  We construct an analytic continuum model to describe the electronic structure
and the electron-phonon interaction in twisted bilayer graphenes with arbitrary
lattice deformation. Starting from the tight-binding model, we derive the
interlayer Hamiltonian in the presence of general lattice displacement, and
obtain a long-wavelength continuum expression for smooth deformation. We show
that the continuum model correctly describes the band structures of the
lattice-relaxed twisted bilayer graphenes. We apply the formula to the phonon
vibration, and derive an explicit expression of the electron-phonon matrix
elements between the moir\'{e} band states and the moir\'{e} phonon modes. By
numerical calculation, we find that the electron-phonon coupling and phonon
mediated electron-electron interaction are significantly enhanced in low twist
angles due to the superlattice hybridization.
