Is the concept of “average shape” legitimate, for a mixture of inclusions of diverse shapes?

In heterogeneous materials, inhomogeneities (pores, cracks, particles) usually represent a mixture of diverse shapes. In modeling the effective elastic and conductive properties, they are often replaced by identical spheroidal shapes – for which exact solutions are available – that intend to represent certain “average shape” (for cracks, for example, the circular shape is routinely used). We find that, in anisotropic cases of non-random orientations, such a replacement cannot generally be done (with the exception of flat cracks). In other words, the concept of “average spheroid’ is not legitimate, and its use may lead to large errors in predicting the effective properties – information on shape distribution is needed. In cases of overall isotropy, the replacement is possible in certain cases; even in these cases, however, the aspect ratio of the “average shape” may not be intuitively obvious.