Micro to Macro: Composites via Homogenization

A composite material is generally taken to be a material consisting of two or more constituentsor phases [96]. These heterogeneities can include voids, fibers and stiff or soft inclusions andcan vary in size. Any material, human-made or natural, can thus be considered heterogeneousat a particular scale. The ongoing improvement of technology over the past century has led tothe rapid development of heterogeneous materials, and the need to understand the interactionsbetween these heterogeneities. In the past, experimental test methods would be conducted ona series of material samples to determine its "effective properties" [41]. This approach hasbecome less appealing in recent times though due to the number of resources required, such astime and money.Conversely, to accurately model the microstructure at the macroscopic level, each constituentin the heterogeneous structure would need to be modeled explicitly. While a hypothesis on themicrostructural behavior at the macro scale is not required in this case, the method is oftenimpractical due to the enormous difference in length scales between the heterogeneities andthe macroscopic sample. Moulinec and Suquet [69] used Fast Fourier Transforms, with imagesof the microstructure, to reduce the size of the meshing, but still required computers with highmemory capabilities. The application of micro modeling is thus limited to localizedphenomena cases, where analysis of the microstructure is required. These include contactproblems [27, 112], microstructural damage [73] and micro-cracking [40]. A parameter fittingtechnique was implemented by Geers [20] and Meuwissen [62] to obtain the effective propertyof a material's microstructure. By fitting material parameters to experimental data, one canobtain the macroscopic strain-energy function of the material. Though successful, thisapproach is tedious and, despite being optimized by Gendy and Saleeb [22] and Ogden et al.[72], requires large volumes of experimental data.An