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### it could be both

*In reply to Yes, with a caveat - the*

dimensional analysis says that you should compare the size of SGP to both size of crack, or size of plastic zone. It is the same as my paper on Paris' law https://www.sciencedirect.com/science/article/pii/S002250960800149X

I tend to agree with you that fatigue is more promising to see effects of SGP, since cracks and plastic zones are both much smaller: for example these 2 interesting papers you mention discuss possible interpretation of DK-th of the material, although of course as in my original message one in practical terms tends to measure things (DK-th) rather than estimate them from dislocation properties or SGP ones.

Anyway, the path to celebrity is very random! Paul Paris became (almost) instantly famous (infact, after 3 rejected papers), for postulating a very simple fitting equation of crack growth, which, as I am say, one measures independently on any other previous knowledge! In retrospect, this was initially rejected because the rest of the croud was attempting to relate crack advance with plasticity quantities --- without much success. In these respects, this is yet another explanation of what I am saying: successful models should be simple, and more effective than previous equivalent ones.

### Congratulations!

*In reply to Fully funded PhD position in the Department of Mechanical Engineering at Clemson University*

Glad to see your start at Clemson!

### Yes, with a caveat - the

*In reply to yes, we almost agree*

Yes, with a caveat - the length scale that matters is the size of the plastic zone, not the size of the crack. So, if you have a large crack, then gradient effects will be most relevant for a regime of remote loads up to 40 MPa*m^1/2 (roughly). But if you are looking at what happens at 180 MPa*m^1/2 then gradient effects are negligible. In other words, it matters in fatigue [1,2], brittle fracture [3,4], environmentally assisted cracking [5,6] but not in ductile fracture.

[1] https://www.sciencedirect.com/science/article/pii/S1359646201009484

[2] https://www.sciencedirect.com/science/article/pii/S0142112318304493

[3] https://www.sciencedirect.com/science/article/pii/S0022509697000185

[4] https://www.sciencedirect.com/science/article/pii/S0022509618307890

[5] https://www.sciencedirect.com/science/article/pii/S1359645416305183

[6] https://www.sciencedirect.com/science/article/pii/S0360319915315913

### yes, we almost agree

*In reply to Let's summarize*

We seem to converge: if one were interested to what happens at the really close crack tip, for sure we need many details.

My guess however is that they matter only in limited number of cases. Turning back to the example of indentation, SGP is crucial if you want to adress very small indentations, but if you model standard macroindentation, DESPITE something happening at the sharp tip of the indenter may be NOT correctly modelled if you avoid SGP, that doesn't matter.

So I guess SGP also would tend to matter if crack is very small, in which case I was already inclined to think (even without SGP) that the resistance would tend to theoretical strength, as the cohesive model says anyway. For large cracks, I think it is unlikely.

Is this correct?

### Let's summarize

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

Yes Mike, I am afraid that I am not following you now. Let me try to summarize:

- Conventional continuum models do not predict very high crack tip stresses in a physically-realistic setting for metals: the infinitesimally deformation theory is not suitable for strains larger than 5-10% (and strains are undoubtedly higher ahead of a crack) and a non-zero crack tip radius exists (even if at the atomic scales, all cracks are blunted). E.g., von Mises plasticity predicts stresses that are at most 4-5 times the yield stress.

- Crack tip stresses cannot be "validated with experiments" (we can't measure stress). But experimental evidence suggests that crack tip stresses must be higher than those predicted by conventional continuum models. There are numerous experiments that show atomic decohesion with clean fracture surfaces in the presence of plasticity. Since the stress level required to trigger atomic decohesion is on the order of 10 times the yield stress, there is obviously something wrong/missing in conventional continuum models. This motivated the development of SSV and other models.

- Gradients of plastic strain confined into a small volume lead to a stress elevation. This has been observed in numerous micro-scale experiments (torsion, bending, etc.) and is associated with the effect of geometrically necessary dislocations (GNDs). These GNDs can be measured ahead of cracks and the situation in the plastic zone ahead of the crack tip is similar to that of nano-indentation (small plastic zone in bulk material). These dislocation hardening effects can be captured by using either Discrete Dislocation Dynamics (DDD) or Strain Gradient Plasticity (SGP) and lead to a crack tip stress elevation.

- This stress elevation predicted by DDD or SGP over 0.1-20 microns ahead of the crack can be key in the understanding and modelling of many damage mechanisms. One example is atomic decohesion in the presence of plasticity, as mentioned above, and another example is hydrogen embrittlement, where the concentration of hydrogen depends exponentially on the hydrostatic stress (so, getting your stresses right is important). I have particularly focused on the latter: we have written papers about this, we have shown a good agreement with experiments, and our models are now being implemented in industry.

I am afraid that there is nothing else that I can say about this topic. But I am very grateful for your interest and will be happy to chat at large about it if we meet physically - maybe at ICTAM in Milano?

### perhaps I am not able to clarify

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

When I use nanoindentation, of course I am testing very small amount of materials, of course SGP is a good idea to follow the reinforcement with scale.

Similarly, when I test very small specimen, say, in torsion or bending.

But when I have a crack in a bulk material the situation is "nothing new" with respect to the entire literature and testing so far conducted for fracture mechanics: simply, you are using SGP at the crack tip, but the crack tip had always been under very high stress. So, the entire literature (and by this I am NOT saying only LEFM) which has worked so far and validated with experiments, cannot be wrong.

Here is where you have to put some care -- evidence why we need a new model for an old problem. Of course there are cases where LEFM may not be sufficient, but it is not just LEFM which I am talking about. Perhaps it would be nice to be more specific, of course industry may not be happy with present situation nor standards. But again, you have to be specific. Besides, the example of environemtal assisted cracking doesn't sound to me the SGP cup of thea. Even if you do find clear application, you then need to convince industry, scientific literature and standards, that the additional effort is worth. Like with everything :)

### LEFM does the engineering job in many cases, agreed

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

I agree Mike, there are many engineering applications where the use of linear elastic fracture mechanics (LEFM) provides a useful and conservative estimate for engineering practice. But there are many other applications where this is not the case (from plate tearing to environmentally assisted cracking). In fact, the use of a variety of fracture assessments that go beyond LEFM is becoming rather frequent in industry, as recognized by several standards.

### I guess the point is: when do we really need to know details?

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

Emilio, I am sure a better understanding of what happens very close to crack tip may be useful in some situations (the next question is "which situation"?).

In the standard fracture situation, which I teach to undergrad students, what I mean is that I measure toughness of a material, and I apply KI<KIc. In KIc, there are clearly many details which we don't know, but which we measure. One problem is to satisfy conditions of LEFM.

Which situation therefore really benefits of SGP theories? I am sure you know some examples. There are some theories which try to use "stress" instead of KI, perhaps to unify the concepts and the transition from short to long cracks, and these require "critical distance" concepts, or cohesive laws. Another theory is Weibull theory, which shows the crucial problems we discussed. By the way, Weibull theory normally starts from assuming KIc is a deterministic quantity, and the cracks are statistically distributed, so strength depends on the latter. In reality, also KIc is statistically distributed...

In turn in some of your papers, you use cohesive laws, but also plasticity and SGP. This means a large number of parameters. In the end, we need to measure these parameters from experiments. So, again, we need to show why we should do this, what do we gain with respect to the standard process KI<KIc?

### A more rich description of crack tip deformation

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

Hi Mike, thank you for your interest. Let me try to address your questions:

1) Yes, if one decides to model a sharp crack and use a small strain constitutive model that predicts a singularity at the crack tip (strain gradient plasticity, von Mises plasticity, elasticity, etc.), then you run into trouble with a Weibull-type analysis.

2) Strain gradient plasticity captures the extra storage of dislocations that is intrinsically associated with plastic strain gradients (GNDs). Large GND densities have been consistently measured ahead of cracks and inclusions. And a stress elevation with gradients of plastic strain is observed in numerous micro-scale experiments (bending, torsion, etc.) as well as other similar problems (e.g., nano-indentation). These strain gradient effects lead to a stress elevation ahead of the crack tip that length-independent constitutive models cannot capture.

I do not know what are the "general results of fracture mechanics" that have been validated but of course that cannot include crack tip stresses (which cannot be measured). The stress elevation predicted by SGP theories is not only supported by the experimental evidence that I have listed before but also by discrete dislocation dynamics simulations (DDD). The implications can be profound: it enables us to rationalize brittle fracture in the presence of plasticity (J2 plasticity falls very short of achieving the theoretical lattice stress) [1-2], and can be a key ingredient in modelling many damage mechanisms (e.g., hydrogen embrittlement where the critical distance is within microns to the crack tip [3]).

One could say that it is a way of enriching your continuum formulation to capture crack tip deformation mechanisms that can otherwise only be captured by using more computationally expensive methods such as DDD (at the cost of introducing an additional parameter for this dislocation hardening, which comes out naturally in DDD).

[1] https://www.sciencedirect.com/science/article/pii/S0022509697000185

[2] https://www.sciencedirect.com/science/article/pii/S0022509618307890

[3] https://www.sciencedirect.com/science/article/pii/S1359645416305183

### ok. So when devil is in the details?

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

Thanks for your reply. So if you had used SGP in the form you have used in other recent papers with Fleck (if I remember well), you would have occurred into this problem of the singular paradox: if singularity of stress is s==r^-p and Weibull modulus is m, you get the volume integral as Int r^1-mp (see https://www.sciencedirect.com/science/article/pii/S0020768305005226)

Anyway, the use of Weakest Link statistics is a little dubious when one applies it to very small regions.

I am not an expert of SGP, but while I fully understand that they mimic the real strengthening due to dislocations less dense in small regions, I don't understand fully the implication of SGP in fracture mechanics, which is what probably you are exploring. The general results of fracture mechanics certainly have been validated for long time: so when is SGP (and its complications) adding really greater insigth / modelling capability?

### Hi Mike - thank you for your

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

Hi Mike - thank you for your kind words and interest. You raise an interesting point indeed. In this case, the results are not sensitive to the mesh because we do not have a singular stress field. Note that we are using a different strain gradient plasticity model that the one that showed an inner K-field close to the crack tip, and we are also using a finite strain framework and a crack with an initial blunting radius (á la McMeeking). You still get local hardening and a stress elevation (relative to conventional plasticity) but your solution is not singular anymore.

Best,

Emilio

### Interesting but have you explored mesh sensitivity?

*In reply to Gradient-enhanced statistical analysis of cleavage fracture*

Hi Emilio nice paper as usual.

However a quick question: I am sure you know that weibull theory leads to a paradox when applied to a singular field and so if you were to refine the mesh results would never converge since your sgp model return to a singular stress near crack tip.

### I am looking forward to your rockets

*In reply to Dear Jiawei,*

Dear Philipp,

Thank you for the brief answer. I am really looking forward to your rockets! Then what is the potential application of soft rockets?

Jiawei

### please see also the suppl. mat. of the PRL paper

*In reply to Shear-Induced Anisotropy in Rough Elastomer Contact*

Interestingly (but we reported only part of this and hidden in the Suppl. Mat.) we found that rough surfaces we measured (standard metal ones used to mold the soft material) had power law PSD, but with strange non-fractal features (apparent fractal dimension lower than two, clusters of contact much more regular than what commonly described), quite far from the common ideal fractal most of the modern community describes, including gaussian heigths, but highly non gaussian slopes and curvatures. So PSD is power law but this doesn't mean we can use the common results. But we haven't gone deep into this.

### Dear Jiawei,

*In reply to Very excited research field*

Dear Jiawei,

Thank you for your commet. The snap-through instability can be used to design a pressure release valve, that opens at a critical pressure. Thus, it could be used in a rocked-like robot, that propells by ejecting a stream of air, when it reaches a certain internal pressure.

In my opinion, one of the important next steps is to find new designs for the control system that allows miniaturization and simpler fabrication.

Best regards

Philipp

### Very excited research field

*In reply to Journal Club for May 2019: Using Instabilities to Control Soft Robots*

Dear Philipp,

The field of soft robots is really exciting and may unleash many new potentials. Thank you for sharing such a wonderful work. I still remember the early project you told me about soft rockets, can the rockets be powered by the pneumatic system you developed?

In your mind, what will be the future direction of this field? What are you planning to further explore this field? Is there anything in this field remain unexplored? Thank you!

Best wishes,

Jiawei

### Dear Nicholas,

*In reply to Miniaturization?*

Dear Nicholas,

I agree with you that the valves are currently relatively bulky. The physical principles behind the valve scale to smaller dimensions, but fabrication becomes difficult. It is no problem to scale down the bistable membrane, but we currently do not have a good way to scale down the tubing. Changes in design that allow the use of straight tubing will solve some of these issues. However, there is a general problem when scaling down valves for pneumatic robots: it reduces air-flow and, thus, limits the speed and the size of robots that can be controlled. Other researchers have shown that with microfluidics control can be implemented on a very small scale, but these fluidic circuits can only control very small robots.

Best

Philipp Rothemund

### Link to our new paper

*In reply to Preferential debonding during demolding of a sandwiched elastomeric layer *

Link to our new paper outlining some strategies to steer interfacial debonding to a desired interface during a demolding process when the two interfaces in a 3-layer structure are equally weak. The article can be downloaded for free until July 5th.

Will appreciate feedback.

### Title updated

### Dear Ruobing,

*In reply to Untethered soft robots*

Dear Ruobing,

I agree with you, that pneumatic soft robots (at least at the moment) require bulky control and air supply when they are untethered. Our work helps to reduce the number of rigid parts by integrating valves into the soft structure. Also if all pneumatic control and air supply could be integrated into the soft structures, pneumatic soft robots would likely require some electronic components when they are intended to navigate in unknown territory (e.g. optical sensors). Magnetic, chemical, and light operated actuators have been demonstrated untethered, but they require the actuator to be close to their energy source (for example an electromagnet).

I believe that the best candidates for untethered operation are electrically operated actuators (e.g., dielectric elastomer actuators, HASEL actuators, thermal actuators), because batteries and control can be miniaturized, even when high voltages are required.

Best

Philipp

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