Hi,
I don't expect to "repair" this as it seems to be an inherent problem, but I have had the Samsung Galaxy S3 for about 3 - 4 years now, and it has always had a Samsung branded (bought from a reputable shop) which came from an older Samsung phone, and around Feb it started saying that the SD card was unreadable and it wasn't mounted, the phone is running 4.1.1 stock ROM, never modded this one.
So, I thought it was the SD Card as it wouldn't read it on a PC etc, so I put in a 32GB Kingston class 10 (I had it for a camera) and it seemed to be OK, anyway, 2 weeks later it started saying the same thing about the SD card being unmounted, this Kingston card had worked fine for the last 6 months, and now it had stopped working, odd, I took it back to the shop and said it stopped working and they replaced it as it was less than a year old, few more weeks went by and the same thing happened again! I started noticing that the battery had now drained where as it would before be on around 75% at the end of the day, it was now on 4%!! I replaced the battery, the SD card still didn't work, so I ignored it, the battery continued to drain to 4% each day for a few days, I took the SD card out and now the battery is down to 75% again at the end of the day.
I am guessing some kind of short in the SD controller is causing this to fry the SD cards and drain the battery? It's not a massive issue i'll just leave the thing without a card it has 16gb internal memory which is plenty anyway, just thought I would share my experience in case anyone else had similar issues!