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Gradient-enhanced statistical analysis of cleavage fracture

Emilio Martínez Pañeda's picture

I hope some of you will find this work interesting. We present a new Weibull framework for cleavage fracture that incorporates strain gradient plasticity and can estimate the three parameters of Weibull-type models without any prior assumptions. A post-print is available at

Gradient-enhanced statistical analysis of cleavage fracture

Emilio Martínez-Pañeda, Sandra Fuentes-Alonso, CovadongaBetegón

European Journal of Mechanics - A/Solids 77, 103785 (2019)


We present a probabilistic framework for brittle fracture that builds upon Weibull statistics and strain gradient plasticity. The constitutive response is given by the mechanism-based strain gradient plasticity theory, aiming to accurately characterize crack tip stresses by accounting for the role of plastic strain gradients in elevating local strengthening ahead of cracks. It is shown that gradients of plastic strain elevate the Weibull stress and the probability of failure for a given choice of the threshold stress and the Weibull parameters. The statistical framework presented is used to estimate failure probabilities across temperatures in ferritic steels. The framework has the capability to estimate the three statistical parameters present in the Weibull-type model without any prior assumptions. The calibration against experimental data shows important differences in the values obtained for strain gradient plasticity and conventional J2 plasticity. Moreover, local probability maps show that potential damage initiation sites are much closer to the crack tip in the case of gradient-enhanced plasticity. Finally, the fracture response across the ductile-to-brittle regime is investigated by computing the cleavage resistance curves with increasing temperature. Gradient plasticity predictions appear to show a better agreement with the experiments.


Mike Ciavarella's picture

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.

Emilio Martínez Pañeda's picture

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.


Mike Ciavarella's picture

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

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?

Emilio Martínez Pañeda's picture

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).



Mike Ciavarella's picture

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?

Emilio Martínez Pañeda's picture

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.

Mike Ciavarella's picture

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 :)

Emilio Martínez Pañeda's picture

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? 

Mike Ciavarella's picture

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?

Emilio Martínez Pañeda's picture

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.


Mike Ciavarella's picture

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

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.

Emilio Martínez Pañeda's picture

Yes, I agree. And we have explored both the L_sgp/a and L_sgp/Rp non-dimensional groups in our recent JMPS paper ( Where L_sgp is the gradient length scale, a is the crack length and Rp is the plastic zone size. And I wish we didn't need L_sgp in our models but you need a length scale parameter if you want to capture a size effect. Nevertheless, I share your Occam's razor philosophy as well.

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