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.
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