(Abridged) We explore 7 clusters from LoCuSS at z~0.2 with spectra of 1965 cluster members from the ACReS Hectospec survey covering a region which corresponds to about three virial radii for each cluster. We measure fluxes of five emission lines of cluster members enabling us to unambiguously derive O/H gas metallicities, and also SFRs from extinction corrected Halpha fluxes. We compare our cluster galaxy sample with a field sample of 705 galaxies at similar redshifts observed with Hectospec. We find that star-forming cluster and field galaxies show similar median specific SFRs in a given mass bin, but their O/H values are displaced to higher values at projected radii of R<R200 compared with galaxies at larger radii and in the field. The comparison with metallicity-SFR-mass model predictions with inflowing gas indicates a slow-quenching scenario in which strangulation is initiated when galaxies pass R~R200 by stopping the inflow of gas. The metallicities of cluster members inside R200 are thereby increasing, but their SFRs are hardly affected for a period of time, because these galaxies consume available disk gas. We use the fraction of star-forming cluster galaxies as a function of clustercentric radius compared to predictions from the Millennium simulation to constrain quenching timescales to be 1-2Gyrs. This is consistent with a slow-then-rapid quenching scenario. Slow quenching (strangulation) starts when the gas inflow is stopped when the galaxy passes R200 with a phase in which cluster galaxies are still star-forming, but they show elevated metallicities tracing the ongoing quenching. This phase lasts for 1-2Gyrs, meanwhile the galaxies travel to denser inner regions of the cluster, and is followed by a rapid phase: a rapid complete quenching of star formation due to the increasing ram-pressure towards the cluster center which can also strip the cold gas in massive galaxies.
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