Interaction of Nonlinearities in FEA
At the University I was taught to first check the plastic capacity of the cross-sections. Then I was taught to…24 October 2022
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!
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:
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!.
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!
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:
And now you know, how to calculate true stress and true strain from engineering values. Let’s wonder then… should you care!
I think this is the main question of this lesson. Firstly, let’s look at what we are missing in terms of “accuracy”.
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!
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:
Thank you for reading, and have a wonderful day!
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