Advances in Mathematical Modelling of Composite Materials

The following sections are included:

• Foreword

• Contents

The article is devoted to the investigation of elastic, thermoelastic and dynamic behaviour of the important class of the so-called matrix composites. They represent a homogeneous matrix containing a random array of filling particles. A generalized version of the effective (self-consistent) field method is developed and its various implementations are reviewed in depth. The basic idea of the method is to treat every inclusion as isolated, but undergoing an effective external field, generated by the surrounding inclusions. Here, in distinction to the traditional forms of the effective field method, this field is considered, however, to be random and special techniques are proposed for calculating its statistical moments. The performance of the method is illustrated for composites, containing inclusions of various shapes (spheres, ellipsoids, long and short fibers, cracks, etc.). Explicit formulae for the overall elastic and thermoelastic properties of such composites are given and then compared to experimental data. The dynamic case (monochromatic wave propagation through the composite) is addressed as well. In the long wave approximation the propagation velocities and attenuation factors of the elastic waves in composites are derived and analyzed.

A general method of placing variational bounds on the effective properties (scalar conductivity and bulk and shear elastic moduli) of random heterogeneous solids is exposed in detail, together with various implementations and specific results. The method utilizes the Hashin-Shtrikman variational principle and trial fields which are obtained upon truncating the functional series expansion for the polarization fields in the solids. It is shown how the earlier variational procedures due to Hashin and Shtrikman, Milton and Phan-Thien and Willis find natural places in the proposed general scheme. Special attention is paid to dispersions of spheres and explicit analytic results for c²-bounds on their effective properties are given in terms of the radial distribution function. Other results discussed as well include, e.g., the fact of nonexistence of certain Gaussian-like random media and an exact c²-relation for the effective conductivity of a dispersion of spheres which contains absolutely convergent integrals only.

The generalization of the Hashin-Shtrikman variational principles to nonlinear problems is reviewed. The methodology employs a comparison material and allows the construction of bounds for the overall energy of any composite whose behaviour can be described in terms of a convex potential function. The development here is in terms of a nonlinearly elastic composite, characterized by a convex energy density function, which is subjected to geometrically linear strains. As examples, the nonlinear Hashin-Shtrikman structure is applied, using a homogeneous comparison material, to composites comprising a nonlinear matrix reinforced by rigid inclusions or weakened by cavities. Recent devlopments in the field have included the use of a linear composite as comparison material. This leads to bounds on the overall energy of the nonlinear composite which involve bounds on the energy of the linear comparison composite. Some results that have been obtained for two phase composites are described. One drawback of the nonlinear Hashin-Shtrikman principles has been the need to choose the comparison material so that it exhibits linear be-haviour. This generally means that at most one new bound, either upper or lower, can be found and in some cases no new bound can be obtained. Methods are now being developed which rely on a nonlinear comparison material and which have the potential to produce two new bounds. The new methodology is outlined in the context of a nonlinear dielectric problem and some results for a periodic composite are described.

The paper presents an advanced unified approach to the study of the processes of inelastic deformation and failure in a class of composites consisting of ductile matrices strengthened by parallelly aligned stiff continuous fibres. The approach governs classes of problems involving loading conditions of thermal and/or mechanical type. It incorporates a relatively simple model of plastification of the matrix phase of a unit composite cell consisting of a single fibre with a perfectly bonded matrix coating. The approach is unified in the sense that it reduces a relatively large variety of real mechanical problems of inelastic behaviour and failure of such composites to particular cases of a certain general mathematical plasticity problem. The model problems of uniform matrix cooling and longitudinal extension of an axially symmetric unit composite cell are studied in detail under certain simplifying assumptions. The results derived are shown to compare favourably with existing experimental data and theoretical estimates. Useful conclusions and estimates are also derived with respect to the influence of the fibre-matrix interactions and/or of existing longitudinal matrix cracks on the critical conditions of failure of the composites considered.

The following sections are included:

• Introduction

• A Model System

• Transformation Theory

• Complex Relations

• Summary and Conclusions

• Acknowledgements

• References

As a consequence of microstructural heterogeneities, fluctuations in the mechanical properties of materials are observed in experiments. This requires a probabilistic approach to relate the microstructure to the overall properties, and to predict scale effects in the fluctuations of properties. In this paper, the problem of the strength of materials is addressed, and various models of random structures developed for fracture statistics are introduced. As any fracture criterion is sensitive to microstructural heterogeneities, such as flaws with low strength, or defects including a local stress concentration, large effects of small scale heterogeneities are observed for fracture phenomena. The approach combines the selection of appropriate fracture criteria and random structure models. It enables us to predict the probability of fracture of heterogeneous media under various loading conditions.

The following sections are included:

• Series on Advances in Mathematics for Applied Sciences