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 

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

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.

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?