  We investigate the influence of gauge fields induced by strain on the
supercurrent passing through the graphene-based Josephson junctions. We show in
the presence of a constant pseudomagnetic field ${\bf B}_S$ originated from an
arc-shape elastic deformation, the Josephson current is monotonically enhanced.
This is in contrast with the oscillatory behavior of supercurrent (known as
Fraunhofer pattern) caused by real magnetic fields passing through the
junction. The absence of oscillatory supercurrent originates from the fact that
strain-induced gauge fields have opposite directions at the two valleys due to
the time-reversal symmetry. Subsequently there is no net Aharonov-Bohm effect
due to ${\bf B}_S$ in the current carried by the bound states composed of
electrons and holes from different valleys. On the other hand, when both
magnetic and pseudomagnetic fields are present, Fraunhofer-like oscillations as
function of the real magnetic field flux are found. We find that the Fraunhofer
pattern and in particular its period slightly change by varying the
strain-induced gauge field as well as the geometric aspect ratio of the
junction. Intriguingly, the combination of two kinds of gauge fields results in
two special fingerprint in the local current density profile: (i) strong
localization of the Josephson current density with more intense maximum
amplitudes; (ii) appearance of the inflated vortex cores - finite regions with
almost diminishing Josephson currents - which their sizes increases by
increasing ${\bf B}_S$. These findings reveal unexpected interference
signatures of strain-induced gauge fields in graphene SNS junctions and provide
unique tools for sensitive probing of the pseudomagnetic fields.
