the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 04 Aug 2026
Seasonal increase of the mixed-mode fracture toughness of faceted-crystal weak layers
Abstract. Mixed-mode fracture toughness of natural snow weak layers has rarely been measured directly, yet it is a key parameter governing dry-snow slab avalanche release on inclined, snow-covered slopes. Most prior estimates relied on assumed elastic properties of slab and weak layer rather than direct measurements. Here, we present field-based fracture experiments investigating the seasonal evolution of the mixed-mode fracture toughness of a natural weak layer composed of faceted crystals. We introduce a novel method that combines digital image correlation of snow displacement fields with inverse modeling to derive effective elastic properties of both the weak layer and the overlying slab. These experimentally constrained elastic properties provided more robust estimates of weak layer fracture toughness than previously possible. Results show a clear increase in fracture toughness over the season from 0.29 ± 0.042 J m-2 to 0.67 ± 0.06 J m-2 in mode I and from 0.10 ± 0.022 J m-2 to 1.0 ± 0.1 J m-2 in mode II. The effective elastic modulus of the weak layer evolved from approximately 0.63 ± 0.094 MPa at the beginning of the measurement period to 0.40 ± 0.046 MPa later in the season, while slab elasticity increased from 16 ± 1 MPa at the start to 40 ± 2 MPa. Furthermore, micro-computed tomography and manual snow profiles revealed that, although the weak layer density remained nearly constant, the specific surface area decreased and the mean local ice and pore thickness increased, as did the grain size. These findings suggest that the increases in both mode I and mode II fracture toughness are associated with grain size growth and growth of the bonds between the grains. The study further demonstrates that displacement-constrained inverse modeling provides robust snow elastic properties, improving fracture-mechanical estimates of avalanche release processes.
Competing interests: At least one of the (co-)authors is a member of the editorial board of The Cryosphere.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-3554', Anonymous Referee #1, 19 Oct 2026
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RC2: 'Comment on egusphere-2026-3554', Pascal Hagenmuller, 28 Oct 2026
Summary:
Knowledge of the fracture toughness of snow weak layers is essential to understand and predict dry-snow slab avalanche release, yet direct measurements of this quantity remain scarce. The authors conducted a large number of mixed-mode fracture tests (an extension of the propagation saw test) on a tilting device that allows the slope angle to be controlled, while the displacement of the sample side wall was monitored with a camera and digital image correlation. The tests were repeated at different slope angles and at three stages in the life (20 days here) of a single weak layer of faceted crystals at a site above Davos. Of these tests, 74 were processed with DIC and 41 were retained for the full inverse analysis. The authors assume that the slab–weak layer system behaves elastically and that the dimensional reduction of the system to a Timoshenko beam on a Winkler foundation applies, i.e. that the Python package weac is applicable. They fitted the slab and weak-layer elastic moduli so that weac reproduces the displacement field measured before crack propagation, and then used these quantities to invert the measured critical cut lengths into weak-layer fracture toughness. The authors observed an increase of the slab elastic modulus (16 to 41 MPa), a decrease of the weak-layer elastic modulus (0.63 to 0.40 MPa) and an increase of its fracture toughness (0.29 to 0.67 J/m2 in mode I, 0.10 to 1.0 J/m2 in mode II). In addition, a few tomographic measurements were conducted on the weak layer, but they are presented as rather qualitative and remain inconclusive.
General assessment:
The authors present very interesting data from systematic and, as far as possible, controlled mixed-mode fracture tests performed on the same weak layer (presumably) during 20 days. This represents an important amount of fieldwork and can fill a gap in current knowledge. Inverting the fracture toughness from the critical cut length by fitting the measured elastic displacement field, rather than assuming a density-elasticity relation, is, in principle, a step beyond what has been done so far. However, the conclusions and the main messages of the paper are overstated and not supported by the data presented. (a) In particular, the seasonal increase of fracture toughness (title), which sounds plausible, may be a direct consequence of the decreasing weak-layer elastic modulus, and the latter appears non-physical to me. (b) Moreover, the discussion and conclusions on the microstructural evolution captured by tomography seem far-fetched, as they are not supported by consistent data in sufficient number. (c) The claimed superiority of fitting an effective modulus on the displacement fields recorded before crack propagation, over using an established density-elasticity parametrization, is asserted but not demonstrated. I therefore recommend either accurate and convincing counter-arguments and discussion, or a profound cutting and rewording of the main messages.
Main comments:
a) The central claim, a "pronounced seasonal increase of fracture toughness », is not supported. It is probably conditional on the fitted weak-layer modulus trajectory. In the weac inversion, the measured critical cut length constrains only a combination (equifinality) of weak-layer compliance E_wl and fracture energy w_f. At a given critical cut length r_c, slab properties and loads, a decrease in E_wl necessarily translates into an increase in the inverted w_f. Here, the derived E_wl decreases and may automatically lead to an increase of w_f. Moreover, a drop in the modulus of a weak layer that strengthens under isothermal metamorphism and overburden seems physically counterintuitive. The part of the signal carried by the measured increase of critical cut lengths is also entangled with systematic differences in slab thickness (Series III has thinner slabs). The season also corresponds here to ~20 days. Please repeat the inversion with E_wl constrained across the three series and report how much of the « seasonal » increase of w_f survives. Unless it largely does, the title, abstract, main text, and conclusions must be reworded.
b) The microstructural data are irrelevant. Section 4.3 honestly qualifies the micro-CT results as "indicative rather than conclusive". However, these limitations do not percolate to the abstract and conclusions, which state examples as findings ("the weak layer density remained nearly constant", toughening "consistent with" bond growth). Besides, the representativeness issue is more severe than acknowledged. The weak layer is only weakly expressed in the stratigraphy, making the exact sampled horizon uncertain from one extraction to the next. The site exhibits spatial variability (spatial and temporal sampling are also correlated). The manually selected ROI (approx. 1 cm3), taken once or twice per date from different locations of a variable layer whose identification is itself uncertain, cannot represent the layer, nor its temporal evolution. The microstructural patterns shown in Fig. A3 obviously show this inconsistency. Moreover, the reported grain-size increase (0.5-0.75 mm to 1.25-1.5 mm within three weeks, apparently from manual measurements that are not described) is difficult to reconcile with coarsening kinetics at these temperatures, and rather suggests that successive samples probed different material than an evolving one. At this point, I recommend removing the microstructural data and interpretation.
c) The authors claim but do not demonstrate that fitting the elastic properties is superior to using a density parametrisation. The inverse derivation of the elastic properties is presented as a decisive step beyond assumed density–elasticity relations, as if the fit delivered the true moduli. The manuscript does not demonstrate this, and several elements indicate the opposite. (i) For the slab, the fit does not emancipate from a density parametrisation at all: it fits the exponent gamma of the very same power law E = E0(rho/rho_ice)^gamma on three density-cutter measurements per test (afterall it does not deviate from Gerling parametrisation, exponent 4.6 to 5 compared to Gerling’s 4.4). (ii) The fit quality does not support the claim: the model overestimates the horizontal displacements over the entire slab (Fig. 7). (iii) The fitted values are not material quantities: the weak layer is snow, not a material of its own class, yet its fitted modulus corresponds to no snow at the measured density (applied to the density parametrisation of the slab, 0.4 MPa for the weak layer means a weak layer of density of 110 kg/m3, instead of 265 kg/m3). In effect, the two free parameters act as a sink that absorbs everything the model does not represent, such as the substrate compliance absorbed into point Winkler elasticity (here, a 10 cm slab over a 10 cm compliant weak-layer/substratum is particularly challenging for the underlying assumptions). A comparison to the simple density parameterisation would help quantify the added value of the methodology. If not, these statements need to be substantially qualified.
d) The reported uncertainties on G_Ic and G_IIc are not consistent with the scatter shown in Fig. 12, and the mixed-mode envelope does not support any conclusion. The fitted envelopes do not pass through the data points or the uncertainty bars (Fig. 12). The scatter is large (R-squared = 0.222). Still, Table 3 reports G_Ic and G_IIc with uncertainties below 10-20 %, and the interaction exponents n and m with no uncertainty at all. With this scatter, the exponent model is over-parameterized: very different envelope shapes (e.g., a circle) would fit the data equally well, and the extrapolated G_IIc depends strongly on the assumed shape. So the statement that G_IIc is smaller than G_Ic for Series I, or larger for Series II and III, cannot be concluded at the reported confidence. This also questions the purpose of the tilting design itself. Tilting does not appear to be the key to the measurement, and a total energy release rate per test would carry the same information with fewer assumptions. Please report uncertainties on n and m, a model comparison against simpler interaction laws. Any conclusion on the relative magnitude of G_Ic and G_IIc should be removed.
e) No SMP during the season? The SMP is said to be used during the field campaign (Figure 1, legend). However, it appears only as a way to suggest spatial homogeneity. Showing the hardening of the slab and the weak layer over time would help to strengthen the conclusions of the paper.
f) Data availability. The publication of the processed DIC displacement fields is a positive point of the paper: I looked at one .nc file as an example (thanks to editor work), the data are well described and in an exploitable format. They describe the DIC but the fields related to the weac fit are absent or set to NaN.
Pascal Hagenmuller
Citation: https://doi.org/10.5194/egusphere-2026-3554-RC2 -
CC1: 'Comment on egusphere-2026-3554', Peng Yue, 02 Nov 2026
This is a public review report, stating that we have no conflict of interest.
Full title: Seasonal increase of the mixed-mode fracture toughness of faceted-crystal weak layers
We are glad to review this manuscript. It is a very interesting work that has carried out a large number of Propagation Saw Test (PST) experiments and studied the fracture toughness under the mixed mode. This paper also obtained some results. Under both compression and shear fracture modes, test the fracture toughness with a natural weak layer composed of faceted crystals. This is an excellent job and the content is also suitable for The Cryosphere. However, we believe that the manuscript still needs improvement. We have provided specific comments as follows:
Specific comments:
- You mentioned the word "seasonal" in the title and content, but it is not a quantitative term, so it is not precise to describe the fracture of the weak layer. Your experimental period is three weeks, but the word "seasonal" is misleading, making people think it is a multi-year seasonal cycle experiment.
- In the Abstract section (5-6 lines), you describe "We introduce a novel method..." However, we believe that this method already exists or there are related measurement techniques available. For instance, DIC measurement technology has been used to track the variation of the displacement field (Bergfeld et al., 2021), the tilting device has also emerged (Adam et al., 2024) and van Herwijnen et al. (2016) has also proposed a method for measuring the effective elastic modulus.
- In Lines 13-14, you said that “the increases in both mode I and mode II fracture toughness are associated with grain size growth and growth of the bonds between the grains.” but in the main text (lines 301-302), You said again that “However, with only four sampling days and no statistical analysis, these observations should be regarded as indicative rather than conclusive.” This is contradictory. What should be obtained in the abstract is a conclusive and definite result.
- In the Introduction section, we feel it is insufficient. The relevant results of some recent super shear cracks of very long PST (Trottet et al., 2022, Bobillier et al., 2024) and the super shear phenomena discovered in field experiments (Simenhois et al., 2023) have not been described at all. The author should, when presenting the current situation, evaluate the super shear model, including its advantages and disadvantages. Since other authors have indicated that PST does not represent the fracture process of real avalanche formation (contact us for discussion), large-scale or extensive experiments may be closer to real avalanches.
- Figure 1. the content of the picture (figure 1b) does not match the description. The picture shows what is like a schematic diagram of PST, but describes it as "field experiments including extracting, tilting and filming the sample". You should use on-site experiment pictures or pictures of the experiment in progress instead to make it clearer. Of course, I can understand what you mean.
- Section 2.2 Mixed-mode fracture test. The model is the standard PST model (1m*0.3m), which is the common PST model we see. However, recent articles have shown that standard PST cannot fully demonstrate the dynamic propagation of cracks, with a PST length of at least 5 meters or more. For super-shear cracks, the supercritical length even exceeds 5 meters. What's your opinion, or what's your evaluation of these results?
- The Result section, you chose thefaceted-crystal weak layer, but most of the previous weak layers were surface hoar or depth hoar.Section 3.1 The critical cut length increases with the increase of the tilt angle and, of course, is related to the direction of the cut weak layer. Both show an increasing trend, but others have reached the same or similar conclusion (Adam et al., 2024).In addition, have you made any new discoveries or found out the dependence between the critical length and the angle?The critical length is also a quantity that I pay close attention to.Because the early anticrack model (Heierli et al., 2008) held that the critical length was independent of the slope angle, later Gaume et al. (2017) obtained through DEM that the critical crack length decreased with the increase of the angle. Subsequently, there were a large number of conclusions, including the super shear model, also supports this conclusion.However, Adam et al. (2024) concluded from the classical PST experiment that the critical crack length increases with the increase of the angle.This is a series of completely opposite conclusions. What are the reasons for these differences?
- We observed that the highest elastic modulus of the snow slab measured in your experiment reached 66 MPa, and the average was also 40 MPa. This is already a rather high elastic modulus of the snow slab. The snow layer might be dense and compacted.The elastic modulus of loose snow or fresh snow is lower than this.van Herwijnen et al. (2016) experiments measured the elastic modulus ranging from 0.08 to 34 MPa, with the elastic modulus mostly concentrated around 5 MPa and generally lower than 10 MPa (van Herwijnen et al., 2016; Fig.8).The elastic modulus in the numerical model is also basically at 5 MPa (Trottet et al., 2022; Bobillier et al., 2024; Bergfeld et al., 2022). What is the reason for this difference? Is it related to the loading device you placed on the upper surface of the snow slab?
- Section 3.3 Mixed-mode interaction law for fracture toughness this relationship has been proposed by Adam et al. (2024), and the relationship between the critical energy release rates of mode I and mode II are also quite similar, which has appeared in the published article. Therefore, the results of these studies are merely increments. What are the differences between you two?
- In the Discussion section, we hope to conduct a comprehensive analysis of the changes in different critical lengths under the angle, such as the different conclusions drawn in the early stage as the angle decreases and in the current stage as the angle increases. This is quite important, and we hope to present a comprehensive analysis.In addition, the reasons for the high elastic modulus of the snow slab layer and whether it is related to loading should be discussed in detail.In terms of section 4.4, limitation analysis, we would like you to analyze the differences between the typical model and the super shear model. Does the claim they made that the short PST cannot fully exhibit crack propagation exist?
Other minor issues:
- The citation of references is not standardized and does not need to be included when applying (e.g.). There are many places in the introduction section that need to be corrected.
- The units are not uniform. For instance, in Picture 3, the unit of length is millimeters, while in the main text, it is meters (89 lines). It is suggested that the author standardize the units.
Peng Yue
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- 1
This study presents field-based mixed-mode fracture experiments on a natural faceted-crystal weak layer, combining a mixed-mode fracture test with digital image correlation (DIC). They combined the displacement field of the experiment with inverse modeling to derive effective elastic properties of both the slab and the weak layer as well fracture toughness in mode I and II. The authors document a clear seasonal increase in mixed-mode fracture toughness over a three-week measurement period, supported by microstructural characterization using micro-computed tomography.
The main novelty and contribution of this paper reside in its experimental methodology and the dataset it provides. By constraining elastic properties directly from displacement fields rather than relying on assumed values, the authors reduce a key source of uncertainty in fracture toughness estimates — an important methodological advance for the field. The resulting dataset constitutes, to my knowledge, the first mixed-mode fracture toughness envelope for a faceted-crystal weak layer, and will serve as a valuable benchmark for future modeling studies.
The manuscript is well-written and the methodology is sound. My main revision is to add a bit of comparison with previous snow mechanical studies to strengthen the discussion. I recommend minor revisions, with the following specific comments to be addressed by the authors.
Specific comments
Line 97: Consider adding to the sentence :” the weak layer was cut, either in the downslope or upslope direction”
Line 139: I would add the symbol E_wl after introducing the weak layer elastic modulus for clarity.
Line 153: Specifying the version package of Scipy and Weac version is enough for scientific reproducibility.
Line 169: It would be helpful to clarify whether the type of distributions (uniform or gaussian) used in the Monte Carlo sampling. This will affect if the sampling is center around the initial guess or explore across the whole range of values. Since the choice of sampling distribution can influence the resulting parameter distribution and the derived confidence intervals, I suggest the authors clarify this point. and would improve the reproducibility of the method.
Line 174: Propagating measurement uncertainties into the physical model is important but I’m wondering where are these uncertainties range come from? Maybe add details on how they were derived.
Line 175: The Poisson’s ratio of the weak layer is stated to be 0.25, but it is unclear whether the same value was assumed for the slab. I suggest clarifying whether the slab Poisson’s ratio was also fixed at 0.25 in Eq.3.
Figure 8: The figure caption refers to “Bergfeld’s density parametrization” for the slab elastic moduli. But it was stated in the methods that the density values shown are derived from the Gerling et al. (2017) scaling law.
Figure 13: Given that the information conveyed by this figure is less significant and well described in the text at Lines 246–248, I question whether it is necessary to include it as a main-text figure. I could also argue that the reduction in SSA is rather small between the Series, maybe tone down a bit the statement in conclusion. However, I do acknowledge a more significant effect on the mean ice thickness between bonds.
The authors may wish to consider whether it could be moved to the Appendix, unless the visual representation provides additional insight beyond what is stated in the text.
Lines 301–302: The statement that the weak layer strengthens progressively over time would benefit from additional references. While the present study focuses on fracture toughness rather than strength, the temporal evolution of weak layer mechanical resistance is well-documented in the snow science literature. Adding references to studies on the strengthening of weak layers over time — for instance, through bond growth — would lend more weight to this claim and could potentially allow the authors to remove the final cautionary sentence of the paragraph, which currently somewhat undermines an otherwise well-supported conclusion.
Line 314: I suggest adding a statement noting that the elastic property values derived in this study are consistent with previously reported values in the literature using different methods. Estimating elastic properties, especially the weak layer, is rather difficult and a comparison will add support to the values reported in this study. This would help contextualize the results and further support the validity of the proposed methodology.