The most common and most likely answer to this crossword clue is the 6 letter word OSSIFY
I've seen this crossword clue in The Daily Telegraph
O
S
S
I
F
Y
More clues leading to the result OSSIFY
- East German fellow has year to stop development
- Get tough and cross if you’re eating it
- Toss if you’re around to become rigid
- Whenever controlled by faceless authoritarian, become rigid
- Cross, if Yorkshire holds become fixed
- Make rigid ship provided with extremely oily outer protection
- Leader taken out in defeat, provided ending in victory, gets to toughen up
- Old saints, if meeting heresy, ultimately become rigid
- Rouse stiffly off and on and become rigid
- You may get it from boss if you become inflexible
Buddy explains!

The straight part
We know that "become rigid" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case become rigid = ossify.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 OSSIFY mean?
OSSIFY verb- To transform (or cause to transform) from a softer animal substance into bone; particularly the processes of growth in humans and animals.
- (animate) To become (or cause to become) inflexible and rigid in habits or opinions.
- (inanimate) To grow (or cause to grow) formulaic and permanent.
- To calcify.