I mean all they did was fight, fight, fight. The amount of armour and weaponry produced for their raids must have been huge. How then is it that we have only found one complete Viking helmet? I find it so hard to believe as there must've been hundreds of thousands if not over a million produced. Why then are archaeological finds from other cultures from that period (the Byzantines, Francians) yet so little from the Vikings?
We have a lot of weapons from the viking age, but indeed very few helmets -- this is a bit surprising, as one would expect well-equipped warriors to invest in a helmet if at all possible, so why don't they survive? There are a few archaeological and social reasons to consider.
Most of our swords, spears, and shields come from persons' graves, or else from water deposits. Both of these archaeological 'contexts' are what archaeologists call structured depositions or 'ritual deposits' (ritual, here, meaning that there is some sort of social 'rite' that happened that left these things behind; the ritual need not be a religious rite per se).
When persons were buried, certain kinds of objects were commonly included in the grave (brooches for women, for example), while other kinds of things rarely were. When you compare cemeteries and settlements, you notice that, for example, only certain kinds of pottery (in England) make it into graves, while others are more common among the living. That is, there's a deliberate choice going into what kinds of things are put in the grave. The grave isn't a complete random sampling of everything people had.
Burials of people with weapons have lots of swords, shields, spears -- but very few helmets (Gjermundbu being the notable exception). This is, incidentally, a pattern that holds true across most of the early middle ages. We have tens of thousands of weapons from graves between c. 400 - 1000, but only a few dozen helmets. Perhaps helmets were exceptionally rare, but it seems more likely that helmets were simply not the sort of thing you used in the burial rite.
Likewise, we find a lot of weapons deposited in water, often lakes or rivers (and often near crossing points in those rivers). These include swords, TONS of spears, axes...but not helmets. Again, these deposits are structured / seem to result from rituals. Like burial, the helmet doesn't seem to have been as important in the ritual.
So...why not helmets, too? This is actually a really hard question to answer, because we're still struggling to explain why weapons of any kind were buried in the grave or cast into watercourses. Scholars used to think that weapons were placed in the grave so warriors could fight with them in valhalla, but there's actually no evidence to support this theory and it's largely fallen out of favor (especially since Christians buried their dead with weapons for hundreds of years, too; Effros 2002). Now, most scholars agree that burial with weapons was a way to show off the social status of the family who was burying the dead -- it showed (literally, in the case of cremation burials) that they had money to burn, and it made the dead person look like the kind of important person that his or her heirs wanted him or her to be remembered as. You buried corpses with weapons so everyone would remember what a great warrior they'd been, and to encourage them to support you -- the heirs -- on the basis of this legacy (Halsall 2010). The problem here, though, is that you'd think that anyone wanting to make the corpse look like a great warrior would include a helmet in the grave. And yet, the number of burials where you find a helmet with the corpse is extremely rare. There are the 'princely graves' in Valsgärde and Uppsala from a century before the Viking Age, which include a lot of helmets. There are also lots of fragments of pre-viking helmets from Gottland, but there the helmets themselves aren't being placed in the grave. If social status is being claimed in the burial, the helmet isn't part of this ritual display.
And something else is missing too, which is mail coats. We have 1 in a grave in early medieval England, and only a handful in Scandinavia. Why not?
My own research focuses on the burial of spears, and what I'm finding is that the burial of weapons with the dead wasn't just about the identity of the corpse, but also about the identity of the weapons buried with the human body. Spears and swords, like the warriors who used them, earned reputations and were (in the early medieval imagination) physically altered by shedding blood (Welton 2016). They actually absorbed some of the properties of the blood they shed into their metal, which could both make them physically and 'magically' stronger and also taint them / make them more prone to violence in the future. Texts talk about weapons with a lot of ambiguity: they are really important, but they also cause a lot of trouble. They have 'agency', that is they have the ability to do stuff on their own, to influence humans to make different kinds of decisions than the people would without interacting with the weapons. Hence, many of the weapons that are put in graves are broken or 'killed' before they are buried -- the burial rite, and the disposal of weapons in water, was a way to stop these weapons from getting out of control (Lund, in Carter et al. 2010)
In contrast to weapons, armor is described differently. In the English poem Beowulf (which was being sung during the Viking Age, but may have been written a century before it started), armor is described as part of the body that wears it (Bazelmans 1999). The poet blurs the lines between helmets and faces: Beowulf and his warriors' faces are 'hard under their helms,' as though the act of putting on a hard helmet can 'harden' your resolve / courage. Beowulf is recognized by his armor, not his body or his face. Grendel's actual skin is armor (Cavell 2014). When Beowulf goes swimming, he doesn't take his armor off. It's just part of who he is. In contrast, weapons in Beowulf are separable from the body -- a sword is stolen from a corpse, and causes no end of trouble. Another sword is loaned (and breaks!). A spear is described as continuing on the path along which it was thrown after its owner had died, a metaphor for how violence is difficult to control once it's set in motion. But armor stays with the body: it's personal, and it doesn't appear to be changed by the blood that its user sheds (unlike weapons, whose properties are tied to the violence acts they commit).
So, I think that there's a very good reason to bury weapons, and also a good reason not to bury helmets and armor. Weapons get tainted by bloodshed, and they need to be replaced every couple of generations so they don't get out of control. If thinking about it as a magical curse helps, I don't think that's entirely inaccurate: killing people made weapons cursed, and it's good to swap out a cursed sword or spear frequently so you don't have all that animus in your life. In contrast, armor takes on the positive qualities of the person who wears it -- Beowulf leaves his mail behind after he dies, and his loyal follower Wiglaf gets it. The armor kept alive the best qualities of the man, and allowed Wiglaf to take up the role of protector that Beowulf had occupied. Weapons are cursed and preserve negative associations with past violence, but armor is blessed with the positive properties of people in the past, and so you keep it around for generations, repairing or reforging it so that this 'magic' could continue to benefit the living.
Sources / for further reading:
Bazelmans, Jos. By Weapons Made Worthy: Lords, Retainers and Their Relationship in Beowulf (Amsterdam University Press, 1999).
Carver, M. O. H., Alexandra Sanmark, and Sarah Semple, eds. Signals of Belief in Early England: Anglo-Saxon Paganism Revisited. Oxford ; Oakville, CT: Oxbow Books, 2010.
Cavell, Megan. “Constructing the Monstrous Body in Beowulf.” Anglo-Saxon England 43 (2014): 155–81. doi:10.1017/S0263675114000064,.
Effros, Bonnie. Caring for Body and Soul: Burial and the Afterlife in the Merovingian World. University Park, Pa: Pennsylvania State University Press, 2002.
Halsall, Guy. Cemeteries and Society in Merovingian Gaul: Selected Studies in History and Archaeology, 1992-2009. Brill’s Series on the Early Middle Ages, v. 18. Leiden ; Boston: Brill, 2010.
Reynolds, A. and Semple, S. 2011. Anglo-Saxon non-funerary weapon depositions, in S. Brookes, S. Harrington and A. Reynolds (eds) Studies in Early Anglo-Saxon Art and Archaeology: Papers in Honour of Martin G. Welch, 40–48, Oxford: Brit. Archaeol. Rep. Brit. Ser., 527.
Welton, Andrew J. “Encounters with Iron: An Archaeometallurgical Reassessment of Early Anglo-Saxon Spearheads and Knives.” Archaeological Journal 173, no. 2 (July 2, 2016): 206–44. doi:10.1080/00665983.2016.1175891.