To make sore or irritated, especially by friction or rubbing; to chafe.
From Old French en- (to cause to be) + chafe (to rub raw). The word entered English in the 16th century and combines the prefix en- with chafe, meaning to warm by rubbing or to cause soreness.
This is a word Shakespeare's characters might have used to complain about tight armor or uncomfortable clothing—it shows how older English was more specific about different types of discomfort, where 'enchafe' meant specifically the pain from rubbing, not just any irritation.
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