Broad and Finn will miss the second One-Day International with knee and shin injuries while Tim Bresnan may miss part of the series as his wife is expected to give birth in the next week.
Rankin, who has played 37 ODIs and 15 Twenty20 Internationals for Ireland, made his debut for England Lions in 2011 and chose to focus on playing for Warwickshire and England last year.
If selected for England in either of the remaining two matches, in Southampton on Sunday (2nd June) and Trent Bridge on Wednesday (5th June), he would join compatriots and former team-mates Ed Joyce and Eoin Morgan in having represented both England and Ireland.
He also joins Warwickshire team-mates Ian Bell, Jonathan Trott and Chris Woakes in the current squad.
England lost the opening match at Lord's by five wickets.
© Cricket World 2013
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