For each of the $8$ isotropy classes of elastic materials, we consider a homogeneous random field taking values in the fixed point set $mathsf{V}$ of the corresponding class, that is isotropic with respect to the natural orthogonal representation of
a group lying between the isotropy group of the class and its normaliser. We find the general form of the correlation tensors of orders $1$ and $2$ of such a field, and the fields spectral expansion.
We establish spectral expansions of homogeneous and isotropic random fields taking values in the $3$-dimensional Euclidean space $E^3$ and in the space $mathsf{S}^2(E^3)$ of symmetric rank $2$ tensors over $E^3$. The former is a model of turbulent fl
uid velocity, while the latter is a model for the random stress tensor or the random conductivity tensor. We found a link between the theory of random fields and the theory of finite-dimensional convex compacta.
