We need pictures. Take a couple from underneath too. Then take a couple of the white substance. Does it come out with pee or poop? Is it creamy or gritty?
How long have you had the tortoise, and do you know its age?
I know it's a lot of questions, but we need info to be able to help you.
If I'm thinking of the right object, yoga mats are fairly soft things. Why would that have been able to hurt the tortoise. Seems like it would be like hitting something with a blanket. I truly doubt your tortoise has been injured by a yoga mat. But the question begs to be asked, "How is it possible for a yoga mat to get inside the tortoise's enclosure in order to hit the tortoise?" You do know, of course, that tortoises belong either outside in their own yard, or inside in they own enclosure.
If this truly is a legitimate question, and you really are worried about your tortoise, then please post pictures of where the tortoise lives, and maybe even the 'white substance' you're seeing.