An experiment: the blog-article

This week, the TU Delft scientist Rolf Hut published an intriguing idea: instead of publishing journal articles, scientists should use blogs to disseminate their findings. In (of course) a blog post, he explains why he thinks this would be a good idea. Blog posts, Hut argues, are more readable than traditional articles. Modern practices of archiving data and methods in collective repositories allow a blog post to be much more streamlined than an article. A blog post can present only the really interesting analysis, and just refer to various repositories for the raw data and methods.

Hut goes on to call on scientists to take up the challenge and publish their next article in the form of a blog. I think Hut has an intriguing idea, but I’m not entirely sure how big the benefits will be. As a scientist, that means there’s only one thing to do: perform an experiment!

Rather than publish some new work, I decided to try turning one of my recent conference papers into a blog, to see how that would look. I should also note that Hut suggests having an editor or colleague review the blog before posting, and that I’m skipping that step here. The paper I’m using is the paper I wrote for the 21st European Conference on Fracture. You can find it here. This paper was published under open access and I am the owner of the copyright (together with my co-authors). Legally I should note that this blog post is published under the CC BY-NC-ND 4.0 license, as I will be reusing figures from my article. Morally I wish to emphasise that this post is a re-write of an existing article, not an original contribution. I also wish to acknowledge the contribution of my co-authors to the original article: R.C. Alderliesten, and R. Benedictus.

Finally a note to my non-expert readers; since the idea of this experiment is to see whether a blog post could replace a scientific article, this post will be aimed at people who already have some pre-knowledge of my field. I hope you will find it readable nevertheless, but if not, I hope to write some more accessible blog posts soon.

Now, without further ado, let me present:

Characterising resistance to fatigue crack growth in adhesive bonds by measuring release of strain energy

Introduction

Ever since the first investigations into fatigue crack growth (FCG) in adhesive bonds (Roderick et al. 1974, Mostovoy & Ripling 1975), models for FCG prediction have been based on the Paris relationship. Although many different models have been published, at their core you usually find a power-law relationship between the strain energy release rate (SERR, G) and the crack growth rate da/dN, i.e:

\frac{da}{dN}=CG_{max}^n or \frac{da}{dN}=\Delta G^n (1)

where C and n are curve fitting parameters. This curve fitting is a problem, because as we’ve argued elsewhere (Pascoe et al 2013) this means FCG prediction models are based on empirical correlations, rather than on an underlying physical theory of crack growth. This means that the limits of validity of the predictive models are unclear. It also means that large amounts of test-data are needed to calibrate the models. This means that using these models for real world structures requires large and expensive test campaigns.

One area where these issues can be seen is that of the so-called R-ratio effect. Various approaches to modelling the effect of the R-ratio can be found in literature:

  • Use both G_{max} and \Delta G in the equation This can be found in the works of Hojo et al. (1987, 1994), Atodaria et al. (1997, 1999a,b) and Khan(2013);
  • Explicitly include R in the equation This was the solution favoured by Allegri et al (2011);
  • Use a modification of the Priddle / Hartman-Schijve equation This has been suggested by Andersons et al. (2004) and Jones et al. (2012, 2016).

The problem with all these approaches is that they are all phenomenological, just ensuring that the graphs have the right shape, rather than reflecting the actual physical processes. This means that the R-ratio effect is still not well understood, and that test campaigns to calibrate these models must test the full range of R-ratios.

In this work we use measurements of energy dissipation to gain more understanding of the physics of crack growth (Pascoe et al. 2015, 2016b), with the ultimate goal of producing models for FCG that are based on an underlying conception of the physical mechanisms involved.

Methodology and data

We performed fatigue crack growth experiments on double cantilever beam (DCB) specimens. The specimens consisted of two arms of aluminium 2024-T3, bonded with FM94 epoxy film adhesive. Tests were performed under displacement control, according to the protocol described in chapter 3 of (Pascoe, 2016a) or in short form in (Pascoe et al., 2016b). These references also describe the data analysis performed to calculate the SERR and the energy dissipation.

Experiments were performed at 4 R-ratios: R = 0.036, R = 0.29, R = 0.61 and R = 0.86.

The raw and processed data from these experiments has been made available online in three parts: Part 1, Part 2, Part 3.

Results

Figure 1 shows the crack growth rate plotted in the traditional manner, i.e. as a function of the linear elastic fracture mechanics (LEFM) parameters G_{max} and \left(\Delta\sqrt{G}\right)^2.

LEFM-dadN
Figure 1: Crack growth rate as a function of G_{max} (left) and \left(\Delta \sqrt{G}\right)^2 (right). There is a clear R-ratio effect visible in both cases. Originally published in http://www.sciencedirect.com/science/article/pii/S2452321616300117 Re-used under the CC-BY-NC-ND 4.0 license

As expected, there is an R-ratio effect, because it is impossible to uniquely characterise a load cycle with only 1 parameter. In other words, each combination of G_{max} or \Delta \sqrt{G} and R represents a different load cycle. That the crack growth rate is then also different should not be surprising. This point is discussed in more detail in chapter 4 of (Pascoe, 2016a).

In contrast, if the crack growth rate is plotted against the amount of energy dissipated in each cycle, dU/dN. The R-ratio largely disappears. This can be seen in figure 2

dUtotdN-dadN
Figure 2: Crack growth rate as a function of energy dissipation, originally published in http://www.sciencedirect.com/science/article/pii/S2452321616300117 Re-used under the CC-BY-NC-ND 4.0 license

With the exception of one outlier, the data seems to collapse along one line. However, careful examination will show that there is still a small R-ratio effect. Apparently the same energy dissipation does not always correspond to the same crack growth rate.

To investigate this further, let us examine the amount of energy that was dissipated during each experiment, during the cycle in which the crack growth rate was equal to 10^-4 mm/cycle. This is shown in figure 3.

Gmax-dUdN-1e-4-corrected for web
Figure 3: Energy dissipation as a function of G_{max} (left) and \left(\Delta\sqrt{G}\right)^2 (right) for a fixed crack growth rate of 10^-4 mm/cycle. Originally published in http://www.sciencedirect.com/science/article/pii/S2452321616300117 Re-used under the CC-BY-NC-ND 4.0 license

Each point in figure 3 corresponds to a different experiment. It is clear that although the crack growth rate was equal to 10^-4 mm/cycle in each case, the amount of energy that was dissipated to produce this crack growth is very different. The figure also shows that there is a very strong linear correlation between the amount of energy that was dissipated and the maximum load (i.e. G_{max}). The higher the maximum load, the more energy had to be dissipated in order to create 10^-4 mm of crack growth. In other words, at higher maximum load the resistance to crack growth was higher.

In order to quantify the resistance to crack growth, we introduce the term G*, defined as:

G^* = \frac{1}{w}{dU/dN}{da/dN} (2)

where w is the specimen width. G* represents the amount of energy required per unit of crack growth. It can be interpreted as the average SERR in a fatigue cycle, but it should be noted that it is not in general equal to the mean value of the applied SERR cycle.

To further investigate the relationship between the resistance and the applied load, figure 4 shows G^* as a function of G_{max} (left) and \left(\Delta\sqrt{G}\right)^2 for the full range of test data.

Gmax-Gstar
Figure 4: G* as a function of G_{max} (left) and \left(\Delta\sqrt{G}\right)^2 Originally published in http://www.sciencedirect.com/science/article/pii/S2452321616300117 Re-used under the CC-BY-NC-ND 4.0 license

In the right panel there is a clear R-ratio effectwhich is not present in the left panel. This implies that G*, i.e. the resistance to crack growth, is a function of the maximum load only. If it also depended on the load range there also would be an R-ratio effect in the left panel.

The resistance to crack growth is of course only half the story. The amount of crack growth that occurs during a cycle will not only depend on the resistance, but also on the amount of energy that was available. To investigate this, we plotted the amount of energy dissipation in the cycle for which the G* value was equal to 0.7 mJ/mm^2, for each of the experiments. This is shown in figure 5.

 

Gstar-fix-0_7
Figure 5: Energy dissipation as a function of G_{max} (left) or \left(\Delta \sqrt{G}\right)^2 (right). Originally published in http://www.sciencedirect.com/science/article/pii/S2452321616300117 Re-used under the CC-BY-NC-ND 4.0 license

It is clear that if G* is fixed, the amount of energy dissipation in the cycle shows a strong correlation with \Delta \sqrt{G}. However, note that in this case the relationship is non-linear. The reason for this non-linearity is as yet unclear. Nevertheless, by the first law of thermodynamics, the amount of dissipated energy must equal the amount of energy available for crack growth. Thus the amount of available energy correlates strongly to the load range, and not to the maximum load.

Conclusions

The results presented in this blog post show:

  • Measuring energy dissipation can be a useful technique to characterise fatigue crack growth. It has allowed us to avoid the false appearance of an R-ratio effect due to using a non-unique load cycle definition. It has also allowed us to separate out the amount of energy required, and the amount of energy available, for crack growth. I.e. it allows the resistance and the crack driving force to be examined separately.
  • The resistance to (fatigue) crack growth is linearly correlated to the maximum load (G_{max}). The reason for this is not yet clear, but a likely hypothesis is that at higher maximum loads more dissipative mechanisms are activated that do not contribute to crack growth. E.g. plastic deformation that is not in the line of the crack advance, or formation of voids that do not link up to the (main) crack tip.
  • The amount of energy available for crack growth shows a power-law correlation to the load range. The load range is ultimately what governs how much work is performed on the specimen during a fatigue cycle. Since the load range governs how much energy is put into the system, it makes sense that is also governs how much energy is available for crack growth. The key question for future research is to understand the relationship between the (far-field) applied cyclic work, and the amount of energy that is actually available for crack growth.

References

  • Allegri, G., Jones, M. I., Wisnom, M. R., & Hallett, S. R. (2011). A new semi-empirical model for stress ratio effect on mode II fatigue delamination growth. Composites Part A, 42(7), 733-740. Link
  • Andersons, J., Hojo, M., & Ochiai, S. (2004). Empirical model for stress ratio effect on fatigue delamination growth rate in composite laminates. International Journal of Fatigue, 26(6), 597-604. Link
  • Atodaria, D. R., Putatunda, S. K., & Mallick, P. K. (1997). A Fatigue Crack Growth Model for Random Fiber Composites. Journal of Composite Materials, 31(18), 1838-1855. Link
  • Atodaria, D. R., Putatunda, S. K., & Mallick, P. K. (1999a). Delamination Growth Behavior of a Fabric Reinforced Laminated Composite Under Mode I Fatigue. Journal of Engineering Materials and Technology, 121(3), 381-385. Link
  • Atodaria, D. R., Putatunda, S. K., & Mallick, P. K. (1999b). Fatigue crack growth model and mechanism of a random fiber SMC composite. Polymer Composites, 20(2), 240-249. Link
  • Hojo, M., Tanaka, K., Gustafson, C. G., & Hayashi, R. (1987). Effect of stress ratio on near-threshold propagation of delamination fatigue cracks in unidirectional CFRP. Composites Science and Technology, 29(4), 273-292. Link
  • Hojo, M., Ochiai, S., Gustafson, C.-G., & Tanaka, K. (1994). Effect of matrix resin on delamination fatigue crack growth in CFRP laminates. Engineering Fracture Mechanics, 49(1), 35-47. Link
  • Jones, R., Pitt, S., Bunner, A. J., & Hui, D. (2012). Application of the Hartman-Schijve equation to represent Mode I and Mode II fatigue delamination growth in composites. Composite Structures, 94(4), 1343-1351. Link
  • Jones, R., Kinloch, A. J., & Hu, W. (2016). Cyclic-fatigue crack growth in composite and adhesively-bonded structures: The FAA slow crack growth approach to certification and the problem of similitude. International Journal of Fatigue, 88, 10-18. Link
  • Khan, R. (2013). Delamination Growth in Composites under Fatigue Loading. PhD Thesis, Delft University of Technology, Delft. Link
  • Mostovoy, S., & Ripling, E. J. (1975). Flaw Tolerance of a Number of Commercial and Experimental Adhesives. In L.-H. Lee (Ed.), Adhesion Science and Technology (pp. 513-562). New York: Plenum Press.
  • Pascoe, J. A., Alderliesten, R. C., & Benedictus, R. (2013). Methods for the prediction of fatigue delamination growth in composites and adhesive bonds – A critical review. Engineering Fracture Mechanics, 112-113, 72-96. Link
  • Pascoe, J. A., Alderliesten, R. C., & Benedictus, R. (2015). On the relationship between disbond growth and the release of strain energy. Engineering Fracture Mechanics, 133, 1-13. Link
  • Pascoe, J. A. (2016a). Characterising fatigue crack growth in adhesive bonds. PhD thesis, Delft University of Technology. Available from repository.tudelft.nl from October 2016
  • Pascoe, J. A., Alderliesten, R. C., & Benedictus, R. (2016b). Characterising resistance to fatigue crack growth in adhesive bonds by measuring release of strain energy. Procedia Structural Integrity, 2 (ECF 21), 80-87. Link
  • Roderick, G. L., Everett, R. A., & Crews Jr, J. H. (1974). Debond Propagation in Composite Reinforced Metals. Hampton, VA: NASA, NASA TM X-71948. NTRS link
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