Pressure increase in a flexible membrane due to an external force.

[sitting_duck]

Homework Statement

An inflatable membrane is filled with an ideal gas at a pressure of 0.3 bar gauge. The gas is at 40 deg C and the volume of the membrane is 100m^3.

The membrane is constrained on all sides and bottom, and a force of F acts on the top of the membrane.

The material the membrane can be considered non-elastic.

What is the increase in pressure inside the membrane due to the external force?

Homework Equations

Pressure=Force/Area

The Attempt at a Solution

I am not looking for a numerical solution, just clarifying the relationship between the external force and the pressure inside the membrane.

The pressure exerted by the external force is Force/Area...however is this the total contact surface of the membrane or the area of the section the force is being applied to?

If the material is to be considered non-elastic then there can't be an increase in pressure as there isn't any decrease in volume inside the membrane...ideal gas law.
However if the material were elastic, what would the relationship be?

I would greatly appreciate any help with this problem.

 

[BvU]

It just says the membrane is inelastic. i.e. the area is fixed. The volume doesn't have to be fixed...

Spoiler

Check if your picture isn't unnecessarily restrictive

 

[sitting_duck]

Is it possible to find a realistic solution, asuming the membrane is inelastic?

This isn't really coursework problem, but part of a research project. Therefore I am looking for the best way to model this.

 

[haruspex]

Is it possible to find a realistic solution, asuming the membrane is inelastic?

Yes. I assume the 0.3 bar is the excess pressure over atmospheric. Without the imposed force F, that pressure is balanced by tension in the membrane. When the force is applied, the membrane may collapse somewhat. The inelasticity means that as soon as the membrane starts to collapse there is no longer any tension in it, so the internal pressure is all that's left to balance F and the external atmosphere.

 

[sitting_duck]

Yes 0.3 bar is the excess pressure over atmospheric. So what you are saying is that the external force plus atmospheric pressure is balancing the internal pressure.

So how do I resolve the external force into a pressure? P=F/A, but which area? The total contact surface area or just the area at the top?

 

[BvU]

Think of one of these jumping cushions they let toddlers jump around on. If still uncertain, work out your relevant equation making an assumption about the area. Who knows, perhaps you are in luck and the area drops out of the calculations to get the pressure !

You know: small feet on small area or the same weight on a 1 m² board. Not so deep a dent in the top surface, but it still might lead to the same pressure increase...

 

[haruspex]

Yes 0.3 bar is the excess pressure over atmospheric. So what you are saying is that the external force plus atmospheric pressure is balancing the internal pressure.

Yes

So how do I resolve the external force into a pressure? P=F/A, but which area? The total contact surface area or just the area at the top?

Think about the interface between the force and the membrane. Given the pressure, what else do you need to know to figure out the force the membrane exerts on the load?

 

[sitting_duck]

If I think of it like a car tyre on the ground. The weight of the car, divided by the number of wheels, subsequently divided by the contact area of the tyres is equal to the pressure of the tyres.

So the pressure inside the membrane is equal to the force divided by the area the force is being applied, i.e the area of the top surface.

 

[haruspex]

If I think of it like a car tyre on the ground. The weight of the car, divided by the number of wheels, subsequently divided by the contact area of the tyres is equal to the pressure of the tyres.

So the pressure inside the membrane is equal to the force divided by the area the force is being applied, i.e the area of the top surface.

Convinces me.