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# Comments

### fixed length scale

*In reply to Griffith should not bother*

In the pf-czm, the length scale can also be fixed to match the damage bandwidth (provided it is available), without sacrifying the critical strength.

### I agree

*In reply to Fracture is unsolved problem*

yes, I totally agree. This is what Prof. Bazant said in his speech for Timoshenko's medal when I was a visiting scholar in NU with him.

### I don't disagree with your

*In reply to cohesive fracture*

I don't disagree with your first statement, although I think that it is relatively easy to build generalized Ambrosio Tortorelly models such that the internal length to strength relation leads to a much smaller length.

I think that a classical confusion is in the applicability of brittle models in nominally brittle situations. Very few materials are perfectly brittle. For nominally brittle materials, the brittle approximation only makes sense at a large enough scale (typically quantified by the size of the structure compared to that of the cohesize or internal length. So even though concrete at the scale of a dam can probably be seen as a brittle material, at the scale of Winkler's experiments, I have strong doubts. So using this experiment to "demonstrate" that phase-field models, seen as approximation of brittle fracture, cannot properly account for nucleation is a dubious claim in my opinion.

In order to study nucleation in L-shaped sample, I would actually recomment the problem from Figure 2(c) in Gómez, F. J., Elices, M., Berto, F., and Lazzarin, P. (2009). Fracture of V-notched specimens under mixed mode (I+II) loading in brittle materials. Int. J. Fracture, 159(2):121–135 is a much better choice that Winkler's experiment.

### Fracture is unsolved problem

*In reply to Sure. But I do not think any*

Fracture is unsolved problem for a century. I doubt we will solve it soon :))) We are not alone in suffering. Fluid mechanicians are stuck with turbulence...

### Sure. But I do not think any

*In reply to By the way, you do not lose*

Sure. But I do not think any phase-field model with a a single damage variable is able to model various failure mechanisms simultaneously, like the compressive one you mentioned. If needed, it can be refined to include additional damage variables, but this is another story.

### Griffith should not bother

*In reply to Well, it is a bit more*

Griffith should not bother you. His theory was pioneering and I appreciate that, yet it contradicts experiments. Cracks are not "mathematical" - they have finite thickness and the characteriistic length should not go to zero.

### By the way, you do not lose

*In reply to fixed length scale*

By the way, you do not lose convergence in your theory because it is not realy phase-field in the sense of damage mechanics. Its a strange animal :))) If it is a sort of CSM then you seemengly need extra criteria for crack nucleation, branching, arrest... Your tension strength is one of such criteria. Do not be obsessed with it! :))) For example, a concrete cube in uniaxial compression fractures without having any tension stresses!

### Well, it is a bit more

*In reply to Well, then the characterisitc*

Well, it is a bit more complicated than this... Although Gamma convergence to teh Francfort-Marigo generalized Griffith energy requires that \ell converges to 0, Sicsic, P. and Marigo, J.-J. (2013). From gradient damage laws to Griffith’s theory of crack propagation. J. Elasticity, 113(1):55–74. showed that a propagating crack satisfies G=G_c for any \ell "small" compared to the domain size (back to the idea that Griffith's approximation makes sense at a scale larger than the cohesize length). This can actually be seen in Figure 1 of Tanné, E., Li, T., Bourdin, B., Marigo, J.-J., and Maurini, C. (2018). Crack nucleation in variational phase-field models of brittle fracture. J. Mech. Phys. Solids, 110:80–99.

### cohesive fracture

*In reply to Figure 3*

Dear Prof. Bourdin,

Thank you for your comments. The AT1 model is indeed able to deal with crack nucleation for most **brittle fracture**, provided the phase-field length scale is related to Irwin's characteristic length *l*ch. In this case, the phase-field length scale is generally small enough not polluting the crack path. However, this is **NOT** always the case even for brittle fracture -- sometimes it is not possible to reconcile both needs to match the critical strength and to be small enough; see the example of indentation fracture [Strobl and Seelig, 2020].

About concrete, unless the structure size is exetrmely large, e.g., dams, concrete cannot be brittle and some quasi-brittle behavior is inevitable. For such cohesive fracture, I think it is not easy to fit both crack paths and peak loads for the standard AT1/AT2 models. If I am wrong, please let me know.

Anyway, we benifit a lot from variational phase-field model for fracture you, Francfort and Marigo pioneered over 20 years ago. And we are basically engineering oriented and might be not so rigorous sometimes. Thank you for your great contributions to this community.

[Strobl and Seelig, 2020] Strobl, M., Seelig, T., 2020. Phase field modeling of hertzian indentation fracture. Journal of the Mechanics and Physics of Solids 143, 104026.

*In reply to Let me ask it simpler. Can*

Yes, provided that the regularization parameter \ell be chosen as sigma_c = \sqrt{3G_c E/8\ell} for the AT1 model.

The essence of [Tanné et al 2018] is to illustrate the dual nature of the phase-field approach: an approximation of Griffith criterion as \ell approaches 0, which inherits from botyh Griffith's strenght and weakness, and a gradient damage model properly accounting for strength and toughness (and the transition from one to the other) when \ell is fixed, given by the relation above. Note that the standard expression for the cohesive length is \ell_c = K_Ic^2/sigma_c^2, so up to a rescaling, the regularization parameter \ell is nothing but the standard cohesive legth, and the Griffith approximation makes sense when the size of teh structire is very large compared to this lebgth.

### Let me ask it simpler. Can

*In reply to I mean that for the AT1 model*

Let me ask it simpler. Can you create the stress-strain curve with a limit point (indicating strength) without introducing the strength explicitly as an additional constraint?

### I mean that for the AT1 model

*In reply to Thanks. Do you mean that*

I mean that for the AT1 model, the one dimensional response consists of an elastic phase (d=0) until a critical stress \sigma_c. Past sigma_c, a solution with d=cst (corresponding to negative rademing) satisfying the _first_ order optimality conditiuon can be constructed but can be shown to be "unstable" in teh sense that it does not satisfy second order optimality conditions. Further analysis sows that necessarily, in this condition, the maximum value of d must be 1, and a _stable_ solution (corresponding to the classical AMbrosio Tortorelli optimal profile for Gamma convergence) can be constructed.

For the AT2 model, the situation is slightly different as the elastic limit is 0. The soltion with homogeneous damage d=cst is stable in the positive hardening phase, but as before unstable in the negative hardening phase. Again, biffurcation towards fully localized state takes place.

I think that the exposition in Marigo, J.-J., Maurini, C., and Pham, K. (2016). An overview of the modelling of fracture by gradient damage models. Meccanica, 51(12):3107–3128. is quite clear (although technical).

### You impose this strength as

*In reply to yes, there is when sigma = ft*

You impose this strength as an extra constraint, and it does not come out of the constitutive law as in damage mechanics. No material instability.

### Figure 3

*In reply to Journal Club For April 2021: Variational phase-field modeling of brittle and cohesive fracture*

I strongly disagree with the statement that Figure 3 from ref 34 "demonstrates" that standard phase-field models cannot capture nucleation properly. As a matter of fact, [Tanné et al 2018], conveniently ignored in this post, makes a compelling case that this is not the case. If anythiong (and as noted in [34]), Figure 3 highlight the fact that the L-shaped experiment in concrete from Winkler thesis is a poor choice for the validation of a brittle fracture model, as representing concrete as a brittle material in a sample of 50cm x 50cm is inconsistent with the idea of a material length scale of the order of K_{Ic}/ \sigma_c, which ranges between 1 and 10 cm in typical concrete if memory serves me right.

Note that [Kumar et al., 2020] proposes an alternative approach to nucleation that does not require the introcudtion of a cohesize zone model, although at teh cost of teh variational nature of the problem.

References:

[Kumar et al., 2020] Kumar, A., Bourdin, B., Francfort, G. A., and Lopez-Pamies, O. (2020). Revisiting nucleation in the phase-field approach to brittle fracture. J. Mech. Phys. Solids, 142:104027.

[Tanné et al 2018] Tanné, E., Li, T., Bourdin, B., Marigo, J.-J., and Maurini, C. (2018). Crack nucleation in variational phase-field models of brittle fracture. J. Mech. Phys. Solids, 110:80–99.

### yes, you are right this time.

*In reply to Well, if it is CZM then you*

yes, you are right this time. Theoretically, things are more complicated and localization can occur anywhere. But numerically there always exist numerical errors and it is not necesary to introdue explicitly the imperfection as in other methods -- numerical errors trigs the localization interior somewhere if both ends are constrined to be free of damage (otherwise, localization occurs at either end).

### Hi,

*In reply to Well, if it is CZM then you*

Hi,

I did not read the thread in much detail, but I think that the confusion comes from the fact that the variational phase field theory is variational by nature (unless its variational nature has been destroyed by some ill-advised modification). Because the total energy is non-convex, the first order optimality conditions: stability with respect to displacement, i.e. static equilibrium, and stability with respect to the phase field variable, i.e. a "smoothed" version of $G=G_c$) are onle _necessary_ conditions. The long series of papers initiated by the work of Pham, Marigo and Maurini in 2010 studied higher order optimality conditions to highlight the fact that above a given loading, the solution with no damage d=0 or uniform damage (d = cst) is no longer a stable critical point of the phase field energy, and that _full_ localisation (which can be interpreted as crack nucleation) of the damage variable takes place.

### Thank you for the note,

*In reply to BFGS, quasi-Newton*

Thank you for the note, Emilio. You are right that BFGS monolithic algorithm is promising in solving the governing equations of PFMs.

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