  We investigate experimentally the mechanical response of a monolayer of
bi-disperse frictional grains to an inhomogeneous shear perturbation across the
jamming transition. We inflate an intruder inside the packing and use
photo-elasticity and tracking techniques to measure the induced shear strain
and stresses at the grain scale. We quantify experimentally the constitutive
relations for strain amplitudes as low as 0.001 and for a range of packing
fractions within 2% variation around the jamming transition. At the transition
strong nonlinear effects set in : both the shear modulus and the dilatancy
shear-soften at small strain until a critical strain is reached where effective
linearity is recovered. The dependencies of the critical strain and the
associated critical stresses on the distance from jamming are extracted via
scaling analysis. We check that the constitutive laws, when applied to the
equations governing mechanical equilibrium, lead to the observed stress and
strain profiles. These profiles exhibit a spatial crossover between an
effective linear regime close to the inflater and the truly nonlinear regime
away from it. The crossover length diverges at the jamming transition.
