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If the original anchor shields have failed you'll need to install new anchors like the ones above. The anchors must be set into holes drilled in the underside of the Corian. You need to take great care that the holes stop at least 1/4" before coming through the top of the counter. You can buy a depth collar that attaches to your drill bit to limit the depth of the hole.
I would also put a dab of epoxy into each hole before inserting the anchor shield. When inserting the shield screw the bolt into the shield all the way but first put some grease on the bolt threads so the epoxy doesn't bond to the bolt.
You will need at least 2 of the kits linked since there should be at least 2 clips for each side of the sink. When you're ready to set the sink do a dry run to make sure your bolts aren't too long. Once your sure the bolts are the right length run a bead of clear silicone on the top of the sink flange where it will bond to the underside of the counter top. Be sure to clean off any old caulking first
If you check the spec sheet on the web page I linked, note that the clips are pictured installed upside down. The open side of the broad U shaped clip goes towards the sink