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The role of plastic strain gradients in the crack growth resistance of metals [code included]

Emilio Martínez Pañeda's picture

I hope some of you will find our recent paper in the Journal of the Mechanics and Physics of Solids interesting. The Abaqus user element (UEL) subroutine developed to implement strain gradient plasticity theories with energetic and dissipative higher order stresses (Gudmundson 2004, Fleck and Willis 2009) can be downloaded from

The role of plastic strain gradients in the crack growth resistance of metals

Emilio Martínez-Pañeda, Vikram S. Deshpande, Christian F. Niordson, Norman A. Fleck

Journal of the Mechanics and Physics of Solids, 126, pp. 136-150 (2019)


Crack advance from short or long pre-cracks is predicted by the progressive failure of a cohesive zone in a strain gradient, elasto-plastic solid. The presence of strain gradients leads to the existence of an elastic zone at the tip of a stationary crack, for both the long crack and the short crack cases. This is in sharp contrast with previous asymptotic analyses of gradient solids, where elastic strains were neglected. The presence of an elastic singularity at the crack tip generates stresses which are sufficiently high to activate quasi-cleavage. For the long crack case, crack growth resistance curves are predicted for a wide range of ratios of cohesive zone strength to yield strength. Remarkably, this feature of an elastic singularity is preserved for short cracks, leading to a severe reduction in tensile ductility. In qualitative terms, these predictions resemble those of discrete dislocation calculations, including the concept of a dislocation-free zone at the crack tip.


Shailendra's picture


Very nice work!

It is a wonderful initiative on your part to make the codes accessible to the public. This will accelerate future investigations and provide interesting directions. 




Emilio Martínez Pañeda's picture

Thank you, Shailendra. I hope that you find the code useful.

LIU WANG's picture

Thank you for sharing. They are very helpful

Emilio Martínez Pañeda's picture

Thank you Liu. If you encounter any problems while using the codes do not hesitate to drop an e-mail.

Mike Ciavarella's picture

...particularly the fact that with SGP we return to the square root singularity.  However, a limitation perhaps is that one needs material constants for hardening (there are models with dozens of them), material constants for SGP (one or more length scales), material constants for the cohesive model (although perhaps the fracture energy is what matters most).   With all these material constants, on one hand it is possible to fit any behaviour, on the other hand, we loose the simplicity and the convenience of modelling at all. Personally, I never liked HRR and elasto-plastic fracture mechanics, like anything that I don't find easy to explain to undergraduate students.

I like mostly Griffith brittle fracture, and full plastic model, and in between, some sort of fitting (or "asymptotic matching").

One hopes to keep a good trade-off, how many constants you have introduced?

Emilio Martínez Pañeda's picture

Thank you Mike. You are right, returning to the square root singularity is indeed an interesting finding. 

Regarding the number of parameters. In its simplest version, the model requires two elastic constants (Young's modulus and Poisson's ratio), two conventional plastic properties (yield stress and hardening exponent), two fracture parameters (cohesive strength and fracture energy) and one material length scale for strain gradient plasticity. All of them can be measured experimentally except for the cohesive strength, which can be chosen on physical grounds (e.g., theoretical lattice strength if we are modelling atomic decohesion). I agree with your vision and I try to keep it a simple as possible (Occam's razor principle) but sometimes the physical picture is complicated and we need to capture those mechanisms if we want to be predictive. As you say, it is a trade-off at the end of the day.

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