Fundamental Equations

There is a need for theoretical and computational tools that provide macroscopic relations for a composite continuum, starting from a description of the composite microstructure. The outlook for this viewpoint is particularly bright, given current trends in high-performance parallel supercomputing. This book is a step along those directions, with a special emphasis on a collection of mathematical methods that together build a base for advanced computational models. Consider the important example of the effective bulk properties of fiberreinforced materials consisting of fibers of minute cross section imbedded in a soft elastic epoxy. The physical properties of such materials is determined by the microstructure parameters: volume fraction occupied by the fibers versus continuous matrix; fiber orientations; shape of the fiber cross sections; and the spatial distribution of fibers. Hashin notes that “While for conventional engineering materials, such as metals and plastics, physical properties are almost exclusively determined by experiment, such an approach is impractical for FRM (fiber-reinforced materials) because of their great structural and physical variety,” The analysis of warpage and shrinkage of reinforced thermoset plastic parts provides yet another example of the important role played by computational models. The inevitable deformation of the fabricated part is influenced by the interplay between constituent material properties, the composite microstructure and macroscopic shape of the component. Computational models play an important role in controlling these deformations to minimize undesired directions that lead to warpage and shrinkage. The strength, stiffness, and low weight of these materials all result from the combination of a dispersed inclusion of very high modulus imbedded in a relatively soft and workable elastic matrix. It thus appears reasonable, as a first approximation, to consider a theory for the distribution of rigid (infinite modulus) inclusions in an elastic matrix, reserving the bulk of our efforts for the study of the role of inclusion microstructure. A framework for computational modeling has been established for materials processing, using models of microstructure with simplified rules for the motion of the inclusions.