The most common and most likely answer to this crossword clue is the 14 letter word RESPONSIBILITY
I've seen this crossword clue in The Times
R
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Buddy explains!

The straight part
We know that "Duty" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case duty = responsibility.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 RESPONSIBILITY mean?
RESPONSIBILITY noun- The state of being responsible, accountable, or answerable.
- The state of being liable, culpable, or responsible for something in particular.
- A duty, obligation or liability for which someone is held accountable.
- The obligation to carry forward an assigned task to a successful conclusion. With responsibility goes authority to direct and take the necessary action to ensure success.
- The obligation for the proper custody, care, and safekeeping of property or funds entrusted to the possession or supervision of an individual.