Stress intensity factors for ring-shaped crack surrounded by spherical inclusions

Interaction of a ring-shaped crack with inhomogeneities such as inclusions is analyzed for the resulting three-dimensional stress field. Considered for the composite solid with a given volume fraction of inclusions are the two cases of (a) spherical voids and (b) spherical inclusions with elastic moduli different from the matrix. A ring-shaped crack is initiated at the equator of one of the voids or inclusions. A three-phase model is used to examine the interaction between the crack and surrounding inhomogeneities. Finite element method is then applied to calculate the stress intensity factor for different configurations. The effects of volume fraction of inhomogeneities, relative size of crack to inclusions, and material constants on crack behavior are discussed.     

The influence of porosity on the elastic response of homogeneous and structural composite media

Continuum theories of composites are employed to analyze the influence of inclusions and porosity on the elastic response of both homogeneous and laminated composite media. The general model analyzed consists of a periodic array of two perfectly bonded laminates; one of which consists of an elastic homogeneous material while the other is made up of a periodic array of cylindrical elastic inclusions that are distributed in another elastic matrix material. Several specific models are deduced as special cases. In all cases, porosity is simulated in the limit as the properties of the inclusions identically vanish. It is demonstrated that porosity plays a major role in the geometric dispersion of such media; in particular, it increases the arrival and rise times (spreading) of a propagating transient pulse. For the special case of elastic inclusions in a homogeneous matrix media, the present results correlate very well with existing experimental data and other approximate analyses.     

The homogenization method for the study of variation of Poisson's ratio in fiber composites

Summary Materials with specific microstructural characteristics and composite structures are able to exhibit negative Poisson's ratio. This fact has been shown to be valid for certain mechanisms, composites with voids and frameworks and has recently been verified for microstructures optimally designed by the homogenization approach. For microstructures composed of beams, it has been postulated that nonconvex shapes (with reentrant corners) are responsible for this effect. In this paper, it is numerically shown that mainly the shape, but also the ratio of shear-to-bending rigidity of the beams do influence the apparent (phenomenological) Poisson's ratio. The same is valid for continua with voids, or for composites with irregular shapes of inclusions, even if the constituents are quite usual materials, provided that their porosity is strongly manifested. Elements of the numerical homogenization theory and first attempts towards an optimal design theory are presented in this paper and applied for a numerical investigation of such types of materials. Received 11 March 1997; accepted for publication 12 September 1997     

BEM for simulation of a 2D elastic body with randomly distributed circular inclusions

Based on our 2D BEM software THBEM2 which can be applied to the simulation of an elastic body with randomly distributed identical circular holes, a scheme of BEM for the simulation of elastic bodies with randomly distributed circular inclusions is proposed. The numerical examples given show that the boundary element method is more accurate and more effective than the finite element method for such a problem. The scheme presented can also be successfully used to estimate the effective elastic properties of composite materials. Project supported by the National Natural Science Foundation of China (No. 19772025).     

含夹杂复合材料宏观性能研究

本文综述并评价了有关含夹杂复合材料的有效弹性模量研究的代表性工作，包括自洽理论，微分法，Ｅｓｈｅｌｂｙ－Ｍｏｒｉ－Ｔａｎａｋａ法，Ｈａｓｈｉｎ和Ｓｈｔｒｉｋｍａｎ的变分法等。指出上述理论由于没有充分考虑复合材料内部的微结构特征，如夹杂的形状、几何尺寸、分布和夹杂间的相互影响，在夹杂的体积份数较大，如大于０．３时已不能有效地预报复合材料的有效弹性模量，随后介绍了近来才发展起来的一种新方法─—相关函数积分法，理论与实验的结果的比较表明，该方法在夹杂体积份数较大时仍然有效。     

The effect of microstructural morphology on the elastic,inelastic, and degradation behaviors of aluminum–alumina composites

Micromechanical models with idealized and simplified shapes of inhomogeneities have been widely used to obtain the average (macroscopic) mechanical response of different composite materials. The main purpose of this study is to examine whether the composites with irregular shapes of inhomogeneities, such as in the aluminum–alumina (Al–Al₂O₃) composites, can be approximated by considering idealized and simplified shapes of inhomogeneities in determining their overall macroscopic mechanical responses. We study the effects of microstructural characteristics, on mechanical behavior (elastic, inelastic, and degradation) of the constituents, and shapes and distributions of the pores and inclusions (inhomogeneities), and thermal stresses on the overall mechanical properties and response of the Al–Al₂O₃ composites. Microstructures of a composite with 20% alumina volume content are constructed from the microstructural images of the composite obtained by scanning electron microscopy (SEM). The SEM images of the composite are converted to finite element (FE) meshes, which are used to determine the overall mechanical response of the Al–Al₂O₃ composite. We also construct micromechanics model by considering circular shapes of the inhomogeneities, while maintaining the same volume contents and locations of the inhomogeneities as the ones in the micromechanics model with actual shapes of inhomogeneities. The macroscopic elastic and inelastic responses and stress fields in the constituents from the micromechanics models with actual and circular shapes of inhomogeneities are compared and discussed.     

Differential geometrical method in elastic composite with imperfect interfaces

I.IntroductionWhethertheinterfacesofcompositematerialsareperfectornotwillaffectitsmacromechanicaloreffectivepropertiesimportantly.Butsofar,almostallofthestudiesontheeffectivepropertiesofcompositematerialsarebasedontheassumptionthattheinterfacesareperfectl"2].Infact,thisisnotappropriateforallinterfaces[31.Thusthestudiesonmechanicalpropertyofcompositematerialswithimperfaceintert'acehavebeenconsideredrecentlyinsomeliteratures.Hashin16]hasextendedtheelasticextremumprinciplesofminimumpotentialandm…     

Thermomechanical constitutive equations for the dynamic response of ceramics

The behavior and failure of brittle materials is significantly influenced by the existence of inhomogeneities such as pores and cracks. The proposed constitutive equations model the coupled micro-mechanical response of these inhomogeneities through evolution equations for scalar measures of porosity, and a “density” function of randomly oriented penny-shaped cracks. A specific form for the Helmholtz free energy is proposed which incorporates the known Mie–Grüneisen constitutive equation for the nonporous solid. The resulting thermomechanical constitutive equations are valid for large deformations and the elastic response is hyperelastic in the sense that the stress is related to a derivative of the Helmholtz free energy. These equations allow for the simulation of the following physical phenomena exhibited by brittle materials: (1) high compressive strength compared with much lower tensile strength; (2) inelastic deformation due to growth and nucleation of cracks and pores instead of due to dislocation dynamics associated with metal plasticity; and (3) loss of integrity (degradation of elastic moduli) due to damage accumulation. The main features of the model are demonstrated by examples of cyclic loading in homogeneous deformation and by a simulation of a dynamic plate-impact experiment on AD85 ceramic. The theoretical predictions of the model are in excellent agreement with the dynamic experimental data.     

Structural interfaces in linear elasticity. Part II: Effective properties and neutrality

The model of structural interfaces developed in Part I of this paper allows us to analytically attack and solve different problems of stress concentration and composites. In particular, (i) new formulae are given for effective properties of composite materials containing dilute suspensions of (randomly oriented) reinforced elliptical voids or inclusions; (ii) a new definition is proposed for inclusion neutrality (to account for the fact that the matrix is always ‘overstressed’, and thus non-neutral in a classical sense, at the contacts with the interfacial structure), which is shown to provide interesting stress optimality conditions. More generally, it is shown that the incorporation of an interfacial structure at the contact between two elastic solids exhibits properties that cannot be obtained using the more conventional approach of the zero-thickness, linear interface. For instance: contrary to the zero-thickness interface, both bulk and shear effective moduli can be optimized for a structural interface; effective properties higher that those possible with a perfect interface can be attained with a structural interface; and neutrality holds with a structural interface for a substantially broader range of parameters than for a zero-thickness interface.     

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.     

Size- and shape-dependent effective conductivity of porous media with spheroidal gas-filled inclusions

Porous media containing gas-filled inclusions embedded in a solid phase constitute an important class of natural or artificial materials of both theoretical and practical interest. In these materials, thermal conductivity is one of the most important properties. In a variety of situations of practical interest, when the characteristic size of gas-filled inclusions is comparable with the mean free path of gas molecules and when the slip flow regime is considered, the behavior of gas near solid surfaces cannot be described by classical thermal conductivity equations. In fact, the boundary conditions at the solid surfaces must be modified by considering that the temperature and normal heat flux simultaneously suffer a discontinuity. The first purpose of the present work is to develop an efficient and accurate micromechanical model capable of estimating the effective conductivity of porous materials while taking into account the discontinuities of the temperature and normal heat flux across solid surfaces and the non-spherical form of gas-filled inclusions. The second purpose of the present work is to study the dependencies of the effective conductivity on the size and shape of gas-filled inclusions. By applying the micromechanical model based on the differential scheme and by using the solution results obtained for auxiliary dilute problem accounting for modified boundary conditions on surface solids, the closed-form expression for the effective conductivity is obtained. Numerical results are provided to illustrate the dependence of the effective conductivity on the size and shape of gas-filled inclusions in the case of randomly oriented inclusions.     

Lamb mode stresses as means of identifying voids in the adhesive zone of bilaminates

Lamb wave technique has emerged as a reliable tool in the nondestructive testing of laminated plates. Some current studies to identify the specific Lamb modes that can characterize different kinds of defects in layered plates using Lamb waves have shown that the modes for which high stresses and low displacements occur in the interface indicate the presence of defects like pores or voids whereas the modes for which the displacements are high show the presence of harder inclusions. In this context this paper tests an earlier analytical model developed to facilitate NDT of porosity in the adhesive zone of bilaminates.The model tested treats the pore infested thin adhesive region as a linear elastic material with voids (LEMV). For certain parametric values of the LEMV adhesive layer the influence of these voids on dispersion and stresses carried by the first few Lamb modes in glass/glue/glass (G/g/G) bilaminate is traced in the range 0–10 MHz. The frequency–phase velocity points experimentally obtained by Kundu and Maslov are seen to fall very close to the present dispersion. The stresses traced using the present model in G/g/G plate at these experimentally tallied points show an easily discernable rise in the central region of adhesive, as observed by Kundu and Maslov.The model appears to be useful as a good first approximation to detect voids in adhesive zone of composite structural elements.     

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.     

Frequency dependence of elastic (SH-) wave velocity and attenuation in anisotropic two phase media

The propagation and attenuation of elastic waves in a random anisotropic two-phase medium is studied using statistical averaging procedures and a self-consistent multiple scattering theory. The specific geometry and orientation of the inhomogeneities (second phase) are incorporated into the formulation via the scattering matrix of each inhomogeneity. The anisotropy of the composite medium is due to the specific orientation of the non-symmetric inclusions. At low frequencies, analytical expressions are derived for the effective wave number in the average medium as a function of the geometry and the material properties and the angle of orientation of the inclusions. The results for the special cases of oriented cracks may find applications in geophysics and material science. The formulation is ideally suited for numerical computation at higher frequencies as evidenced by the results presented for composites reinforced by fibers of elliptical cross section.     

Stress concentration in incompressible multicomponent materials

A method for determining the effective elastic constants and the factors of stress concentration in microstructural elements is proposed for nonlinear incompressible multicomponent composite materials randomly reinforced with spheroidal inclusions with an arbitrary ratio of the longitudinal and lateral dimensions. Use is made of the Mori-Tanaka scheme that has, as a first approximation, the result of calculation of the elastic characteristics based on a model taking account of two-point statistical moment functions of arbitrarily high order. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 108–114, March, 2000.     

Rate form of the Eshelby and Hill tensors

Expressions are derived for the rates of change of the S and P tensors for transformed homogeneous inclusions in an anisotropic comparison medium undergoing prescribed changes of its elastic moduli. General results are obtained for ellipsoids and then reduced to yield explicit expressions in terms of the Stroh eigenvalues for cylindrical and disk-shaped inclusions in anisotropic solids and for spherical inclusions in isotropic solids. Applications are illustrated by solving the rate problem for an inhomogeneity in a large volume of a comparison medium, which is shown to be readily adaptable to standard averaging techniques for predictions of rates of change of overall moduli of composite materials experiencing evolution of phase moduli.     

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i) is positive definite only when the discrepancy tensor is negative defined; (ii) the non-local material symmetries are the same of the discrepancy tensor, and (iii) the non-local effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthotropic matrix.     

On the theory of elastic shells made from a material with voids

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.