She told him she'd spent all her savings but he DIDN'T BAT AN EYELID.
NOT BAT AN EYELID: To show no sign of surprise or worry when something unexpected happens.
These discussions may well OPEN THE DOOR TO a peaceful solution.
OPEN THE DOOR TO SOMETHING: To make something possible.
When she started borrowing my clothes without asking, I had to PUT MY FOOT DOWN.
PUT YOUR FOOT DOWN: To use your authority to stop something happening.