  The isotropic elastic moduli closest to a given anisotropic elasticity tensor
are defined using three definitions of elastic distance, the standard Frobenius
(Euclidean) norm, the Riemannian distance for tensors, and the log-Euclidean
norm. The closest moduli are unique for the Riemannian and the log-Euclidean
norms, independent of whether the difference in stiffness or compliance is
considered. Explicit expressions for the closest bulk and shear moduli are
presented for cubic materials, and an algorithm is described for finding them
for materials with arbitrary anisotropy. The method is illustrated by
application to a variety of materials, which are ranked according to their
distance from isotropy.
