I realize it work like this in warband due to food shortage, but I still think fatigue should be included in things that make moral waiver.
Now about attrition, food shortage is not The same as attrition due to climate. I am suggesting that troops suffer attrition or May even desert if exposed to extreme weather for long periods of time. Of course they Will desert you if you have no food, but in medieval times many soldiers died due to cold weather, this would perhaps make lords think twice before starting a campaign or besieging a castle at the start of winter.