  In this work, the cross sections of $\phi$ photoproduction are systematically
investigated with the two gluon exchange model and the pomeron model. The
results obtained show that the theoretical values of the two models agree well
with the experimental data. Since the mass radius and mechanical properties of
the proton are encoded in the scalar gravitational form factor of the momentum
energy tensor, based on the differential cross sections of $\phi$
photoproduction at near-threshold predicted by the two gluon exchange model,
the average mass radius of the proton is derived as $\sqrt{\left\langle
r^{2}\right\rangle_\mathrm{m} }=0.78 \pm 0.06$ fm. As a comparison, we directly
extract the proton mass radius from the experimental data of $\phi$
photoproduction to obtain an average value of $0.80\pm0.05$ fm, which is very
close to the result given by the two gluon exchange model. Taking a similar
approach, we extracted the average value of $m_{D}$ to obtain the distribution
of the pressure and shear force contributed within the proton, and compared the
results with other groups. The results of this paper may provide necessary
theoretical information for the subsequent in-depth study of the internal
structure and properties of proton.
