The most common and most likely answer to this crossword clue is the 13 letter word CALCIFICATION
I've seen this crossword clue in The Guardian
C
A
L
C
I
F
I
C
A
T
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Buddy explains!
The straight part
We know that "Hardening" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case hardening = calcification.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does CALCIFICATION mean?
CALCIFICATION noun- The process of change into a stony or calcareous substance by the deposition of lime salt; -- normally, as in the formation of bone and of teeth; abnormally, as in calcareous degeneration of tissue.