We present a partitioning of the elastic constants of a crystal into atomic contributions by using the atomic basin concept inherent to Bader's Quantum Theory of Atoms in Molecules. The partition is made by following the evolution of the cell volume and the atomic basin volumes under appropriately defined cell deformations. The method is carefully examined, including internal consistency checks. The transferability of atomic contributions between different crystals is determined by obtaining and comparing the oxygen contribution to the elastic constants of a selection of cubic oxides that includes the rock-salt, perovskite, antifluorite, and cuprite crystal families.