Hi Paulo,
I think I can explain why you're having troubles with JA (short for Jerusalem Artichokes).
You don't say which named variety you planted so may I safely assume those were store-bought tubers? If so they probably come from an icelandic
greenhouse or better from continental europe and so they are of a variety optimized for giving the maximum yield under optimal growing conditions which at the very least means frost-free days until mid November.
You see, JA are really short-day plants which means they usually flower AND tuberize only when they receive less than 12 hours of light (past September, 23 to be precise).
Under long days JAs perform vegetative grow and store sugars into their stems, under short days they translocate such sugars into tubers. Flowering terminates the accumulation of sugars and marks the onset of a senescence syndrome where stored sugars and proteins are dismantled and used to enlarge overwintering tubers.
Note that tuberization and flowering are actually independent processes and tuberization may start before flowering for particularly late flowering varietis: in this case sugars are directly sent to tubers while photosynthesis is still at peak rate!
The most productive JA varieties under temperate climates of
course tend to flower very very late and tuberize under 12hrs so that they can maximize sunlight usage until late fall.
For this precise reasons your JA stand no chances of producing anything outside in Iceland: their aerial parts are killed by frosts before they begin to form any tuber.
However, there is considerable genetic variability in this genus: there are early flowering varieties in full bloom with 14 hours day lengths, two of them growing as weedy clones near my place (Italy, by the way) and there are even dayneutral cultivars which actually flower independent of daylength, depending only on plant development stage.
These flower about 4-5 month earlier than anything else.
Before giving up I suggest you look for dayneutral cultivars: there are actually three of them probably within your reach, they are "stampede" from the US, "bianca" from europe and "dagnøytral" grown in Norway and at my place.
Dagnøytral in particular is the only JA variety I know of that actually produces acceptably sized tubers in Norway. It is worth giving it a try.
I see you strive to reach self sufficiency so maybe you could be interested in a research
project on jerusalem artichokes: my group aims at developing a sub-arctic adapted JA variety.
Anything flowering earlier than the current dagnøytral is acceptable, moreover dwarf varieties stand a better chance of delivering in long days or even continuous light with very short growing seasons since plant maturity may occur relatively fast.
Would you be interested in participating? (if so PM) If you want, I may send you a couple of dagnøytral tubers for testing next growing season, just make sure to bury them so they don't dry and die.
Don't misunderstand my intentions: I'd like to hear if dagnøytral may already deliver in Iceland for a very precise reason. One day I'd like to move there and start my own organic farm without expensive greenhouses for the main crops.
thank you for reading!