  We show that, as it happens in graphene, elastic deformations couple to the
electronic degrees of freedom as pseudo gauge fields in Weyl semimetals. We
derive the form of the elastic gauge fields in a tight-binding model hosting
Weyl nodes and see that this vector electron-phonon coupling is chiral,
providing an example of axial gauge fields in three dimensions. As an example
of the new response functions that arise associated to these elastic gauge
fields, we derive a non-zero phonon Hall viscosity for the neutral system at
zero temperature. The axial nature of the fields provides a test of the chiral
anomaly in high energy with three axial vector couplings.
