Analysis of Composite Materials with Random Microstructure

ix Czech Technical University in PragueKlokner InstituteAbstract Analysis of Composite Materials with Random Microstructureby Jan Zeman In this thesis, a comprehensive approach to the modelling of composite materials witha microstructure exhibiting various types of disorder is proposed. In particular, a fibrousgraphite-epoxy composites with fibers randomly distributed within a transverse plane sec-tion and plain weave fabric composites are addressed as representatives of such materialsystems. Two different modelling strategies are examined. The first one assumes a welldefined geometry of the microstructure arrangement and specific boundary conditions. Inthis framework, the complicated real microstructure is replaced by a substantially simplerperiodic unit cell, which statistically resembles the real microstructure in a suitable sense.Periodic distribution of such unit cells is explored and the finite element method is calledto carry out the numerical computation. The theoretical basis for the second approach arethe Hashin-Shtrikman variational principles. The random character of the fiber distribu-tion is incorporated directly into the variational formulation employing certain statisticaldescriptors.A number of statistical descriptors suitable for the microstructure characterizationof a random medium is examined first. Several methods for their determination areproposed and tested for a simple theoretical model of a microstructure. Additionally,a validity of various statistical hypotheses usually accepted for a random heterogenousmedium is checked for the real microstructure represented here by the graphite fiber towembedded into the polymer matrix. Next, a stochastic optimization algorithm based onefficient combination of genetic algorithm essentials with basic concepts of the simulatedannealing method is introduced.Suitable optimization procedure formulated in terms of the selected statistical de-scriptors is proposed to derive the desired unit cell. Several numerical experiments areperformed to demonstrate the applicability of the selected optimization method to thiscomplicated optimization problem. A number of numerical studies is performed to quan-tify individual unit cells. The objective is to identify a number of particles required forspecific problems to provide a sufficiently accurate representation of the behavior of realcomposites. A standard problem of deriving the effective mechanical properties is con-sidered first. A general approach permitting either strain or stress control is pursued. Itis observed that the unit cell consisting of five fibers only provides reasonably accurateestimates of the macroscopic properties. Similar conclusions apply to the thermal, linearand non-linear viscoelastic problems considered next.In certain applications the finite element approach used with the unit cell analysis mayprove to be unnecessary expensive. In such a case, one may appreciate well-known effec-tive medium theories where applicable. Here, the most widely used variational principlesof Hashin and Shtrikman extended to account for the presence of various transformationfields defined as local eigenstrain or eigenstress distributions in the phases are revisited.