  The harmonic product of tensors---leading to the concept of harmonic
factorization---has been defined in a previous work (Olive et al, 2017). In the
practical case of 3D crack density measurements on thin or thick walled
structures, this mathematical tool allows us to factorize the harmonic
(irreducible) part of the fourth-order damage tensor as an harmonic square: an
exact harmonic square in 2D, an harmonic square over the set of so-called
mechanically accessible directions for measurements in the 3D case. The
corresponding micro-mechanics framework based on second---instead of
fourth---order damage tensors is derived. An illustrating example is provided
showing how the proposed framework allows for the modeling of the so-called
hydrostatic sensitivity up to high damage levels.
