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Inverse problem: Stress-strain curve from load-displacement curve of CT specimen

Joe Kelleher's picture

I have a load-displacement curve for a compact tension specimen (side-notched small square plate), that has been plastically loaded in tension, to leave residual stress around the notch. I want to derive the stress-strain curve for the sample material from this load-displacement curve. Naturally the load-displacement curve is geometry dependent.

If I know a candidate stress-strain curve, I can easily simulate the plastic loading in Abaqus to calculate the load-displacement curve for that stress-strain curve. But I want to go the other way! One thing I've tried with reasonable success is to us Python's scipy.optimize.fmin function to iteratively solve the finite element model for different stress strain curves (specifically, different constants in the Swift model for a stress-strain curve). Then, I calculate the least-squares difference between the displacements in the model result and my experimental displacements. By minimising this quantity, I get an estimate for the stress-strain parameters. But this approach seems to get stuck on local optima, as the result is different depending on which starting solution you give it.

Does anyone have any suggestions please? Is this a well-known problem?



Hi Joe,

your problem is our daily task. 

Transfer your load-displacement curve into a true stress - true plastic strain curve. Then try to approximate this curve either with Swift or other hardening laws like Hocket Sherby, Voce or Gosh, depending on the curve characteristic. Keep in mind, that not only the approximation of the available data range (material dependent, lets say  up to 20% eq. plastic strain) , but also the extrapolation for higher strains in a physical way is important.

If you need more information, don't hesitate to contact me. at gmail com


Martin J. Gross

Joe Kelleher's picture

Hi Martin, thanks for your reply. Unfortunately I don't understand what you mean by "Transfer your load-displacement curve into a true stress - true plastic strain curve." - doing this is the core issue I'm facing. In a simple tensile test it's just a case of dividing force and displacement by cross-sectional area and length (with appropriate true stress correction). What does one do when there is a non-uniform distribution of stress and strain in the sample?

I chose the Swift hardening law as it is simple, but unlike a simple power law relation, it still gives a non-zero stress at zero plastic strain. In effect this is the yield stress, and must be realistically matched if the elastic part of the load-displacement relation is to be fitted.



Depending on the amount of plastic strain, you could try to find an analytical solution for the stress distribution using the deformation theory of plasticity and then try to back out the load-displacement curves.  Some work like that has been done for mode III but I'm not sure whether there's anything for mode I.  People often use Neuber's rule to get an approximate stress state.

I haven't looked at the literature myself but you could try papers such as

"Plastic stress-strain history at notch roots in tensile strips under monotonic loading" by Ralph Papirno, Experimental Mechanics, Volume 11, Number 10 / October, 1971,446-452.

You can also check out this paper.


-- Biswajit 

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