A simple method for calculating interaction of numerous microcracks and its applications

The effects of microcrack interaction on the failure behavior of materials present one problem of considerable interest in micromechanics, which has been extensively argued but has not been resolved as yet. In the present paper, a simple and effective method is presented based on the concept of the effective field to analyze the interaction of microcracks of a large number or of a high density. To determine the stress intensity factors of a microcrack embedded in a solid containing numerous or even countless microcracks, the solid is divided into two regions. The interaction of microcracks in a circular or elliptical region around the considered microcrack is calculated directly by using Kachanov’s micromechanics method, while the influence of all other microcracks is reflected by modifying the stress applied in the far field. Both the cases of tensile and compressive loading are considered. This simplified scheme may yield an estimate for stress intensity factors of satisfactory accuracy, and therefore provide a potential tool for elucidating some phenomena of material failure associated with microcracking. As two of its various promising applications, the above scheme is employed to investigate the size effects of material strength due to stochastic distribution of interacting microcracks and to calculate the effective elastic moduli of elastic solids containing distributed microcracks. Some conventional micromechanics methods for estimating the effective moduli of microcracked materials are evaluated by comparing with the numerical results. Only two-dimensional problems have been considered here, though the three-dimensional extension of the present method is of greater interest.     

Mathematical Models and Methods of the Mechanics of Stochastic Composites

The basic approaches used in mathematical models and general methods for solution of the equations of the mechanics of stochastic composites are generalized. They can be reduced to the stochastic equations of the theory of elasticity of a structurally inhomogeneous medium, to the equations of the theory of effective elastic moduli, to the equations of the theory of elastic mixtures, or to more general equations of the fourth order. The solution of the stochastic equations of the elastic theory for an arbitrary domain involves substantial mathematical difficulties and may be implemented only rather approximately. The construction of the equations of the theory of effective moduli is associated with the problem on the effective moduli of a stochastically inhomogeneous medium, which can be solved by the perturbation method, by the method of moments, or by the method of conditional moments. The latter method is most appropriate. It permits one to determine the effective moduli in a two-point approximation and nonlinear deformation properties. In the structure of equations, the theory of elastic mixtures is more general than the theory of effective moduli; however, since the state equations have not been strictly substantiated and the constants have not been correctly determined, theoretically or experimentally, this theory cannot be used for systematic designing composite structures. A new model of the nonuniform deformation of composites is more promising. It is constructed by performing strict mathematical transformations and averaging the output stochastic equations, all the constants being determined. In the zero approximation, the equations of the theory of effective moduli follow from this model, and, in the first approximation, fourth-order equations, which are more general than those of the theory of mixtures, follow from it     

On the stiffness of materials containing a disordered array of microscopic holes or hard inclusions

We study the macroscopic mechanical behavior of materials with microscopic holes or hard inclusions. Specifically, we deal with the effective elastic moduli of composites whose microgeometry consists of either soft or hard isolated inclusions surrounded by an elastic matrix. We approach this problem by taking the stiffness of the inclusion phase to be a complex variable, which we eventually evaluate at the soft or hard limits. Our main result states that there is a certain class of non-physical, negative-definite values of the elastic moduli of the inclusion phase for which the effective tensor does not have infinities or become otherwise singular.We present applications of this result to the estimation of effective moduli and to homogenization theorems. The first application involves using complexanalytic methods to obtain rigorous and accurate bounds on the effective moduli of the high-contrast composites under consideration. We also discuss the variational estimates of Rubenfeld & Keller, which yield a complementary set of bounds on these moduli. The best bounds are given by a combination of the analytical and variational results. As a second application, we show that certain known theorems of homogenization for materials with holes are simple consequences of our main result, and in this connection we establish corresponding new theorems for materials with hard inclusions. While our rederivation of the homogenization theorems for materials with holes can be closely related to other known constructions, it appears that certain elements provided by our main result are essential in the proof of homogenization for the hard-inclusion case.     

Estimate of effective elastic moduli with microcrack interaction effects

A method is developed for estimating the effects of microdefect interaction on the effective elastic properties of heterogeneous solids. An effective medium is defined to calculate the global effective elastic moduli of brittle materials weakened by distributed microcracks. Each microcrack is assumed to be embedded in an effective medium, the compliance of which is obtained from the dilute concentration method without accounting for interaction. The present scheme requires no iteration; it can account for microcrack interaction with sufficient accuracy. Analytical solutions are given for several two- and three-dimension problems with and without anisotropy.     

Interaction between dislocation and two circular cylindrical inclusions

In this paper, the elastic field of the infinite homogeneous medium with two circular cylindrical inclusions under the action of a screw dislocation is investigated and the corresponding analytical solution is obtained. Here, the conformal mapping and the theorem of analytical continuation are used. From the results obtained, it can be seen that the elastic field depends on the shear moduli of individual phases, the geometric parameters of the system, and the position and relative slip of the screw dislocation. In addition, the corresponding specific cases are also considered in this paper when two circular cylindrical inclusions are tangent to each other and they are holes and/or rigid inclusions. Finally, numerical results are illustrated to show the interaction between the screw dislocation and two circular cylindrical inclusions.     

Theory of the elastic moduli of a heterogeneous medium

It is shown that the theory of random functions permits the expansion of the effective tensor X~jkl for the elastic moduli with respect to correlation functions and that it leads in the second approximation in the Voigt-Reuss scheme to values that lie to one side of the Xijkl, while in the third approximation it brackets the latter. The analysis is used to refine the Hashin limits to the elastic moduli for a mechanical mixture of isotrcpic components and polycrystalline aggregates of cubic structure.There are two methods for calculating the effective elastic moduli of heterogeneous solids: virial expansion [2] (as a power series in the concentration of one of the components) and the method of correlation functions [2] (expansion with respect to relative fluctuation of the elastic moduli). Identical results should be obtained in the two cases if all terms are incorporated, but great mathematical difficulties restrict one to the lowest approximations. The first approximation in the virial method gives better results when the concentration of one component is low, while the method of correlation functions gives better results when the fluctuations in the elastic moduli are small and the concentrations are similar.Methods have been developed for determining the upper and lower bounds in both approaches, and various schemes of averaging are used for this purpose in the correlation-function method. The upper bound is established by renormalizing the equation of equilibrium, while the lower one is found by renormalizing the equation of incompatibility. The range of the bracketing can be reduced by means of higher approximations. The range can be reduced in the limit to zero, which implies passing from an approximate effective tensor to the true one, which relates the means in stress and strain over the material. Here we show that the two methods of renormalization give identical results when all terms of the series are summed.If the tensor has a Gaussian distribution, the moment functions of odd order are zero, while the even ones are expressed via combinations of the binary functions [3]. However, a mechanical mixture of several components is not Gaussian, and the odd moments are not zero. Splitting of the higher-order correlation functions is possible also for mechanical mixtures having determinate phase interfaces, but this involves various simplifying assumptions. A derivation is given for a moment of arbitrary order, which allows one to formulate the conditions under which such splitting is possible. The results are used in calculating the exact value of the effective bulk modulus for a medium with a homogeneous shear modulus.We are indebted to V. V. Bolotin for a discussion.     

计算微裂纹损伤材料有效模量的一种简单方法

给出了一种基于Taylor模型的有效介质方法。用以计算微裂纹相互作用对有效本构关系的影响,该方法假设每一个微裂纹位于一种有效介质之中,该有效介质的弹性模量由不考虑微裂纹相互作用的Taylor模型计算、和自洽方法相比,这种方法计算简单,而且结果更准确。     

Elastic moduli of the piezoelectric cochlear outer hair cell membrane

The outer hair cell is a specialized cell in the mammalian cochlea, believed to amplify incoming sound waves. This amplification is associated with the outer hair cell's electromotility, a unique cellular phenomenon of voltage-dependent length changes. Outer hair cell properties can be described in terms of the piezoelectric relationships, and the elastic moduli are a key part of them. We revisit the problem of estimating the elastic moduli of the outer hair cell composite membrane (wall) where two methods have previously been proposed. We analyze the two methods, while taking into account experimental ranges of the measured parameters. We have shown that cell stiffness is the critical parameter that determines the difference between the method predictions, and we have found a range of stiffness where the results are reasonably close. The elastic moduli corresponding to this range can be recommended for estimation of the characteristics of the piezoelectric model.     

Elastic properties of inhomogeneous transversely isotropic rocks

The problem to determine the effective elastic moduli and velocities of elastic wave propagation in transversely isotropic solid containing aligned spheroidal inhomogeneities (solid grains, vugs and micro-cracks) has been solved using the self-consistent scheme known as effective medium approximation (EMA). Since a solution of so-called one-particle problem is a base for each self-consistent method, we solved this problem as a first step for spheroidal inhomogeneity in a transversely isotropic medium. In contrast to the known solution of this problem by Lin and Mura we obtained the expressions for the strain field inside inclusion in the explicit form (without quadratures). The obtained solution was used then in the symmetric variant of the EMA where each component of the system was considered as spheroid with its own aspect ratio. This approach was applied to simulate the properties of the rocks containing isolated pores and micro-cracks. For connected fluid-filled pores we used the anisotropic variant of the Gassmann theory. The results of the calculations, obtained for the effective elastic moduli, have been compared with the experimental data and theoretical simulations of the other authors. Unlike many other rock mechanics theories, EMA approximation gives correct elastic moduli values even in the nondilute concentration of inhomogeneities. The comparison of the experimental data for oriented crack system with the EMA predictions indicates their good correspondence.     

Bounds for effective elastic moduli of inhomogeneous solid bodies

In connection with the extensive use of various kinds of inhomogeneous materials (glass, carbon and boron reinforced plastics, cermets, concrete, reinforced materials, etc.) in technology, there arises a need to calculate the elastic properties of such systems. Here in each case it is necessary to work out specific methods for finding both elastic fields and effective moduli. Since, as a rule, such methods do not take into account the character of distribution of inhomogeneities in space, which is reflected on the form of the central moment functions [1], they can be referred to a single class and, consequently, can be obtained by a common method [2], In the given paper, by means of the method of solution of stochastic problems for microinhomogeneous solid bodies proposed in the work of the author [2], we find elastic fields and effective moduli in an arbitrary approximation. Depending on the choice of parameters, the latter form bounds within which there lie the exact values of the effective moduli. It is shown that the conditions used earlier for finding these parameters [3] are not the best ones. The effective elastic moduli of an inhomogeneous medium are calculated, and bounds, narrower than the bounds formed in [3], are found for them.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhniki, No. 5, pp. 144–150, September–October, 1973.     

Effective elastic moduli of an inhomogeneous medium with cracks

In the present paper, the effective elastic moduli of an inhomogeneous medium with cracks are derived and obtained by taking into account its microstructural properties which involve the shape, size and distribution of cracks and the interaction between cracks. Numerical results for the periodic microstructure of different dimensions are presented. From the results obtained, it can be found that the distribution of cracks has a significant effect on the effective elastic moduli of the material. The project supported by the National Education Committee for Doctor     

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

Bounds on the overall properties of composites with ellipsoidal inclusions

A new model is put forward to bound the effective elastic moduli of composites with ellipsoidal inclusions. In the present paper, transition layer for each ellipsoidal inclusion is introduced to make the trial displacement field for the upper bound and the trial stress field for the lower bound satisfy the continuous interface conditions which are absolutely necessary for the application of variational principles. According to the principles of minimum potential energy and minimum complementary energy, the upper and lower bounds on the effective elastic moduli of composites with ellipsoidal inclusions are rigorously derived. The effects of the distribution and geometric parameters of ellipsoidal inclusions on the bounds of the effective elastic moduli are analyzed in details. The present upper and lower bounds are still finite when the bulk and shear moduli of ellipsoidal inclusions tend to infinity and zero, respectively. It should be mentioned that the present method is simple and needs not calculate the complex integrals of multi-point correlation functions. Meanwhile, the present paper provides an entirely different way to bound the effective elastic moduli of composites with ellipsoidal inclusions, which can be developed to obtain a series of bounds by taking different trial displacement and stress fields.     

Size-dependent effective elastic moduli of particulate composites with interfacial displacement and traction discontinuities

This work aims at estimating the size-dependent effective elastic moduli of particulate composites in which both the interfacial displacement and traction discontinuities occur. To this end, the interfacial discontinuity relations derived from the replacement of a thin uniform interphase layer between two dissimilar materials by an imperfect interface are reformulated so as to considerably simplify the characteristic expressions of a general elastic imperfect model which is adopted in the present work and include the widely used Gurtin–Murdoch and spring-layer interface models as particular cases. The elastic fields in an infinite body made of a matrix containing an imperfectly bonded spherical particle and subjected to arbitrary remote uniform strain boundary conditions are then provided in an exact, coordinate-free and compact way. With the aid of these results, the elastic properties of a perfectly bonded spherical particle energetically equivalent to an imperfectly bonded one in an infinite matrix are determined. The estimates for the effective bulk and shear moduli of isotropic particulate composites are finally obtained by using the generalized self-consistent scheme and discussed through numerical examples.     

Size-dependent effective thermoelastic properties of nanocomposites with spherically anisotropic phases

Composites made of semi-crystalline polymers and nanoparticles have a spherulitic microstructure which can be reasonably represented by a spherically anisotropic volume element. Due to the high surface-to-volume ratio of a nanoparticle, the particle-matrix interface stress, usually neglected in determining the effective elastic moduli of particle-reinforced composites, may have a non-negligible effect. To account for the latter in estimating the effective thermoelastic properties of a composite consisting of nanoparticles embedded in a semi-crystalline polymeric matrix, this work adopts a coherent interface model for the nanoparticle-matrix interface and proposes an extended version of the classical generalized-self consistent method. In particular, Eshelby's formulae widely used to calculate the elastic energy change of a homogeneous medium due to the introduction of an inhomogeneity are extended to the thermoelastic case. The nanoparticle size effect on the effective thermoelastic moduli of the composite are theoretically shown and numerically illustrated.     

Closed-form solutions for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix

This paper presents two different analytical methods to investigate the magneto-mechanical coupling effect for piezomagnetic inhomogeneities embedded in a non-piezomagnetic matrix. First, the magnetoelastic solution is expressed in terms of magnetoelastic Green's function that can be decoupled into elastic Green's function and magnetic Green's function. Second, the problem is analyzed by the equivalent inclusion method, and then, the formulation of the inhomogeneity problem can be decoupled into an elastic problem and a magnetic inhomogeneity problem connected by some eigenstrain and eigenmagnetic fields. For the piezomagnetic composites with a non-piezomagnetic matrix, these two solutions are completely equivalent each other though they are obtained by means of two different methods. Moreover, based upon the unified energy method, the effective magnetoelastic moduli of the composites are expressed explicitly in terms of phase properties and volume fractions. Then the dilute and Mori–Tanaka schemes are discussed, respectively. Finally, the calculations are made to predict the effective magnetoelastic moduli and illustrate the performance of each model.     

A continuum theory for isotropic two-phase elastic composites

The paper presents the effective stiffness theory for isotropie two-phase elastic composites. The theory predicts dispersion of longitudinal and transverse plane time-harmonic travelling waves. The limiting phase velocities at vanishing wave numbers serve in the determination of the elastic moduli of the equivalent homogeneous isotropic medium. These elastic moduli are compared with the effective moduli defined statically.     

Iteration method in linear elasticity of random structure composites containing heterogeneities of noncanonical shape

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically inhomogeneous random set of heterogeneities of arbitrary shape. The general integral equations connecting the stress and strain fields in the point being considered with the stress and strain fields in the surrounding points are obtained for the random fields of heterogeneities. The method is based on a recently developed centering procedure where the notion of a perturbator is introduced and statistical averages are obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. Effective elastic moduli and the first statistical moments of stresses in the heterogeneities are estimated for statistically homogeneous composites with the general case of both the shape and inhomogeneity of the heterogeneities moduli. The explicit new representations of the effective moduli and stress concentration factors are built by the iteration method in the framework of the quasicristallite approximation but without basic hypotheses of classical micromechanics such as both the EFH and “ellipsoidal symmetry” assumption. Numerical results are obtained for some model statistically homogeneous composites reinforced by aligned identical homogeneous heterogeneities of noncanonical shape. Some new effects are detected that are impossible in the framework of a classical background of micromechanics.     

A continuum micromechanical theory of overall plasticity for particulate composites including particle size effect

Classical continuum micromechanics cannot predict the particle size dependence of the overall plasticity for composite materials, a simple analytical micromechanical method is proposed in this paper to investigate this size dependence. The matrix material is idealized as a micropolar continuum, an average equivalent inclusion method is advanced and the Mori–Tanaka's method is extended to a micropolar medium to evaluate the effective elastic modulus tensor. The overall plasticity of composites is predicted by a new secant moduli method based on the second order moment of strain and torsion of the matrix in a framework of micropolar theory. The computed results show that the size dependence is more pronounced when the particle's size approaches to the matrix characteristic length, and for large particle sizes, the prediction coincides with that predicted by classical micromechanical models. The method is analytical in nature, and it can capture the particle size dependence on the overall plastic behavior for particulate composites, and the prediction agrees well with the experimental results presented in literature. The proposed model can be considered as a natural extension of the widely used secant moduli method from a heterogeneous Cauchy medium to a micropolar composite.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.