To help decrease the likelihood of tearing during a workout, consider;
-Learning how to properly grip a barbell, kettlebell and pull up bar.
-Chalk is not magic. If your hands and the bar are dry, you just want to apply a dusting of chalk, if any. I mean, so little that you look at your hands, and you may not be able to tell that you just put chalk on them. If your hands and/or the barbell contain a lot of moisture, you just want to apply a dusting of chalk to help with your grip. Where many go wrong, is 90 seconds later, you slather on more chalk. At this point, you’re creating the perfect storm for friction, and tears. A little chalk may be beneficial, too much chalk will lead to tearing.
-Use a towel to dry your hands during a workout, not more chalk.
Ideally, you should be proactive with your hand care and a small amount of maintenance can have great benefits. You want your hands to be smooth; removing rough and lumpy skin will decrease the likelihood of tearing your hands. A pumice stone, a PedEgg, or a corn/callus shaver after a shower when your skin is soft will pay great dividends.
Leather hand grips or gloves are another option to consider. These may take some time getting comfortable using, but can help tremendously if used properly.
If you do CrossFit long enough, there will come a time when tears will happen; caring for them properly at this point can save you a lot of time, and pain. First, clean it with soapy water, hydrogen peroxide or iodine. If it’s a small tear, or blister, keep the top layer of skin intact. On the other hand, if it’s a large tear, you will want to remove that torn piece of skin with sterilized scissors. The final step is to keep it covered and moist. You can use some kind bandage, with Vitamin E or sterilizing ointment to help keep it clean and moist. The moisture will keep it from becoming dry, and likely cracking and slowing the healing as you start to use your hand.
I also recommend New-Skin as a good alternative for a tear. It includes anti-bacterial qualities and creates a synthetic skin to help protect the tear, and speed healing.
As with most health issues, the key is to be proactive. Take good care of your hands to limit the tears, and allow youself to continue to train on your own terms.