With a default Linux Mint installation, do the following after connecting the new hard disk to your computer, we will assume the new HDD is /dev/sda:

1. sudo pvcreate /dev/sda
2. sudo vgextend mint-vg /dev/sda
3. sudo lvextend -l +100%FREE /dev/mint-vg/root
4. sudo cryptsetup resize /dev/mapper/mint–vg-root
5. Enjoy!

Source: https://blog.jamiebaldanza.org/2014/01/23/extend-encrypted-lvm-by-adding-new-physical-disk/