Elastic field and effective moduli of periodic composites with arbitrary inhomogeneity distribution

This paper develops a semi-analytic model for periodically structured composites, of which each period contains an arbitrary distribution of particles/fibers or inhomogeneities in a three-dimensional space. The inhomogeneities can be of arbitrary shape and have multiple phases. The model is developed using the Equivalent Inclusion Method in conjunction with a fast Fourier Transform algorithm and the Conjugate Gradient Method. The interactions among inhomogeneities within one computational period are fully taken into account. An accurate knowledge of the stress field of the composite is obtained by setting the computational period to contain one or more structural periods of the composite. The effective moduli of the composite are calculated from average stresses and elastic strains. The model is used to analyze the stress field and effective moduli of anisotropic composites that have cubic symmetry. It shows that the bulk and shear moduli predicted by the present model are well located within the Hashin-Shtrikman bounds. The study also shows that the stress field of the composite can be significantly affected by the distribution of inhomogeneities even though the effective moduli are not affected much.     

Effect of a Minimal Spacing between Inhomogeneities on the Effective Moduli of Isotropic Plates with Circular Inhomogeneities

An interaction factor is used to characterize the effect of the interactions among inhomogeneities on the effective moduli. Many circular samples of plates are generated using a random number generator. When randomly generating a sample, a minimal spacing between inhomogeneities is set. Then, simulations using FEM are carried out to check the effect of the minimal spacing on the interaction factor. FEM results show that the interaction factor decreases as the minimal spacing increases.     

Variational bounds for the effective electroelastic moduli of piezoelectric composites with electromechanical coupling spring-type interfaces

The present work focuses on variational bounds for the effective electroelastic moduli of multiphase piezoelectric composites with thin piezoelectric interphase. Both the inhomogeneities and the matrix are assumed to be piezoelectric and transversely isotropic. The piezoelectric interphase is modeled as the spring-type interface with electromechanical coupling. The inhomogeneities are assumed to be spheroidal so that the reinforcement geometry is able to range from thin flake to continuous fiber. The effective properties of the piezoelectric composite with interfacial imperfection are defined and the principles of minimum internal energy and enthalpy are derived. These principles are applied to analytically obtain the upper and lower bounds for the effective electroelastic moduli. Unlike the Voigt–Reuss-type bounds for perfect interface, the present bounds depend not only on the material properties and volume fraction, but also on the interface parameters, inhomogeneity shape and orientation. An example of a two-phase composite is given for detailed discussion, where dependence of the electroelastic moduli and their bounds on the inhomogeneity shapes and orientations as well as the interface properties is provided and discussed. To qualitatively account for the dependence, analysis based on two possible mechanisms, i.e., the simple mixture rule of composite and the weakening effect by imperfect interface, are also provided.     

Ultrasonic wave propagation in two-phase media: Spherical inclusions

The scattering theory, recently developed via the extended method of equivalent inclusion, is used to study the propagation of time-harmonic waves in two-phase media of elastic matrix with randomly distributed elastic spherical inclusion materials. The elastic moduli and mass density of the composite medium are determined as functions of frequencies when given properties and concentration of the spheres and the matrix. Velocities and attenuation of ultrasonic waves in two-component media are determined. An averaging theorem that requires the equivalence of the strain energy and the kinetic energy between the effective medium and the original matrix with inhomogeneities is employed to derive the effective moduli and mass density. The functional dependency of these quantities upon frequencies and concentration provides a method of data analysis in ultrasonic evaluation of material properties. Numerical results for effective moduli, velocity and/or attenuation as functions of concentration of spherical inclusion material, or porosity, are graphically displayed.     

Elastic solids with high concentration of arbitrarily oriented multiphase particles

Summary The effective properties of elastic solids are strongly linked to their interacting micro-constituent phases. For materials containing dilute distributions of single-phase inhomogeneities, the overall behavior can be estimated in a straightforward manner. But in the non-dilute case, due to the complex inter-particle and particle-matrix interactions the treatment is rather involved. When the particles are heterogeneous, not only become the mentioned interactions more complex, but must properly account for the intra-particle interactions as well. The present work addresses an analytical approach to determine the overall moduli of elastic solids containing random distributions of arbitrarily oriented ellipsoidal heterogeneities at high concentrations. The approach is based on the extension of the equivalent inclusion method (EIM) to interacting multi-inhomogeneities. The long and short range interaction effects are intrinsically incorporated by the eigenstrain field. In the process, the average of the associated disturbance strain is computed within a representative volume element (RVE) using a superposition scheme. For verification of the proposed theory several theoretical estimates, experimental results, and bounds for the problems which have been obtained in the literature are reexamined. Consideration of more complex scenarios further demonstrates the efficacy of the proposed theory.     

The use of boundary strip method (BSM) for evaluation of the transverse mechanical properties of fibrous composites

The boundary strip method (BSM) is applied for evaluation of the transverse mechanical properties of fibrous composites with random and periodical fiber distributions. This special semi numerical method helps find the link between the microscopic behavior of the composite material and its macroscopic response in a rather detailed manner, enabling definition of stress and strain magnitudes at each point of the cross section. Here, specifically statistical model based on the boundary strip method, is used for assessment of the transverse effective moduli of fibrous composites. Random fiber distributions are compared with periodic fiber distributions having square or hexagonal array arrangements. Those are the common models used nowadays and modeled by the finite element or the boundary element. A comparison with the bounds of the polarization extremum principles is conducted too. The influence of the randomly distributed fibers on the transverse effective moduli is investigated and a good correlation is found between the results of the present model and the lower bound of the polarization extremum principles.     

Boundary element analysis of nanoinhomogeneities of arbitrary shapes with surface and interface effects

In this paper, a boundary element method (BEM) is proposed to analyze the stress field in nanoinhomogeneities with surface/interface effect. To consider this effect, the continuity conditions along the internal interfaces between the matrix and inhomogeneities are modeled by the well-known Gurtin–Murdoch constitutive relation. In the numerical analysis, the interface elastic moduli and the geometry of the nanoscale inhomogeneity are varied to show their influence on the induced stress field. The interaction between nanoscale inhomogeneities and the effect of different geometric shapes of inhomogeneities, including ellipse, triangle, and square are also investigated for different interface material parameters. It is shown that the elastic field can be greatly influenced by the interfacial energy and geometry of nanoscale inhomogeneities. The proposed BEM formulation is very general, including the complete Gurtin–Murdoch model and is further convenient for arbitrary shapes of inhomogeneity.     

A general unified treatment of lamellar inhomogeneities

Consider a lamellar inhomogeneity embedded in an unbounded isotropic elastic medium. When the elastic moduli of the lamellar inhomogeneity are zero it is a crack, if its elastic moduli are infinite it is an anticrack, and when its elastic moduli are finite it is called a quasicrack. Based on the Eshelby’s equivalent inclusion method (EIM), the present paper develops a unified approach for determination of the exact closed-form expressions for modes I, II, and III stress intensity factors (SIFs) at the tips of lamellar inhomogeneities under a remote applied polynomial loading.     

Percolation, Fractal Behavior, and High-T c Superconductivity

Hole suppliers like Sr in doped La₂CuO₄ are mainly randomly distributed. Assuming that the holes are dislocated over a few lattice constants away from the Sr atom, the conducting areas form randomly distributed circles in the CuO₂ layer planes. Conductivity and also superconductivity can occur only when these circles touch each other and form percolation clusters. Mobile holes are accompanied by diffusing d-electrons. Their spin direction is no longer localized on distinct places, and antiferromagnetism breaks down. The phase diagram of high-T _c superconductors is discussed on the basis of a modified continuum percolation model for which the centers of each circle are located on lattice points. The inhomogeneities due to the random hole distributions lead to broad peaks instead of sharp singularities in the static and dynamic response functions.     

Percolation, Fractal Behavior, and High-Tc Superconductivity

Hole suppliers like Sr in doped La₂CuO₄ are mainly randomly distributed. Assuming that the holes are dislocated over a few lattice constants away from the Sr atom, the conducting areas form randomly distributed circles in the CuO₂ layer planes. Conductivity and also superconductivity can occur only when these circles touch each other and form percolation clusters. Mobile holes are accompanied by diffusing d-electrons. Their spin direction is no longer localized on distinct places, and antiferromagnetism breaks down. The phase diagram of high-T _c superconductors is discussed on the basis of a modified continuum percolation model for which the centers of each circle are located on lattice points. The inhomogeneities due to the random hole distributions lead to broad peaks instead of sharp singularities in the static and dynamic response functions.     

Numerical analysis of interaction and coalescence of numerous microcracks

The deformation and failure behaviors of brittle or quasi-brittle solids are closely related to interaction and propagation of stochastically distributed microcracks. The influence of microcrack interaction and evolution on the mechanical properties of materials presents a problem of considerable interest, which has been extensively argued but has not been resolved as yet. In the present paper, a novel numerical method is used to calculate the effective elastic moduli and the tensile strength, and to simulate the failure process of brittle specimens containing numerous microcracks. The influences of some crack distribution parameters reflecting the non-uniform spatial concentration, size and orientation distributions are examined. The effective elastic moduli and the tensile strength of brittle materials exhibit different dependences on microcrack interaction. For example, two microcrack distributions that lead to the identical effective elastic moduli may cause a pronounced difference in the tensile strengths and failure behaviors of materials. By introducing two criteria for microcrack growth and coalescence in terms of Griffith’s energy release rate, the above numerical method is extended to simulate the coalescence process of microcracks that results in a fatal crack and the final rupture of a specimen.     

Order and disorder in heterogeneous material microstructure: Electric and elastic characterisation of dispersions of pseudo-oriented spheroids

The paper deals with the electrical and elastic characterisation of dispersions of pseudo-oriented ellipsoids of rotation: it means that we are dealing with mixtures of inclusions of different eccentricities and arbitrary non-random orientational distributions. The analysis ranges from parallel spheroidal inclusions to completely random oriented inclusions. A unified theory covers all the orientational distributions between the random and the parallel ones. The electrical and micro-mechanical averaging inside the composite material is carried out by means of explicit results which allows us to obtain closed-form expressions for the macroscopic or equivalent dielectric constants or elastic moduli of the overall composite materials. In particular, this study allows us to affirm that the electrical behaviour of such a dispersion of pseudo-oriented particles is completely defined by one order parameter which depends on the given angular distribution. Moreover, the elastic characterisation of this heterogeneous material depends on two order parameters, which derive from the orientational distribution. The theory may be applied to characterise media with different shapes of the inclusions (i.e. spheres, cylinders or planar inhomogeneities) yielding a set of procedures describing several composite materials of great technological interest.     

Effective thermoelastic moduli and stress concentrator factors in nanocomposites

Summary Nanocomposites are modeled as a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random field of spheroid nanofibers with prescribed random orientation. An estimation of the effective thermoelastic properties of NC was performed by the effective field method (see Buryachenko, [10]) taking into account the random orientation of nanofibers as well as justified selection of spatial correlations of fiber location. The independent justified choice of shapes of inclusions and correlation holes provides the matrix of effective moduli which is symmetric (in contrast to the Mori-Tanaka approach). One estimates also the effective tensor of thermal expansion and stress concentrator factors depending on the orientation of the fiber being considered as well as on the justified choice of the shape of correlation holes, concentration and orientation distribution functions of nanofibers.     

Elastic Moduli of Composites with Rigid Elliptical Inclusions

Effective elastic moduli of 2-D solids with randomly located perfectly rigid elliptical inclusions are derived using non-interacting approximation. This approximation constitutes the basic building block for various approximate schemes in micromechanics of materials with interacting inhomogeneities. Anisotropy due to non-random orientation of inclusions is investigated. The results are obtained in closed form and compared against existing solutions.     

Elastic moduli for a class of porous materials

Summary The effective elastic moduli for a class of porous materials with various distributions of spheroidal voids are given explicitly. The distributions considered include the unidirectionally aligned voids, three-dimensionally and two-dimensionally, randomly oriented voids, and voids with two types of biased orientations. While the 3-d random orientation results in a macroscopically isotropic solid, the porous media associated with the other arrangements are transversely isotropic. The five independent elastic constants for each arrangement, as well as the two for the isotropic case, are derived by means of Mori-Tanaka's mean field theory in conjunction with Eshelby's solution. Specific results for long, cylindrical pores and for thin cracks with the above orientations are also obtained, the latter being expressed in terms of the crack-density parameter. Before we set out the analysis, it is further proven that, in the case of long, circular inclusions, the five effective moduli of a fiber composite derived from the Mori-Tanaka method coincide with Hill's and Hashin's lower bounds if the matrix is the softer phase, and coincide with their upper bounds if the matrix is the harder.With 5 Figures     

A new homogenization method based on a simplified strain gradient elasticity theory

Homogenization methods utilizing classical elasticity-based Eshelby tensors cannot capture the particle size effect experimentally observed in particle–matrix composites at the micron and nanometer scales. In this paper, a new homogenization method for predicting effective elastic properties of multiphase composites is developed using Eshelby tensors based on a simplified strain gradient elasticity theory (SSGET), which contains a material length scale parameter and can account for the size effect. Based on the strain energy equivalence, a homogeneous comparison material obeying the SSGET is constructed, and two sets of equations for determining an effective elastic stiffness tensor and an effective material length scale parameter for the composite are derived. By using Eshelby’s eigenstrain method and the Mori–Tanaka averaging scheme, the effective stiffness tensor based on the SSGET is analytically obtained, which depends not only on the volume fractions and shapes of the inhomogeneities (i.e., phases other than the matrix) but also on the inhomogeneity sizes, unlike what is predicted by the existing homogenization methods based on classical elasticity. To illustrate the newly developed homogenization method, sample cases are quantitatively studied for a two-phase composite filled with spherical, cylindrical, or ellipsoidal inhomogeneities (particles) using the averaged Eshelby tensors based on the SSGET that were derived earlier by the authors. Numerical results reveal that the particle size has a large influence on the effective Young’s moduli when the particles are sufficiently small. In addition, the results show that the composite becomes stiffer when the particles get smaller, thereby capturing the particle size effect.     

Approximate Evaluation for Effective Elastic Moduli of Cracked Solids

A system of equations for effective elastic moduli of 2-D cracked solids is presented by combining the energy balance equation proposed by Shen and Yi (2000) with the integral equations which control the problem of an infinite solid with a finite number of cracks in a sub-region. Then, using Kachanov's method (Kachanov, 1987) for the solutions of the integral equations, 2-D effective bulk and shear moduli for solids with randomly distributed cracks are evaluated.     

Micromechanical effective elastic moduli of continuous fiber-reinforced composites with near-field fiber interactions

A higher-order micromechanical framework is presented to predict the overall elastic deformation behavior of continuous fiber-reinforced composites with high-volume fractions and random-fiber distributions. By taking advantage of the probabilistic pair-wise near-field interaction solution, the interacting eigenstrain is analytically derived. Subsequently, by making use of the Eshelby equivalence principle, the perturbed strain within a continuous circular fiber is accounted for. Further, based on the general micromechanical field equations, effective elastic moduli of continuous fiber-reinforced composites are constructed. An advantage of the present framework is that the higher-order effective elastic moduli of composites can be analytically predicted with relative simplicity, requiring only material properties of the matrix and fibers, the fiber?Cvolume fraction and the microstructural parameter ??. Moreover, no Monte Carlo simulation is needed for the proposed methodology. A series of comparisons between the analytical predictions and the available experimental data for isotropic and anisotropic fiber reinforced composites illustrate the predictive capability of the proposed framework.     

Numerical design of thin-walled structural members on account of their strength

The numerical design and optimization problems for the reinforced thin-walled structural members on account of a local strength criterion are considered. Two types of the elements of construction are analysed in detail: wafer plates and honeycomb-like shells. The optimal design concerns the following factors: the values of the effective stiffness moduli; the minimization of the weight; the ability to sustain the required ultimate strains. The computations are based on the application of the general homogenized shell model. The effective computational algorithm for a global minimum search for the function of several variables under the restrictions and the computer software are developed and applied. The asymptotically correct results for the effective stiffness moduli and the local stress distributions, which are available in the framework of the general homogenized shell model, ensure that the gain of optimization will not be overlapped by modelling errors. The effectiveness and advantages of the developed approach is illustrated by several numerical examples. The optimal design computational algorithm is extended to the case of fatigue failure analysis.