numerical results show that the inclusions near contact surface could cause significant alterations in the contact pressure distribution. The stiff inclusions could toughen the surrounding material and reduce the internal stresses while the soft inclusions could increase the subsurface stresses.     

Applications of the method of complex functions to dynamic stress concentrations

The complex function method used in the solution of static stress concentration around an irregularly shaped cavity in an infinite elastic plane is generalized to the case of dynamic loading. This paper presents the solutions of two dimensional elastic wave equations in terms of complex wave functions, and general expressions for boundary conditions for steady state incident waves. Dynamic stresses around a cavity of arbitrary shape are then expressed in series of complex ‘domain functions’, the coefficient of the series can be determined by truncating a set of infinite algebraic equations. Results of dynamic stress concentration factors for circular and elliptical cavities are given in this paper.     

饱和土中的任意形状孔洞对弹性波的散射

根据Biot波动理论建立了求解饱和土中任意形状孔洞对弹性波散射问题的复变函数方法.首先通过引入位移势函数把稳态条件下的Biot波动方程解耦为势函数所满足的Helmholtz方程.利用分离变量方法即得到Helmholtz方程完备的通解.根据所得位移势函数的通解,可得骨架位移、流体相对骨架的位移、应力和孔压的表达式.通过保角变换方法,把物理平面上的孔洞映射到像平面上单位圆.利用土骨架和流体的边界条件,即可确定波函数展开式中的未知系数.给出了一些数值结果.     

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.     

线夹杂的力学模型及其界面应力分析

采用材料力学的直杆和梁的变形假定，对平面线夹杂问题提出了一种能同时考虑夹杂两侧法向应力和剪应力间断的新的力学模型，然后通过集中力作用的Ｋｅｌｖｉｎ解答，求得了单夹杂问题的基本解。文中还导出了夹杂两侧的界面应力公式。最后对夹杂端点的应力强度因子及界面应力作了计算，结果令人满意     

On solutions of the problem in stresses with the use of Maxwell stress functions

The spatial problems of elasticity are mainly solved in displacements [1, 2], i.e., the Lamé equations are taken as the initial equations. This is related to the lack of general solutions for the system of basic equations of elasticity expressed in stresses. In this connection, a new variational statement of the problem in stresses was developed in [3, 4]; this statement consists in solving six generalized equations of compatibility for six independent components of the stress tensor, while the three equilibrium equations are transferred to the set of boundary conditions. This method is more convenient for the numerical solution of problems in stresses and has been tested when solving various boundary value problems. In the present paper, analyzing the completeness of the Saint-Venant identities and using the Maxwell stress functions, we obtain a new resolving system of three differential equations of strain compatibility for the three desired stress functions φ, ξ, and ψ. This system is an alternative to the three Lamé equilibrium equations for three desired displacement components u, v, w and is simpler in structure. Moreover, both of these systems of resolving equations can be solved by the new recursive-operator method [5, 6]. In contrast to well-known methods for constructing general solutions of linear differential equations and their systems, the solutions obtained by the recursive-operator method are constructed as operator-power series acting on arbitrary analytic functions of real variables (not necessarily harmonic), and the series coefficients are determined from recursive relations (matrix in the case of systems of equations). The arbitrary functions contained in the general solution can be determined directly either from the boundary conditions (the obtained system of inhomogeneous equations with a right-hand side can also be solved by the recursive-operator method [6]) or by choosing them from various classes of analytic functions (elementary, special); a complete set of particular solutions can be obtained in the same function classes, and the coefficients of linear combinations of particular solutions can be determined by the Trefftz method, the least-squares method, and the collocation method.     

无限域分层介质的基本解

本文从三维弹性力学最基本的平衡方程和本构关系出发，推导出状态传递微分方程，在求解状态传递微分方程时，建议了一种对指数矩阵进行分解的方法，避免了直接解法可能导致状态变量的发散的问题，引入了无穷远处的状态为量为有限值的条件，推导出上，下无限层表面的位移与应力关系式，再根据状态传递方程，可得出层状介质任意点的应力和位移的值，此结果可直接退化到无限域经典的Kelvin解。     

Interacting elliptic inclusions by the method of complex potentials

An accurate series solution has been obtained for a piece-homogeneous elastic plane containing a finite array of non-overlapping elliptic inclusions of arbitrary size, aspect ratio, location and elastic properties. The method combines standard Muskhelishvili’s representation of general solution in terms of complex potentials with the superposition principle and newly derived re-expansion formulae to obtain a complete solution of the many-inclusion problem. By exact satisfaction of all the interface conditions, a primary boundary-value problem stated on a complicated heterogeneous domain has been reduced to an ordinary well-posed set of linear algebraic equations. A properly chosen form of potentials provides a remarkably simple form of solution and thus an efficient computational algorithm. The theory developed is rather general and can be applied to solve a variety of composite mechanics problems. The advanced models of composite involving up to several hundred inclusions and providing an accurate account for the microstructure statistics and fiber–fiber interactions can be considered in this way. The numerical examples are given showing high accuracy and numerical efficiency of the method developed and disclosing the way and extent to which the selected structural parameters influence the stress concentration at the matrix–inclusion interface.     

含非完整界面涂层夹杂中螺型位错的弹性分析

研究了含非完整界面圆形涂层夹杂内部一个螺型位错在夹杂、涂层与无限大基体材料中产生的弹性场.运用复变函数函数方法,获得了三个区域复势函数的解析解答.利用求得的应力场和Peach-Koehler公式,得到了作用在螺型位错上位错力的精确表达式.主要讨论了两个非完整界面对位错力的影响规律.结果表明,涂层界面对夹杂内部螺型位错的吸引力随着界面粘结强度的弱化而变大.界面非完整程度增加削弱材料弹性失配对位错力的影响.在一定条件下,非完整界面可以改变夹杂内位错与涂层/基体系统之间的引斥干涉规律,并使位错在夹杂内部产生一个稳定或非稳定的平衡点.     

The material force acting on a screw dislocation in the presence of a multi-layered circular inclusion

In this paper, we study the interaction of a screw dislocation with a multi-layered interphase between a circularly cylindrical inclusion and a matrix. The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different shear moduli. The number of layers may also be arbitrary. Continuity of traction and displacement across all interfaces is assumed. We extend Honein et al.’s solution of circularly cylindrical layered media in anti-plane elastostatics to the case where all the singularities reside inside the inclusion core. The solution to this heterogeneous problem is given explicitly, for arbitrary singularities, as a rapidly convergent Laurent series, whose coefficients are expressed in terms of those of the complex potential of a corresponding homogeneous problem with the same singularities. We then consider the two particular cases of a screw dislocation, where, in the first instance, the dislocation resides inside the matrix, while, in the second instance, it is located in the inclusion core. In both instances, the Peach–Koehler force acting on the dislocation is calculated explicitly as a rapidly convergent series. We present several examples, where the effect of the layers on the material force is examined.     

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