  This study investigates the occurrence of acoustic topological edge states in
a 2D phononic elastic waveguide due to a phenomenon that is the acoustic
analogue of the quantum valley Hall effect. We show that a topological
transition takes place between two lattices having broken space inversion
symmetry due to the application of a tunable strain field. This condition leads
to the formation of gapless edge states at the domain walls, as further
illustrated by the analysis of the bulk-edge correspondence and of the
associated topological invariants. Although time reversal symmetry is still
intact in these systems, the edge states are topologically protected when
inter-valley mixing is either weak or negligible. Interestingly, topological
edge states can also be triggered at the boundary of a single domain if
boundary conditions are properly selected. We also show that the static
modulation of the strain field allows tuning the response of the material
between the different supported edge states.
