Geometric effective modeling of proton pressure and shear forces

The spatial distributions of pressure and shear forces inside the proton, encoded in the energy–momentum tensor and its gravitational form factors, have become accessible through lattice-QCD calculations and deeply virtual Compton scattering (DVCS) measurements. While these approaches provide detailed numerical information, closed analytic expressions for the corresponding mechanical profiles remain largely unavailable. In this work, we introduce a geometric effective model for the mechanical structure of the proton that yields analytic expressions for the isotropic pressure p(r) and shear force distribution s(r). The model is constructed at the level of the spatial energymomentum tensor and is designed to be complementary to QCD, without modifying its underlying dynamics. We show that mechanical stability, including the von Laue condition and a negative D–term, emerges naturally within this effective description. The resulting pressure and shear profiles reproduce the characteristic features observed in lattice-QCD calculations and DVCS extractions, including a repulsive central pressure and a surface–peaked shear force, with quantitative agreement at the 5–10% level across the confinement region. The model further provides a natural definition of the mechanical confinement radius from the extremum of the shear distribution. Finally, we discuss phenomenological implications and experimentally testable signatures in DVCS observables and future electron–ion collider measurements. The present approach is not intended as a replacement for QCD, but as a compact analytic parametrization of proton mechanical observables that enables direct comparison with existing experimental and lattice results.