Raid 1 replace failed disk – partition.

Prepare a new disk/partition. Depending on the partition type it may be necessary to clone the functional disk which can be done with ddrescue.

After the cloning you should replace the UUID of the new disk as it is and exact copy of the original disk.

Add the new partition with mdadm, replace the md0 to the appropriate array and the sda1 to the appropriate partition.
mdadm –manage /dev/md0 –add /dev/sda1

To see the result type:
cat /proc/mdstat