  A basis is derived from elastic plate theory that quantifies equilibrium and
dynamic deformation and electron diffraction patterns of twisted bilayer
graphene (TBG). The basis is derived by solving in-plane and out-of-plane
normal modes of an unforced parallelogram elastic plate. We show that a
combination of only a few basis terms successfully captures the relaxed TBG
structure with and without an underlying substrate computed using atomistic
simulations. The results are validated by comparison with electron diffraction
experiments. A code for extracting the elastic plate basis coefficients from an
experimental electron diffraction image accompanies this paper. TBG dynamics
are also studied by computing the phonon band structure from atomistic
simulations. Low-energy phonons at the $\Gamma$ point are examined in terms of
the mode shape and frequency. These modes are captured by simple elastic plate
models with uniformly distributed springs for inter-layer and substrate
interactions.
