Search Google Images on anatomy of a book. Some of the images will show the hinges of a book. Where a hardbound book's front or back cover meets the spine is the hinge. It is the pivot on which a cover opens and closes. In a hardcover book, the cloth or other material covering the boards and forming in part the backstrip has a hollow groove look. That's the hinge. Inside of a hardcover book, front and back, half of a sheet of somewhat stiff and strong paper is pasted to the cover and to the inner hinge of the backstrip. This paper is called the front or back endsheet. The half of it thus pasted is called the pastedown endsheet. The other half is called the free endsheet. A book has a cracked or broken hinge when its front or back endsheet has one or more small or large tear or break along where the pastedown and free endsheets meet along the book's front or back hinge. If an endsheet is completely broken apart, then the book's cover is detached. Less commonly, the cover itself is visibly broken, split, cracked, or torn (whatever such term seems most fitting to the describer) along its hinge. For examples, search Google Images on book spine OR cover broken OR split OR cracked OR torn. In my description of such a defect as a cracked hinge, I would include a measurement in inches or centimeters and would show the defect in a photo. Minor instances can be mended (though not like new) with book paste applied with a toothpick.