Getting deformation energy from DIC (in progress)

Lanning, W. R. and Muhlstein, C. L. (In preparation) The energetic interpretation of strain fields around a propagating crack in ductile thin sheets

Digital image correlation (DIC) can measure the surface deformation of materials from sequences of images, essentially telling is when and where material is changing shape. However, our ability to measure force is limited since we can only place sensors at the specimen boundaries. Thus, we are faced with a problem: can we compute the energy that is stored, absorbed, and released during fracture with from nothing but experimentally measured data (no simulations or constitutive laws)? Spoiler: we can!

Error propagation through DIC analysis (in progress)

Collins, J. G., Lanning, W. R., and Muhlstein, C. L. (In preparation) A Monte Carlo strain field mining methodology to identify and minimize image boundary-induced errors in interpolated fields

Digital image correlation (DIC) can be a very flashy technique for measuring strain. It produces attractive color-coded spatial distributions of surface strains that jazz up any conference presentation. But when it comes time to analyze that data and use it as the basis of further computations (stress, stress intensity factor, work density, etc.), how far can we trust the output?

This leads us to an interesting problem: the interpolation strategies we use to fill in the strains between discrete measurements can have unexpected consequences. If we apply anything higher-order than linear interpolation, such as a parabolic or cubic scheme, we get much prettier displays of the data, but this can also introduce errors and artifacts.

The main thrust of this manuscript is James Collins’ work probing the propagation of random errors through DIC strain analyses. He then used those error metrics to design analyses which were less sensitive to random errors. Unfortunately, I can’t divulge exactly how and why his approaches worked until this goes to press, but it is very clever! However, I can talk a little more about my contributions to error propagation through DIC analyses.

My contributions to this paper relate to mathematical fundamentals underlying interpolation schemes. When we pick a particular fitting method, we are implicitly assigning a model to the data, and that model requires a certain amount of input. It takes two points to define a line, three to define a parabola, four to define a cubic function, and so on. Adding more points than required lets us build up confidence statistics in the fitted model. Thus, we can look at the features of an interpolated strain field from DIC and immediately know which features we can trust most – those that cover enough area to encompass many data points. Those features amount to calculations which have multiple measurements to back them up.

If you ever want to be a pest at a conference, you can seek out DIC-based talks and see if they know how to tell analysis artifacts from actual material deformation. Look at any features of a strain field (maxima, minima, periodic strains, etc.) and compare them to the spatial distribution of the points which were tracked by the DIC analysis. If the strain falls below a certain threshold, it is very likely to be an artifact of the interpolation scheme. Here is a (very) rough calculation to find out of a strain is more likely to come from the propagation of error through the analysis than due to actual material deformation:

$$ \epsilon < \frac{m_{interp order}}{n_{trackedpoints}} \times \frac{\Delta_{tracking resolution}}{ d_{\epsilon feature} } $$

Also, beware any claim that “DIC measures strain at the resolution of the image.” The fact that image resolution is measured in length units while strain is unitless should be a dead giveaway how wrong that notion is. DIC might be able to measure displacement at the resolution of the image, or even at sub-pixel levels with a good tracking algorithm. However, strain is derived from some kind of fit to the data, and strain resolution should be at least judged in terms of the quality of the fit (i.e. the distribution of residuals).

A novel way of looking at plastic zones

Javaid S.S., Lanning, W.R., and Muhlstein, C.L. (2019) The development of zones of active plasticity during mode I steady-state crack growth in thin aluminum sheets, Engineering Fracture Mechanics, Volume 218

Traditionally, the “plastic zone” around a crack tip is the greatest extent of plastic deformation, a high-water mark of irreversible material flow. This makes sense in the context of high-strength materials since the plastic zone in such materials grows in a very limited fashion around a crack tip which moves very little before the specimen fractures. But in highly ductile materials, where crack growth occurs in a continuous process rather than a single burst, the story is different.

Thin ductile sheets exhibit stable crack growth in Mode I loading. The crack propagates as the specimen is pulled apart perpendicular to the crack face. In such a case, analyses such as the essential work of fracture (EWF), can compute the average energy needed to propagate a crack. However, EWF also applies the traditional concept of a static plastic zone while the crack tip is moving. It makes more sense to use a definition of the plastic zone that accounts for a moving crack tip and changing distribution of strain.

If the plastic zone (PZ) describes the maximum extent of plasticity over an entire process, we can define the zone of active plasticity (ZAP) as the region where plastic deformation is occurring at any moment in time. In this study, S.S. Javaid applied the ZAP concept from my dissertation to double-edge notch tensile specimens like those used in EWF experiments. He was able to measure the changing ZAP extent and shape over the course of the experiments using DICTograFer, the digital image correlation (DIC) suite I developed at Georgia Tech.

A new way to measure crack growth Resistance in Thin Ductile Sheets

Lanning, W. R., Johnson, C., Javaid, S. S., and Muhlstein, C. L. (2019) Mode I steady-state crack propagation through a fully-yielded ligament in thin ductile metal foils, Theoretical and Applied Fracture Mechanics, Volume 101, Pages 141-151

Many publications report shockingly low fracture toughnesses in ultra-thin metal specimens. However, those reports are based on fracture toughness measurements which attempt to apply conventional deformation models to the ultra-thin systems while simultaneously blaming exotic deformation mechanisms for the unusual results. By strategically comparing different specimen geometries on correlation plots of crack growth driving force vs. normalized crack length, I demonstrated that LEFM analyses produce plausible-looking, but entirely artificial fracture toughness measurements. Crack propagation in ductile thin sheets is actually controlled by critical stress (alternatively viewed as a work density gradient), which indicates that thin ductile sheets converge to steady-state crack propagation after the initiation and transition stages of crack growth.

In this publication, I was still calling my approach a “plastic collapse” analysis because it used the same axes and parameters as its namesake in Broek’s excellent fracture mechanics book. But I moved on from that nomenclature in later work because it was a bit misleading. The classic plastic collapse analysis used the critical net section stresses from a population of notched specimens while my analysis was an in-situ measurement of the stress as a crack propagated through a single specimen.

A cautionary tale about applying fracture mechanics in thin sheets

Lanning, W. R., Johnson, C., Javaid, S. S., and Muhlstein, C. L. (2017) Reconciling fracture toughness parameter contradictions in thin ductile metal sheets. Fatigue Fract Engng Mater Struct, 40: 1809–1824

Be careful using linear-elastic fracture mechanics on ductile metals! Conventionally, one might use a K to measure fracture toughness then use that value of K along with the yield strength to determine whether the measurement was valid. But low K measurements in thin sheets do not necessarily mean the plastic zone was small, and this has major ramifications for recent reports of ultra-low fracture toughness thin films and nanowires.

Using DIC to predict bonded joint strength

Collins, J. G., Dillon, G. P., Strauch, E. C., Lanning, W. R., and Muhlstein, C. L. (2016) Correlating bonded joint deformation with failure using a free surface strain field mining methodology. Fatigue Fract Engng Mater Struct, 39: 1124–1137

My mentor, James Collins, discovered that the surface strain maps of bonded joints could be used to predict the strength of the joint even if you were not looking at the part of the joint that failed!

I developed the math related to screening the strain field features used in the analysis. Look for the portion of the paper where the mean value theorem is invoked. The core concept is that our ability to confidently calculate strain increases with the number of displacement measurements and the length scale over which those measurements were taken. A small number of measurements taken close together can only be used to find large strains. A large number of measurements spread over a large area could be used to find smaller strains. The exact scaling effect can be computed from a principle similar to the standard error, modified for spatial measurements.

How do the layers in MLCCs affect their strength?

Lanning, W. R. and Muhlstein, C. L. (2013), Strengthening mechanisms in MLCCs: residual stress versus crack tip shielding. J. Am. Ceram. Soc., 97: 283-289

Some previous publications reported that multilayer ceramic capacitors (MLCCs) with denser electrode arrays were more resistant to cracking than those with fewer electrodes. They attributed the strengthening effect to direct interactions between the crack tip and electrodes. However, I found that the capacitors were actually strengthened by residual stresses resulting from the tape-casting and sintering process used to manufacture MLCCs. This suggests that other composite systems could be strengthened by strategic use of thermal expansion mismatch between the metal reinforcement and ceramic matrix.

This work also addresses a common belief that because the dielectric breakdown strength and mechanical strengths of MLCCs can have similar distributions, both strengths are controlled by similar flaw populations. However, we found that mechanical failures initiate at the specimen surfaces. Electrical breakdown must occur between the electrodes in the interior of the device. So the flaw populations may be similar. I.e. perturbations at the electrode/dielectric interface initiate dielectric breakdown and surface pores initiate mechanical fracture. But the failures are segregated to different regions of the device, so the failure-initiating flaw populations are segregated spatially, and the electrical and mechanical failures cannot be controlled by the same flaw populations.