That can't possibly be true. There are only be on the order of 1000 day/night charge/use cycles in the life of a battery and that number is too small for high-cycle fatigue to appear. Low cycle fatigue can happen but only with extreme strains where it's obviously deformed.

Also, if you could make batteries last longer just by using more finely stranded wire, somebody would have done that.

Not only that, but a crack doesn't lower resistance by much because the length of the current path is so short that the narrower cross-section hardly affects resistance over the whole length of a conductor.