Consider a linear elastic heterogeneous material under an external load. In the macroscopic scale, denoted by D, the effective material properties can be obtained by certain weighted averages of the microstructure at the microscopic scale, denoted by d. To use homogenization, the length scales satisfy

D d or the ratio of the two size scales satisfies

= d/D 1 Here, we set up two coordinates: x is a global coordinate at the macroscale, where the

elastic fields change slowly; and y is a local coordinate, shown in Figure 8.1, which are related to each other through the scale ratio as

y = x/

As schematically illustrated in Figure 8.2, when the heterogeneousmaterial is subjected to an applied load, the elastic fieldmay continuously change at themacroscale slowly. However, at the microscale, the local elastic field may fluctuate rapidly due to the material change in the heterogeneous microstructure. Considering the periodicity of the microstructure, the fluctuation of local elastic field in the y coordinate will follow a certain pattern but its average still follows the trend in the x coordinate. Therefore, the local field can be written in terms of both x and y. The perturbation of the local field and material properties will be periodic, say f (x, y) = f (x, y + dδ) with f representing a general local field function.