MD80 FLT CREW NOTES VIBRATION AT V1 AND SUSPECTS TIRE FAILURE WHICH IS LATER CONFIRMED. FLT CREW CONTINUES TO DESTINATION BEFORE DECLARING AN EMERGENCY AND LANDING UNEVENTFULLY. POST FLIGHT REVEALS DAMAGE TO FLAP.

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APCHING V1 ON RWY ZZZ; WE NOTICED AN UNUSUAL VIBRATION THAT CONTINUED THROUGHOUT THE TKOF AND GEAR RETRACTION. IT WAS ACCOMPANIED BY STRONGER THAN NORMAL SMELL OF RUBBER. WITHIN A MIN; FLT ATTENDANT #4 CALLED FROM THE BACK TO INFORM US OF RUBBER SMELL AND STRONG VIBRATION IN BACK OF THE AIRPLANE. WE HAD NO OTHER UNUSUAL INDICATIONS IN THE COCKPIT AND CONTINUED TOWARDS ZZZ1. WITHIN 15 MINS; CTR CALLED TO INFORM US OF A RELAY FROM TWR ABOUT POTENTIAL TIRE PIECES FROM OUR AIRPLANE LEFT ON THE RWY. THIS NOW CONFIRMED OUR SUSPICION OF A POSSIBLE TIRE FAILURE ON OUR AIRPLANE. WE CONTINUED TO ZZZ1; SINCE THERE WAS NO REASON TO RETURN. WE INFORMED OUR DISPATCHER OF THE CIRCUMSTANCES; OBTAINED A PHONE PATCH TO MAINT AND WENT THROUGH THE ABNORMAL PROC ON 'SUSPECTED TIRE DAMAGE/FAILURE.' APCHING ZZZ1 WE DECLARED AN EMER AND ASKED FOR ARFF EQUIP TO STAND BY. FO WAS FLYING AND TOUCHED DOWN VERY SOFTLY; WHICH IMMEDIATELY RETURNED ABNORMAL VIBRATIONS AND CONFIRMED THAT WE HAD MAIN WHEEL ISSUES. WE PULLED OFF THE RWY AND THE FIRE CHIEF CONFIRMED TIRE TREAD SEPARATION ON L INBOARD MAIN. WE REQUESTED ESCORTED TAXI TO THE GATE. UPON CLOSER LOOK; TIRE FAILURE WAS CONFIRMED; ALONG WITH A SIZABLE HOLE IN THE TRAILING EDGE PORTION OF THE L INBOARD FLAP. IT IS OUR SUSPICION THAT THIS HOLE WAS PUNCTURED ON LNDG IN ZZZ1 DUE TO FURTHER TREAD SEPARATION AND NOT ON TKOF SINCE NOBODY MENTIONED ABNORMAL NOISE OR VIBRATION DURING OUR FLT.