Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear

Summary An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.     

Neutrality of the elliptic inhomogeneity in the case of non-uniform loading

In this paper, we consider the problem of a single elliptic elastic inhomogeneity embedded within an infinite elastic matrix in antiplane shear. In particular, we examine the (stress) neutrality of this inhomogeneity when a non-uniform stress field is prescribed in the surrounding matrix. Since it is known that neutral elastic inhomogeneities do not exist when the inhomogeneity is assumed to be perfectly bonded to the matrix, the method presented here is based on the assumption of imperfect interface and the appropriate choice of the (single) interface parameter (characterizing the imperfect interface) to achieve the desired neutrality. Specifically, neutrality is established for specific (polynomial) classes of prescribed states of stress in the surrounding matrix. The results in this paper affirm the feasibility of designing a neutral elastic inhomogeneity by controlling the (imperfect) interface parameter describing the inhomogeneity-matrix interface.     

Influence of imperfect interface on the elastic moduli of syntactic foams

Syntactic foams are manufactured by dispersing microspheres in a polymeric matrix, and the macroscale material properties of these foams are estimated by analyzing a periodic distribution of the inclusions. The analysis in the simplest form, further assume that the inclusions are perfectly bonded to the matrix material. It has been shown in a previous study [P.R. Marur, Mater. Lett. 59 (2005) 1954–1957.] that analytical model overestimated the experimentally determined elastic moduli, and that the morphology of particle distribution has negligible influence on the elastic moduli. In this paper, the assumption of perfect adhesion between the inclusion and the matrix is relaxed to allow for possible localized slip and separation at the particle interface. The analytical results obtained considering imperfect interface well agree with the measured elastic modulus reported in the literature.     

Interaction between screw dislocations and inclusions with imperfect interfaces in fiber-reinforced composites

A three-phase composite cylinder model is utilized to study the elastic interaction between screw dislocations and embedded multiple circular cross-section inclusions (fibers) with imperfect interfaces in composites. By means of complex variable techniques, the explicit solutions of stress and displacement fields are obtained. With the aid of the Peach–Koehler formula, the explicit expressions of image forces exerted on screw dislocations are derived. The equilibrium positions of the appointed screw dislocation near one of the inclusions are discussed for variable parameters (interface imperfection, material mismatch and dislocation position) and the influence of the nearby inclusions and dislocations is also considered. The results show that, if the inclusion is stiffer than the matrix and the magnitude of the degree of interface imperfection reaches the certain value, a new equilibrium position for the screw dislocation in the matrix can always be produced in comparison with the previous solution (the perfect interface). The effect of elastic constants of the inclusion on the image force and the equilibrium position of the appointed screw dislocation is weak when the interface imperfection is strong. It is also seen that the magnitude of the image force exerted on the appointed dislocation caused by multiple inclusions is always smaller than that produced by a single inclusion. The impact of the closer dislocations on the mobility of the appointed dislocation is very significant.     

Effect of inclusion shape on the effective elastic moduli for composites with imperfect interface

Summary Bounds on the effective elastic moduli for isotropic composites consisting of randomly oriented spheroidal inclusions with imperfect matrix-inclusion interface are proposed based on Hashin's extremum principle. Phenomenally, these bounds are the first-order ones for such composites, and contain the effect of the size and shape of inclusions, and the elastic properties of constituent phases and interfaces. In the limit cases, these bounds reduce to those known ones. The effect of inclusion shape and interface imperfection on the bounds is discussed with some numerical results for a WC/Co metal-matrix composite.     

Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding

We consider an anisotropic elastic inclusion of arbitrary shape embedded inside an infinite dissimilar anisotropic elastic medium (matrix) subjected to a uniform antiplane shear loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical.     

Modeling fatigue crack nucleation at primary inclusions in carburized and shot-peened martensitic steel

The mechanics of high cycle fatigue crack nucleation (formation of a stable crack that can grow away from the influence of the notch root of the inclusion) at subsurface primary inclusions in carburized and shot-peened martensitic steel subjected to cyclic bending is investigated using three-dimensional (3D) finite element (FE) analysis. FE models are constructed using a voxellation technique to address the shape, size, and distribution of primary inclusions within clusters. The critical depth for fatigue crack nucleation is predicted considering the gradient in material properties arising from carburization, prestrain and compressive residual stress distribution due to shot peening, and the gradient of applied bending stress. The influence of inclusion shape and interface condition (intact or debonded) with the matrix on the driving force for fatigue crack nucleation is examined. It is observed that the inclusion shape has minimal influence on the predicted results while the effect of the interface condition is quite significant. For partially debonded interfaces, the predicted critical depth from surface for fatigue crack nucleation agrees qualitatively with experimental observations.     

Stress concentration due to a hemispherical surface inclusion

Fatigue cracks often initiate at surface inclusions, and sometimes appear at inclusions within the bulk. To clarify the relative efficiency of these crack sources, an approximate solution for the elastic stress of a hemispherical surface inclusion is provided and compared with the existing result of a spherical interior inclusion. The approximate solution is obtained from the Eshelby solution for the elastic field of an ellipsoid inclusion by introducing the Green's function of an elastic half-space. The numerical calculated results indicate that the stress concentration of a surface inclusion is higher when the inclusion is harder than the matrix, while that of an embedded inclusion is higher when the inclusion is softer than the matrix.     

Numerical investigation of the stress field of particulate reinforced polymeric composites subjected to tension

The aim of the present study is the detailed numerical investigation of the stress/strain distribution in polymeric matrix composites reinforced with spherical inclusions, using the finite element method (FEM). Perfect adhesion between the matrix and the inclusions was assumed and from the computed stress/strain profiles of the system, debonding initiation and propagation can easily be predicted. Analytical models available in the literature may predict the stress/strain distribution within the inclusion and along the matrix/inclusion interface, while the FEM may yield results for the whole composite, including within the inclusions. Three typical volume fractions of the composite were examined and the results were justified by the analytical predictions of other researchers. The numerical results show that the matrix starts to debond from the inclusions at angular distance forty‐five degrees and as the applied load increases the debonding zone gradually extends. Copyright © 2005 John Wiley & Sons, Ltd.     

求解具有弱界面的复合材料有效性能的近似模型与边界元法

基于中间层模型, 建立了用于描述具有弱界面的多夹杂复合材料的三层嵌入式模型。首先, 给出了计算多夹杂复合材料有效性能的公式, 随后将三层嵌入式模型分成两种体系得到了相应的细观力学近似方法。此外, 针对中间层模型进一步给出了快速多极边界元的基本列式。最后, 通过算例验证所提出的两种方法的正确性及有效性, 并进一步考察了界面性能对复合材料整体宏观性能的影响。比较结果发现, 细观力学近似模型形式简单, 易于编程计算, 适用于快速预测具有弱界面的复合材料有效弹性性能。而数值算法能够快速有效直观的考察复合材料内部受弱界面影响后的应力分布。算例分析结果表明, 当中间层厚度较大而弹性模量较低时, 会使得夹杂不再具有增强效果。并且, 随着中间层厚度的增加, 基体内承担的最大应力也随之增加, 从而容易导致复合材料在界面处产生破坏。     

任意分布多个椭圆形刚性夹杂的反平面问题

对于硬夹杂与软基体的复合材料,考虑夹杂间的相互影响,采用坐标变换和复变函数的依次保角映射方法,构造任意分布且相互影响的多个椭圆形刚性夹杂模型的复应力函数,同时满足各个夹杂的边界条件,利用围线积分将求解方程化为线性代数方程,推导出了在无穷远双向均匀剪切,椭圆形刚性夹杂任意分布的界面应力解析表达式,算例分析给出了单夹杂模型与多夹杂模型的夹杂形状对界面应力最大值的影响规律,并进行了对比,描绘出了曲线。     

The influence of nonmetallic inclusions on the mechanical properties of steel: A review

The literature regarding the influence of nonmetallic inclusions on the mechanical properties of steel is reviewed, with critical comments on various studies. A brief discussion of inclusion rating methods and a synopsis of the effects of applied stress on inclusions in an isotropic, elastic matrix are presented. The parameters considered are tensile strength, impact strength, reduction of area, fatigue properties and fracture toughness. It is concluded that in many applications, the type of inclusions are more important than the total content and as matrix strength increases, the notch effect of inclusions becomes more significant. Also, mechanical properties can be influenced by any one or a combination of the following inclusion parameters; shape, size, quantity, interspacing, distribution, orientation, interfacial strength, and physical properties relative to the matrix.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

Binary inclusion model for the overall elasticity of imperfectly bonded composites

Summary. A micromechanical approach is developed to estimate the overall elastic moduli of composite materials with imperfectly bonded spherical fillers. The randomly dispersed particles are assumed to satisfy linear interfacial conditions where both tangential and normal interface displacement discontinuities are linearly related to the respective surface tractions. Using the generalized version of Eshelbys equivalent inclusion method proposed by Furuhashi et al. [6] the analysis of the heterogeneous medium reduces to the study of a corresponding homogeneous medium containing spherical inclusions with a proper distribution of eigenstrain and Somigliana dislocation fields. Based on the estimated pair-wise average of strain fields in two interacting imperfect fillers embedded in the homogeneous infinite matrix, the ensemble phase volume average of field quantities has been evaluated within a representative volume element containing a finite number of imperfect particles. For the case of a constant radial distribution function, results are in reasonable agreement with those based on the generalized self-consistent method and composite sphere assemblage proposed by Hashin [11].     

Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.     

Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix

Based on the complex variable method, this paper addresses the plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is loaded by in-plane mechanical loads while the inclusions are applied by anti-plane electric loads at infinity. The general solutions are first derived for the complex potentials both in the matrix and inside the inclusions, and then numerical results are presented to show the effects of applied electric field, inclusion arrays and material properties on the electroelastic fields around the inclusions. It is shown that the inclusion arrays have a significant influence on the stress distribution at the interface between the matrix and piezoelectric inclusions.     

A circular inclusion with imperfect interface in finite plane elastostatics

We consider a circular elastic inclusion embedded in a particular class of harmonic materials subjected to remote uniform stresses. The imperfect interface can be rate dependent as well as rate independent. First, we study the situation in which both rate-depending slip and diffusional relaxation are present on the sharp inclusion-matrix imperfect interface. It is found that in general, the internal Piola stresses within the inclusion are spatially non-uniform and decay with two relaxation times. Interestingly, the average mean Piola stress within the circular inclusion is time independent. Some extreme cases for the imperfect interface are discussed in detail. Particularly, we find a simple condition leading to internal uniform Piola stresses that decay only with a single relaxation time. Second, we investigate a rate-independent spring-type imperfect interface on which normal and shear tractions are proportional to the corresponding displacement jumps. It is found that in general, the internal Piola stresses are intrinsically non-uniform. A special kind of the spring-type interface leading to internal uniform Piola stresses is also found.     

Bounds on the effective thermal conductivity of composites with imperfect interface

A new model is developed to bound the effective thermal conductivity of composites with thermal contact resistance between spherical inclusions and matrix. To construct the trial temperature and heat flux fields which satisfy the necessary interface conditions, the transition layer for each spherical inclusion is introduced. For the upper bound, the trial temperature field needs to satisfy the thermal contact resistance conditions between spherical inclusions and transition layers and the continuous interface conditions between transition layers and remnant matrix. For the lower bound, the trial heat flux field needs to satisfy the continuous interface conditions between different regions. It should be pointed out that the continuous interface conditions mentioned above are absolutely necessary for the application of variational principles, and the thermal contact resistance conditions between spherical inclusions and transition layers are suggested by the author. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with imperfect interfaces are rigorously derived. The effects of the size and distribution of spherical inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and does not need to calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides an entirely different way to bound the effective thermal conductivity of composites with imperfect interface, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds are finite when the thermal conductivity of spherical inclusions tends to ∞ and 0, respectively.