  New corrections to General Relativity are considered in the context of
modified $f(R)$ gravity, that satisfy cosmological and local gravity
constraints. The proposed models behave asymptotically as $R-2\Lambda$ at large
curvature and show the vanishing of the cosmological constant at the flat
spacetime limit. The chameleon mechanism and thin shell restrictions for local
systems were analyzed, and bounds on the models were found. The steepness of
the deviation parameter $m$ at late times leads to measurable signal of
scalar-tensor regime in matter perturbations, that allows to detect departures
form the $\Lambda$CDM model. The theoretical results for the evolution of the
weighted growth rate $f\sigma_8(z)$, from the proposed models, were analyzed.
