# Elastic energy stored in a balloon

• Diana27
In summary, when I squeeze the balloon, it changes its shape from a sphere to a pancake. The stored elastic energy is a function of the surface area of the balloon.f

#### Diana27

Hi, I have a balloon filled with glass beads (exercise-stress ball). When I squeeze the balloon, it changes its shape. Let's assume the initial shape is a sphere with radius R0 and thickness h0 and the final shape is like a pancake (cylinder R1 + half torus R2 with thickness h1). Can you help me to calculate the elastic energy stored in the balloon once I squeeze the balloon, please. Thanks in advance.

Do you think it will be the same regardless of the strength/elasticity of the balloon material, which you have not mentioned?

Do you think it will be the same regardless of the strength/elasticity of the balloon material, which you have not mentioned?

I do not think it will be the same, as I'm stretching the material. I'm assuming a rubber balloon material.

A first thought was that the energy stored in the material could be a simple function of surface area. However there is an easy counter-example. Consider a spherical rubber balloon enclosing an incompressible fluid such as water. Holding the rest of the balloon stationary, deflect a small patch of the balloon's surface tangentially. This requires an energy input but does not change either surface area of the balloon or enclosed volume.

So a complete solution will require either simplifying assumptions or a careful description of the deflection.

I do not think it will be the same, as I'm stretching the material. I'm assuming a rubber balloon material.
And do you think all rubber has the same elasticity/strength?

And do you think all rubber has the same elasticity/strength?
Yes

Yes
Then you would do well to do a bit more research.

Look, I'm asking you these leading question trying to get you to arrive at the obvious conclusion that your "problem statement" is so ill defined that it's approximately like asking "how high is up?" and expecting a meaningful answer.

Why are you interested in calculating the stored elastic energy in the membrane for this situation? You are talking about a pretty complicated kinematic and material stress analysis problem. Certainly, the details of the loading are going to be important, the packing response of the glass beads within the membrane is going to play a role, and the deformational behavior of the rubber to the kinematics of the local in-plane strain environment is going to be important. Modeling this problem properly would be very challenging, to say the least. The question is, is it really worth the effort?

A first thought was that the energy stored in the material could be a simple function of surface area. However there is an easy counter-example. Consider a spherical rubber balloon enclosing an incompressible fluid such as water. Holding the rest of the balloon stationary, deflect a small patch of the balloon's surface tangentially. This requires an energy input but does not change either surface area of the balloon or enclosed volume.

So a complete solution will require either simplifying assumptions or a careful description of the deflection.

This kind of assumptions are correct?: Why are you interested in calculating the stored elastic energy in the membrane for this situation? You are talking about a pretty complicated kinematic and material stress analysis problem. Certainly, the details of the loading are going to be important, the packing response of the glass beads within the membrane is going to play a role, and the deformational behavior of the rubber to the kinematics of the local in-plane strain environment is going to be important. Modeling this problem properly would be very challenging, to say the least. The question is, is it really worth the effort?

I should say It worth, it's part of a research I'm starting and I want to be sure that I'm handling this problem in a proper way.
Could you help me, how can I start?

Thanks

#### Attachments

This is a very complicated problem that has very little practical applicability. Are you sure that you want to work on this, given that there are many other simpler problems that have practical applicability?

I would start by developing the strain energy equation for the rubber as a function of the three principal stretches. The rubber deformations in your application may be large, so you can't use the small strain approximations to describe the rubber elasticity behavior, and you can't use a one dimensional version because the local deformations are going to be 3D.

The behavior the the glass bead gravel inside the ball is going to be complicated, so you can start out by researching the rheological behavior of non-consolidated granular solids.

Temporarily, before including the glass bead behavior in the model, you should consider assuming there is air inside. At least then the behavior of the material inside the ball would be simple to include. You can switch to granular beads later.

You also need to start formulating the stress equilibrium equations for the rubber cover, treating either as a membrane or a shell. A membrane is, of course easier to solve.

This is a very complicated problem that has very little practical applicability. Are you sure that you want to work on this, given that there are many other simpler problems that have practical applicability?

I would start by developing the strain energy equation for the rubber as a function of the three principal stretches. The rubber deformations in your application may be large, so you can't use the small strain approximations to describe the rubber elasticity behavior, and you can't use a one dimensional version because the local deformations are going to be 3D.

The behavior the the glass bead gravel inside the ball is going to be complicated, so you can start out by researching the rheological behavior of non-consolidated granular solids.

Temporarily, before including the glass bead behavior in the model, you should consider assuming there is air inside. At least then the behavior of the material inside the ball would be simple to include. You can switch to granular beads later.

You also need to start formulating the stress equilibrium equations for the rubber cover, treating either as a membrane or a shell. A membrane is, of course easier to solve.

Thank you very much for your help, this information is helpful to me.