Micro to Macro: Composites via Homogenization

                            A composite material is generally taken to be a material consisting of two or more constituents
or phases [96]. These heterogeneities can include voids, fibers and stiff or soft inclusions and
can vary in size. Any material, human-made or natural, can thus be considered heterogeneous
at a particular scale. The ongoing improvement of technology over the past century has led to
the rapid development of heterogeneous materials, and the need to understand the interactions
between these heterogeneities. In the past, experimental test methods would be conducted on
a series of material samples to determine its "effective properties" [41]. This approach has
become less appealing in recent times though due to the number of resources required, such as
time and money.
Conversely, to accurately model the microstructure at the macroscopic level, each constituent
in the heterogeneous structure would need to be modeled explicitly. While a hypothesis on the
microstructural behavior at the macro scale is not required in this case, the method is often
impractical due to the enormous difference in length scales between the heterogeneities and
the macroscopic sample. Moulinec and Suquet [69] used Fast Fourier Transforms, with images
of the microstructure, to reduce the size of the meshing, but still required computers with high
memory capabilities. The application of micro modeling is thus limited to localized
phenomena cases, where analysis of the microstructure is required. These include contact
problems [27, 112], microstructural damage [73] and micro-cracking [40]. A parameter fitting
technique was implemented by Geers [20] and Meuwissen [62] to obtain the effective property
of a material’s microstructure. By fitting material parameters to experimental data, one can
obtain the macroscopic strain-energy function of the material. Though successful, this
approach 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 technique
replaces the complex microstructure with one that is statistically homogeneous at the
macroscopic level. The replacement of the complex microstructure with one that is
homogeneous overcomes the need for complex meshing. Also, depending on the chosen
homogenization theory, the mechanical response can be investigated with no prior knowledge
of the material, giving a first approximation on the stress distribution at the micro level. These
characteristics surpass the limitations posed by the other methods, making it the preferred
choice when modeling composites. There are several different homogenization approaches, of
which the most common are briefly discussed below. For a more comprehensive review, we
refer the reader to Nemat-Nasser and Hori [71].