On the equivalent inclusion method and impotent eigenstrains

The equivalency equations and the nature of the solution are investigated when an inhomogeneity under applied stresses is simulated by an inclusion with eigenstrains. The equivalency equations by which the equivalent eigenstrain is obtained becomes singular when the inhomogeneity is void and the applied stress has a form of polynomials of coordinates of degree one. The solutions of the system of the equivalency equations are not uniquely determined. Explicit expressions are given for the impotent eigenstrains which do not generate any stress field throughout a material.This research was supported by the U.S. Army Research Office under Grant No. DAAG 29-77-G-0042 to Northwestern University.On leave of absence of Meiji University, Tokyo.     

A variational form of the equivalent inclusion method for numerical homogenization

Due to its relatively low computational cost, the equivalent inclusion method is an attractive alternative to traditional full-field computations of heterogeneous materials formed of simple inhomogeneities (spherical, ellipsoidal) embedded in a homogeneous matrix. The method can be seen as the discretization of the Lippmann–Schwinger equation with piecewise polynomials. Contrary to the original approach of Moschovidis and Mura, who discretized the strong form of the Lippmann–Schwinger equation through Taylor expansions, we propose in the present paper a Galerkin discretization of the weak form of this equation. Combined with the new, mixed boundary conditions recently introduced by the authors, the resulting method is particularly well-suited to homogenization. It is shown that this new, variational approach has a number of benefits: (i) the resulting linear system is well-posed, (ii) the numerical solution converges to the exact solution as the maximum degree of the polynomials tends to infinity and (iii) the method can provide rigorous bounds on the apparent properties of the statistical volume element, provided that the matrix is stiffer (or softer) than all inhomogeneities. This paper presents the formulation and implementation of the new, variational form of the equivalent inclusion method. Its efficiency is investigated through numerical applications in 2D and 3D elasticity.     

Special properties of Eshelby tensor for a regular polygonal inclusion

When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.The project supported by the National Natural Science Foundation of China (10172003 and 10372003) The English text was polished by Keren Wang.     

Interacting cracks and ellipsoidal inhomogeneities by the equivalent inclusion method

Based on the Eshelby's equivalent inclusion method (EIM) and Hill's theorem on discontinuities of elastic fields across the interfaces, a theory for the determination of the stress intensity factors (SIFs) of arbitrarily oriented interacting cracks under non-uniform far-field applied stress (strain) is developed. As shown in this investigation the EIM proposed by Moschovidis and Mura can be extended for treatment of such problems, but their formulations are quite cumbersome and computationally inefficient. An alternative analytical approach is proposed that is computationally more efficient, and unlike the method of Moschovidis and Mura can easily handle complex problems of interacting inhomogeneities and cracks. It is seen that as the interaction between the inhomogeneities becomes stronger, this method yields results that are closer to the solutions reported in the literature than the solutions obtained using the extended EIM of Moschovidis and Mura, which is developed herein. Problems involving combinations of interacting elliptic and penny shape cracks and inhomogeneities are excellent candidates for demonstration of the accuracy and robustness of the present theory, for which the previous EIM produces less accurate results. Due to the limitations imposed on the existing methods, every reported treatment has been tailored for a certain category of problems, and only uniform far-field loadings have been remedied. In contrast, the present theory is more general than the previously reported theories and it encompasses interacting cracks having a variety of geometries subjected to non-uniform far-field applied stress (strain); moreover, it is applicable to modes, I, II, III, and mixed mode fracture.     

Validation of the tangent formulation for the solution of the non-linear Eshelby inclusion problem

An extension of the Eshelby problem for non-linear viscous materials is considered. An ellipsoidal heterogeneity is embedded in an infinite matrix. The material properties are assumed to be uniform within the ellipsoid and in the matrix. The problem of determining the average strain rate in the ellipsoid in terms of the overall applied strain rate is solved in an approximate way. The method is based on the non-incremental tangent formulation of the non-linear matrix behavior [Acta Metall. 35 (1987) 2983]. In the present work this approximate solution is verified with a good agreement by comparing to finite element calculations for various inclusion shapes and loading conditions.     

The determination of an equivalent elastic modulus for bodies with cracks by boundary element method

The equivalent elastic modulus of cracked bodies with orderly distributed cracks was computed with the boundary element method. A practical self-consistent scheme has been proposed in consideration of the mutual interaction effects of the cracks. The influence of friction coefficients and orientation of cracks has been investigated. Some computational examples have been given, and the results show that the proposed method is adequate and the scheme is efficient.This project is supported by the National Natural Science Foundation of China.     

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.     

Interaction of fiber and transformation toughening

B small scale bridging (SSB) assumption, a theoretical study is made of the interaction between the fracture toughening effects of transforming particles and crack bridging fibers which are aligned in the direction perpendicular to the crack surfaces. The fibers bridging the crack are assumed to undergo large amounts of slipping inside the matrix. It is found that the interaction can be synergistic over a parametric range of material properties of the ceramic composite system. The Mode-1 plane-strain fracture toughness of the ceramic composite system is determined in terms of fiber strength, transformation toughening parameters and interaction parameters. The results obtained here are qualitatively similar to those for the interaction between the fracture toughening effects of transforming particles and crack-bridging ductile particles by A and B (1988a, J. Mech. Phys. Solids 36, 581).     

Modeling elasto-plastic indentation on layered materials using the equivalent inclusion method

This paper develops a fast semi-analytical model for solving the three-dimensional elasto-plastic contact problems involving layered materials using the Equivalent Inclusion Method (EIM). The analytical elastic solutions of a half-space subjected to a unit surface pressure and a unit subsurface eigenstrain are employed in this model; the topmost layer is simulated by an equivalent inclusion with fictitious eigenstrain. Accumulative plastic deformation is determined by a procedure involving an iterative plasticity loop and an incremental loading process. Algorithms of the fast Fourier transform (FFT) and the Conjugate Gradient Method (CGM) are utilized to improve the computation efficiency. An analytical elastic solution of layered body contact (O’Sullivan and King, 1988) and an indentation experiment result involving a layered substrate (Michler et al., 1999) are used to examine the accuracy of this model. Comparisons between numerical results from the present model and a commercial FEM software (Abaqus) are also presented. Case studies of a rigid ball loaded against a layered elasto-plastic half-space are conducted to explore the effects of the modulus, yield strength, and thickness of the coating on the hardness, stiffness, and plastic deformation of the composite body.     

损伤拉索的等效弹性模量及其参数分析

损伤拉索会出现线形松弛、应力水平降低的情况,必然会影响拉索的等效弹性模量。本文首先引入损伤程度、位置及范围3个参数,用以描述拉索损伤形态的特征,建立损伤拉索索力和线形计算公式,采用数值方法计算了损伤拉索弦向等效弹性模量精确数值,并和经典的等效弹性模量公式的计算结果进行了比较分析,分析了考虑损伤时两种不同计算方法结果的误差。计算表明,对于500m弦向长度以内的损伤拉索,拉索的弦向长度Lc越大,倾角越小,等效弹性模量的损失越大,并且应用割线模量公式计算的误差也越大,当Lc=500m时,损伤拉索相对误差值在2.5%~4.5%之间。弦向应变越小,等效弹性模量损失越大,弦向应变在[0.001,0.004]内,应用割线模量公式计算的相对误差小于3.5%。损伤程度及损伤范围对引用等效弹性模量公式的误差影响较大,倾角对等效弹模公式相对误差的影响也不容忽视。弦向长度、弦向应变、倾角和损伤程度参数都是通过改变拉索的松弛程度进而影响等效弹性模量的数值以及公式的误差。     

Stochastic homogenization analysis on elastic properties of fiber reinforced composites using the equivalent inclusion method and perturbation method

This paper describes a methodology for evaluation of influence of microscopic uncertainty in material properties and geometry of a microstructure on a homogenized macroscopic elastic property of an inhomogeneous material. For the analysis of the stochastic characteristics of a homogenized elastic property, the first-order perturbation method is used. In order to analyze the influence of microscopic geometrical uncertainty, the perturbation-based equivalent inclusion method is formulated. In this paper, an analytical form of the perturbation term using the equivalent inclusion method is provided.As a numerical example, macroscopic stochastic characteristics such as an expected value or variance of the homogenized elastic tensor of a unidirectional fiber reinforced plastic, which is caused by microscopic uncertainty in material properties or geometry of a microstructure, are estimated with computing the first order perturbation term of the homogenized elastic tensor. Compared the results of the proposed method with the results of the Monte-Carlo simulation, validity, effectiveness and a limitation of the perturbation-based homogenization method is investigated.     

A numerical application of the Eshelby theory for geobarometry of non-ideal host-inclusion systems

Meccanica - In the complex geodynamic processes occurring at convergent plate margins, rocks can be subducted at depth into the Earth experiencing metamorphism. A mineral inhomogeneity entrapped...     

The non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion

The problem of an anisotropic ellipsoidal inclusion which undergoes a stress-free transformation strain (in the sense of J.D. Eshelby) is considered, and the following theorem is proved: If an ellipsoidal region in an infinite anisotropic linear elastic medium undergoes, in the absence of its surroundings, a stress-free transformation strain which is a polynomial of degree M in the position coordinates x_t, then the final stress and strain state in the transformed inclusion, when constrained by its surroundings, is also a polynomial of degree M in x_t.     

Mechanical interaction between spherical inhomogeneities: an assessment of a method based on the equivalent inclusion

This paper assesses the ability of the Equivalent Inclusion Method (EIM) with third order truncated Taylor series (Moschovidis and Mura, 1975) to describe the stress distributions of interacting inhomogeneities. The cases considered are two identical spherical voids and glass or rubber inhomogeneities in an infinite elastic matrix. Results are compared with those obtained using spherical dipolar coordinates, which are assumed to be exact, and by a Finite Element Analysis. The EIM gives better results for voids than for inhomogeneities stiffer than the matrix. In the case of rubber inhomogeneities, while the EIM gives accurate values of the hydrostatic pressure inside the rubber, the stress concentrations are inaccurate at very small neighbouring distances for all stiffnesses. A parameter based on the residual stress discontinuity at the interface is proposed to evaluate the quality of the solution given by the EIM. Finally, for inhomogeneities stiffer than the matrix, the method is found to diverge for expansions in Taylor series truncated at the third order.     

基于应变梯度理论的微尺度蜂窝等效模量计算方法

本文提出了一种新的能够计及尺度效应的微纳米蜂窝等效模量的计算方法。将一种单参数应变梯度理论引入到本构方程当中,并基于能量等效原理推导了蜂窝面内等效模量地计算公式。算例分析表明,本文方法能够有效地计及尺度效应对蜂窝等效模量的影响。尺度效应与胞壁厚度和长度的值都有关,当胞壁厚度较小时,尺度效应显著,本文方法预测的模量会明显高于传统方法;而当胞壁厚度较大时,尺度效应变得微弱乃至可以忽略不计。但如果胞壁的长度/厚度比很大,则面内等效模量会趋近于0,此时是否考虑尺度效应意义不大。     

平面任意形状等剪切模量异质夹杂问题的解析解

本文研究任意形状夹杂域在受到远端均匀荷载和均匀本征应变作用下的弹性场问题,其中基体和夹杂的材料不同但具有相同的剪切模量。利用等效理论将远端均匀荷载引起的扰动转化为等效均匀本征应变的作用,再利用K-M势函数表达扰动场问题的界面连续条件;借助于黎曼映射定理,用洛朗多项式将平面光滑闭合曲线外部区域映射到单位圆外部区域,借助柯西积分公式和Faber多项式求解了等剪切本征应变下夹杂和基体的K-M势函数的显式解析解,其中考虑了夹杂相对于基体的刚体位移。将得到的结果与相关文献的结果进行对比,表明了本论文的方法和结果是有效的和正确的。     

The Eshelby tensor and the theory of continuous distributions of inhomogeneities

The purpose of this note is to reaffirm the fact that there exists a natural connection between Noll’s theory of inhomogeneities and the Eshelby tensor. One way to expose this connection consists in allowing the inhomogeneity pattern to evolve in time and then exploring the thermodynamic implications.     

Finite element analysis of the fatigue crack growth rate in transformation toughening ceramics

The fatigue crack growth rate in the zirconia tetragonal polycrystal is analyzed through the finite element method. In order to achieve this purpose, a continuum based constitutive law for materials subjected to phase transformations has been suitably implemented into a commercial finite element code. The fatigue crack growth in a notched beam, subjected to a cyclic four points bending load, has been investigated through a sensitivity analyses with respect to the two most relevant constitutive parameters: one accounting for the amount of the transformation strain and one accounting for the activation energy threshold. The fatigue crack growth rate typical of transforming materials is characterized by two distinct stages: at the beginning of the crack propagation process, the crack growth rate exhibits a negative dependency on the applied stress intensity factor; thereafter, a linear positive dependency is observed. This two stage process is well caught by the finite element model presented in this paper. Moreover, the response of the computational analyses has shown that the strength of the transformation process is determinant for the crack growth process to be arrested.