The underside of my sink in my kitchen is crumbling off onto my cleaning supplies. What do I do to stop that and what is it so I can make sure it doesn't return?
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The white material that is flaking off looks like an insulation/sound deadening coat applied by the sink manufacturer. It wasn't supposed to come off. Unfortunately, I don't think there is a quick & easy solution.
The white material looks friable (it breaks up into dust). I see the white dust on your garbage disposal unit. If the sink is over a couple decades old, have a flake of the material looked at by a lab that deals with asbestos. Wet a paper towel, dab up a sample and drop both the sample and the paper towel into a zip-lock baggy. The lab will just look at the sample with a microscope, so it shouldn't be expensive. Often they will take a look at no charge.
If it contains asbestos have an asbestos abatement contractor remove the sink. Again, don't assume their service is super expensive - abatements are very common.
If the material doesn't contain asbestos, here is a suggestion: Remove everything that is stored under the sink to enable your access. Wear respiratory and eye protection - a good dusk mask and goggles. Scrape the material off the bottom of the sink. It looks like it may come off fairly easy. The tool that comes to mind is a paint scraper. A dust management tip is to wet the white material with a spray bottle before you scrap it off. I would clean up with wet paper towels rather than a broom or a vacuum, just for better dust management.
After the bulk of the insulation is removed you could coat the bottom of the sink with an encapsulant, or even another sound deadening product.
This looks like sound-deadening material (a dense, rubbery paint). If so, its only purpose is to reduce the noise of pots and pans banging in the sink, or to tone down the racket of a garbage disposal. It can safely be removed, covered with a sound-deadening mat, or overcoated with a rubberized paint.
As Jack asks, is there some chemical or abrasion causing the flaking?
See Soundproof Central for more suggestions.