We do not, I think, disagree on the physics (in any event, this stuff is much closer to your day job than mine, I last thought seriously about crack propagation in a final year undergraduate module thirty years ago).
My contention is quite limited: it's simply that a crack or split in a helmet is not necessarily evidence that it's failed.
There are plenty of scenarios where a crack would indicate a failure or degradation of performance. But there are also plenty of scenarios where a crack does not allow you to deduce anything about its performance.
I said at the start of this exchange that the test for a helmet having done something useful is the presence of crushing, not the absence of a crack (or conversely, failure is indicated by the absence of crushing, not the presence of a crack). I broadly stand by that. In particular circumstances, it may be possible to deduce from the location of the crack that the ability of the helmet to absorb energy by crushing was impaired. But in general, to say, as has been done on this thread, "it cracked/split so it can't have done any good" is surely unhelpful. "It didn't crush so it can't have done much good" is usually also applicable and much more helpful.
I realise I am fast becoming obsessive on this narrow point and apologise to everyone who doesn't share
@McWobble's and my interest in material properties.
Well, given the materials I look at in my job (gels and structured liquids), it would be somewhat unusual to say the least to see cracks appear in them...
You are thinking too much about the crack and too little on the reasons - or more correctly, the
forces behind the crack. A crack is usually the result of a shear or tensile stress. In circumstances where the impact vector is normal to the helmet surface, and it is supported underneath the contact point by an unyielding object with a greater Young's modulus and yield strength than the helmet material, the only force will be a compressive one. This, however, is unrealistic.
This assumes that the helmet is a uniform spherical shell. Most helmets have vents, and all have attachment points for straps. These will introduce shear stresses and tensile stresses at the strap attachments. In reality, a helmet will not fit perfectly. This inevitably introduces areas where the helmet structure is unsupported from below, and thus will experience bending moments, which appear as shear. Of more importance, the majority of impacts are
not normal to the helmet surface: there will be a lateral component which acts along the helmet surface. A shear stress, in other words. It is not sufficient for a helmet to crush progressively under a compressive load: it must also be able to cope with the shear (and tensile) stresses set up by a typical impact (note that the usual tests such as the weight being dropped onto a helmet on an anvil
do not test for shear strength). Cracking indicates that the helmet was subject to shear forces that were higher than it was able to cope with.
The question, of course, is whether this failure matters. I would argue that your critreia for success - the appearance of crushing - is insufficiently rigorous. As I pointed out in my last post, cracking is indicative that the helmet's ability to reduce impact acceleration was compromised. It is not difficult to conceive a scenario where helmet failure in shear reduced the amount of compression, and thus increased the impact forces experienced by the wearer. Such a scenario cannot be considered a success, especially given just how marginal the protection a helmet offers actually is. You need proper fracture analysis, backed up with finite element modelling to determine exactly what effect on the impact forces experienced by the wearer are (not trivial, this!) but it is a vey useful indication that protection was compromised. What is impossible to say is just how much that protection was reduced, not by photographs or even careful measurements.
One of the comments I forgot to add in my last post (and decided not to edit it in afterwards, as it was already far too long and boring [1]) was about the role of the shell. The foam cracking is mitigated if the shell is sufficiently strong in shear to hold the fragments together. The shell needs to be both stiff, strong and be resistant to crack initiation and propagation (a property engineers call toughness. Metals possess it, glass does not.). The polycarbonate shell is inadequate in all these attributes. A thin aluminium shell would perform significantly better, and furthermore would itself deform plastically in impact. The plastic deformation of a metal is akin to it behaving as a viscous fluid and is very effective at dissipating energy - that's how the crumple zones in a car work.
[1] We've had insults, circular arguments but I don't think anyone has gone for the out-and-out bore-them-to-death approach. Well, until now.
Sorry.
