Super-times with superalloys — a wee micromechanics quest
In the first of a series, I’m writing up a few of our papers in a Blog format. I’ll aim to capture a bit of “life behind the scenes” of our work and to give you a flavour of what we found exciting / difficult / and maybe a sneak peak of what lies around the corner.
We’ll kick off with a discussion on “Crystal plasticity modelling and HR-DIC measurement of slip activation and strain localization in single and oligo-crystal Ni alloys under fatigue” by Guan et al. published as Open Access in the International Journal of Plasticity, available online now.
This paper is the latest in our “Nickel campaign”, where we have focused on a few specific microstructures to build up an understanding of fatigue crack initiation in metals. This paper outlines a major step forwards in that we directly combine experiment and simulation to understand deformation fields. We find quantitative agreement between these fields and this means we hope to use future simulations and models to provide new insight on the patterning of plastic deformation in engineering alloys. This major leap forwards for us in understanding how materials deform and also showing that we can capture the details rather well! I love this piece of work, as we’re using these ideas to get a much better grip on how materials ultimately fail.
Using both experiment and simulation, we deformed two single crystal Ni-based superalloys of different crystallographic orientations. This lets us test whether our results are simply a fluke, or we’ve got a good handle on how metals deform. Both our crystals we focus here on deformation within the tensile fibres of the specimen (the bottom edge where the material is pulled along the horizontal axis).
In the experiment, we followed the local strain by tracking how lots of markers ‘move’ when we deform the sample (using a technique called Digital Image Correlation, DIC). We have used silica particles (they are cheap and we can disperse them easily) as our markers and tracked them in the SEM. It was difficult to get the alignment sorted (and I have no idea of the magic that my team did for this alignment, though I know we used hardness markers & also coded our own DIC code to do a 1st pass rigid body translation + rotation to make the images match well).
Zooming in with the high resolution DIC, we see that the striking surface (plastic) strain maps reveal the total accumulated plastic strain and we can see clearly that there is slip in a series of quite ‘sharp’ bands and that with increasing load these get progressively stronger.
Seeing the slip in this region at high resolution is quite pretty — but is also starts to pose some interesting questions! Namely — when we link between experiment and simulation, how do we manage the ‘resolution’ issue of both? We are driving crystal plasticity experiments to focus our resolution and reveal clear slip planes. In the crystal plasticity simulations, we ‘smooth’ this data out across elements. In practice, I think it is likely that the discrete ‘extra’ information about the in plane total strain and spatial distribution of slip, now available with experiments, will be useful in refining the constitutive laws in the simulations (and we would need to more spatially refined methods, such as discrete dislocation dynamics to see individual slip bands).
Simulations + Experiments = #MatSci WIN!!!
Next up, we put the crystal orientation into our simulation. Our simulation is set up in ABAQUS and a physically based crystal plasticity law. This involves a bit of work: first we (or rather, Yongjun) took a full 3D representation of our loading geometry and built our mesh. Then we had a fair amount of work to sort out how to translate the crystal orientation, measured using EBSD, into the simulation (part of this process involved the generation of our EBSD tutorial paper). This crystal orientation is used to rotate the slip systems into the crystal geometry. The ABAQUS crystal plasticity simulation is a very clever calculator — effectively finding the best solution to the problem of accumulating plastic strain (and the elastic deformation) for each time step.
We use Fionn’s lovely constitutive law as it is beautifully simple (and terrifically useful) in taking the local stress state and telling us how much plastic slip can happen. Very briefly, and crudely, I’m going to introduce the essentials:
- The slip law addresses the challenge of “how does the slip rate for each slip system evolve as a function of the local stress state” aka tell me about the plastic strain at each material point.
- This particular law is physically based, as in it is based upon a thermal activation term and the ability to capture when slip is active based upon the resolved shear stress on each slip system.
- The thermal activation term is justified based upon the physics of jumping of dislocations from one pinning obstacle to the next, and link this with whether the resolved shear stress on each slip system is greater than the critical resolved shear stress (i.e. when slip is allowed to happen).
- Slip activity is controlled by the local shear stress, and this is put within a sinh term to give you quite a sharp ‘switch’ to turn on and off slip, depending on the magnitude of the resolved shear stress.
This slip law is repeated across all the slip systems and the finite element package (ABAQUS) finds the best convergence to adhere to equilibrium and compatibility, while also portioning deformation between the elastic and plastic terms. In this formulation, we can add hardening by updating the critical resolved shear stress according to a rule (such as a function of the local plastic strain gradient and total plastic strain) but that’s a story for another day.
If we know how the slip law accommodates the local strain, then we can compare the size, shape and extent of the plastic strain field with an optical view of the slip bands on the surface (we bunged a camera in front of the mechanical test as it was deforming!).
We start with an idea of the active slip system (in both experiment, from the crystal orientation and how they match up with the steps, and in the simulation, from which system accumulates slip). Our next neat observation comes up — we see a beautiful replication of the size and shape of the plastic strain field (and the activity of another slip system in the centre of the beam) in both experiment and simulation. At this point you might be asking (especially if you are a Mechanical Engineer…): “Why is the slip field asymmetric, if the loading is symmetric?”. Well — we do apply symmetric loading, but the crystal response with anisotropic plastic flow, as but in single crystals elastic and plastic deformation are anisotropic!
Sticking grains together:
At this point, I’m quite excited: we’ve seen that experiments can capture what we see in simulation (and vice versa…). Next up, we ask the question, does this work in polycrystals? So we had a play! The problem with polycrystals is that we don’t usually know the sub-surface microstructure and life is a bit tricky. Grain structures are 3D monsters — and looking under the covers (well surface) is an expensive and time consuming task. This means that it is usually tricky to directly replicate experiment with simulation.
Fionn had a trick up his sleeve for this task! He suggested that we take a special kind of polycrystal — one where we have multiple grains in the surface, but let’s extrude those subsurface. This is ‘trivial’ to do in simulation, but normally a bit trickier in experiments. However, we’re playing with Ni-based superalloys, and loads of lovely work has gone into controlling their microstructures and we can use a directionally solidified (DS) bar, more usually employed to reduce grain boundary creep in a turbine blade.
With the careful craftsmanship of Yongjun, EBSD experiments were performed to map the surface grain structure and reproduce (by hand I think!) these as ABAQUS meshes. We repeated the cyclic loading experiment and simulation for this new microstructure.
The simulation revealed that during cyclic loading we see that different grains deformed differently. This may sound a bit obvious (different grains have different crystal orientations, and so the resolved stresses on different slip systems are different, and so the accumulated plastic strain must be different), but it’s really rather important. If we think about fatigue, materials tend to ‘rip themselves apart’ when there is lots of heterogenous deformation — i.e. that some regions are being pulled apart faster than their neighbours. A very simple story is that regions deforming lots tend to see and exhaustion of plasticity and therefore cracking happens (the more complex story is that now think that fatigue crack initiation is more about the energy release rate, but I’ll leave that to a future blog post). What’s striking in this simulation (Figure 18 from the paper), is that we see that the accumulated plastic strain within different points can vary by a factor of 4. We also see that the hardening rates (the slopes of these curves) are also quite different — this is really important to get to grips with in understanding failure.
Beyond pretty pictures — towards quantitative understanding:
We have the full field DIC maps and the full field simulations, so lets compare! Our pretty pictures look quite reasonable — there’s a bit of high frequency noise in the DIC (this is an experimental artefact due to ROI size, overlap and the strain calculations themselves) — but the fields look pretty good. You can see lots of red bits in simulation and that match red bits in experiment, blue bits in both and all the rainbow in between. I can wave in front a picture and get really excited and try to persuade you. But I could probably do this too if I played any two pictures and puts some funky colour scales — we’re not talking about a quantitative comparison yet… Oh and this is also a tensor problem (so lots of fields) and we can pick and choose where point while we jump up and down waving like an excited monkey.
So we took a few line out from both experiment and simulation and compared. The line graphs are plotted in Figure 17. They the red lines show the crystal plasticity predictions. Black lines show the experiment. We pick a few lines across a few grains and plot a variety of the tensor terms. Judge for yourself how good it agrees (I think I disagree with our lovely reviewer, who asked us to tone down our language a few notches…).
Summary (aka why this work is not only a bit of fun…)
In this work we’ve shown that we can join together simulations and experiment and we can match the deformation rather well. We’ve built up our story with single crystals and demonstrated that this works really well in polycrystals.
There are a few things that might be worth a look at: Ni-based superalloys are used quite often at temperature (though engines also start off cold!); and in the polycrystal case, the grain boundaries are simply regions that adhere to compatibility and equilibrium
Our combined experiment and simulation toolkit gives us confidence in our models and has helped us hone the quality of our experiments. Now we can use these with even more vigour to try to ask questions like: “What sort of microstructures do we really want?”; “Can we predict where cracks are going to form?” “Can we predict when they will form?”.
In addressing these challenges, we can fly forward in in our quest to make aeroplanes zoom faster / lighter / cheaper / more efficiently / with less fuel consumption / to more exotic destinations and with fewer CO2 emissions (aka saving the world with Materials Science!).
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You can find more of our work over at: http://www2.expmicromech.com/publications