What is this part called in English?

Hello! I got a banged up iPhone 5S 32 GB silver edition from a customer I do tech support for. She said it was dead. I said it is just a flesh-wound (referring to the Black Knight from Monthy Python. Don't know if she got it). Here is an image of the front

And here is of the back

My question is what is the back called? Not the display-part but where the logic board and stuff are placed. I tried search for "case" and similar key-words but I got result for the phone case, those things you put upon the phone to protect it. My thought is to buy a new or refurbished part and repair it. I have already tested the iPhone with a spare screen I have at home. And it works. Seems "only" it will need a new thing for the logic board and such and a new screen.