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# Flaw sensitivity

I now include this paper in my course on fracture mechanics.

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## Comments

## Zhigang, interesting this parallel between ductility definitions

Dear Zhigang, your paper is very interesting and I need to find time to read it in more details. However, one figure strikes my attention, and that is the plot which compares the work of rupture in tensile tests, and GIc in fracture. I have often warned my students that there are many "qualitative" definition of a ductile behaviour --- the elongation in tensile test (or the work under the curve, which can be divided by the surface of the section of the specimen), a high KIc in a fracture mechanics test, and a high resilience in a Charpy pendulum. Clearly the three are related, and all are energy per unit surface, and none is strictly related to the work of adhesion of the Griffith criterion, which perhaps is a lower bound to GIc, but not so clearly, as your paper shows. Any further comments from this perspective?

## Re: Zhigang, interesting this parallel between ductility defini

Indeed, if we consider a flaw-less sample, glass has the largest work of fracture of all materials (Fig. 5). Are you referring to this figure?

## yes, interesting that the highest work to rupture is glass!

Yes, Zhigang, that Ashby map is very interesting, and I am not sure I have seen it before

-- is it in any of Ashby's papers or books?

In textbooks, particularly for metals, they write measure the tensile curve and

say "ductile" usually when there is "high elongation" , say A>5%. Not defining the work per unit volume.

How can glass have 3 orders of magnitude higher W than steel as in Fig5?

In materials elastic up to rupture (as glass is), then W=1/2 E eps_max^2.

I guess for steels one should rather take a perfectly plastic estimate W=sigmaY eps_max_steel so it has nothing

to do with E, and

W_glass / W_steel= 1/2 E_glass eps_max_glass^2 / ( sigmaY eps_max_steel ) = 1000 or

E_glass / sigmaY = 500 eps_max_steel / eps_max_glass^2

Take realistically E_glass = 200GPa and sigmaY=400MPa,

eps_max_steel = eps_max_glass^2

But are eps<<1? I would guess eps_max_glass=1%, so eps_max_glass^2 = 10^-4

so this means eps_max_steel = 10^-4 ??

I must be missing something and making some terrible mistake!

## Flaw sensitivity

We constructed Fig. 5 ourselves. The figure was reported for the first time in this paper. The procedure to create this figure was explianed in the paper. In particular, for glass, we assumed thoeretical atomic strength in the calculation, as explained in the paper.

## ok, thanks. But...

Thanks, that makes sense, with theoretical strength, one has sigma=E and eps near 1, in which case my equation gives also eps_max_steel=1 which makes also sense.

I wonder however why you cannot theoretically think the same of steel. Surely if you were to drawn extremely thin rods of steel, you would also eliminate defects and dislocations. Indeed, you certainly know Hutchinson, Fleck, Willis and others talk of strain gradient plasticity, although it is not clear to me the limit for very thin scales. The ones I know about focus on modeling collective behavior of dislocations (mostly GNDs) and as far as I know do predict infinite strength when the strain gradient is infinite. But that seems like a funny prediction. The strength shouldnt actually exceed the homogeneous dislocation nucleation stress. I think SGP is not designed to predict behavior in the dislocation source-limited regime (although there might be versions I dont know about that do).

So your Fig.5 probably wants to make some points that I need to read more carefully about, but it is not necessarily the only plot one can make. In some respects, you may be mixing apples with potatoes?

## Some response

Dear Zhigang, Thank you for highlighting this work in your course!

Dear Mike, Thank you for bringing this great discussion.

For real solid materials, there're some intrinsic material properties to characterize the fracture at very small scale hardly affected by geometry, such as strength typically. And we in this work regard the work to rupture (in the unit of J/m^3) as an energetic fracture parameter in the small scale, that is hardly affected by geometry.

To produce there values for broad materials, we found the fracture of glass is very sensitive to flaws, and experimental reports does not reach that small scale. Also different from metals, the fracture of the glass involves very little inelastic energy dissipation, so we turned to the consideration of using the theoretical bonding energy as a representive of the work to rupture of glass.

For steels (and other ductile metals), the rupture involves much plastic work, and the theoretical estimation is not straightforward. So we relied on flaw-insensitive experimental data. We used yield stress instead of E in the estimation of dutile fracture, as stated in section 5, which is same as your thoughts.

## teach flaw sensitivity in class

In class (ES 247 Fracture Mechanics), I demonstrated your basic experiment using rubber bands with and without cut. I then briefly described your paper. There is a homework problem.

Probelme 20. Flaw sensitivity

Read the following paper. Chao Chen, Zhengjin Wang, and Zhigang Suo, Flaw sensitivity of highly stretchable materials, Extreme Mechanics Letters 10, 50-57 (2017).

(a) Explain Fig. 1

(b) Explain Equation 5

(c) Explain Fig. 5

## the devil is in the details

Zhigang

that there is different "flaw sensitivity" in different materials is sure and is not new. Your paper has the merit to have used some concepts of non linear fracture mechanics. However, Fig.5 is really not convincing to me, and saying that glass is the most "ductile" of all materials is really raising some eyebrows and would confuse even the most intelligent of the Harvard students :)

I rather prefer simpler Ashby maps on this with on axis fracture thoughness vs. strength , where inclined lines correspond to size of "process zone" or "plastic zone", and in that case also an indication of "flaw sensitivity". You will find indeed glass as the most sensitive to defects. This is rather more reasonable. Of course your definition of flaw sensitivity is not so different if you take

G/W = KIc^2/ (E sy eps-f)

whereas Ashby's one is KIc^2/sy^2.

To convince me that your Fig.5 map is more appropriate, you should indicate me why substituting sy with E eps-f improves the correlation, and this is not simple. This returns to my original question that ductility is often also referred to from eps-f.

The devil is in details....

You can find Asbhy maps in one of the figures of this paper

On notch and crack size effects in fatigue, Paris’ law and implications for Wöhler curves

https://www.fracturae.com/index.php/fis/article/view/2009

where you can see also a discussion of flaw sensitivity in fatigue which perhaps is also of interest for your course.

## interesting discussion

Very interesting discussion - of course don't blame me for being a little lazy in studying details of your paper, which, I repeat, is very interesting.

If I were you, however, when you define the work of rupture at small scale, I would use the theoretical strength and the covalend bond energy for all materials [by the way, does it vary much?], because size effects and sensitivity to flaws are not unique to glass. The theoretical strength calculation you did for glass valid for all materials to compute the work of rupture for very small scales.

What changes is that in what you call "plastic" materials there should be two transitions and not 1 as you are assuming.

It is very similar situation of Hurtado and Kim model for friction with dislocations. http://rspa.royalsocietypublishing.org/content/455/1989/3363.short

"A micromechanical dislocation model of frictional slip between two asperities is presented. The model suggests that when the contact radius is smaller than a critical value, the friction stress is constant, of the order of the theoretical shear strength, in agreement with reported atomic force microscope (AFM) friction experiments. However, at the critical value there is a transition beyond which the friction stress decreases with increasing area, until it reaches the second transition where the friction stress gradually becomes independent of the contact size. "

So you have really two transitions and two plateaux... Maybe we should write a comment note together :)

In fact, close to theoretical strength for steel has been reported also at nano-scale

https://www.nature.com/articles/srep36810

regards, mike

## We noticed the discrepancy

Hi Mike,

Thanks for raising these good questions.

We noticed the discrepancy between our model and Ashby's model (Dugdale's length). I'd like to point some differences. First the Dugdale's model requires small scale yielding at crack and linear elasticity in other part. This strong assumption will not hold when flaw size reduces to critical length. While the length of flaw sensitivity we proposed is based on the crack-tip singularity of strain energy density W~G/r, which is not limited to specific nonlinear material model, as long as we assume the material doesn't unload before fracture, similar to the consideration in HRR theory.

Also, our model focuses on the sensitivity of rupture strain instead of rupture stress. For nonlinear materials the sensitivities of these two are different. For instance, a ductile material may fail at nearly a constant stress close to yield strength, but would still be sensitive to flaws in terms of the failure strain, since the stress-strain curve becomes flat.

I think we all agree that glass is not ductile :) The work to rupture (work under the stress-strain curve) does not indicate ductility. A non-ductile but stiff material can also have a high value of work to rupture. And according to our plot, glass has a very small length of flaw-sensitivity in nanometers, which means the material is very brittle. Of course we didn't sample every material in but some typical ones.

I also agree theoretic strength may not vary too much for covalent solids (not include polymers), as the other example Alumina in the plot. But this might not be necessarily true for many others. Polymers need to count the volume fraction of chains, which will significantly reduce the work density. And the theoretic strength for ductile materials should be determined by the energy barrier to flow instead of bond strength. I would expect significant difference because of these different mechanisms.

Hope this could address some of your concerns.

## nice answers

Yes, you answer some points, and we are getting into technical details as it is important --- devil is in details. Your contribution is certainly useful, science makes progress by new ideas.

So your length scale comes from ratio

Rnew= G/W = KIc^2/ (E sy eps-f)

whereas more traditionally, RAshby=KIc^2/sy^2.

So the ratio of the two proposals is

K= Rnew/RAshby = sy /(E eps-f)

Normally, low sy have high eps-f (strain to failure), and hence this ratio K is likely to vary largely.

So the two length scales are not going to be the same at all!

To convince that your R is better than Ashby, you need some quantitative evidence, not just qualitative arguments.

Can you provide such quantitative evidence please?

## Seeking clarification

Hi Chao,

I am reading this article with great interest. Could you please clarify a couple of points for me?

1. During experiments, the width was kept at least 5 times the depth of cut. As you test c-values from 0.05 to 50mm, that means the sample width was at least 0.25mm to 250mm. I would rather have the width fixed (corresponding to the maximum cut) and vary the size of the cut to categorize the cuts ranging from 'small flaw' to 'large flaw'. Is there a reason to vary the width?

2. During the numerical simulations, I presume the plot annotated Gent in Fig. 3b is a fitted-curve to the results of simulations conducted at various cut values. Wherein for each simulation, you estimate the stretch-to-rupture by comparing the energy release with the fracture energy. Am I correct?

Thanks,

Narinder

## Re:

Hi Narinder,

Thank you. About your concerns:

1. Yes, we tried large samples. We kept the width at least 5 times larger than cut to minimize the finite width effect. Within this constrain I have no issue if you keep a constant width, and only the length of edge cut matters.

2. Yes, the details are descripted in the numeric part in section 4.

## are there more rigorous geometrical conditions

Hi Chao,

while hoping for replies on my previous question :), I imagine Narinder wanted to ask you if there are standards for testing fracture thoughness in rubber like materials, like there are for metals etc. As you know, the size of the specimen, do NOT depend only on width or thickness being much greater than cut, but rather these quantities must be much greater than the expected plastic radius, which means they depend on material properties KIc and strength, which you don't know a priori by the way. With non-linear fracture mechanics, I don't think there can be anything simpler --- more complex perhaps, but not simpler.

So your tests, as your extension to "other materials", are perhaps good zero-th order guess, but miss some quantitative rigour.

Thanks, Mike

## Re

Hi Mike,

We obviously don't have experimental support for other materials other than the two elastomers we studied:) The primary focus was fracture of nonlinear elastomers. Our bold extension of this length to other materials is truely based on the theoretical consideration as you pointed out. The involved two parameters, fracture energy and work-to-rupture (also called toughness in industry), applies to all materials. The generality really encouraged us. In contrast, KIc requires linear elastic fracture (with ssy), and sigma_y requires yielding, which are difficult for many materials other than metals. That's why we are so excited to introduce this length scale even without evidence from other materials. We do believe this length scale revealed new insight for nonlinear fracture.

I would be super happy to see any follow-up about how bad (or good) this length is for specific materials, including metals :)

## you may find these notes useful

Here you find some notes that are useful. Notice they start by defining "toughness" what you call "work of rupture", while "fracture toughness" as a material property that you reach only under plane strain conditions, and therefore ASTM gives the size of the specimen big enough something like W,B>2.5 (KIc /sigmaY)^2.