A systematic study of the heavy-ion fusion-cross-section enhancement at sub-barrier energies is carried out. The asymptotic energy shift introduced in a previous paper as a measure of this enhancement is considered from a theoretical point of view. It is claimed that this energy shift is composed of two terms: One of them is related to the bulk properties of nuclear matter, and the other corresponds to deviations depending on the specific nuclear structure of the collision partners. We show that the former can be approximately described by the neck-formation model for fusion and the latter is frequently a consequence of static deformation or vibrational excitation of the projectile and/or the target. A comparison of the neck-formation effects with those arising from neutron flow suggests that there is a connection between these two mechanisms. (orig.).