What could be the relation between a bicycle's inner tube getting punctured and the pressure of the tube.
Are they directly proportional? I.e, if the pressure of Bicycle tube is higher then it has higher chance of getting punctured.
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It's not going to be directly proportional. Leaving aside the quality of the tyre and what it will be rated for, the shape of the graph is likely to be a U shape:
It's not going to be as simple as that, but might work as a general rule.
The pressure has little effect on flats caused by sharp objects. Tire construction prevails here.
Pressure does have effect a big effect on pinch flats. The more pressure in your tire, the less it will deform when you hit a bump (or the very evil train tracks). If if can't deform enough to pinch the tube, no flat. As the pressure decreases, the amount of force required to pinch also decreases. I'll drop some pressure for rainy wet roads, but I'll be more careful about things that go bump when I do.
That said, don't exceed the manufacture's (both tire and rim) max or min pressure. Worth keeping in mind that 110 lbs put in during the cool morning will be quite a bit higher as the sun beats halfway through a four hour ride.
Conversely, your tube can lose pressure for many reasons as well. Fill it during a flat change with CO2 during a hot day and you may notice it being really low the next morning.
Tire pressure, should be checked prior to each ride.
This is an excerpt from https://mast.queensu.ca/~peter/grade12/MHF4U-1/15.pdf It is a 12th grade textbook explaining how pressure works. "The students have a tendency to think of the tire as a balloon, with the air being pushed out because of the elastic force of the tube as it contracts. But when the tube is imprisoned inside the tire, it doesn’t stretch as it gets filled. If it did, it would be weaker. So it’s not like a balloon at all. In fact it’s better to think of the tire as a rigid structure made of hard plastic with air pressure inside."
Having said that, understand that the reason everyone is talking about the structure of tube instead of the pressure. The Pressure is just there to keep the tube from collapsing. The real answer is that the puncture has a direct correlation to the ability of the tube to retain pressure, but the pressure cannot make the puncture happen or make it worse. The math gets terribly complex for this forum, in my opinion.
The biggest thing to remember is that the pressure is air that has been compressed so to the point where the flimsy tube presses out to the actually inner wall of the tire. The puncture, releases the pressure at a rate directly proportional to the puncture size relative to the pressure currently in the tube, meaning the pressure reduces as the total pressure in the tube reduces. The air inside the tube is trying to equalize with the pressure outside the tube.
I really think the question you are trying to answer is, can higher air pressure increase the risk of a puncture, to which the answer is... sort of but only in that the high pressure, COULD, stretch the tube to the point of weakness, but that only happens if the tube is allowed to expand and the pressure is WAY over the recommended pressure for a tube inside a tire.
In the text, it is talking about a car tire, which is tubeless, so I had to explain that the same principles that protect a car tire from stretching and losing integrity, the tire the tube is in is too rigid for the air pressure you are going to be able to pump into the tube without grossly over-inflating on purpose. The short answer is, that if you over-inflate a tube, outside of the tire to the point it starts to deform, then yes. but if you are just pumping it with a hand pump inside the tube, you are fine.
So while there is truth in what you are thinking, the threshold for over-inflation and the limits of most humans ability to pump a tire that far, there are tons of other parts of that tube that will fail before a puncture, like the valve stem or seam. But having a tube under-inflated causes the same uneven wear on your tread, make it harder to pedal through certain terrain and put more load on you.
The more of the math problems I did for this questions, the more like a scientist I felt and I would hate to make a simple math error and ruin the whole explanation.
If you took a balloon and inflated it until it’s hard, you could very easily pop it with the point of a biro. On the other hand, if you only half inflated the balloon so it's still soft, you could prod It repeatedly and it wouldn't burst. You'd have to really apply force to get that biro into the balloon.
From a scientific basis, giraffes have evolved to have very soft spongy lips with a thick skin so they can chomp berries out of thorny trees. If we carried that fact into a wheel, then a softly inflated tube with a good quality tyre would be best at stopping punctures.
I don't believe a tube inflated to maximum PSI has a magical ability to push out bits of glass and thorns, sounds like nonsense to me, and flies in the face of giraffes lips which have evolved for that purpose!
A softer tyre might get pinch flats, but I've never experienced one ever. I've had many thorns and cuts in my tyres though, and usually always after I've topped up the pressure. Coincedence? Probably not. I like riding higher pressure though.