  In this work, we present a theoretical research on the lattice relaxations,
phonon properties, and relaxed electronic structures in magic-angle twisted
bilayer graphene (TBG). We construct a continuum elastic model in order to
study the lattice dynamics of magic-angle TBG, where both in-plane and
out-of-plane lattice displacements are take into account. The fully relaxed
lattice structure calculated using such a model is in quantitative agreement
with experimental measurements. Furthermore, we investigate the phonon
properties in magic-angle TBG using the continuum elastic model, where both the
in-plane and out-of-plane phonon modes are included and treated on equal
footing. We identify different types of moir\'e phonons including in-plane
sliding modes, soft out-of-plane flexural modes, as well as out-of-plane
breathing modes. The latter two types of phonon modes exhibit interesting
monopolar, dipolar, quadrupolar, and octupolar-type out-of-plane vibration
patterns. Additionally, we explore the impact of the relaxed moir\'e
superlattice structure on the electronic band structures of magic-angle TBG
using an effective continuum model, which shows nearly exact agreement with
those calculated using a microscopic atomistic tight-binding approach. Our work
lays foundation for further studies on the electron-phonon coupling effects and
their interplay with $e$-$e$ interactions in magic-angle TBG.
