Nanocrystalline materials show exceptional mechanical properties relative to their coarse-grained varieties. Because the volume fraction of grain boundaries in nanocrystalline materials can be as large as 30%, the mechanical properties of nanocrystalline materials are significantly influenced by this amorphous grain boundary phase. For example, the elastic modulus has been shown to decrease by 30% for nanocrystalline metals and more than 50% for nanocrystalline ionic materials. This is because the amorphous grain boundary regions are less dense than the crystalline grains, and thus have a larger volume per atom, . Assuming the interatomic potential, , is the same within the grain boundaries as in the bulk grains, the elastic modulus, , will be smaller in the grain boundary regions than in the bulk grains. Thus, via the rule of mixtures, a nanocrystalline material will have a lower elastic modulus than its bulk crystalline form.