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 alternative method to obtain
the effective properties is homogenization. This techniquereplaces the complex microstructure
with one that is statistically homogeneous at themacroscopic level. The replacement of the
complex microstructure with one that ishomogeneous overcomes the need for complex meshing. Also
depending on the chosenhomogenization theory the mechanical response can be investigated with
no prior knowledgeof the material giving a first approximation on the stress distribution at
the micro level. Thesecharacteristics surpass the limitations posed by the other methods
making it the preferredchoice when modeling composites. There are several different
homogenization approaches ofwhich the most common are briefly discussed below. For a more
comprehensive review werefer the reader to Nemat-Nasser and Hori [71].
