The claw on a dog, at least, have a soft pulp centre with the hard outer encapsulating it. The hard exterior is much like human nail, I suppose (I'm not an expert, but familiar with dogs).

The first link below gives an explanation about how to correctly trim claws, with a diagam of the claw:

"Det gör ont på hundar men när klon (eller klokapsel som det egentligen heter) väl är av brukar det vara ok däremot så finns det infektionsrisk om det är öppet in i pulpan."

So it seems refers to the hard outer around the pulp. They can (relatively) easily be broken in vigourous activity, so I could imagine this being relevant in your context.

I did find this referred to as the 'nail shell' in English - see the second link. I suspect just calling it 'nail' and 'claw' is common too and would be clear enough in context.

I would like to be able to verify that the construction of the claw is the same in rodents, to cap off this suggestion .. But I hope this is helpful without!