# Deformation Plasticity. Eulerian strain

According to the manual Eulerian strain e is used for deformation plasticity.

¿Is that minus sign in the Eulerian Strain formula right? (formula 364 manual 2.20)

I have a model in which a simple rectangular bar is stretched from 1m to 1.5m.
According to the formula my Eulerian strain should be :

E Euler= .5/1*[1-.5/2]=.375 m/m

But I’m obtaining CCX RESULT = .625 m/m which maybe is a coincidence but corresponds to ;

E Euler= .5/1*[1+.5/2]=.625 m/m

¿Could some developer check which sign has been implemented to compute the Strains?

Regards

I have checked the Taylor expansion of the logarithmic strain and the formula is right. There is a minus there. I don’t understand that longitudinal Strain exx= .625 m/m ?¿?

inp File attached. It is mainly a uniaxial tensile test with frictionless support to reach a uniaxial Stress state.

UPDATED : Deformation Platicity file.

1. It’s better to use the preformatted text block when pasting whole input files - easier to scroll with that additional slider.

2. You are talking about the deformation plasticity but your model doesn’t use it. Yet ?

3. The formula for Eulerian strain is indeed correct. From a Polish book:

1. Are you sure that what you output corresponds to the Eulerian strain ? According to the documentation it does but only for the deformation plasticity:

• E [TOSTRAIN (real),TOSTRAII (imaginary)]: strain. This is the total Lagrangian strain for (hyper)elastic materials and incremental plasticity and the total Eulerian strain for deformation plasticity.

1. Source-code examination can give some answers. For example, this file seems to be important: https://github.com/Dhondtguido/CalculiX/blob/9b4bb3cc387e873e9075638f56bb3b3f70f23d3c/str2mat.f
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Hi Calc_em and thanks again. I promise I will be quiet for some days after this post. You help me a lot and now I can see what is going on.

I have different materials on my model as I’m doing some tests. Some of them are hyperelastic.
Seems ccx triggers nonlinear analysis even if the materials are not used or even if one is computing over an elastic material. The Strain shown on the GUI is the Lagrangian Strain.

I have isolate the *Deformation Plasticity material model and now my Strain is showing exactly the same result as your formula.

Final Length l=1.5m
Initial Length l0=1m

e= .277 m/m

I have no idea what the formula 364 shown on the ccx 2.20 manual refers too but it doesn’t seem the Eulerian Strain. Anyway the fact that the change in sign matches perfectly with the Lagrangian strain seems a very weird coincidence. ¿Isn’t it?

Initial length L =1m
ﬁnal length L + ΔL = 1.5m

E Euler (-) = 0.375 m/m ¿¿?¿
E Euler (+) = 0.625 m/m ¿¿?¿ (Same as Lagrangian strain for (hyper)elastic materials and incremental plasticity)

I don’t mind at all, I’m always glad to do the research about CalculiX and help others. And you’ve also helped me many times before.

This formula is in the chapter specifically about the deformation plasticity and it’s mentioned that:

For all practical purposes, the Eulerian strain coincides with the logarithmic strain.

Then there’s an explanation that the difference between them is around 1.3% for engineering strain of 20%. Theoretically, this particular material model should output Eulerian strain. Let’s also keep in mind that in results total strain = elastic strain + plastic strain. For comparisons with theory you may need to extract the elastic part.

This diagram showing the differences between various strain measures for uniaxial tension (taken from the aforementioned Polish book written by I. Kreja) might be interesting for this discussion:

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Hi Calc_em and thanks for that phenomenal graph. So, much clear now.
I have been able to get exactly those same values on a ccx model and compare against hand formula.
Hope this will work the same with shear strains.

Regarding Manual formula 363 ccx 2.20:
E_logarithmic = LN [ 1+ ΔL/L] Is expanded up to the ﬁrst three terms of a Taylor series around ΔL/L=0

LN [ 1+ X]= X – X^2/2 + X^3/3 +… (manual takes just the first two terms not three as it says.)
E_log ≈ ΔL/L – ΔL/L ^2/2 + …= ΔL/L [1- ΔL/L/2] which is the formula shown on the manual.

-If I use two terms = .5/1*[1-.5/2]= .375 m/m (Do not agree with the result)
-If I use three terms = .5 – .5^2/2 + .5^3/3 = .4166 m/m (Exact value should is E_logarithmic = 0.405 m/m)

What manual calls Eulerian Strain is just the second order approximation of the logarithmic Strain.
They may be similar for small strains, but at large stretches one can see there is a big difference. I don’t think that should be called Eulerian Strain.

Regards

you have to take into account that despite using same symbols, Eulerian strain and Lagrangian strain aren’t calculated using same quantities: for Lagrangian you use material coordinates as you did but for Eulerian you have to use spatial coordinates hence exx=(0.5/1.5)x(1-0.5/(2x1.5))=0.2778

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I understand from this post that the file you shared it’s not the one using *deformation plasticity

Wait a minute ?¿?¿. I thought it was clear to me. jajaja.

aah, ok. that’s just a “coincidence”. Eulerian can be computed as Logarithmic but using the final length instead of initial length. That’s a nice curiosity but I don’t think it is convinient mixing or relate with formula 363 on ccx 2.20 manual as it clearly states:

“For a tensile test specimen, with initial length L …”

Naming that second order approximation of the logarithmic strain as Eulerian Strain could be confusing and might lead to errors. I would add a 0 subscript to the formula in the manual to differentiate.

I will fix the posted file with the deformation version.

The formulas from that book are as follows:

logarithmic (natural) strain is different from Euler strain.

Euler is measured with respect to the deformed state as you see in formulas posted by Calc_em.

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Here’s a complete set of formulas related to that plot, for full clarity:

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Yes , I completely agree. That’s why I suggest not to mix them in the manual. As it is written now, it seem like the Eulerian Strain is a second degree approximation of the Logarithmic Strain and it is not.

Note: Logarithmic and natural strain is not the same. Just learn it from Calc_em graph. I think you mean logarithmic (true). I can belive how good is that graph. If I had to deduce that from the many references out there I could spent a month on still don’t have thinks clear. Thanks again Calc.

it depends on your reference, maybe it’s not a consolidated name…Green-Lagrange strain is also known as Saint Venant strain…https://ntrs.nasa.gov/api/citations/20050180483/downloads/20050180483.pdf

Hi,
the formula for the Eulerian strain in the manual is unfortunately wrong, the second term should have -3/2 instead of -1/2. So the Eulerian strain is not as close to the logarithmic strain as I had hoped for.
I took this opportunity to change the formulation in CalculiX for deformation plasticity from Eulerian strain to logarithmic strain, as in Abaqus. Will be available in the next CalculiX version (summer 2023).

Best Greetings,

Guido

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