A generalized stress correction scheme for the Maxwell elasto-brittle rheology: impact on the fracture angles and deformations

Plante, Mathieu; Tremblay, L. Bruno

doi:https://doi.org/10.5194/tc-15-5623-2021

Articles | Volume 15, issue 12

https://doi.org/10.5194/tc-15-5623-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/tc-15-5623-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 15, issue 12

Research article

|

10 Dec 2021

Research article |

| 10 Dec 2021

A generalized stress correction scheme for the Maxwell elasto-brittle rheology: impact on the fracture angles and deformations

Mathieu Plante and L. Bruno Tremblay

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Final revised paper (published on 10 Dec 2021)
Preprint (discussion started on 02 Feb 2021)

Interactive discussion

Status: closed

RC1:
'Review of tc-2020-354', Anonymous Referee #1, 18 Mar 2021

Review of “A generalized stress correction scheme for the MEB rheology: impact on the fracture angles and deformations” by Plante and Tremblay (tc-2020-354).

This manuscript is a generally well written and easy to follow description of an extension to the MEB model of Plante et al (2020), but also has implications for other MEB implementations. This extension addresses two problems of the original model: large (numerical) error growth, which may be model-code specific) and too large fracture angles (which is probably not model-code specific. The new scheme allows to specify a more general correction of stress states that exceed the Mohr-Coulomb failure criterion. Sensitivity experiments in uniaxial compression illustrate that the parameterization indeed reduces the error growth and also reduces the fracture angles towards more realistic values. From the sensitivity experiments a preferred parameter set is determined. This is a very useful addition to the development of MEB rheology (and code) and should be published subject to minor revisions.

My main point of critique is that the manuscript is missing a bit of general introduction and a clear problem statement. To my mind, the manuscript can be improved by taking the reader more by the hand than is done. This only requires a few sentences here and there or maybe an additional paragraph, e.g.

(1) what do we expect from a “brittle” model in contrast to a “granular material”. The concept of “granular material/flow” is used often in the text, but it is not clear (from the text) how the brittle part of the model relates to that. Do we expect that a brittle model represents a granular material properly?

(2) state the issues with sea models and MEB in particular that are addressed in this paper in separate paragraphs. Now the angle-issue is mentioned in the middle of a paragraph that is introduced by: “The damage parameterization is relatively new, …”

(3) discuss if the new scheme can be also useful for other implementations of MEB (e.g. neXtSim)

There are some technical issues (figure referencing and captions) detailed below.

The points below sometimes repeat my main points.

page 1

Abstract: General background and (more importantly) a clear problem statement is missing from the abstract. E.g., the problem of too large angles is not stated and the error growth is also only mentioned as the target of the new parameterization. It would help to have more context here already (1-2 extra sentences).

l3: any correction path: unclear “any”

l18: significantly: repetition

l22: the presence of and deformations along LKFs

page 2

l30: Hunke, 2001: not sure if this is an appropriate reference (for what)?

l44: “The fracture angle simulated by the MEB and standard VP models” It will be easier to follow, if you dedicate a separate paragraph (or at least an introductory sentence to this paragraph) to the fracture angles as a problem statement before describing what VP and MEB models do wrong.

ll60 I think that the problem statement is not clear enough. Unless you are very familiar with the details of the implementation of MEB models, it’s not clear where Plante et al (2020) had numerical difficulties and if this is specific to their implementation. It should be clear if this will also be of value for, e.g. neXtSIM, or Dansereau et al.

Also the fracture angle problem is somewhat buried in the introduction and should be more prominent, because the paper devotes a large part to this.

page 3

l62: (Sulsky and Peterson, 2011) fix parentheses

l81: (Plante et al., 2020) fix parentheses

page 4

l96: is -> in

l104: “resulting in dominant elastic component”? not clear, something missing?

page 5

ll117: maybe put \mu, \phi, c into Fig1 for better illustration?

page 6

eq 16: where does the “some algebra” start from? Maybe add a little more explanation here to guide the reader.

l148: “something that is not possible in the standard parameterization otherwise \Psi …” please rephrase.

page 7

l166: on -> of

page 8

l208: asymmetry factor: not immediately clear why this measures error. I assume that you expect perfectly symmetric solutions about the center line, but I think that this needs to be explained.

The same is true for “damage activity”, what do you want to use this for and how does this diagnostic achieve that.

page 8

eq.26/27. the notation is a bit unusual and looks a little like (pseudo-) code. Why not use standard indexing as one would expect in a maths text?, e.g. \left(\sigma_{II}\right)_{n_x-i,j}

page 9

l235: 0.29 N/m? units?

l243: “mostly elastic with divergence along the fracture line” Where do we see that divergence? In Fig3 I mostly see negative divergence = convergence.

l248/9 The references to figure 4 are not correct. There is no Fig 4i, then it’s not clear from the caption, what we are seeing in color (damage?). It would help to add the timing in the plot (maybe top right or bottom left of rhs column).

page 10

l251: here and everywhere else: Units should NOT be in italics).

l254: 10−6 Nm−2 (unit not in italics): in 4.1 it was 1e-8!! In Fig5 it seems to be 1e-8 as well.

l254:are -> is

l254: “damage error amplification ratio R” maybe refer to equation 23 here?

l258: indicate -> indicates

In Figure 6 the panels for eps_asym and R_max are exchanged wrt to Figure 5. Why confuse the reader?

l263: “the production of” could be removed

l264: I would argue for \gamma \ge 0 the improvement is significant (including 0). But the asymmetry also grows for \gamma > 0 and only for values > 45 it seem to stay low. Why not discuss that here?

page 11

l286: “Based on these results, we suggest the use of a correction path that is normal to the yield criterion (\gamma = arctan \mu, see black points in Fig. 9).”

my say, \gamma = \phi in this case, (isn’t it)?

page 12

l331: “are robust to the exact” -> are not sensitive to the exact, are robust with respect to the exact …

l335: reach -> reaches

l335: elastic wave are however no-longer -> elastic waves, however, are no longer …

l344: in -> is

page 13

l349: the uniaxial -> a uniaxial

l351: not sure if “post-fracture” (or pre-fracture) is grammatically correct. I would use “after (and before) fracture” in most places in this manuscript

l352: “contrary to laboratory experiments of granular materials and satellite observations of sea ice.” A short discussion about to what extent we expect granular behavior in an MEB model seems in place (not here in the conclusions but somewhere in the introduction?), in order to understand if this is an encouraging or a discouraging result

l361: “the production of” remove, see above

page 21

Figure3: miximum -> maximum/minimum?

page 22

Figure 4. What are the meaning and the units of the color scale? Is this for the control simulation only?

page 27

Fig9: “The theoretical fracture angle from the Mohr-Coulomb and Roscoe theories are indicated by dashed and dash-dotted lines for reference.” something like this, could also be useful in Fig7.

Citation: https://doi.org/10.5194/tc-2020-354-RC1
- AC2: 'Reply on RC1', Mathieu Plante, 22 Jun 2021
  
  The comment was uploaded in the form of a supplement: https://tc.copernicus.org/preprints/tc-2020-354/tc-2020-354-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/tc-2020-354-AC2
RC2:
'Comment on tc-2020-354', Anonymous Referee #2, 31 Mar 2021

The comment was uploaded in the form of a supplement: https://tc.copernicus.org/preprints/tc-2020-354/tc-2020-354-RC2-supplement.pdf

Citation: https://doi.org/10.5194/tc-2020-354-RC2
- AC3: 'Reply on RC2', Mathieu Plante, 22 Jun 2021
  
  The comment was uploaded in the form of a supplement: https://tc.copernicus.org/preprints/tc-2020-354/tc-2020-354-AC3-supplement.pdf
  
  Citation: https://doi.org/10.5194/tc-2020-354-AC3
RC3:
'Comment on tc-2020-354', Anonymous Referee #3, 23 Apr 2021

In this paper the authors introduce a modification of the MEB rheology in the form of a generalized damage parameterisation. They then proceed to test this new parameterisation using an idealised uniaxial loading setup. They find that the new parameterisation influences the resulting fracture angle, bringing it in the range of observations. The paper is well written and clear, using good English and sentence structure, and a logical flow from section to section and paragraph to paragraph.

The introduction of a modification of the MEB rheology is a niche topic, but potentially an important one and certainly one relevant for publication in the Cryosphere. As it stands, the paper has some faults I would like the authors to address. I expect they can do this adequately and that the resulting work will be fit for publication in the Cryosphere.

Major comments:

It is not clear why the authors are proposing this addition to the MEB. Is it numerics or physics, or something else? You say something general at the start, but it’s vague and really only says what your modification does, not why you want to do it in the first place. This point should be crystal clear and guide the entire paper. Ideally the authors should say something like: “we want to introduce this scheme because we know it represents better the physics (and is incidentally better for the numerics). We see this by looking at the fracture angles (or some other measure)”. Such a statement at the top would make this paper very strong. An admittedly overly harsh evaluation of the current state is that the authors change something for dubious reasons and get a different response – so why should we care? Is this the right result, but for the wrong reasons? I don’t think that’s a fair assessment, but unless the motivation is clearer it will be the impression a critical reader gets.

Related to this lack of clear focus, I find it difficult to understand why you do the experiments that you do, so reading sections 4 and 5 is more demanding of the reader than it need be.

I also question the fact that the authors don’t introduce heterogeneity into their model. They even point out themselves that it “is responsible for much of the brittle material behaviour in progressive damage models” and indicate that the residual errors are not important in a heterogeneous field – which is what MEB is supposed to give. This choice needs to be much better justified than is currently done.

Finally, there's almost a hostile tone towards the MEB rheology in the discussion and conclusion section. The authors are practically gleeful in pointing out various faults of the model that are not relevant to the modifications they propose. It is of course fine to point out the faults of MEB - which apparently are plentiful – but the way it is done here borders on un-professional, in my opinion.

Minor comments:

L16: The formulation makes it sound as if leads and LKFs are interchangeable, but they are not.

L120: Shouldn’t the cohesion be a function of resolution (see Weiss, 2007)? If that’s the case, how do you get the same value from large scale and the lab?

L135: What’s the physical justification for proposing this generalised stress correction?

L222: Mohr-Coulomb and Roscoe theories both concern granular materials, but hear we’re dealing with the fracturing of a solid. Are they still valid? Please elaborate.

L248: There’s a lot of information in figure 4 and the reader needs more help in deducing why you created it and what it’s supposed to tell us.

L276: A reference to the contrasting results is needed.

L291: A reference for what is typical for granular material is needed (a textbook will suffice).

L316: This entire paragraph is a bit up-side-down to me. You start by saying the MEB is not good enough, for various reasons (begging the question of why you use it in the first place, actually) - and then you say how your new addition will not save it. A more natural way to write this is to first say that although the decohesive stress tensor can do some things it cannot fix everything, including etc.

L329: This paragraph is off topic, discussing experiments not introduced before and not relevant to the introduction of the decohesive stress tensor. Please remove.

L353: Now I’m confused, did you want to solve the ridging problem by introducing the decohesive stress? Again, a more natural way to present your results would be to first state what works and then what remains.

Citation: https://doi.org/10.5194/tc-2020-354-RC3
- RC4:
  'Amendment to RC3', Anonymous Referee #3, 26 Apr 2021
  
  Please consider this an amendment to my review, RC3.
  I now realise that the paper is fundamentally flawed and should be rejected/withdrawn.
  The authors propose that super-critical stress be relaxed onto the failure envelope via a path different from the shortest distance to the origin in (σ_I, σ_II) space - as per figure 1. In order to do so they propose calculating the damage factor Ψ via equation (17) or (25, which has a typo). This, however, will not have the desired effect, because Ψ scales d, which then scales all the components of σ equally. Please consider equations (12), (8), (9), and (5) to see how changing d affects σ.
  In other words, as long as d is a scalar, and not a tensor, then any form of Ψ will reduce the stress towards the origin - Ψ only determines how fast this happens.
  For a more concrete example consider the stress change σ' -> σ_c in figure 1a. In order to change the stress in this manner, we need to reduce the shear stress but increase (in absolute value) the normal stress. This cannot be done by increasing d.
  
  A generalised stress correction scheme, therefore, requires d to be a tensor, so that different components of the stress can change differently. This is not done here, and the scheme proposed simply does not do what the authors claim it does. Instead of taking a different path to the failure envelope, the proposed scheme takes the same path as the standard scheme but increases the damage too little for the stress to cease being super-critical. The proposed scheme is thus not usable and a paper discussing it is not warranted.
  
  Citation: https://doi.org/10.5194/tc-2020-354-RC4
  - AC1: 'Reply on RC4', Mathieu Plante, 29 Apr 2021
    
    It is correct that the stress correction Ψ is a scalar, but the reviewer missed the fact that we use a decohesive stress tensor (σ_D) to bring the stress back onto the yield curve, and which depends on the correction path angle (see Eqs 18-20, discussion on L150-161, Fig 1, and derivation below).
    Therefore, the corrected normal stress invariant reads:
    σ_Ic = Ψσ'_I + σ_ID ,
    rather then σ_Ic = Ψσ'_I , which was the reviewer’s concern.
    The components of the decohesive stress tensor are then retrieved using the yield criterion, such that:
    (c - σ_IIc )/μ = Ψσ'_I + σ_ID ,
    or,
    σ_ID = (c - σ_IIc )/μ - Ψσ'_I ,
    Using the relation σ_IIc= Ψσ'_II(by construction, see Fig. 1), we get Eq. 19 (where there was a typo):
    σ_ID = (c - Ψ(σ'_II + σ'_I) ) /μ ,
    In which Ψ depends on the correction path angle. We can provide a more complete derivation in this thread if desired.
    
    Citation: https://doi.org/10.5194/tc-2020-354-AC1
- AC4: 'Reply on RC3', Mathieu Plante, 22 Jun 2021
  
  The comment was uploaded in the form of a supplement: https://tc.copernicus.org/preprints/tc-2020-354/tc-2020-354-AC4-supplement.pdf
  
  Citation: https://doi.org/10.5194/tc-2020-354-AC4

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Reconsider after major revisions (further review by editor and referees) (05 Aug 2021) by Yevgeny Aksenov

AR by Mathieu Plante on behalf of the Authors (16 Sep 2021) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (18 Sep 2021) by Yevgeny Aksenov

RR by Anonymous Referee #1 (30 Sep 2021)

ED: Publish subject to technical corrections (01 Nov 2021) by Yevgeny Aksenov

Dear Authors,

On behalf of the journal I am pleased to inform you that the manuscript tc-2020 "A generalized stress correction scheme for the MEB rheology: impact on the fracture angles and deformations" by Mathieu Plante and L. Bruno Tremblay can be published after the technical correction listed in the second round of reviews and given at the end of this message. Please follow the journal recommendations how to proceed with the corrections of the manuscript. Thank you very much for choosing to publish your research in TCR.

Yevgeny Aksenov (TCR editor)

Second review of “A generalized stress correction scheme for the MEB rheology: impact on the fracture angles and deformations” by Plante and Tremblay

The author adequately replied to all of my comments, in particular, my main concern of general introduction and problem statement has been addressed. During rereading the revised manuscript I noted a few typos and made a few suggestions, listed below.

Typos/suggestions:

page 1
l3 suggestion: The conventional MEB

l12: longer-term without hyphen? better: in the long term

page 2
l32 scheme -> schemes

l39: framework -> frameworks, please check the sentence, what does “that influence” refer to? Diversity, sea-ice theologies?

l48: produces -> produce

page 3
l67: have -> has

l72: suggestion: other neXtSIM specific characteristics, such as the Lagrangian numerical scheme,

page 6
l150: eq: 12, should this be \left(\frac\sigma_{xx}-\sigma_{yy}}{2}\right)^2

page 9
l235: shouldn’t that be \sigma_{I}’ < 0?

l249: (Ringeisen et al., 2019) fix parentheses

page 10
l268: “of the solution vector”, isn’t is the residual vector?

l279: (nx,ny) really picky, but I would still write $(n_x,n_y)$ one $(i,j)$ in the text.

page 11
l290: analog -> analogous

page 12
l355: I think you mean this: amplification ratio R (see Eq. 27) that reaches ~20 in the control simulation (Fig. 5c).

page 15
l420: increase->increased?

Section 7 conclusion with upper case C

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AR by Mathieu Plante on behalf of the Authors (05 Nov 2021) Author's response Manuscript

Short summary

We propose a generalized form for the damage parameterization such that super-critical stresses can return to the yield with different final sub-critical stress states. In uniaxial compression simulations, the generalization improves the orientation of sea ice fractures and reduces the growth of numerical errors. Shear and convergence deformations however remain predominant along the fractures, contrary to observations, and this calls for modification of the post-fracture viscosity formulation.