The drain on a Corian sink can be replaced just like any other sink drain. This is a simple do-it-yourself job. All it takes is a little time, the right parts and a pair of tongue and groove channel-lock pliers. Having a helper on hand will be useful when it comes time to install the flange from the top.
Remove the old drain under the sink. Clear everything out from under the sink and place a large bowl or small bucket under the drain to catch water and anything else that may be in the drain trap.
Loosen the nuts holding the sections of drain pipe together and remove the pipes. If the nuts are especially stubborn, try spraying a lubricant on the threaded joints and letting it work for a couple of hours. That should free them up enough to get them loose with the pliers.
Remove the drain plug assembly. The type with a slot that works the pivot rod can be removed by turning the plug counterclockwise and lifting. The other kind with a complex pivot rod assembly has a spring clip that must be pinched and slid back on the pivot rod. Flex the clevis strap (the perforated strip beneath the pull-up rod) back as you slide the spring clip. Remove the cap that holds the pivot rod to the tailpiece and remove the pop-up and rod. Remove the jamb nut just beneath the opening in the sink with the channel-lock pliers and remove the tailpiece from the flange.
Place a small piece of wood, a bit of scrap 2-by 4-for example, under the flange and hit the board gently with a hammer to loosen the flange. Remove the old flange and carefully scrape the old silicone or plumber's putty from around the drain opening with a plastic putty knife or rubber spackling knife.
Install the new drain and flange. Have a helper on hand to hold the drain in place from underneath while setting and tightening the new flange.
Place a length of plumber's putty on the under side of the flange before setting it. Use plumber's putty rather than silicone sealant on a Corian sink. It seals just as well as silicone and is easier to remove without damaging the sink if the drain or flange needs to be replaced again. Make sure the length of putty goes all the way around the flange for a good seal.
Press the flange in place and tighten it from under the sink. It should be snug, but if you tighten it too much, it will crack the Corian sink. Install the new drain plug assembly before installing the new drain pipe and trap. Wrap the threads of the new drain pipe and trap with Teflon tape or joint compound for a secure leak-free connection.