  We present a tight-binding investigation of strained bilayer graphene within
linear elasticity theory, focusing on the different environments experienced by
the A and B carbon atoms of the different sublattices. We find that the
inequivalence of the A and B atoms is enhanced by the application of
perpendicular strain $\epsilon_{zz}$, which provides a physical mechanism for
opening a band gap, most effectively obtained when pulling the two graphene
layers apart. In addition, perpendicular strain introduces electron-hole
asymmetry and can result in linear electronic dispersion near the K-point. When
applying lateral strain to one layer and keeping the other layer fixed, we find
the opening of an indirect band gap for small deformations. Our findings
suggest experimental means for strain-engineered band gaps in bilayer graphene.
