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Define 'brittle' .... without cut and paste ?
More evasion perchance? I'm very glad the the Institute of Mechanical Engineers adopts the very sensible precaution of not awarding CEng status to just anyone with a pajero...
A brittle material is one which does not have any mechanism of crack stopping. Which means, once a crack is initiated, it will continue to grow - unless all stress is removed. A tough material does possess a crack stopping mechanism - in metals, this is because dislocations have mobility in all six degrees of freedom (though admittedly I only add that as I know it will annoy Sazzaa).
Now let's revisit this little gem:
The crack is indicator that the force on the styrene has exceeded its shear strength along that particular stress line . That does not mean that the styrene stops deforming through the rest of it's structure away from that point. The thing about a fall is it is normally associated with a tumble and that means that the lid will also continue to give a benefit with secondary impacts.
Firstly, your chemistry: styrene is a rather gloopy liquid at room temperature. Its polymer, polystyrene is a solid. It's important to know the difference, isn't it? Because it would take a rather gifted muppet to try and make a helmet out of styrene...
Polystyrene is a brittle material. In other words, it behaves in an elastic fashion until its strength (tensile, shear or compressive, depending on the applied stress) is exceeded when it will crack. Chemical bonds are broken in this process, and new surfaces created: both absorb energy. The word "deform" unless qualified, means elastic strain. Elastic deformation absorbs almost no energy whatsoever, so will not provide any protection whatsoever. Your claim that it will is wrong. (Tip: accurate definitions are everything in science.)
The expanded polystyrene used in lids can fail in one of two modes. It can crush, where a lot of very small cracks appear within the bulk of the structure, or it can crack, where a few large cracks propagate thoughout the bulk of the structure. Bulk cracks such as those absorb little energy so do little to mitigate impact forces. The aim of the helmet designer is to ensure that the helmet will crush rather than crack. Worse, failure in brittle materials cannot be predicted accurately. If you don't believe me, try to guess how many peices will a glass shatter into when you drop it? This unpredictability means it is impossible to guarantee that a helmet will provide any more protection once it has failed. The fundamentals of materials science contradict your claims.