Representative Elementary Volume and Averaging

Assume that the center of the REV is located at x and x is any arbitrary point within the REV. The mean value of a function for this REV is defined by 

As shown in Fig. 5.2 the mean value / (jr, r) depends on the REV i.e., if the representative size I is too small, the mean value f(x,t) represents only the material at the center, whereas if the size I is too large (e.g., the case of an inhomogeneous medium) the mean value converges to another limit from its original mean value f(x,t). We note that there are upper and lower limits for the representative size, depending on the microscale geometrical properties of the material. 

Ichikawa and A.P.S. Selvadurai, Transport Phenomena in Porous Media, 

5 Classical Theory of Diffusion and Seepage Problems in Porous Media 

Let g(x, t, x) be another dependent variable specified in dv. Since we have 

---