M. Sejnoha and J. Zeman
A number of statistical descriptors suitable for the microstructure characterization of a random media are examined first. Several methods for their determination are proposed and tested for some simple theoretical models of microstructures. Additionally, a validity of various statistical hypotheses usually accepted for a random heterogenous media is checked for the real microstructure represented here by the graphite fiber tow embedded into the polymer matrix.
Suitable optimization procedure formulated in terms of selected statistical descriptors is proposed to derive the desired unit cell. A variety of stochastic optimization algorithms is examined to solve this problem. Judging from our experience with similar optimization problems, the genetic algorithms based solution techniques are explored. This study suggests that the augmented simulated annealing method, which effectively combines the essentials of genetic algorithms with the basic concept of the simulated annealing method, is superior to other approaches. Applicability of such unit cells is tested for polymer systems. Nevertheless, other systems such as ceramic or metal matrix systems may also benefit from the present work.
A number of numerical studies are performed to quantify individual unit cells. The objective is to identify a number of particles required for specific problems to provide a sufficiently accurate representation of the behavior of real composites. A standard problem of deriving the effective mechanical properties is considered first. A general approach permitting either strain or stress control is pursued. It is observed that the unit cell consisting of five fibers only provides reasonably accurate estimates of the macroscopic properties. Similar conclusion follows from the thermal and viscoelastic problems considered next.
In certain applications the finite element tool used with the unit cell analysis may prove to be unnecessary expensive. In such a case, one may appreciate well known effective medium theories where applicable. Here, the most widely used variational principles of Hashin and Shtrikman extended to account for the presence of various transformation fields defined as local eigenstrain or eigenstress distributions in the phases are revisited. Random character of fibers arrangement is described here by the two-point probability function. When used with the Hashin-Shtrikman variational principles this function provides sufficient information for obtaining bounds on the effective material properties of real composites with statistically homogeneous microstructures. The Fourier transform is successfully implemented when solving the resulting integral equations. This allows an arbitrary choice of the reference medium so that often encountered anisotropy of individual phases creates no obstacles in the solution procedure.