Quantitative/Qualitative Measurement of local strain analysng the microstructure

Dear Bharadwaj,

1. If you don't mark any grid lines before the experiment, and "just go about" your experiments, is there a way to measure local strains?...

...LOL!

If you mean measurement, then, by definition, no!

2. You may, however, estimate the local strains, if you know the initial geometry in detail.

For the estimation process, you would have to assume some constitutive relationship for your material, and further, also assume some specifics of how the material relieves the elastic strains upon unloading.

In principle, this would be an inverse problem, and if you have big enough strains to take the specimen beyond the initial (mostly/assumedly) linear region (and thus into the nonlinear and plastic deformation region), then it would also be a problem having both: constitutive nonlinearity and irreversibility of deformation. And, I am not at all sure how the unloading part would be treated in analytical solutions even if it were feasible to formulate this as an inverse problem.

However, rather than treating it as an in-principle inverse problem, practically speaking, you could get away by using stepping through a computer simulation. You could try solving for both the loading and the unloading effects at relatively fine increments of displacements, and comparing the resultant (plastically deformed) global shape (i.e. the one remaining after unloading in the simulation) with what your experimental specimens actually show. That as-unloaded displacement field which gives the best possible geometrical fit to the experimentally observed shape, is the solution you would be looking for. (Essentially, such a stepping through would be an explicit approach of simulation, and this trial-and-error explicit approach would be practically much more easier than writing an inverse-problem solver even in simulation. You could use the existing software for the trial-and-error approach.) Using this solution displacement field, you could then easily get the elastic and plastic parts of the local strain-field.

It would still be only an estimate, because: (i) it would be based on a simulation (thus subject to discretization and other errors), and (ii) you would have made assumptions about the constitutive and unloading behaviors.

You would in all probability have to use a computer simulation because (i) your geometry probably would not be amenable to a closed-form analytical solution, (ii) and even if it were, you are probably already getting into the plastic deformation region. In any case, I am not clear how people analytically address the situation if the plastic strains are small enough that the unloading part cannot be ignored. That's why I focused on the getting the estimate using computer simulation.

3. I wrote all the above because you are a student and probably doing research.

If you were from industry, I would have advised you to strike a very remarkable friendship for a day with your materials manager, thereby get another set of specimens made of the same material, mark these with the initial grid lines, and...

Hey, your report would have been ready by now. You could even be found trying to impress it on your boss how you really deserve an immediate promotion.

But, you (still) are a student, and so...

4. ... And so... (No, I am not yet done! (LOL!))

If the plastic deformation is big enough that the elastically unloaded one can be ignored, then an experimental way to get an estimate of the plastic part of the strain would be to use microscopy and stereology.

The essential idea is to treat the microscopic features as if they themselves supplied a grid.

Use some undeformed material from the same lot, and apply stereological principles to get statistical distributions of various features of microstructure such as grain size, orientation of the surface area, etc. Experimentally measure the same for your deformed specimens, at various locations, and infer the local plastic strains.

5. Apart from it all, you can always experimentally measure the residual strains even if you don't know the initial configuration. That part is routinely done. Indeed, even if they call it the residual stress measurement, what they actually measure is only the residual strain.

6. The differences between the results obtained via point numbers 4. and 5. are academically interesting, and for reasons of the falling interest in typing further, are left as an exercise to the reader. ("Perfection is what(ever) you get at the point of exhaustion.") Here are some hints, though: think of (i) the scale i.e. the size of the area over which measurements are made and the physical parameters to which the measurement principle is more sensitive, (ii) the sensitivity to the local and immediately surrounding inhomogeneity.

7. Hope this helps. And, BTW, this entire answer applies to a situation involving only the simplest monotonic loading like a tension test---and not one involving fatigue.

Hope this really helps.

Best,

--Ajit

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