I use a weller 80watt, sp80l, with the large tip but I'm on and off the cell in less than a few sec, even if I don't get the job done (rarely) - come back to it later after it cools.

40sec on a cell is not good and most prob did damage the cells. Your prep or something is wrong, possible the lack of flux and or roughing/cleaning the cell terminals. Even grease from fingerprints can mess this up, its gotta be clean to start with!