The phrase "become rigid" can be replaced with a number of synonyms depending on the context. One such synonym is "harden," which can refer to a physical material becoming stiff or a person becoming inflexible in their beliefs or behavior. Another synonym is "solidify," which describes the process of something becoming solid or firm. "Stiffen" is a similar synonym that refers to a material or body part becoming stiff or rigid. "Freeze" can also be used to describe something becoming rigid, particularly when referring to temperatures below zero. Additionally, "petrify" is a more dramatic synonym that refers to something becoming completely stiff or rigid.