Calculating deformation energy

[UVW]

Hi all,

I've been trying to understand how to calculate the elastic energy of a deformed object. For example, if I have a box, and I push on it and it deforms, how can I calculate this deformation energy?

I don't know much about elasticity, but I have read a little about strain energy and find the following equation promising:
http://en.wikipedia.org/wiki/Strain_energy

However, I keep coming across different types of strain tensors, and principle strain, and shear strain, and all kinds of things, and I just don't know how to put it together. I would imagine that you simply take a lot of dot products between forces and displacements, but I don't know how to take care of this with these tensors and such.

So for the box example, could you help me understand how I could go about calculating this energy? Thanks!

 

[Andrew Mason]

Hi all,

I've been trying to understand how to calculate the elastic energy of a deformed object. For example, if I have a box, and I push on it and it deforms, how can I calculate this deformation energy?

I don't know much about elasticity, but I have read a little about strain energy and find the following equation promising:
http://en.wikipedia.org/wiki/Strain_energy

However, I keep coming across different types of strain tensors, and principle strain, and shear strain, and all kinds of things, and I just don't know how to put it together. I would imagine that you simply take a lot of dot products between forces and displacements, but I don't know how to take care of this with these tensors and such.

So for the box example, could you help me understand how I could go about calculating this energy? Thanks!

I don't think there is any simple way to determine a formula based on first principles in order to calculate the deformation energy of a body.

Generally speaking, the energy used in deforming the body is converted to heat. The internal energy of the deformed body is the same as the undeformed body. So the work done ends up as heat (using the first law of thermodynamics: Q = ΔU + W → Q = W if ΔU=0).

The easiest way to measure the energy of deformation (ie. the work done in achieving deformation) might be to measure the heat flow.

AM

 

[AlephZero]

Generally speaking, the energy used in deforming the body is converted to heat. The internal energy of the deformed body is the same as the undeformed body. So the work done ends up as heat (using the first law of thermodynamics: Q = ΔU + W → Q = W if ΔU=0).

A thermodynamicist might want to count it all as heat energy, but a mech engineer would prefer to work with stresses and strains.

There are two basic ways to find the elastic internal energy. One is to find the work done to deform the body, which is the OP's

you simply take a lot of dot products between forces and displacements.

That doesn't involve any tensors - forces and displacements are vectors.

If the material behavior is linear (i.e. there is no permanent plastic deformation, etc), the other way is to integrate the strain energy density over the volume of the body. The SED is either $\frac 1 2 \sigma_{ij}\epsilon_{ij}$ in tensor notation, or $\frac 1 2 \sigma^T \epsilon$ writing stresses and strains the engineer's way, as vectors of length 6. That fact that "engineers' shear strain" is twice "tensor shear strain" conveniently makes the two formulas give the same answer. (The engineer's formula includes each shear strain once, but the tensor formula includes the two equal off-diagonal terms separately).

However, I keep coming across different types of strain tensors, and principle strain, and shear strain, and all kinds of things.

If you ask a specific question about what is confusing you, you will probably get some help here, but that statement is too vague to answer apart from the general advice to find one good source of teaching material on continuum mechanics and work through it systematically, rather than jumping around from one website to another.

 

[UVW]

Actually, this is specific enough for me already! Thanks very much to both of you!

 

[PJC01]

Hello,

Calculating deformation energy can be a complex process, but there are a few key concepts that can help you understand the basics. First, let's define what we mean by deformation energy. Deformation energy is the energy required to deform an object from its original shape to its deformed shape. This energy is stored in the object as potential energy and is released when the object returns to its original shape.

To calculate this energy, we need to consider two main factors: the force applied to the object and the displacement of the object. In the case of your box example, the force would be the force you apply to the box to deform it, and the displacement would be the amount the box moves or deforms from its original shape.

To calculate the deformation energy, we can use the equation for strain energy that you mentioned in your post. This equation takes into account the force and displacement, as well as the elastic modulus of the material, which is a measure of its stiffness. This equation can be applied to different types of strain, including shear strain and principle strain, as long as we have the necessary values for each component.

In the case of a box, we would need to know the force applied to the box, the amount of displacement or deformation, and the elastic modulus of the material the box is made of. This information can then be plugged into the strain energy equation to calculate the deformation energy.

It's important to note that this is a simplified explanation and that there may be other factors to consider depending on the specific situation. Additionally, there are more advanced methods for calculating deformation energy, such as using finite element analysis or stress-strain curves. It may be helpful to consult with a materials science or engineering expert for a more in-depth understanding.

I hope this helps clarify the process of calculating deformation energy. Keep exploring and learning, and don't hesitate to seek out additional resources and guidance. Good luck!