Elphick missed Town's warm-weather training trip to Dubai to have a scan on the injury, picked up when he was tackled by Ryan Ledson in Saturday's 3-1 defeat to Preston North End.
Elphick will go under the knife on Monday, at which point it might be possible to put a timescale on his return.
“Gutted is an understatement, especially as I had no control over the situation and it was something so reckless,” he tweeted after the news was announced.
Manager Danny Cowley commented: “He was a big part of this team both as a player and a character.
“He is a resilient and determined player so I have no doubt he will return fitter and stronger than ever.”
Former Brighton and Hove Albion, Bournemouth, Reading and Hull City defender Elphick joined from Aston Villa in the summer. Despite a difficult start to his Terriers career which saw him at fault for the opening goal in Huddersfield's first game of the season, the 32-year-old has started all but two Championship games this season.