Use of True Stress/True Strain 2

[prost]

I recently read the thread
Nonlinear FEA with Von Misses Plasticity in 17-4 PH900 Stainless Steel

and have a question about the original poster's use of the 'true stress' vs. 'true strain'. Why are true stress and true strain used instead of engineering stress and engineering strain? If the FE software really uses true stress and true strain, then what is the constitutive relation between this true stress and true strain tensors?

Is this normal practice, for the FE software to demand input of true stress and true strain as the material curve?

 

[gurmeet2003]

prost,

True stress and true strain are required only for a non-linear analysis. When the strains become large the engineering stress-strain does not represent the physics properly.

The assumption in the engineering stress/strain is that the orignial cross-sectional area does not change much( only true for small strain mostly in the elastic zone). Once yielding occurs the strain and stress values have to be based on the instantanious area of crossection and the length.

Gurmeet

 

[rb1957]

and so you'd need material stress/strain based on true stress/strain ... whereas most data are presented in terms of engineering stress/strain. but that's a simple enough translation, no?

 

[prost]

The translation is what bothers me. You compute true stress and true strain using a specious assumption, that your relationship between engineering stress and true stress is known--you are assuming the coupon necks down in a well defined way, but in reality, that assumption degrades quickly as the stress/strain curve continues to bend over. the primary assumption is the volume remains the same as you deform it; in an actual test, you can't get the gage section to deform this way (that is, the gage section necks non uniformly).

How are true stress/true strain related to the classical continuum mechanics stress/strain measures? Stress/strain measures are complementary, based on a common coordinate system. You have material coordinates and spatial coordinate systems; you would pair Cauchy stress with Eulerian (AKA Almansi) strain, and 2nd Piola Kirchoff Stress with Lagrange (AKA Green) strain.

Ok, that's one problem I have with the use of the true stress/true strain, because I don't really know what they are as they are defined. Secondly, what is the constitutive relationship you are using to relate true stress to true strain? I am familiar with linear elasticity, and with Ramberg Osgood (power law), but both of those were derived with engineering stress and strain. OK, I will accept that you could compute coefficients to fit a power law in true stress to a true strain, akin to the way Ramberg Osgood is used to fit engineering stress and strain. But why would you use true stress and true strain instead of engineering stress and engineering strain? What's the advantage of using true stress/strain over using engineering stress/strain in the power law curve fitting?

 

[rb1957]

i'd've thought (not that i've thought much about it) that
engineering stress = load/(original area) and
true stress = load/(true area)
then ...
engineering strain = (change in length)/(original length)
but then what's true strain, if it isn't the same as eng'g strain ?
then with the constant volume assumption, if you know the strain, you know the true area ...
(original area)*length = (true area)*length*(1+strain)
this is leading to saying that the translation from eng'g stress/strain to true stress/strain is "just" ...
at some strain, true stress = eng'g stress*(1+strain)

 

[crisb]

http://en.wikipedia.org/wiki/Stress_(physics)

 

[crisb]

Trying again!

en.wikipedia.org/wiki/Stress_(physics)

crisb

 

[prost]

Yes, those are the definitions I have found in my mechanics of materials book, by Bauld, using the same definition as the Wiki entry:
A0*l0=A*l (constant volume, so you are assuming a plastic deformation)

sT=true stress, eT=true strain. sE=engineering stress, eE=engineering strain.

sT=sE*(1+eE)
eT=ln(1+eE)

The plasticity assumption seems a stretch--what about for non plastic, or mildly plastic, deformation? Probably a very small difference caused by this assumption.

But why even take it this far? Measuring the deformed volume is hard, and you are making a restrictive assumption regarding the volume of your gage section--first that it is constant volume, second, that the deformation is more or less uniform. OK, even accepting that, why do it? Why not using engineering stress/strain, which are easy to calculate and/or measure?

Is it possible that sometimes a power law fits the true stress/strain better than it fits engineering/strain?

 

[EdR]

prost:

As an aside...if you check the differences between "true" and "engineering" strain at the maximum value listed (.07) I got a difference of about 3.6%.....If someone is trying to do computations to an accuracy of this level they are kidding themselves in my opinion.....

Ed.R.

P.S. If I remember correctly the basic computations between stress and strain are the same regardless of whether you are using "true" or "engineering" values.....

 

[prost]

Thank you all for your comments.

Still scratching my head..about half the stuff I do is nonlinear. When you look in MIL-5 at stress-strain curves, those are engr. stress/strain measures, with a Ramberg Osgood power law fit to the data--you don't see the actual data, all you see is the R-O power law curve fit, plus they give you the power 'n' in the R-O relation. Keep in mind, then, that this is in terms of engineering stress/strain.

Further, when you decompose the total strain (eT) into the plastic (eP) and elastic parts (eE), eT=eP+eE, you are again using engineering strain definition, (it's hard to write equations here, but I'll give it a shot):

eE=0.5*[du(i)/dx(j)+du(j)/dx(i)]...where x(k) are x,y,z for k=1,2,3, and u(k) are u,v,w displacements for k=1,2,3.

Computing true stress/strain from a set of tests (or in MIL-5's case, using data already available) and then making a power law curve fit of true stress/strain, seems like you are creating an unnecessary step--just go to MIL-5 and use directly the power law curves in terms of engineering stress/strain. That way you are consistent with the original Ramberg-Osgood nonlinear (plasticity) constitutive relation:

E*eT=sigma+Alpha*(sigma/E)^n

Note 'eT' here is not 'true strain', not the full nonlinear definition of true strain,

0.5*{du(i)/dx(j)+du(j)/dx(i)-[du(k)/dx(i)]*[du(k)/dx(j)]},

(sum on 'k'), valid for arbitrarily large displacements and strains, but the total strain as defined above.

Then there's the definition of 'nonlinear'--is plasticity 'nonlinear'? Yes, but not as nonlinear as say, a Neo Hookean rubber constitutive equation, which is valid for arbitrarily large displacements and strains. You can't say that about plasticity constitutive equations in general, that is, that they are valid for arbitrarily large displacements and strains. But that's a whole 'nuther topic I think.

 

[prost]

nc31--if I would have known what I know now, that the term "structural engineer" means primarily those that work on buildings and bridges, and not on aerospace structures, I would have agonized more about my choice of 'structural engineer' vs. say, 'aerospace engineer' or 'aeronautical engineer'. Knowing what I know now, I would classify myself as an 'aerospace engineer' who does structural analysis, to avoid confusion with the term 'structural engineer'.

 

[prost]

I think I understand well enough to explain it to myself now. My mistake was in forgetting that you can write any constitutive equation you like, with any stress/strain measures. Usually, you try to pair stress/strain measures that complement each other. For instance, you could write the equations of linear elasticity, valid for small strain/small displacement,
as
YM*eX=sigmaX-nu*(sigmaY+sigmaZ)
YM*eY=sigmaY-nu*(sigmaX+sigmaZ)
YM*eZ=sigmaZ-nu*(sigmaX+sigmaY)

where eX,eY,eZ are engineering strain,
e(i,j)=0.5*[du(i)/dx(j)+du(j)/dx(i)], sigmaX, etc. are engineering stress, defined as normal load divided by initial area. YM==Young's modulus, nu==Poisson ratio

This is an extremely useful constitutive relation because it works so well for large numbers of stress states we encounter--many strains we encounter are linearly proportional to the stress.

Or, I could have written a constitutive relation similar to linear elasticity, this time, using 2nd Piola Kirchoff Stress, and Green-Lagrange strain. These would be valid for arbitrarily large displacements and strains, but they look similar to the linear elasticity:

YM*E(1,1)=S(1,1)-nu*(S(2,2)+S(3,3)), etc. where E(1,1) is Exx (normal strain in x direction), S(1,1) is Sxx, 2nd Piola Kirchoff Stress in the x direction.

Just because I write this nonlinear analogy to linear elasticity, doesn't mean it is as useful as linear elasticity. Does it fit any data, or any observations we have made? Not many, in fact there's an analytic solution of the deformation of a bar of this material which gives you counterintuitive results.

I could also choose to model my material with a power law curve fit, which 'looks' like Ramberg Osgood, but uses true stress and true strain instead of engineering stress and engineering strain. It's my choice. The best reason for doing so I think is that I can describe the material's behavior better with the power law that uses true stress and strain, than if I would use engineering stress and strain--using the true stress/strain fits the data better, I guess that's one way to put it.

So I have a question for those of you who are inputting stress-strain curves in terms of true stress and true strain:
If you use true stress and true strain (it matters not whether you put a curve in the FEA, or a table of points on the true stress/true strain curve; the important thing is that you are using true stress/strain) then when you input true stress and true strain material curves, is your FEA solution valid for arbitrarily large displacements and large strains? (I presume most of us have the same solution process-the FEA solver runs through a linear solution, to get the first guess for the nonlinear solution).

From the FEA software's perspective, the software doesn't care how you calculated the true stress/true strain behavior of your material; it just takes the constitutive relation you give it and computes away.

 

[EddyC]

prost,

Maybe I can help you with this, if I am understanding you correctly. Are you talking about the portion of the stress/strain curve beyond the yield point/proportional limit? If so, I usually average the varying portion of the curve above the yield point, calculate the tangent modulus and perform a non-linear analysis. Is this what you are trying to do? Any stress above the yield point will result in some permanent localized distortions. Whether this is of use to you depends upon your failure criteria. I've experimented with this using Nastran. Other FEAs can do the same. Hope this helps you.

 

[prost]

My question ultimately, after I reasoned out why some FEA would want the user to input true stress/strain, is now: if you input true stress/strain for the material constitutive model, what are the kinematics being used by the FEA? Large displacement, large strain? this would mean that the FEA solution is valid for arbitrarily large displacements and large strains. If you know the difference between material and spatial coordinates, and the differences between all the stress/strain measures such as 1st Piola Kirchhoff stress, and Lagrange strain, etc., then you should know what I am talking about. Otherwise, if you don't, it will be difficult for me to clarify.