Three-phase inclusions of arbitrary shape with internal uniform hydrostatic thermal stresses

We investigate the internal thermal stress field of a three-phase inclusion of arbitrary shape which is bonded to an infinite matrix through an interphase layer. The three phases have different thermoelastic constants. It is found that the internal thermal stress field induced by a uniform change in temperature can be uniform and hydrostatic within an inclusion of elliptical or hypotrochoidal shape when the thickness of the interphase layer is properly designed for given material parameters of the three-phase composite. Several examples are presented to demonstrate the solution. The thermal stress analysis of a (Q + 2)-phase inclusion of arbitrary shape with Q ≥ 2 is also carried out under the assumption that all the phases except the internal inclusion share the same elastic constants. It is found that the irregular inclusion shape permitting internal uniform hydrostatic thermal stresses becomes really arbitrary if a sufficiently large number of interphase layers are added between the inclusion and the matrix.     

A circular inhomogeneity with a mixed-type imperfect interface in anti-plane shear

We present a rigorous study of the problem associated with a circular inhomogeneity embedded in an infinite matrix subjected to anti-plane shear deformations. The inhomogeneity and the matrix are each endowed with separate and distinct surface elasticities and are bonded together through a soft spring-type imperfect interphase layer. This combination is referred to in the literature as a ‘mixed-type imperfect interface’ due to the fact that the soft interphase layer (described by the spring model) is bounded by two stiff interfaces arising from the separate surface elasticities of the inhomogeneity and the matrix. The entire composite is subjected to remote shear stresses and we allow for the presence of a screw dislocation in either the inhomogeneity or the matrix. The corresponding boundary value problem is reduced to two coupled second-order differential equations for the two analytic functions defined in the two phases (as well as their analytical continuations) leading to solutions in either series or closed-form. The analysis indicates that the stress field in the composite and the image force acting on the screw dislocation can be described completely in terms of three size-dependent parameters and a size-independent mismatch parameter. Interestingly, in the absence of the screw dislocation, the size-dependent stress field inside the inhomogeneity is uniform. Several numerical examples are presented to demonstrate the solution for a screw dislocation located inside the matrix. The results show that it is permissible for the dislocation to have multiple equilibrium positions.     

On a partially debonded rigid line inclusion penetrating a circular inhomogeneity

We derive closed-form solutions to the mixed boundary value problem of a partially debonded rigid line inclusion penetrating a circular elastic inhomogeneity under antiplane shear deformation. The two tips of the rigid line inclusion are just mutual mirror images with respect to the inhomogeneity/matrix interface, and the upper part of the rigid line inclusion is debonded from the surrounding materials. By using conformal mapping and the method of image, closed-form solutions are derived for three loading cases: (i) the matrix is subjected to remote uniform stresses; (ii) the matrix is subjected to a line force and a screw dislocation; and (iii) the inhomogeneity is subjected to a line force and a screw dislocation. In the mapped ξ-plane, the solutions for all the three loading cases are interpreted in terms of image singularities. For the remote loading case, explicit full-field expressions of all the field variables such as displacement, stress function and stresses are obtained. Also derived is the near tip asymptotic elastic field governed by two generalized stress intensity factors. The generalized stress intensity factors for all the three loading cases are derived.     

Shear of orthogonally reinforced spherofibrous composites

The problem of determining the shear characteristics and interphase stress concentration of fibrous composites with spherical inclusions is examined on the basis of a three-phase model. Stress fields caused by diffusion interaction of phases are neglected. The elastic moduli of the composite are investigated and compared with those obtained from a two-phase model. The general formula for determination of the shear modulus of triorthogonally reinforced compsites is derived using previously investigated relationships for averaged stress fields. The matrix of these compsites contained spherical cavities. The dependence of integral characteristics of three-phase composites on their bulk phase concentration was investigated. The stresses between phases were studied as a function of composite structure.A. A. Blagonravov Machine-Science Institute, Russian Academy of Sciences, Moscow, Russia. Translated from Mekhanika Kompozitnykh Materialov, No. 1, 104–111, January–February, 1997.     

复合材料对称层合板单向拉伸与面内剪切下的三维应力分析

本文给出了单向拉伸与面向剪切载荷下复合材料对称层合板中心区域的应力和应变沿板厚的数值计算分布规律。计算结果表明，在斜交对称辅层的层合板中心区域层间界面附近存在着层间边界层效应。层间界面处纤维走向的突变导致局部的三维应力状态和很强的应力集中。     

Dynamic interaction of an elastic medium with a thin-walled curvilinear piezoelectric inclusion under longitudinal vibrations of a composite

Using the method of matched asymptotic expansions, we obtain models of dynamic interaction of a thin-walled curvilinear piezoelectric inclusion of variable thickness with an elastic isotropic matrix under stationary vibrations of the composite. The elastic system is under conditions of longitudinal shear. Different cases of electric boundary conditions on the surface of the heterogeneity are considered. We propose an algorithm for the construction of boundary layer corrections for refining the behavior of displacements and stresses in the vicinity of the edge of the inclusion for its different shapes.     

An interface crack moving between magnetoelectroelastic and functionally graded elastic layers

A constant crack moving along the interface of magnetoelectroelastic and functionally graded elastic layers under anti-plane shear and in-plane electric and magnetic loading is investigated by the integral transform method. Fourier transforms are applied to reduce the mixed boundary value problem of the crack to dual integral equations, which are expressed in terms of Fredholm integral equations of the second kind. The singular stress, electric displacement and magnetic induction near the crack tip are obtained asymptotically and the corresponding field intensity factors are defined. Numerical results show that the stress intensity factors are influenced by the crack moving velocity, the material properties, the functionally graded parameter and the geometric size ratios. The propagation of the moving crack may bring about crack kinking, depending on the crack moving velocity and the material properties across the interface.     

Free edge stress prediction in thick laminated cylindrical shell panel subjected to bending moment

A thick composite cylindrical shell panel with general layer stacking is studied to investigate the free edge and 3D stresses in the panel which is subjected to pure bending moment. To this aim, a Galerkin based layerwise formulation is presented to discretize the governing equation of the panel to ordinary differential equations. Employing a reduced displacement field for the cylindrical panel, the governing equations for thick panel are developed in terms of displacements and a set of coupled ordinary differential equations is obtained. The governing equations are solved analytically for free edge boundary conditions and applied pure bending moment. The accuracy of numerical results is examined and the distribution of interlaminar and in-plane stresses is studied. The free edge stresses are studied and the effect of radius to thickness ratio, width to thickness ratio and layer stacking on the distribution of stresses is investigated. The focus of numerical results is on the prediction of boundary layer and free edge stress distribution.     

Analytic solution for a circular nano-inhomogeneity with interface stretching and bending resistance in plane strain deformations

In the mechanical analysis of composites containing nano-inhomogeneities, it is customary to consider only the stretching resistance of the inhomogeneity-matrix interface but neglect the bending resistance of the interface. In this paper, we consider a circular nano-inhomogeneity in an infinite elastic plane subjected to an arbitrary uniform remote in-plane loading with both stretching and bending resistance incorporated on the interface. Analytic solutions are obtained for the stress field both inside and outside the inhomogeneity by using an integral-type boundary condition representing the jump in traction across the interface. We show that the presence of interface bending resistance has no influence on the average of the mean stress in the inhomogeneity, and for certain interface stretching and bending rigidities the stress field inside the inhomogeneity can remain uniform regardless of the specific uniform remote loading. Numerical examples are presented to examine the influence of the interface bending resistance on the interfacial tractions imposed on the inhomogeneity and matrix for a uniform remote uniaxial loading. It is found that the introduction of interface bending resistance perturbs the (interfacial) tractions imposed on the inhomogeneity only slightly whether the inhomogeneity is softer or harder than the matrix, while it may influence the (interfacial) tractions imposed on the matrix significantly when the inhomogeneity is much softer than the matrix. Moreover, it is shown that the peak of the interface bending resistance-induced jump in traction across the interface initially increases and then decreases as the inhomogeneity becomes harder (from an initial state in which the inhomogeneity is softer than the matrix).     

The effect of interphase on residual thermal stresses. 2. Unidirectional fiber composite materials

In real composite materials an additional phase may exist between the fiber and the matrix. This phase, commonly known as the interphase, is a local region that results from the matrix bonds with the fiber surface or the fiber sizing. The differing thermal expansions or contractions of the fiber and matrix cause thermally induced stresses in composite materials. In the present study, a four-cylinder model is proposed for the determination of residual thermal stresses in unidirectional composite materials. The elastic modulus of the interphase is a function of the interphase radius and thickness. The governing equations in terms of displacements are solved in the form of expansion into a series [1]. The effective elastic characteristics are obtained using the finite element approach. The effect of the interphase thickness and different distributions of the interphase Young's modulus on the thermal residual stress field in unidirectional composite materials is investigated.For Pt. 1, see [1].Published in Mekhanika Kompozitnykh Materialov, Vol. 33, No. 2, pp. 200–214, March–April, 1997.     

Finite plane deformations of a three-phase circular inhomogeneity-matrix system

We consider finite plane deformations of a three-phase circular inhomogeneity-matrix system in which the inhomogeneity, the interphase layer and the matrix belong to the same class of compressible hyperelastic materials of harmonic-type but with each phase possessing its own distinct material properties. We obtain the complete solution when the system is subjected to general classes of remote (Piola) stress, specifically, remote stress distributions characterized by stress functions described by general polynomials of order n?1 in the corresponding complex variable z used to describe the matrix. As a particular case of the aforementioned analysis, we establish an Eshelby-type result namely that, for this class of harmonic materials, a three-phase circular inhomogeneity under uniform remote stress and eigenstrain, admits an internal uniform stress field when subjected to plane deformations.     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Micromechanical analysis of the stress transfer in single-fiber composite: The influence of the uniform and graded interphase with finite-thickness

A theoretical model is developed to analyze the stress transfer between fiber and matrix through the interphase with finite thickness. The Young's modulus of interphase is assumed to be homogeneous uniform or power-graded along radial direction while other material parameters are constants. The bonds between fiber and interphase as well as between interphase and matrix are perfect. The geometrical equations are strictly satisfied except that the radial displacement gradient with respect to the axial direction is neglected, as its magnitude is much smaller than that of the axial displacement gradient with respect to the radial direction. The equilibrium equations along radial direction are strictly satisfied, while the equilibrium equations along axial direction are satisfied in the integral forms. In addition, both the interfacial displacement and stress continuity conditions as well as stress boundary conditions are enforced exactly. Two coupled 2nd-order ordinary differential equations can be obtained in terms of average axial stresses in fiber and matrix. Finite element analysis (FEA) with refined mesh for single-fiber composite containing uniform interphase with finite thickness is developed to validate the present model. Series of parameter studies are performed to investigate the influence of interphase properties and thickness as well as the fiber volume content and model length on the stress distribution in composites.     

含椭圆形夹杂的压电体平面应变问题

利用复变函数理论,对在无限远处均匀应力和电位移载荷作用下的含有椭圆形弹性夹杂的横观各向同性压电材料,作了力电分析．在该文有限元结果和前人相关理论解的基础上,提了出一个可接受的认为弹性夹杂体内的应力场为常应力场的假设．在采用了不导通电边界条件之后,获得了以复势形式表示的压电基体的和弹性夹杂体内部的应力场解．     

Free-edge stress analysis of general composite laminates under extension, torsion and bending

In this study, based on the reduced form of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates subjects to extension, torsion, and bending. The constant parameters being in the displacement field, which describe the global deformation of a laminate, are appropriately calculated by using the improved first-order shear deformation theory. Reddy’s layerwise theory is subsequently employed for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. Various numerical results are developed for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates and geometric parameters on the boundary-layer stress are studied.     

Stress Distribution in an Elastic Body with a Periodically Curved Row of Fibers

Within the framework of a piecewise homogeneous body model, with the use of exact three-dimensional equations of elasticity theory for anisotropic bodies, a method is developed for investigating the stress distribution in an infinite elastic matrix containing a periodically curved row of cophasal fibers. It is assumed that fiber materials are the same and fiber midlines lie in the same plane. The self-balanced stresses arising in the interphase in uniaxial loading the composite along the fibers are investigated. The influences of problem parameters on these stresses are analyzed. The corresponding numerical results are presented.     

The three-dimensional problem of a thin inclusion in a composite elastic wedge

The three-dimensional problem of a thin rigid elliptic inclusion in the middle of a composite elastic wedge is investigated. The wedge consists of three connected wedge-shaped layers connected by a sliding clamp, in which the layer containing the inclusion is incompressible. The outer faces of the composite wedge are also under sliding-clamp conditions. The inclusion is completely bonded to the elastic medium in the contact region. Using Fourier and Kontorovich–Lebedev transformations, a system of integral equations of the problems are derived for the shear contact stresses. A regular asymptotic method is used to solve this system. Calculations are carried out. The results can be used for calculations on the strength of rubber-metal articles and structures having a corner line.     

圆形界面刚性线夹杂的平面问题

研究了圆弧形界面刚性线夹杂的平面弹性问题．集中力作用于夹杂或基体中的任意点,并且无穷远处受均匀载荷作用．利用复变函数方法,得到了该问题的一般解答．当只含一条界面刚性线夹杂时,获得了分区复势函数和应力场的封闭形式解答,并给出刚性线端部奇异应力场的解析表达式．结果表明,在平面荷载下界面圆弧形刚性线夹杂尖端应力场和裂纹尖端相似具有奇异应力振荡性．对无穷远加载的情况,讨论了刚性线几何条件、加载条件和材料失配对端部场的影响．     

Thermoelastic stability analysis of laminated composite plates: An analytical approach

The paper presents Chebyshev series based analytical solutions for the postbuckling response of the moderately thick laminated composite rectangular plates with and without elastic foundations. The plate is assumed to be subjected to in-plane mechanical, thermal and thermomechanical loadings. In-plane mechanical loading consists of uniaxial, biaxial, shear loadings and their combinations. The temperature induced loading is due to either uniform temperature or a linearly varying temperature across the thickness. The mathematical formulation is based on higher order shear deformation theory (HSDT) and von-Karman nonlinear kinematics. The elastic foundation is modeled as shear deformable with cubic nonlinearity. The thermal and mechanical properties of the composites are assumed to be temperature dependent. The quadratic extrapolation technique is used for linearization and fast converging finite double Chebyshev series is used for spatial discretization of the governing nonlinear equations of equilibrium. The effects of plate parameters and foundation parameters on buckling and postbuckling response of the plate are presented.     

On a finite crack partially penetrating two circular inhomogeneities and some related problems

We investigate a Mode-III finite slit crack partially penetrating two circular inhomogeneities embedded in an unbounded matrix. In order to obtain analytical solutions, it is assumed that the two circular inhomogeneity-matrix interfaces are Apollonius circles with respect to the two crack tips (or equivalently the two crack tips are just mutually image points with respect to each one of the two circular interfaces). Particularly closed-form expressions of the stress intensity factors at the two crack tips are obtained even though only series form solutions to the original boundary value problem can be derived. The loadings considered in this research include: (i) remote uniform anti-plane shearing; (ii) a straight screw dislocation at any position of the three-phase composite system; (iii) a Zener-Stroh crack. The results are verified by comparison with existing solutions. The related problem of a circular hole partially merged in two circular inhomogeneities is also addressed, with closed-form expressions of the stress concentration factors derived.