  Several numerical studies have shown that the electronic properties of
twisted bilayers of graphene (TBLG) and transition metal dichalcogenides (TMDs)
are tunable by strain engineering of the stacking layers. In particular, the
flatness of the low-energy moir\'e bands of the rigid and the relaxed TBLG was
found to be, substantially, sensitive to the strain. However, to the best of
our knowledge, there are no full analytical calculations of the effect of
strain on such bands. We derive, based on the continuum model of moir\'e flat
bands, the low-energy Hamiltonian of twisted homobilayers of graphene and TMDs
under strain at small twist angles. We obtain the analytical expressions of the
strain-renormalized Dirac velocities and explain the role of strain in the
emergence of the flat bands. We discuss how strain could correct the twist
angles and bring them closer to the magic angle $\theta_m\sim1.05^{\circ}$ of
TBLG and how it may reduce the widths of the lowest-energy bands at charge
neutrality of the twisted homobilayer of TMDs. The analytical results are
compared with numerical and experimental findings and also with our numerical
calculations based on the continuum model.
