We investigate the origin of fast shear strength healing induced by mechanical perturbations during slide-release-slide (SRS) experiments using a ring shear apparatus (ACSA, Navier/CERMES, Ecole des Ponts ParisTech, France). A 100-mm-thick annular sample of siliceous sand (0.6 mm mean diameter) is submitted to shear by the mean of a rotating cylinder in a semi-Couette geometry. We explore the role of shear stress perturbations related to small reverse offsets of the loading interface. We show that controlled releases of the shear stress induce shear strength increases when resuming shear load (i.e. the Tightening-up effect of unloading or Tu-effect). However, a threshold of the shear stress perturbation amplitude to get a significant restrengthening is observed. The shear strength increase is shown to be logarithmically related to the amount of imposed reverse offset and linearly to the induced volumetric strain. These results suggest that small perturbations of the contact status (i.e. inelastic strain) in the granular assembly of the gouge interface, have a major influence on the fault restrengthening.