True Stress – True Strain

When I started my FEA adventure at University I was completely unaware, that true stress and true strain existed. I learned about it accidentally by reading some scientific articles, and I instantly felt bad!

The term “true stress – true strain” suggests that the “engineering values” are somehow false. But this is often not the case. Understanding what’s behind this, will make you feel better!

For years I associated “true stress – true strain” with expertise, and even better FEA software. And I dreamed that one day I will be able to use those values as well. You know, to obtain the “truly true” outcomes.

When I finally understood how this works, it made me smile – I think you will smile as well!

If nor true stress – true strain… then what?!?

I think it is only fair to start… at the start!

It’s easy to see that “true stress – true strain” is something “extra”. And this means that there must be something that is “just normal” in comparison.

The “normal” stress-strain relationship is called engineering stress-strain. That would be the typical outcome of a normal tensile test of a specimen. I admit that I’m always tempted to call engineering values “false” (as opposed to “true” values). But realistically they are not false at all!

The fact that the stress-strain relationship is “engineering” and not “true” comes from a certain simplification. That is a simplification that we do when we do the tensile testing itself. Let’s check what that is!

Firstly, what you do in the tensile test, is measure the force that is required to elongate the sample. And of course, you also measure the elongation of that sample. Note, that the above chart (the outcome of the tensile test) is not a stress-strain relationship. It’s the force-elongation chart.

Of course, it’s super simple to “turn that” into the stress-strain curve, right? All you have to do is to obtain the stress and strain from the test. To do that you need:

• To get the stress: divide the force from the test by the area of the specimen

• To get the strain: divide the elongation by the length of the specimen on which you measure such elongation

Both the cross-section area and the “measurement length” of the specimen are constant. So, you can get the chart that will look exactly the same as the one above but is scaled to show the stress-strain relationship.

If you would do those steps – you would get the “engineering stress-strain” relationship. This is what your software is asking you about, unless it states, that you should provide the “true” stress-strain curve.

Well… in most cases, you won’t even use the entire stress-strain relationship. You will build your material model “around this” instead… but let’s not get ahead of ourselves!.

The truth behind the “true” stress!

I think that you already noticed, that the procedure I’ve just described… seems legit!

I mean, I don’t think I would ever come up with what is wrong here on my own. I’ve learned about this for the first time as a student. I was taught this in steel structure lectures as a “curious note”.

In fact, you could design stuff (even with FEA) for your entire life, not know what a “true” stress-strain is, and you would be just fine!

But let’s check what that is anyway 🙂

As you know from Hooke’s law (and if you don’t please read this!)… stuff “shrinks to the sides” as it elongates! Since our specimen is in tension we can expect such a change in shape:

What this means is, that the cross-section area of our specimen shrinks, as the elongation increases!

Since we are measuring the Force (not the stress!) in the experiment, we need the cross-section area to calculate stress. But we shouldn’t really assume that the cross-section is “constant”. This is because the cross-section will be smaller and smaller as the test load increases. And this leads to “higher and higher stresses” (since you divide by a smaller cross-section).

This nuance is the thing that makes the difference between the “engineering” and “true” stress-strain relationship.

And now the kicker – this is a super easy difference to take into account!

Converting to the truth!

It’s actually possible to calculate the “true” stress-strain using the engineering stress-strain values. This is of course a super cool thing (as measuring everything each moment of the test would be a nightmare). So, let’s calculate the “true values” in 4 simple steps:

• Firstly, we have to assume that the volume of the material remains constant. Simply put, we are not “adding” or “removing” mass from reality! Also, nothing “suspicious” is happening without material as we load it! This means that when I take a product of cross-section and length I get the same volume. Even if I take those values during the test. This assumption works only before necking in the specimen starts. After necking, the volume would have to be calculated differently, as the cross-section of the specimen is no longer constant along the length!

• Secondly, we can make some “simple math” (boy I like simple math… compared to difficult math at least :P). I will use a “T” subscript for “true values” and “ENG” for engineering. Of course, the true stress is the applied load by the cross-section “at that time”, so we can start with:

• Thirdly: We have to realize, that the strain is actually the “current elongation” divided by the “current length”. This means, that our specimen is getting “longer and longer” as we go! This means that the “true” strain is a sum of strains we “gather” as the specimen elongates, like this:

• Finally, we don’t have to measure this every second of each test to deal with the sum above, we can simply use… integral. Yea, I know, things spiraled out of control fast – this happens in math quite often. Actually, there is a decent chance this is the only integral on my entire blog! Of course, we won’t solve it… just google for the solution!

And now you know, how to calculate true stress and true strain from engineering values. Let’s wonder then… should you care!

Is it even worth the hassle?

I think this is the main question of this lesson. Firstly, let’s look at what we are missing in terms of “accuracy”.

Please remember:

Below considerations only work before the necking starts in the sample. The good thing is, that in an actual design you cannot go “past necking” since that is a road with no return! So… all is good!

You already know how to calculate “true” values based on engineering stresses and strains. So let’s assume that we are working with “normal” S235 grade steel. Also, let’s say that we are aiming at 2-3% plastic strain to be the “limit” we are willing to accept. This means, that our model will not have higher strains, since we won’t allow it anyway.

And now the kicker! Let’s test how the outcomes would change if I would take true stress-strain into account. Let’s assume that I’m aiming at a “pessimistic” 2% plastic strain. Of course, for the steel S235, the yield stress is 235MPa. So when we have 2% plastic strain, the stress is 235MPa.

The whole thing looks like this:

I guess I don’t have to convince you that the stresses higher by 2%, and minimally lower plastic strains aren’t worth the effort, right?

Of course, the higher the plastic strain the more “off” it becomes! But then you realize, that the stress will be off by the same percent as the plastic strains you have. And I can easily live with 2-3% of stress error.

Not to mention, that the true strains will get lower than engineering strains too. And frankly, it’s the strains and the equilibrium path that I’m checking not stresses in Nonlinear FEA. So if anything, true strains are helping me a little bit!

The true ending!

This post is based on a portion of content that comes from my Nonlinear FEA Masterclass online course. If you liked it, I’m certain you will like the rest of the course as well. So you know – give it a look 🙂

And, as I always do, a few things worth remembering from what you’ve read here:

• True stress – true strain is not dark magic! While it definitely sounds super complicated, realistically speaking it’s just a way of looking at things you already know anyway.
• In case of low plastic strains… straight ignore this! As you saw in the calculations above, for 2-3% of plastic strains in the model, the impact on outcomes is completely negligible. It will however be more and more important, as you get more and more strains in your model. So don’t forget that!
• It’s always a good idea to pay attention to what your solver is expecting you to input. On the same note, notice as well what values you get out of codes or other sources of data. If this is not clearly explained, I would assume that the “default” would be engineering values.
• It is easy to calculate true values from the engineering ones! Assuming of course that you will ever need that… I must admit that so far, I never was in a position where I had to do that. Perhaps one day I will work with materials that have much higher plastic strains (or I will get more “courageous” with materials I’m using now)… then it may play a role. For now, I’m just happy that I understand it.
• If someone is telling you that your design is “wrong” because you haven’t used “true stress”, don’t panic! Check the maximal strains that you have in your model. It is actually likely in structural design, that the change won’t even impact the design. Heck, it may even “help you” as the strains will be lower!

Thank you for reading, and have a wonderful day!