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.     

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.     

A boundary element solution to scattering of elastic SH wave by arbitrarily shaped inclusions embedded in an infinite anisotropic medium

Scattering problems for inhomogeneous bodies are investigated by the integral equation method. The boundary integral equation (BIE) for the scattered displacement field associated with finite inhomogeneities in an anisotropic medium are derived with the help of the generalized Green's identity. The discretization of BIE is based upon the constant element, linear element and quadratic element. Several numerical examples for calculating the scattering displacement, stress and scattering cross section from a cylinder, an interface crack, and two elliptic cylinders are given. Results show that the present method can be advantageously applied to a wide range of scattering problems of elastic waves.     

Dynamic stress concentration around elliptic cavities in saturated poroelastic soil under harmonic plane waves

Potential function and complex function in the elliptic coordinate system are employed to solve the problem of scattering harmonic plane waves by multiple elliptic cavities in water saturated soil medium. The steady state Biot’s dynamic equations of poroelasticity are uncoupled into Helmholtz equations via given potentials. The stresses and pore water pressures are obtained by using complex functions in elliptic coordinates with certain boundary conditions. Finally, the dynamic stresses for the case of two interacting elliptic cavities are obtained and discussed in details via a numerical example.     

A screw dislocation near one open inhomogeneity and another closed inhomogeneity both permitting constant interior stresses

We prove that the interior stresses within both a non-parabolic open inhomogeneity and another interacting non-elliptical closed inhomogeneity can still remain constant when the matrix is simultaneously under the action of a screw dislocation and uniform remote anti-plane stresses.The constancy of interior stresses is realized through the construction of a conformal mapping function for the doubly connected domain occupied by the surrounding matrix.The mapping function is endowed with the information describing the screw dislocation via the incorporation of two specifically defined logarithmic terms.The constant interior stress fields are observed to be independent of the specific open and closed shapes of the two inhomogeneities and the existence of the screw dislocation.In contrast,the existence of the neighboring screw dislocation significantly affects the open and closed shapes of the two inhomogeneities.     

Analytical solution for a reinforcement layer bonded to an elliptic hole under a remote uniform load

A general solution to a reinforced elliptic hole embedded in an infinite matrix subjected to a remote uniform load is provided in this paper. Investigations on the present elasticity problem are rather tedious due to the presence of material inhomogeneities and complex geometric configurations. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacement and stresses in a reinforcement layer and the matrix are derived explicitly in a series form. Some numerical results are provided to investigate the effects of the material combinations and geometric configurations on the interfacial stresses. The results show that there exists an optimum design of a reinforcement layer such that both the magnitude of stress concentration and the interfacial stresses could be fairly reduced.     

Acoustics and dynamics of coaxial interacting vortex rings

Using a contour dynamics method for inviscid axisymmetric flow we examine the effects of core deformation on the dynamics and acoustic signatures of coaxial interacting vortex rings. Both “passage” and “collision” (head-on) interactions are studied for initially identical vortices. Good correspondence with experiments is obtained. A simple model which retains only the elliptic degree of freedom in the core shape is used to explain some of the calculated features.     

Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites

It is well-known that classical continuum theory has certain deficiencies in predicting material’s behavior at the micro- and nanoscales, where the size effect is not negligible. Higher order continuum theories introduce new material constants into the formulation, making the interpretation of the size effect possible. One famous version of these theories is the couple stress theory, invoked to study the anti-plane problems of the elliptic inhomogeneities and inclusions in the present work. The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions. Subsequently, the elastic fields of a single inhomogeneity in conjunction with the Mori–Tanaka theory is employed to estimate the overall anti-plane shear moduli of composites with uni-directional elliptic cylindrical fibers. The dependence of the anti-plane elastic moduli on several important physical parameters such as size, aspect ratio and rigidity of the fiber, the characteristic length of the constituents, and the orientation of the reinforcements is analyzed. Based on the available data in the literature, certain nano-composite models have been proposed and their overall behavior estimated using the present theory.     

Three-Dimensional Dynamic Contact Problem for an Elliptic Crack Interacting with a Normally Incident Harmonic Compression–Expansion Wave

The paper deals with the three-dimensional dynamic problem for an elliptic crack interacting with a normally incident harmonic compression–expansion wave, considering the contact interaction of the crack faces. An asymmetric solution is obtained using an iteration algorithm developed earlier. Numerical results are presented     

Elastic interaction of spherical nanoinhomogeneities with Gurtin-Murdoch type interfaces

A complete solution has been obtained for the problem of multiple interacting spherical inhomogeneities with a Gurtin-Murdoch interface model that includes both surface tension and surface stiffness effects. For this purpose, a vectorial spherical harmonics-based analytical technique is developed. This technique enables solution of a wide class of elasticity problems in domains with spherical boundaries/interfaces and makes fulfilling the vectorial boundary or interface conditions a routine procedure. A general displacement solution of the single-inhomogeneity problem is sought in a form of a series of the vectorial solutions of the Lame equation. This solution is valid for any non-uniform far-field load and it has a closed form for polynomial loads. The superposition principle and re-expansion formulas for the vectorial solutions of the Lame equation extend this theory to problems involving multiple inhomogeneities. The developed semi-analytical technique precisely accounts for the interactions between the nanoinhomogeneities and constitutes an efficient computational tool for modeling nanocomposites. Numerical results demonstrate the accuracy and numerical efficiency of the approach and show the nature and extent to which the elastic interactions between the nanoinhomogeneities with interface stress affect the elastic fields around them.     

On the overall elastic moduli of composites with spherical coated fillers

A micromechanical approach is presented to estimate the overall linear elastic moduli of three phase composites consisting of two phase coated spherical particles randomly dispersed in a homogeneous isotropic matrix. The theoretical method is based on Eshelby’s equivalent inclusion method and its recent extension by Shodja and Sarvestani [J. Appl. Mech. 68 (2001) 3] to evaluate the local field variables in case of double (multi) inhomogeneities. Using Tanaka–Mori theorem [J. Elasticity 2 (1972) 199] and a decomposition of Green’s function integral equation, the pair-wise average phase values of stress and strain in two interacting coated particles are estimated. Following Ju and Chen [Acta Mech. 103 (1994) 103; Acta Mech. 103 (1994) 123] the ensemble phase volume average of stress and strain fields can be evaluated within a representative volume element containing a finite number of coated particles. Comparisons with classical bounds are presented to illustrate the accuracy of the proposed method.     

An efficient approximate numerical method for modeling contact of materials with distributed inhomogeneities

This research explores the influence of distributed non-interpenetrating inhomogeneities on the contact of inhomogeneous materials via a new efficient numerical model based on Eshelby’s Equivalent Inclusion Method. The half-space contact of a sphere with an inhomogeneous material is considered, and the solutions take into account interactions between all inhomogeneities. The efficiency and solution accuracy of the proposed method are demonstrated through comparative studies with those of an existing numerical method and the finite element method. The influence of spatial inhomogeneity orientations on the contact elastic field is investigated and parametric studies are conducted for the effect of arbitrarily distributed inhomogeneities on the stress field of the materials. The significance of the influences of inhomogeneity distribution parameters on the inverse volumetric stress integral is quantified and the corresponding data are fitted into selected several formulas as a step towards understanding the rolling contact fatigue life of the materials.     

On the method of equivalent inclusions in elastodynamics and the scattering fields of two ellipsoidal inhomogeneities

Gurtin’s variational principle is used to derive the equivalence equalions for general linear elastodynamical problems of multiple inhomogeneities. An approximate expression on scattering field of two ellipsoidal inhomogeneities is obtained by the method of equivalent inclusions. And some numerical examples are presented.     

Elastic equilibrium of a half plane containing a finite array of elliptic inclusions

An accurate analytical method has been proposed to solve for stress in a half plane containing a finite array of elliptic inclusions, the last being a model of near-surface zone of the fibrous composite part. The method combines the Muskhelishvili’s method of complex potentials with the Fourier integral transform technique. By accurate satisfaction of all the boundary conditions, a primary boundary-value elastostatics problem for a piece-homogeneous 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 equations and thus an efficient computational algorithm. The theory developed is rather general and can be applied to solve a variety of elastostatics problems. Up to several hundred interacting inclusions can be considered in this way in practical simulations which makes the model of composite half plane realistic and flexible enough to account for the microstructure statistics. The stress concentration factors and effective thermoelastic properties of random structure composites with dilute concentration of fibers are estimated in the vicinity of a free edge. The numerical examples are given showing accuracy and numerical efficiency of the developed method and disclosing the way and extent to which the nearby free or loaded boundary influences the local and mean stress concentration in the fibrous composite.     

基于浸入边界-格子Boltzmann通量求解法的椭圆柱流动特性分析

基于浸入边界-格子Boltzmann通量求解法,开展了雷诺数Re=100不同几何参数下单椭圆柱及串列双椭圆柱绕流流场与受力特性对比研究。结果表明,随长短轴比值的增加,单椭圆柱绕流阻力系数先减小后缓慢上升,最大升力系数则随长短轴比值的增大而减小;尾迹流动状态从周期性脱落涡到稳定对称涡。间距是影响串列圆柱及椭圆柱流场流动状态的主要因素,间距较小时,串列圆柱绕流呈周期性脱落涡状态,而椭圆柱则为稳定流动;随着间距增加,上下游圆柱及椭圆柱尾迹均出现卡门涡街现象,且串列椭圆柱临界间距大于串列圆柱。串列椭圆柱阻力的变化规律与圆柱的基本相同,上游平均阻力大于下游阻力;上游椭圆柱阻力随着间距的变大先减小,下游随间距的变大而增加,当间距达到临界间距时上下游阻力跃升,随后出现小幅度波动再逐渐增加,并趋近于相同长短轴比值下单柱体绕流的阻力。     

Piezoelectric composites with periodic multi-coated inhomogeneities

A new, robust homogenization scheme for determination of the effective properties of a periodic piezoelectric composite with general multi-coated inhomogeneities is developed. In this scheme the coating does not have to be thin, the shape and orientation of the inclusion and coatings do not have to be identical, their centers do not have to coincide, their properties do not have to remain uniform, and the microstructure can be with the 2D elliptic or the 3D ellipsoidal inclusions. The development starts from the local electromechanical equivalent inclusion principle through the introduction of the position-dependent equivalent eigenstrain and electric field. Then with a Fourier series expansion and a superposition procedure, the volume-averaged equivalent eigenstrain and electric field for each phase are obtained. The results in turn are used in an energy equivalent criterion to determine the effective properties of the composite. In this model the interphase interactions in each multi-coated particle and the long-range interactions between the periodically distributed particles are fully accounted for. To demonstrate its wide range of applicability, we applied it to examine the properties of several periodic composites: (i) piezoelectric PZT spherical particles in a polymer matrix, (ii) continuous glassy fibers with thin PZT coating in an epoxy matrix, (iii) spherical PZT particles coated by thick or functionally graded piezoelectric layer, (iv) spheroidal voids coated with a thick non-piezoelectric layer in a PZT matrix, and (v) spherical piezoelectric inhomogeneities with eccentric, non-uniform thickness coating. The calculated results reflect the complex nature of interplay between the properties of core, matrix, and coating, as well as whether the coating is uniform, functionally graded, or eccentric. The accuracy of this new scheme is checked against the double-inclusion and other micromechanics models, and good agreement is observed.     

Inclusions and inhomogeneities in strain gradient elasticity with couple stresses and related problems

At small length scales and/or in presence of large field gradients, the implicit long wavelength assumption of classical elasticity breaks down. Postulating a form of second gradient elasticity with couple stresses as a suitable phenomenological model for small-scale elastic phenomena, we herein extend Eshelby’s classical formulation for inclusions and inhomogeneities. While the modified size-dependent Eshelby’s tensor and hence the complete elastic state of inclusions containing transformation strains or eigenstrains is explicitly derived, the corresponding inhomogeneity problem leads to integrals equations which do not appear to have closed-form solutions. To that end, Eshelby’s equivalent inclusion method is extended to the present framework in form of a perturbation series that then can be used to approximate the elastic state of inhomogeneities. The approximate scheme for inhomogeneities also serves as the basis for establishing expressions for the effective properties of composites in second gradient elasticity with couple stresses. The present work is expected to find application towards nano-inclusions and certain types of composites in addition to being the basis for subsequent non-linear homogenization schemes.     

Contact analysis in the presence of an ellipsoidal inhomogeneity within a half space

Many materials contain inhomogeneities or inclusions that may greatly affect their mechanical properties. Such inhomogeneities are for example encountered in the case of composite materials or materials containing precipitates. This paper presents an analysis of contact pressure and subsurface stress field for contact problems in the presence of anisotropic elastic inhomogeneities of ellipsoidal shape. Accounting for any orientation and material properties of the inhomogeneities are the major novelties of this work. The semi-analytical method proposed to solve the contact problem is based on Eshelby’s formalism and uses 2D and 3D Fast Fourier Transforms to speed up the computation. The time and memory necessary are greatly reduced in comparison with the classical finite element method. The model can be seen as an enrichment technique where the enrichment fields from the heterogeneous solution are superimposed to the homogeneous problem. The definition of complex geometries made by combination of inclusions can easily be achieved. A parametric analysis on the effect of elastic properties and geometrical features of the inhomogeneity (size, depth and orientation) is proposed. The model allows to obtain the contact pressure distribution – disturbed by the presence of inhomogeneities – as well as subsurface and matrix/inhomogeneity interface stresses. It is shown that the presence of an inclusion below the contact surface affects significantly the contact pressure and subsurfaces stress distributions when located at a depth lower than 0.7 times the contact radius. The anisotropy directions and material data are also key elements that strongly affect the elastic contact solution. In the case of normal contact between a spherical indenter and an elastic half space containing a single inhomogeneity whose center is located straight below the contact center, the normal stress at the inhomogeneity/matrix interface is mostly compressive. Finally when the axes of the ellipsoidal inclusion do not coincide with the contact problem axes, the pressure distribution is not symmetrical.     

Contact analysis in presence of spherical inhomogeneities within a half-space

This paper presents a fast method of solving contact problems when one of the mating bodies contains multiple heterogeneous inclusions, and numerical results are presented for soft or stiff inhomogeneities. The emphasis is put on the effects of spherical inclusions on the contact pressure distribution and subsurface stress field in an elastic half-space. The computing time and allocated memory are kept small, compared to the finite element method, by the use of analytical solution to account for the presence of inhomogeneities. Eshelby’s equivalent inclusion method is considered in the contact solver. An iterative process is implemented to determine the displacements and stress fields caused by the eigenstrains of all spherical inclusions. The proposed method can be seen as an enrichment technique for which the effect of heterogeneous inclusions is superimposed on the homogeneous solution in the contact algorithm. 3D and 2D Fast Fourier Transforms are utilized to improve the computational efficiency. Configurations such as stringer and cluster of spherical inclusions are analyzed. The effects of Young’s modulus, Poisson’s ratio, size and location of the inhomogeneities are also investigated. Numerical results show that the presence of inclusions in the vicinity of the contact surface could significantly changes the contact pressure distribution. From a numerical point of view the role of Poisson’s ratio is found very important. One of the findings is that a relatively ‘soft’ and nearly incompressible inclusion – for example a cavity filled with a liquid – can be more detrimental for the stress state within the matrix than a very hard inclusion with a classical Poisson’s ratio of 0.3.