My experience, both as a carpenter and as a crucifixion enthusiast, is that the wood pieces are a figment of someone's imagination. Small pieces of wood like that depicted would have to have a hole drilled through them first, or else, yes, they will split, no matter what you do to the nail. There is absolutely no known evidence that these pieces of wood were ever used for crucifixions, and no basis in physics for them to behave as depicted in the photo.
Yehohnan's ankles were nailed to the cross (one of several variations that the Romans would have used, depending on circumstances and desires), and there were microscopic slivers of wood detected on the nails, but the consensus is far from unanimous as to whether the wood came from the cross itself, or a piece of wood between the ankle and the nailhead. All anyone has agreed to is that the wood slivers were from an olive tree, too soft of a wood to generally be used in a crucifixion. Yehohnan, it seems, was crucified during the Roman conquest of Jerusalem, when Josephus tells us that they denuded the entire surrounding area of trees to make thousands of crosses. In such a circumstance, any tree available would have been pressed into service, and that seems to have been the case here.
To be fair, such a piece of wood COULD have been used to prevent the victim wriggling off the nail head, but the nails I have seen seem to have adequately large heads as it is. I feel that the use of pieces of wood like this is largely fiction.