Calculation of characteristics of the elastic and thermophysical properties of a multiphase composite containing composite or hollow spherical inclusions

Conclusion We generalized the self-consistent method of effective media to the case of a four-phase model consisting of a core, a shell, a binder, and the effective medium. We obtained analytic solutions for the elastic characteristics, coefficient of linear expansion, heat capacity, and thermal conductivity of a multiphase composite containing several types of composite (or hollow) spherical inclusions. In the special case of a composite containing inclusions of just one type, the solutions obtained for the bulk modulus of elasticity K, coefficient of linear expansion a, heat capacity c, and thermal conductivity agree (within the framework of the two-stage approach) with the values found using known solutions for a three-phase model [8]. The first stage entails calculation of the effective characteristics of a spherical composite inclusion, while the second stage involves calculating the analogous characteristics for the composite as a whole.The possibilities of the solutions that were found were illustrated in a calculation of the shear modulus of a composite containing spherical hollow inclusions. It was shown that by assuming a nonaxisymmetric Weibull distribution of the parameter (the ratio of the thickness of the wall of a particle to its radius) it is possible to reach better agreement between the calculations and the experimental data in [4] than when calculations are performed using only the mean value of .The solutions obtained here can be used to find optimum combinations of volume fractions of different types of fillers in multiphase composites.The work was sponsored at the University Iberoamericana in 1994 by the Mexican National Council of Science and Technology (CONACYT).Translated from Mekhanika Kompozitnykh Materialov, Vol. 30, No. 4, pp. 512–519, July–August, 1994.     

Generalized self-consistent method: Modeling and computation of effective elastic properties of composites with composite or hollow inclusions

A new approach to the generalized self-consistent method [1,2] has been developed for problems of the statistical mechanics of composites with composite or hollow inclusions. The approach can reduce the problem of predicting the effective elastic properties of composites to a simpler averaged problem of a single, composite, or hollow inclusion with inhomogeneous elastic surrounding in a homogeneous effective elastic medium. The problem of predicting the effective elastic properties of composites with unidirectional hollow fibers or hollow spherical inclusions are studied by using the new approach.Submitted to the 10th International Conference on Mechanics of Composite Materials, April 20–23, 1998, Riga, Latvia.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 2, pp. 173–183, March–April, 1998.     

Polarization tensors and effective properties of anisotropic composite materials

In this paper, we present a systematic scheme for derivations of asymptotic expansions including higher-order terms, with estimates, of the effective electrical conductivity of periodic dilute composites in terms of the volume fraction occupied by the inclusions. The conductivities of the inclusion and the matrix may be anisotropic. Our derivations are based on layer potential techniques, and valid for high contrast mixtures and inclusions with Lipschitz boundaries. The asymptotic expansion is given in terms of the polarization tensor and the volume fraction of the inclusions. Important properties, such as symmetry and positivity, of the anisotropic polarization tensors are derived.     

On the effective moduli of composite materials: Slender rigid inclusions at dilute concentrations

Summary Using slender-body theory the effective elastic moduli of a certain class of composite materials are determined by analyzing the response of an infinite elastic medium, containing a single slender rigid inclusion, to a given applied strain. Solutions are found as perturbation expansions in the slenderness ratio of the inclusion, , which is small. These then yield the five independent elastic moduli, which characterize the macroscopic state of a dilute dispersion of slender rigid inclusions aligned in a common direction. The increase in rigidity due to the inclusions is found to be of order/ ² ln(2/) for the longitudinal Young's modulus and of order for the other four moduli, where is the volume fraction of inclusions. Although the general theory is restricted to axisymmetric inclusions having ends which are no more blunt than a prolate spheroid, the results are shown to be valid approximations for circular cylinders in certain cases.

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Résumé Cet article donne les résultats d'une application de la théorie des corps élancés à la mécanique des milieux continus. On y analyse la réponse d'un milieu élastique infini, contenant une seule particular rigide et élancée, quand il est soumis à des efforts donnés. La théorie générale est resteinte à une classe de particules possédant une symmétrie de révolution, mais étant au moins aussi pointues que des sphéroïdes allongés. Dans certains cas les résultats peuvent néanmoins être appliqués approximativement à des inclusions cylindriques circulaires. A partir de cette solution, on a calculé les cinq modules d'élasticité indépendants qui caractérisent l'état global d'un milieu à faible concentration de particules, ces dernières étant toutes alignées dans une même direction. Les résultats sont donnés sous forme de développement de perturbation, en fonction du coefficient de forme,K, des particules. L'accroissement de la rigidité dû á leur présence est de l'ordre de pour le module d'élongation de Young, et de l'ordre de pour les quatre autres modules, où représente le rapport du volume des particules au volume total.

     

Calculation of elastic characteristics and creep functions of hollow-sphere plastics by the effective medium method

Conclusion Solutions for the elastic characteristics and the creep functions of a composite containing hollow spherical fillers as applies to the four-phase nucleus/jacket/binder/equivalent-homogeneous-material model are obtained in the study when the method of self-correspondence is used. It is demonstrated that if the two-stage approach (when the elastic characteristics of the nucleus + jacket system, and the composite are calculated in the first and second stages, respectively) yields an exact solution for the bulk modulus K_* of the composite, it is highly approximate when the shear modulus G_* of the composite is determined. The error of determination of G_* increases considerably (by a factor of 2–2.5 when = 0.4) when Kerner's approximate solution (2) is used in lieu of solution (8) for the three-phase model within the framework of the two stage approach. Dzenis and Maksimov [5] establish by comparison with experimental data that the four-phase model provides a rather exact solution for the elastic modulus of a composite when the bulk content of hollow spheres 0.4. It is also demonstrated that use of Kerner's approximate solution (2) within the framework of the two-stage approach in predicting the creep of a composite yields an inadmissibly high error in the region of the principal relaxation transition of the binder from the glassy to the highly elastic state.This work was sponsored at the Iberoamericana University in 1993 by the Mexican National Council of Science and Technology (CONACYT).Translated from Mekhanika Kompozitnykh Materialov, Vol. 30, No. 2, pp. 177–188, March–April, 1994.     

Frequency-dependent properties of multiphase materials: origins,needs, and effective use in geophysics and modern synthetic materials.

Dynamic processes in multiphase materials, e.g., fluid-filled sandstones or bones, are described by models that include frequency-dependent properties. The origin of such properties is introduced as an averaged representation of frequency-dependent microscale motions. In addition to classical frequency-dependence of fluid flow, the influence of weak, high-porosity materials and fluid-fluid interfaces is discussed. The relevant characteristic numbers are contrasted and specific situations are demonstrated, in which frequency dependence has to be considered or not. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effective conductivity of composite materials with random positions of cylindrical inclusions: finite number inclusions in the cell

The effective conductivity of composite materials with random position n ²,n?∈?N, and the cylindrical identical inclusions inside periodic cells is considered. We compare the results for symmetric and nonsymmetric cases of location of the inclusions in the cells and find that a symmetric structure provides a minimum for the effective conductivity among all the structures having n ² inclusions of such conductivity and sizes.     

Explicit jump immersed interface method for virtual material design of the effective elastic moduli of composite materials

Virtual material design is the microscopic variation of materials in the computer, followed by the numerical evaluation of the effect of this variation on the material’s macroscopic properties. The goal of this procedure is an in some sense improved material. Here, we give examples regarding the dependence of the effective elastic moduli of a composite material on the geometry of the shape of an inclusion. A new approach on how to solve such interface problems avoids mesh generation and gives second order accurate results even in the vicinity of the interface. The Explicit Jump Immersed Interface Method is a finite difference method for elliptic partial differential equations that works on an equidistant Cartesian grid in spite of non-grid aligned discontinuities in equation parameters and solution. Near discontinuities, the standard finite difference approximations are modified by adding correction terms that involve jumps in the function and its derivatives. This work derives the correction terms for two dimensional linear elasticity with piecewise constant coefficients, i.e. for composite materials. It demonstrates numerically convergence and approximation properties of the method.      

Analysis of a method for computing the speed of sound in a medium with inclusions

A method for computing the speed of sound in a medium with inclusions with the help of the theory of multiple scattering of a plane wave on a doubly periodic multilayered lattice of transparent particles is proposed.     

The use of the method of boundary states to analyse an elastic medium with cavities and inclusions

The analytical method of boundary states is developed and theoretically substantiated. A corollary of the Weierstrass theorem is proved according to which a function that is harmonic in a bounded, simply connected domain can be approximated by a series of homogeneous harmonic polynomials. A basis of the space of functions that are harmonic outside any neighbourhood of a point is constructed. An algorithm is developed for filling the basis of the space of the states of a multicavity elastic body. The method is used to solve a series of problems of determining of the stress-strain state of an unbounded elastic medium containing spherical cavities or inclusions with different boundary conditions: the boundary of the cavity is free (the Southwell problem), constrained or under conditions of contact with a rigid core. The effect of the width of the intercavity layer on the stress concentration is analysed in a non-axisymmetric problem with two cavities. The form of the relation between the mean-square discrepancy in the boundary conditions of the solution obtained and the number of elements in the basis is indicative of the numerical convergence of the solution of this problem.     

Inverse thermal imaging in materials with nonlinear conductivity by material and shape derivative method

The material and shape derivative method is used for an inverse problem in thermal imaging. The goal is to identify the boundary of unknown inclusions inside an object by applying a heat source and measuring the induced temperature near the boundary of the sample. The problem is studied in the framework of quasilinear elliptic equations. The explicit form is derived of the equations that are satisfied by material and shape derivatives. The existence of weak material derivative is proved. These general findings are demonstrated on the steepest descent optimization procedure. Simulations involving the level set method for tracing the interface are performed for several materials with nonlinear heat conductivity. Copyright © 2011 John Wiley & Sons, Ltd.     

Formation of the structure and effective properties of composite material in the manufacture of shells with cross-ply reinforcement

Conclusion The proposed model of formation of the reinforcing framework in the manufacture of an axisymmetric shell by the CPW method makes it possible to numerically determine the thicknesses and rates of filling the layers, reinforcement angles as a function of the main parameters of the technological process, geometry of the mandrel, and characteristics of the reinforcing filament. It was established that the thicknesses of the layers, reinforcement rates and angles, and, as a consequence, the effective stiffness characteristics of the composite are substantially variable both along the meridian and over the thickness of the shell.Translated from Mekhanika Kompozitnykh Materialov, No. 5, pp. 609–617, September–October, 1992.The authors thank V. A. Frolov for the experimental results.