I have a 1990 240 volvo station wagon(125,000 miles). It starts just fine when cold, or when driven for 10 minutes or less. When I turn the key, there is no sound, but lights and radio work well.If I drive for forty minutes or more it often will not start and must be left for at least 20 minutes, often longer. Then it will start on its own. Jumping it sometimes works but not always. I have left it overnight with my
mechanic but he has not been able to replicate the problem. He says that he is unable to diagnose problem if he can not replicate it. I thought that there was a relay that I replaced in a 1985 volvo when I had a similar problem, but no one knows what I am talking about.
You have what they call a “Hot soak restart” problem. This may have developed because the fuel pump is going out and it is not sending a consistent flow of gas to the injector pump. Simply put, the gas is turning into a vapor before it reaches the injector pump, and, to be effective, gas has to stay in liquid form until it has passed the injectors where it is turned into the proper strength of vapor. Hot soak restart acts just like the car has ran out of gas because it basically has. I bet if you have the fuel pump replaced the problem will stop.
Also, you might have a refrigeation or recirculation valve which might be confused with your thought of it being a relay. Trucks have these refrigeration/recirculation valves I haven’t heard of cars having them.
OR… the more I read your message the more I get confused. Is the problem that the car wont even crank? If so, that is pretty simple too. Have the starter replaced. The solenoid spring in the starter has become too elastic to be effective. The hotter the car gets, the more that spring will expand and give dead battery symptoms. I can’t tell which of these two things is your problem. Will the car not crank after you have cranked it too long because the hot soak restart problem? Even if you crank the battery dead, after you have waited for someone to come and jump start you the gas system would have coolled off enough to start the car. You are going to have to determine which of these two it is for your mechanic - unless he actually drives the car as you do and it strands him somewhere.