Equation Alternatives for Young's Modulus

[LCSphysicist]

Homework Statement

I will post below

Relevant Equations

There is no

I just found a definition to the Young modulus as:

Is this a plausible representation of Y? That is, i know the definition

, i don't think we can say this definition and the first definition is equal.

 

[haruspex]

Homework Statement:: I will post below
Relevant Equations:: There is no

I just found a definition to the Young modulus as:
View attachment 268280
Is this a plausible representation of Y? That is, i know the definition View attachment 268282, i don't think we can say this definition and the first definition is equal.

The second looks invalid in that it should be $\frac{dF}{A}=\frac{YdL}L$. That would make them identical.
Alternatively, maybe they mean $\frac{\Delta F}{A}=\frac{Y\Delta L}L$, which is not exactly the same.

 

[etotheipi]

The second looks invalid in that it should be $\frac{dF}{A}=\frac{YdL}L$. That would make them identical.
Alternatively, maybe they mean $\frac{\Delta F}{A}=\frac{Y\Delta L}L$, which is not exactly the same.

The simple definition of the Young's modulus, $E$, for axial tension that I am familiar with is$$E = \frac{\sigma}{\varepsilon} = \frac{T}{A} \frac{L_0}{\Delta L} \implies \frac{T}{A} = \frac{E \Delta L}{L_0}$$where $\Delta L = L - L_0$. Taking the differentials of both sides, holding $A$ constant, would give$$\frac{dT}{A} = \frac{E}{L_0} d(L - L_0) = \frac{EdL}{L_0}$$which is your first equation. So I think they were just being sloppy using a differential $dL$ when they should have used $\Delta L$

 

[haruspex]

The simple definition of the Young's modulus, $E$, for axial tension that I am familiar with is$$E = \frac{\sigma}{\varepsilon} = \frac{T}{A} \frac{L_0}{\Delta L} \implies \frac{T}{A} = \frac{E \Delta L}{L_0}$$where $\Delta L = L - L_0$. Taking the differentials of both sides, holding $A$ constant, would give$$\frac{dT}{A} = \frac{E}{L_0} d(L - L_0) = \frac{EdL}{L_0}$$which is your first equation. So I think they were just being sloppy using a differential $dL$ when they should have used $\Delta L$

But the continuous and discrete forms are different in principle, no? It could be that Y varies somewhat with strain.

 

[etotheipi]

You're probably right, I really don't know very much about this ! To me it seems like the equations in the OP should be $Y = \frac{L_0}{A} \frac{\partial F}{\partial L}$ and $\frac{F}{A} = \frac{E\Delta L}{L_0}$. I hadn't thought about having a variable Young's modulus. I'm tempted to tag @Chestermiller to see what he thinks!

 

[Chestermiller]

If u(x) is the displacement at material location x along a rod under axial loading F, in terms of Young's modulus, $$F=AY\frac{du}{d x}$$This applies if Y varies with x. If Y is a constant, so that the deformation is homogeneous and u is zero at x =0, then, at x = L, $u(L)=\Delta L$, and the equation integrates to $$F=AY\frac{\Delta L}{L}$$

 

[etotheipi]

I just found a definition to the Young modulus as:

I wondered, do you have a reference for this @LCSphysicist? I thought about it for a little longer but I couldn't make sense of it. But maybe context would help?

 

[LCSphysicist]

I wondered, do you have a reference for this @LCSphysicist? I thought about it for a little longer but I couldn't make sense of it. But maybe context would help?

Oh, yes i have :) i will post a example using it, if it helps:

Imagine a wire, stuck on both ends, in such way that it length remains constant, the initial tension on the wire is F. If we cooled the wire, how much increase the tension?

Ans:

We can suppose the tension is function of the temperature and length, so
F = f(T,L)

(1)
But we know that:

and

not only, we know too:

So

Now, returning to the differential equation (1)

would be the answer, if we suppose A, α and Y approximately constant.

At least to me, if Y satisfy the condition, it makes sense.

Reviewing now, maybe Y was adjust only to this type of question. Anyway, the doubt arise if we can generalize Y as

or not.

 

[etotheipi]

Oh, yes i have :) i will post a example using it, if it helps:

If you don't mind me asking, which textbook is this? Your example makes use of the fact in question but it would be interesting to see where it comes from.

 

[LCSphysicist]

If you don't mind me asking, which textbook is this? Your example makes use of the fact in question but it would be interesting to see where it comes from.

Finn's thermal physics :)
Read this book, i am really liking it, as he construct the things by a logic sequence.

 

[etotheipi]

Okay, I think this is it. There are two different definitions of strain, true strain $\tilde{\varepsilon}$ defined by $d\tilde{\varepsilon} = \frac{dL}{L} \implies \tilde{\varepsilon} = \ln{(1+\frac{\Delta L}{L})} = \ln{(1+\varepsilon)}$, and also the engineering strain, which is defined by the more familiar relation $d\varepsilon = \frac{dL}{L_0} \implies \varepsilon = \frac{\Delta L}{L_0} = \frac{L-L_0}{L_0}$. The engineering strain is much more common, and is used in the definition of the Young modulus $\sigma = Y\varepsilon$. But for infinitesimal strains, $\tilde{\varepsilon} \approx \varepsilon$ and the Young modulus is$$Y = \underbrace{\frac{L_0}{A} \left( \frac{\partial F}{\partial L} \right)_T}_{\text{w/ engineering strain}} \approx \underbrace{\frac{L}{A} \left( \frac{\partial F}{\partial L} \right)_T}_{\text{w/ true strain}}$$So I believe the equation quoted in Finn is valid as an approximation in the limit of small strains.

 

[Chestermiller]

Okay, I think this is it. There are two different definitions of strain, true strain $\tilde{\varepsilon}$ defined by $d\tilde{\varepsilon} = \frac{dL}{L} \implies \tilde{\varepsilon} = \ln{(1+\frac{\Delta L}{L})} = \ln{(1+\varepsilon)}$, and also the engineering strain, which is defined by the more familiar relation $d\varepsilon = \frac{dL}{L_0} \implies \varepsilon = \frac{\Delta L}{L_0} = \frac{L-L_0}{L_0}$. The engineering strain is much more common, and is used in the definition of the Young modulus $\sigma = Y\varepsilon$. But for infinitesimal strains, $\tilde{\varepsilon} \approx \varepsilon$ and the Young modulus is$$Y = \underbrace{\frac{L_0}{A} \left( \frac{\partial F}{\partial L} \right)_T}_{\text{w/ engineering strain}} \approx \underbrace{\frac{L}{A} \left( \frac{\partial F}{\partial L} \right)_T}_{\text{w/ true strain}}$$So I believe the equation quoted in Finn is valid as an approximation in the limit of small strains.

That's certainly true, since at large strains, Young's modulus is not constant for any strain measure, and does not apply in large 3D deformations.

 

[mjc123]

The difference between OP's two definitions is the difference between the tangent modulus and the secant modulus. The former is the slope of the tangent to the stress-strain curve at a given point; the latter is the slope of the line joining that point to the origin. That is, the former is the differential increase of stress with strain, the latter is total stress/total strain. The two are different if the stress-strain curve is non-linear. See diagram.

 

[haruspex]

The difference between OP's two definitions is the difference between the tangent modulus and the secant modulus

AS I wrote in post #2, that is possibly the difference in intent, but the second as written is not valid. Which modulus is intended depends on how one corrects it.

 

[etotheipi]

AS I wrote in post #2, that is possibly the difference in intent, but the second as written is not valid. Which modulus is intended depends on how one corrects it.

Agreed. For that second equation, there are a few viable alternatives. Equation of differentials with the two types of strain, $$\frac{dF}{A} = \frac{Y dL}{L_0}$$ $$\frac{dF}{A} \approx \frac{Y dL}{L}$$ Equation of non-infinitesimals with two types of strain $$\frac{F}{A} = \frac{Y\Delta L}{L_0}$$ $$\frac{F}{A} \approx Y \ln{ \left( 1+\frac{\Delta L}{L_0} \right)}$$I don't know what was intended, but you're right that at the moment the second equation is incorrect.