What kind of material of PCB should I choose to work at -40 degree centigrade?

I've used FR4 at 4K and others have used it at much lower temperatures.

The physical characteristics go somewhat downhill at low temperatures, but board failure such as delamination does not normally occur from mere exposure to cold temperatures. The Charpy tests referred to in your linked answer are a measure of strength of a notched specimen to shock (add a stress riser then whack it with a hammer, basically). If you are in a severe mechanical environment you may have to consider the lower impact strength and use a thicker board or support it better.

Solder joint failure due to differences in coefficient of thermal expansion can be a factor, especially with lead-free solder and things like large BGA packages.

-40°C is just a nippy day in some parts of the world, and -55°C is the lower end of the military temperature range, both limits are within the normal range of epoxy-glass boards, and there are plenty of reasonably-priced components available that have guaranteed specifications at those temperatures (especially -40).

You can use a ceramic substrate: see e.g. here. Of course it will be expensive, and may require special tooling, and I should say that have no personal experience with these. But it is the usual solution for circuits that must operate at extremes of temperature.

If it is not absolutely ruled out by the application, you might also consider using a heated enclosure. This will be less costly and will allow you to use conventional materials and techniques.