A closedform model for layered snow slabs
 ^{1}University of Rostock, Chair of Lightweight Design, Germany
 ^{2}Technical University of Darmstadt, Department of Civil and Environmental Engineering, Institute of Structural Mechanics and Design, Germany
 ^{}These authors contributed equally to this work.
 ^{1}University of Rostock, Chair of Lightweight Design, Germany
 ^{2}Technical University of Darmstadt, Department of Civil and Environmental Engineering, Institute of Structural Mechanics and Design, Germany
 ^{}These authors contributed equally to this work.
Abstract. We propose a closedform analytical model for the mechanical behavior of stratified snow covers for the purpose of investigating and predicting the physical processes that lead to the formation of drysnow slab avalanches. We represent the system of a stratified snow slab covering a collapsible weak layer by a beam composed of an arbitrary number of layers supported by an anisotropic elastic foundation in a twodimensional planestrain model. The model makes use of laminate mechanics and provides slab deformations, stresses in the weak layer, and energy release rates of weaklayer anticracks in real time. The closedform solution accounts for the layeringinduced coupling of bending and extension in the slab and of shear and normal stresses in the weak layer. It is validated against experimentally recorded displacement fields and a comprehensive finite element model indicating very good agreement. We show that layered slabs cannot be homogenized into equivalent isotropic bodies and reveal the impact of layering on bridging with respect to weaklayer stresses and energy release rates. It is demonstrated that inclined propagation saw tests allow for the determination of mixedmode weaklayer fracture toughnesses. Our results suggest that such tests are dominated by mode I when cut upslope and comprise significant mode II contributions when cut downslope. A Python implementation of the presented model is publicly available as part of the "Weak Layer Anticrack Nucleation Model (WEAC)" software package under https://github.com/2phi/weac and https://pypi.org/project/weac.
Philipp Weißgraeber and Philipp Laurens Rosendahl
Status: final response (author comments only)

RC1: 'Comment on tc2022140', Anonymous Referee #1, 02 Nov 2022
Comments to tc2022140
This manuscript with the title of “A closedform model for layered snow slabs” presents a closedform analytical model for the purpose of investigating and predicting the physical processes that lead to the formation of drysnow slab avalanches. The reviewer found this manuscript has significant issues regarding fundamental methodology, the weak layer failure mechanisms, validation and conclusions. Therefore, the reviewer do not recommend publishing this manuscript considering the following points.
 Section 2 in the manuscript presents the proposed model using continuous laminated beam or plate theory to establish a kinematic model of snowpack. This model can only analyse continuous deformation instead of fracture phenomena of snowpack especially in the weak layer. It is well known that the weak layer failure is a mixed mode damage and fracture propagation, which should be a fundamental study in assessing snow avalanches. The reviewer would suggest authors reconsidering the methodology in investigating snowpack fracture using fracture mechanics approach together with laminated plate theory.
 Figure 1 show a schematic model of layered snowpack, in which a weak layer is modeled as an elastic foundation using two springs with stiffness K_{n} and K_{t}. But this manuscript did present how normal and shear stiffness are coupled in the weak layer. It was mentioned bending and extension coupled in snowpack body as a layered beam. But the failure is in the weak layer, studying coupled effects in the weak layer is much more important than the snow slab above the weak layer.
 Figure 4 shows some different snowpack cases with initial cracks. Fig. 4b is an upslope PST. This is a major case regarding snowpack propagation saw test (PST), some researchers have done experimental work and modelling simulation. The following published papers are for authors’ reference:
 Gaume, T. Gast, J. Teran, A. van Herwijnen & C. Jiang, Dynamic anticrack
propagation in snow, Nat. Commun. DOI: 10.1038/s4146701805181w.
 Gaume, A. van Herwijnen, G. Chambon, N. Wever, J. Schweizer, Snow fracture in relation to slab avalanche release: critical state for the onset of crack propagation, The Cryosphere 11 (2017) 217–228, https://doi.org/10.5194/tc112172017.
 Jiye Chen, Blair Fyffe, Dawei Han and Shangtong Yang, Predicting mixed mode damage propagation in snowpack using the extended cohesive damage element method, Theoretical and Applied Fracture Mechanics 122 (2022) 103567, https://doi.org/10.1016/j.tafmec.2022.103567.
Reference c summaries the weak layer’s fracture mechanisms in the upslope PST case. It is well accepted to recognise a mixed mode damage and fracture in the weak layer under compression and bending. This can also be seen from Fig. 4b, the left part of segment 2 is under compression and shear and the right part under tension and shear. The left part is subjected to both compression and bending in the same way in compression. The bending caused tension and compression have opposite effects on the right part. The left part of segment 2 would fail by compressive crushing and shearing. Once the weak layer fails by the mixed failure mode with crushing and shearing, the snow slab moves down due to gravity. Reference c proposed a new mixed mode damage criteria and successfully assessed the weak layer’s failure initiation and propagation. But this manuscript concluded that the upslope PST is a mode I crack (opening). This is questionable. This manuscript has not yet considered the micro damage phenomenon in the weak layer because of the continuous mechanics model used, which cannot reflect the reality of failure in the weak layer. Snow is a porous material with micro crystal construction, which can be easily crushed by compression.
 Equation 19 and Figure 16 present a total energy release rate in the weak layer: G = G_{1} + G_{2}. But this manuscript has not used the G for assessing crack propagation. Theoretically, total G is not total fracture energy criterion G_{c} in the mixed mode case, it contains contributions from G_{1c} and G_{2c} in a coupled model. This important point was missed by this manuscript.
 On page 9, the last sentence: “ Energy release rates obtained using weakinterface kinematics cannot capture very short cracks …. ”. This is a case from the kinematic model. However, if energy release rate obtained by fracture mechanics in weak layer, it can certainly capture any crack length. This has been done in standard fracture benchmarks of laminated composites in literatures.
 Section 3 in this manuscript presents validation. It compared the outcomes from the proposed analytical model with FEM based continuous deformation analysis regarding deformation and stresses. There is no usage of test results in validation. This is also questionable. Firstly, the modelling results from FEM based continuous analysis are not reliable when the FEM mesh with cracks, or without validation by test work, the FEM results would not be accepted to validate other new models regarding fracture issues.
 On page 12, the last paragraph, “ Experimental validations are challenging since direct measurements of stresses are not possible and displacement measurements require considerable experimental effort ”. This is truth. However, above references a and b reported critical initial crack length in PST experimental work, which causes crack propagation and sliding of snow slab above weak layer, and the critical initial crack length related bending moment can be used for validation if the critical load can be provided by the analytical model in this manuscript. But this manuscript failed at the validation in this way. Abovementioned reference c has completed a similar validation.
 In example section, manuscript presents a lot of information about stress and deformation distributions in snow slab in different cases. The reviewer thinks this information is less important in assessing snowpack fracture. The outcomes from this proposed model are not only difficult for validation and also in application for assessing snowpack fracture initiation and propagation. As a closedform solution, it is expected to provide precise failure response which shows a loaddeformation curve. Thus, it can clearly present elastic stage, damage point, damage accumulation and final crack point.
 This manuscript mentioned a basic concept of a failure mode in weak layer: mode I (collapse) or anticrack. Although this terminology was used by some researchers in their early work. The reviewer thinks that this terminology is not in the line of the basic concept of fracture mechanics about mode I: opening crack. The wording of collapse or anticrack would not be a right description of the failure mode in opposite way of mode I. Abovementioned reference c properly suggested a terminology of crushing damage by compression, and proposed a mixed mode damage criteria with crushing damage and shearing crack (mode II) to assess the failure in weak layer under mixed loading.
 A closedfrom should clearly present the failed load and residual strength when crack propagates through the failure response or loaddeformation curve. The reviewer thinks that this manuscript has not reach the purpose of investigating and predicting the physical processes that lead to the formation of drysnow slab avalanches.

AC3: 'Reply on RC1', Philipp Rosendahl, 04 Feb 2023
Dear reviewer, thank you for your comments and suggestions, which help us to improve our paper. Please find our response to each concern and remark of the review below:
Remarks:
1.) Section 2 in the manuscript presents the proposed model using continuous laminated beam or plate theory to establish a kinematic model of snowpack. This model can only analyse continuous deformation instead of fracture phenomena of snowpack especially in the weak layer. It is well known that the weak layer failure is a mixed mode damage and fracture propagation, which should be a fundamental study in assessing snow avalanches. The reviewer would suggest authors reconsidering the methodology in investigating snowpack fracture using fracture mechanics approach together with laminated plate theory.
We agree with the reviewer, that dry snowslab avalanche release and especially the weak layer failure is governed by mixedmode fracture processes. A fundamental problem of fracture mechanics is that the formation of new crack surfaces creates a new continuum mechanics boundaryvalue problem. When solved with finiteelement methods, this either requires solving a new boundaryvalue problem with new mesh or methods of representing the newly boundaries with additional degrees of freedom such as discontinuous ansatz function (XFEM), new field quantities (phase field), degradation of element stiffness instead of removal (damage mechanics and cohesive element approaches). The present model tackles this fundamental challenge by avoiding the need for discretization and numerical concepts altogether by using established concepts of structural mechanics that yield closedform analytical expressions that, in return, allow of nearrealtime solutions of the underlying boundaryvalue problems. This way, they enable fracture mechanical analyses by allowing for the realtime solution of many different boundaryvalue problems represented by different crack states or crack advancement.
We have demonstrated this type of analysis with concepts of fracture mechanics in previous works [Ros2020a, Ros2020b].2.) Figure 1 show a schematic model of layered snowpack, in which a weak layer is modeled as an elastic foundation using two springs with stiffness Kn and Kt. But this manuscript did present how normal and shear stiffness are coupled in the weak layer. It was mentioned bending and extension coupled in snowpack body as a layered beam. But the failure is in the weak layer, studying coupled effects in the weak layer is much more important than the snow slab above the weak layer.
The model considers weaklayer stiffnesses Kn and Kt independently. This allows for the consideration of isotropic weak layers (where Kn and Kt are calculated from Young’s modulus and Poisson’s ratio such that Kn and Kt are coupled through the latter) and for the consideration of anisotropic weak layers whose normal and shear stiffnesses are measured independently.
Weaklayer normal and shear deformations (not stiffnesses) are coupled in the model through their attachment to the slab. That is, weaklayer shear deformations introduce slab bending, which in turn causes weaklayer normal stresses.
3.) Figure 4 shows some different snowpack cases with initial cracks. Fig. 4b is an upslope PST. This is a major case regarding snowpack propagation saw test (PST), some researchers have done experimental work and modelling simulation. The following published papers are for authors’ reference:
a) Gaume, T. Gast, J. Teran, A. van Herwijnen & C. Jiang, Dynamic anticrack
propagation in snow, Nat. Commun. DOI: 10.1038/s4146701805181w.
b) Gaume, A. van Herwijnen, G. Chambon, N. Wever, J. Schweizer, Snow fracture in relation to slab avalanche release: critical state for the onset of crack propagation, The Cryosphere 11 (2017) 217–228, https://doi.org/10.5194/tc112172017.
c) Jiye Chen, Blair Fyffe, Dawei Han and Shangtong Yang, Predicting mixed mode damage propagation in snowpack using the extended cohesive damage element method, Theoretical and Applied Fracture Mechanics 122 (2022) 103567, https://doi.org/10.1016/j.tafmec.2022.103567.
Reference c summaries the weak layer’s fracture mechanisms in the upslope PST case. It is well accepted to recognise a mixed mode damage and fracture in the weak layer under compression and bending. This can also be seen from Fig. 4b, the left part of segment 2 is under compression and shear and the right part under tension and shear. The left part is subjected to both compression and bending in the same way in compression. The bending caused tension and compression have opposite effects on the right part. The left part of segment 2 would fail by compressive crushing and shearing. Once the weak layer fails by the mixed failure mode with crushing and shearing, the snow slab moves down due to gravity. Reference c proposed a new mixed mode damage criteria and successfully assessed the weak layer’s failure initiation and propagation. But this manuscript concluded that the upslope PST is a mode I crack (opening). This is questionable. This manuscript has not yet considered the micro damage phenomenon in the weak layer because of the continuous mechanics model used, which cannot reflect the reality of failure in the weak layer. Snow is a porous material with micro crystal construction, which can be easily crushed by compression.When we speak of mode I anticrack propagation we refer to the symmetric nature of the deformation field [Bro1999]. Because we consider anticracking, mode I corresponds to weaklayer crushing and collapse. This agrees with the reviewer’s understanding of the fracture process described above. Further details of this definition are given in a previous The Cryosphere publication by the authors [Ros2020b]. We will revise the manuscript to make this more clear.
4.) Equation 19 and Figure 16 present a total energy release rate in the weak layer: G = G1 + G2. But this manuscript has not used the G for assessing crack propagation. Theoretically, total G is not total fracture energy criterion Gc in the mixed mode case, it contains contributions from G1c and G2cin a coupled model. This important point was missed by this manuscript.
Of course we agree that mixedmode crack nucleation and propagation is governed by the interaction law of the mode I and mode II fracture toughnesses as discussed [Ber2023]. However, we are not aware of any experimental work by other authors that provides this relation. On the contrary, the magnitude of the mode I and II fracture toughnesses in pure modes are not well defined. We are currently working on new methods to identify mixedmode fracture envelopes of weak interfaces [Ada2022].However, it is not the aim of the present work to establish a novel failure criterion. We have proposed a mixedmode finite fracture mechanics approach for nonlayered slabs in an earlier work [Ros2020b] and we will incorporate the present model for layered slabs in this framework in the future.
5.) On page 9, the last sentence: “ Energy release rates obtained using weakinterface kinematics cannot capture very short cracks …. ”. This is a case from the kinematic model. However, if energy release rate obtained by fracture mechanics in weak layer, it can certainly capture any crack length. This has been done in standard fracture benchmarks of laminated composites in literatures.
We agree with the reviewer that this comprises a limitation of the model. This is the reason why we indicate this as the limitations within the chosen modeling approach. In the present case, the chosen kinematics do not capture extremely short cracks well. However, many established failure analysis concepts that use stress, strain, strainenergy density evaluation at a distance or the concept of finite fracture mechanics, do not make use of the localized quantities so that this limitation does not necessarily render a problem.6.) Section 3 in this manuscript presents validation. It compared the outcomes from the proposed analytical model with FEM based continuous deformation analysis regarding deformation and stresses. There is no usage of test results in validation. This is also questionable. Firstly, the modelling results from FEM based continuous analysis are not reliable when the FEM mesh with cracks, or without validation by test work, the FEM results would not be accepted to validate other new models regarding fracture issues.
The use of FEM results to compare two solution methods of the same boundary value problem of structural mechanics models is common practice [Zie2005]. When composed carefully, FE models are very well suited for boundaryvalue problems including cracks, notches, or other stress concentrators [Kun2013].
The FEM model used in the present work has been conducted carefully, including a mesh sensitivity analysis to rule out mesh effects. Since the focus of the present work is the development of a realtime closedform model for layered snowpacks and since we use FEM just for validation purposes, we do not believe the work would benefit from a detailed description of the FE model. Instead, we will add a reference to a previous publication, that discusses the FE model in detail – with the only difference being the slab layering.Moreover, we validate the displacement field produced by our model using fullfield DIC displacement measurements in Fig 7. (anonymous reviewer #2 agrees).
7.) On page 12, the last paragraph, “ Experimental validations are challenging since direct measurements of stresses are not possible and displacement measurements require considerable experimental effort ”. This is truth. However, above references a and b reported critical initial crack length in PST experimental work, which causes crack propagation and sliding of snow slab above weak layer, and the critical initial crack length related bending moment can be used for validation if the critical load can be provided by the analytical model in this manuscript. But this manuscript failed at the validation in this way. Abovementioned reference c has completed a similar validation.
Thank you for this suggestion. We will incorporate an estimate of the weaklayer (crushing) fracture toughness that can be identified from PST data in our revised manuscript.8.) In example section, manuscript presents a lot of information about stress and deformation distributions in snow slab in different cases. The reviewer thinks this information is less important in assessing snowpack fracture. The outcomes from this proposed model are not only difficult for validation and also in application for assessing snowpack fracture initiation and propagation. As a closedform solution, it is expected to provide precise failure response which shows a loaddeformation curve. Thus, it can clearly present elastic stage, damage point, damage accumulation and final crack point.
The stress distributions and energy release rates of snowpacks with intact and collapsed weak layers are fundamental for the analysis of weaklayer failure [Sch2003, Sch2016]. We agree that a closedform analytical solution providing a precise failure response of a slabweaklayer configuration is of great interest. This is precisely what our model aims at with great efficiency. We provide it as public sourcecode to other researchers on github.com.Whether brittle failure phenomena such as weaklayer fracture really show damage accumulation is worth a long discussion. For instance, does damage accumulate in the weak layer if many skiers travel over the same point? Does that mean we should avoid welltraveled slopes – which is certainly counterintuitive?
9.) This manuscript mentioned a basic concept of a failure mode in weak layer: mode I (collapse) or anticrack. Although this terminology was used by some researchers in their early work. The reviewer thinks that this terminology is not in the line of the basic concept of fracture mechanics about mode I: opening crack. The wording of collapse or anticrack would not be a right description of the failure mode in opposite way of mode I. Abovementioned reference c properly suggested a terminology of crushing damage by compression, and proposed a mixed mode damage criteria with crushing damage and shearing crack (mode II) to assess the failure in weak layer under mixed loading.
As already discussed in point 3.), we have used the usual way to consider symmetry of the displacement field to distinguish different loading modes of a crack [Bro1999]. This has been done by many researchers that studied anticracks, e.g. in snow [Hei2008, Gau2018], in compaction bands of limestone [Ste2005], pressure (dis)solution in rocks [Fle1981]. We follow the highly influential thoughts of Heierli et al. [Hei2008] in our denomination “collapse” but agree that crushing is an accurate description of the phenomenon, too.
10.) A closedfrom should clearly present the failed load and residual strength when crack propagates through the failure response or loaddeformation curve. The reviewer thinks that this manuscript has not reach the purpose of investigating and predicting the physical processes that lead to the formation of drysnow slab avalanches.
We agree with the reviewer that some of these features would be of interest in a failure model. The present model is not a failure model.
References used in the response:
[Ada2022] Adam V., Bergfeld B., van Herwijnen A., Weißgraeber P., Rosendahl P. L. (2022) Modified propagation saw tests for analyses of weaklayer fracture properties (https://www.igsoc.org/wpcontent/uploads/2022/10/procabstracts_79.html#A3837)
[Baz2003] Bažant, Z. P., Zi, G., & McClung, D. (2003). Size effect law and fracture mechanics of the triggering of dry snow slab avalanches. Journal of Geophysical Research: Solid Earth, 108(B2).
[Ber2023] Bergfeld, B., van Herwijnen, A., Bobillier, G., Rosendahl, P. L., Weißgraeber, P., Adam, V., ... & Schweizer, J. (2023). Temporal evolution of crack propagation characteristics in a weak snowpack layer: conditions of crack arrest and sustained propagation. Natural Hazards and Earth System Sciences, 23(1), 293315.
[Bro1999] Broberg, K. B. (1999). Cracks and fracture. Elsevier.
[Fle1981] Fletcher, R. C., and D. D. Pollard (1981), Anticrack model for pressure solution surfaces, Geology, 9, 419–424.
[Gau2018] Gaume, J., Gast, T., Teran, J., Van Herwijnen, A., & Jiang, C. (2018). Dynamic anticrack propagation in snow. Nature communications, 9(1), 3047.
[Hei2008] Heierli, J., Gumbsch, P., & Zaiser, M. (2008). Anticrack nucleation as triggering mechanism for snow slab avalanches. Science, 321(5886), 240243.
[Kun2013] Kuna, M. (2013). Finite elements in fracture mechanics. Solid Mechanics and Its Applications, 201, 153192.
[Ros2020a] Rosendahl, P. L., & Weißgraeber, P. (2020). Modeling snow slab avalanches caused by weaklayer failure–Part 1: Slabs on compliant and collapsible weak layers. The Cryosphere, 14(1), 115130.
[Ros2020b] Rosendahl, P. L., & Weißgraeber, P. (2020). Modeling snow slab avalanches caused by weaklayer failure–Part 2: Coupled mixedmode criterion for skiertriggered anticracks. The Cryosphere, 14(1), 131145.
[Sch2003] Schweizer, J., Bruce Jamieson, J., & Schneebeli, M. (2003). Snow avalanche formation. Reviews of Geophysics, 41(4).
[Sch2011] Schweizer, J., van Herwijnen, A., & Reuter, B. (2011). Measurements of weak layer fracture energy. Cold Regions Science and Technology, 69(23), 139144.
[Sch2016] Schweizer, J., Reuter, B., Van Herwijnen, A., & Gaume, J. (2016, October). Avalanche release 101. In Proceedings ISSW (pp. 111).
[Ste2005] Sternlof, K. R., Rudnicki, J. W., & Pollard, D. D. (2005). Anticrack inclusion model for compaction bands in sandstone. Journal of Geophysical Research: Solid Earth, 110(B11).
[Zie2005] Zienkiewicz, O. C., & Taylor, R. L. (2005). The finite element method for solid and structural mechanics. Elsevier.

CC1: 'Comment on tc2022140', Xingyue Li, 08 Nov 2022
This study develops an analytical model for the investigation of mechanical behavior of a stratified snow cover over a weak layer. The snow cover is considered as an arbitrarily layered beam solved by laminate mechanics, while the weak layer is modeled as a set of springs attached to the bottom of the snow cover. The model is firstly verified with the experimental data and the numerical results from finite element modeling, and then adopted to investigate the factors affecting slab release, including layering, bridging and slope angle. The key novelty of this study to the reviewer is the consideration of the layered snow, compared to the previous study conducted by the authors. The outcomes from this study could offer useful information for understanding the failure behavior of stratified snow over a weak layer, and thus give relevant information on slab avalanche release. However, there are some concerns to be clarified as detailed below.
1) Consideration of the crack: As described in the methodology, the model considers the weak layer as springs with normal and shear stiffnesses, and can handle different scenarios such as the ones with partially collapsed weak layer in Fig. 4. But it is not clear how the crack in the weak layer is considered and whether the scenarios in Fig. 4 are predefined. If the scenarios are predefined, how to determine these different initial conditions in practice? If not, please clarify the triggering of the crack. For example, what is the criteria to trigger the crack? Do the springs have certain shear and normal strength, above which they break? Please also discuss the propagation of the crack with time.
2) Comparison with the results from a homogeneous equivalent layer: In the results 4.1, the current model has been compared with the homogeneous model by Monti et al. (2015). It is stated that “Both concepts are benchmarked against the stiffnesses computed using finite element analyses”, please clarify whether “both concepts” are the current model and the model by Monti et al. (2015). If yes, as they have been benchmarked already, why the model by Monti et al. (2015) does not have consistent stiffness with the FEA in Fig. 10? In addition, please clarify that if the correct equivalent stiffnesses are implemented to a homogeneous model such as that by Monti et al. (2015) or the previous model by the authors (Rosendahl and Weißgraeber, 2020a), will the homogeneous models give good prediction on the mechanical behavior of the slab?

AC1: 'Reply on RC2', Philipp Rosendahl, 04 Feb 2023
Dear reviewer, thank you for your comments and suggestions, which help us to improve our paper. Please find our response to each concern and remark of the review below:
This study develops an analytical model for the investigation of mechanical behavior of a stratified snow cover over a weak layer. The snow cover is considered as an arbitrarily layered beam solved by laminate mechanics, while the weak layer is modeled as a set of springs attached to the bottom of the snow cover. The model is firstly verified with the experimental data and the numerical results from finite element modeling, and then adopted to investigate the factors affecting slab release, including layering, bridging and slope angle. The key novelty of this study to the reviewer is the consideration of the layered snow, compared to the previous study conducted by the authors. The outcomes from this study could offer useful information for understanding the failure behavior of stratified snow over a weak layer, and thus give relevant information on slab avalanche release. However, there are some concerns to be clarified as detailed below.
1) Consideration of the crack: As described in the methodology, the model considers the weak layer as springs with normal and shear stiffnesses, and can handle different scenarios such as the ones with partially collapsed weak layer in Fig. 4. But it is not clear how the crack in the weak layer is considered and whether the scenarios in Fig. 4 are predefined. If the scenarios are predefined, how to determine these different initial conditions in practice? If not, please clarify the triggering of the crack. For example, what is the criteria to trigger the crack? Do the springs have certain shear and normal strength, above which they break? Please also discuss the propagation of the crack with time.
The model can be used to study both slabs on intact weak layers and slabs where the weak layer has partially failed. The weak layers are modeled with a socalled weakinterface approach as it has been used in many works to study structural situation with stiffer structures being supported by foundations of significantly lower stiffness [Het1946]. Such weakinterface models can also be used to study locally failed foundations by removing the support effect of the interface [Kre1992, Len2001]. The present model can then be used to establish failure models that require the stress distribution in the weak interface and the energy release rate of crack configurations [Ste2015]. We have proposed a mixedmode finite fracture mechanics approach for nonlayered slabs in an earlier work [Ros2020b] and we will incorporate the present model for layered slabs in this framework in the future.
We thank the reviewer for pointing out that the description of the boundary conditions in both scenarios (slabs on intact weak layers and slabs where the weak layer has partially failed) should be improved. We will revise this section to make it more clear.
2) Comparison with the results from a homogeneous equivalent layer: In the results 4.1, the current model has been compared with the homogeneous model by Monti et al. (2015). It is stated that “Both concepts are benchmarked against the stiffnesses computed using finite element analyses”, please clarify whether “both concepts” are the current model and the model by Monti et al. (2015). If yes, as they have been benchmarked already, why the model by Monti et al. (2015) does not have consistent stiffness with the FEA in Fig. 10? In addition, please clarify that if the correct equivalent stiffnesses are implemented to a homogeneous model such as that by Monti et al. (2015) or the previous model by the authors (Rosendahl and Weißgraeber, 2020a), will the homogeneous models give good prediction on the mechanical behavior of the slab?
Thank you for pointing out this ambiguity. What we mean is that we compare the model by Monti et al. against finite element analyses and also compare the present model against FE analyses. We will improve the formulation in the revised manuscript to make this more clear.
Monti et al. have not focused on stiffnesses but instead looked at stresses within the snowpack, where the agreement was not 100% but deemed satisfying by the authors. We can only speculate that they were not aware of the stiffness discrepancies.
If slabs are layered, there is no one equivalent stiffness. Instead, layered slabs always respond differently to bending or tension. When there are only small differences in the stiffness of all layers of the slab, the use of an equivalent stiffness can provide satisfactory results because the error from neglecting layering effects is small [Ber2023]. We will include this point in our revised manuscript.
References used in the response:
[Ber2023] Bergfeld, B., van Herwijnen, A., Bobillier, G., Rosendahl, P. L., Weißgraeber, P., Adam, V., ... & Schweizer, J. (2023). Temporal evolution of crack propagation characteristics in a weak snowpack layer: conditions of crack arrest and sustained propagation. Natural Hazards and Earth System Sciences, 23(1), 293315.
[Het1946] Hetényi, M., & Hetbenyi, M. I. (1946). Beams on elastic foundation: theory with applications in the fields of civil and mechanical engineering (Vol. 16). Ann Arbor, MI: University of Michigan press.
[Kre1992] Krenk, S. (1992). Energy release rate of symmetric adhesive joints. Engineering Fracture Mechanics, 43(4), 549559.
[Len2001] Lenci, S. (2001). Analysis of a crack at a weak interface. International Journal of Fracture, 108(3), 275290.
[Ste2015] Stein, N., Weißgraeber, P., & Becker, W. (2015). A model for brittle failure in adhesive lap joints of arbitrary joint configuration. Composite Structures, 133, 707718.
[Ros2020b] Rosendahl, P. L., & Weißgraeber, P. (2020). Modeling snow slab avalanches caused by weaklayer failure–Part 2: Coupled mixedmode criterion for skiertriggered anticracks. The Cryosphere, 14(1), 131145.  AC4: 'Reply on CC1', Philipp Rosendahl, 04 Feb 2023

AC1: 'Reply on RC2', Philipp Rosendahl, 04 Feb 2023

RC2: 'Comment on tc2022140', Anonymous Referee #2, 09 Nov 2022
This study develops an analytical model for the investigation of mechanical behavior of a stratified snow cover over a weak layer. The snow cover is considered as an arbitrarily layered beam solved by laminate mechanics, while the weak layer is modeled as a set of springs attached to the bottom of the snow cover. The model is firstly verified with the experimental data and the numerical results from finite element modeling, and then adopted to investigate the factors affecting slab release, including layering, bridging and slope angle. The key novelty of this study to the reviewer is the consideration of the layered snow, compared to the previous study conducted by the authors. The outcomes from this study could offer useful information for understanding the failure behavior of stratified snow over a weak layer, and thus give relevant information on slab avalanche release. However, there are some concerns to be clarified as detailed below.
1ï¼Consideration of the crack: As described in the methodology, the model considers the weak layer as springs with normal and shear stiffnesses, and can handle different scenarios such as the ones with partially collapsed weak layer in Fig. 4. But it is not clear how the crack in the weak layer is considered and whether the scenarios in Fig. 4 are predefined. If the scenarios are predefined, how to determine these different initial conditions in practice? If not, please clarify the triggering of the crack. For example, what is the criteria to trigger the crack? Do the springs have certain shear and normal strength, above which they break? Please also discuss the propagation of the crack with time.
2ï¼Comparison with the results from a homogeneous equivalent layer: In the results 4.1, the current model has been compared with the homogeneous model by Monti et al. (2015). It is stated that “Both concepts are benchmarked against the stiffnesses computed using finite element analyses”, please clarify whether “both concepts” are the current model and the model by Monti et al. (2015). If yes, as they have been benchmarked already, why the model by Monti et al. (2015) does not have consistent stiffness with the FEA in Fig. 10? In addition, please clarify that if the correct equivalent stiffnesses are implemented to a homogeneous model such as that by Monti et al. (2015) or the previous model by the authors (Rosendahl and Weißgraeber, 2020a), will the homogeneous models give good prediction on the mechanical behavior of the slab?

AC1: 'Reply on RC2', Philipp Rosendahl, 04 Feb 2023
Dear reviewer, thank you for your comments and suggestions, which help us to improve our paper. Please find our response to each concern and remark of the review below:
This study develops an analytical model for the investigation of mechanical behavior of a stratified snow cover over a weak layer. The snow cover is considered as an arbitrarily layered beam solved by laminate mechanics, while the weak layer is modeled as a set of springs attached to the bottom of the snow cover. The model is firstly verified with the experimental data and the numerical results from finite element modeling, and then adopted to investigate the factors affecting slab release, including layering, bridging and slope angle. The key novelty of this study to the reviewer is the consideration of the layered snow, compared to the previous study conducted by the authors. The outcomes from this study could offer useful information for understanding the failure behavior of stratified snow over a weak layer, and thus give relevant information on slab avalanche release. However, there are some concerns to be clarified as detailed below.
1) Consideration of the crack: As described in the methodology, the model considers the weak layer as springs with normal and shear stiffnesses, and can handle different scenarios such as the ones with partially collapsed weak layer in Fig. 4. But it is not clear how the crack in the weak layer is considered and whether the scenarios in Fig. 4 are predefined. If the scenarios are predefined, how to determine these different initial conditions in practice? If not, please clarify the triggering of the crack. For example, what is the criteria to trigger the crack? Do the springs have certain shear and normal strength, above which they break? Please also discuss the propagation of the crack with time.
The model can be used to study both slabs on intact weak layers and slabs where the weak layer has partially failed. The weak layers are modeled with a socalled weakinterface approach as it has been used in many works to study structural situation with stiffer structures being supported by foundations of significantly lower stiffness [Het1946]. Such weakinterface models can also be used to study locally failed foundations by removing the support effect of the interface [Kre1992, Len2001]. The present model can then be used to establish failure models that require the stress distribution in the weak interface and the energy release rate of crack configurations [Ste2015]. We have proposed a mixedmode finite fracture mechanics approach for nonlayered slabs in an earlier work [Ros2020b] and we will incorporate the present model for layered slabs in this framework in the future.
We thank the reviewer for pointing out that the description of the boundary conditions in both scenarios (slabs on intact weak layers and slabs where the weak layer has partially failed) should be improved. We will revise this section to make it more clear.
2) Comparison with the results from a homogeneous equivalent layer: In the results 4.1, the current model has been compared with the homogeneous model by Monti et al. (2015). It is stated that “Both concepts are benchmarked against the stiffnesses computed using finite element analyses”, please clarify whether “both concepts” are the current model and the model by Monti et al. (2015). If yes, as they have been benchmarked already, why the model by Monti et al. (2015) does not have consistent stiffness with the FEA in Fig. 10? In addition, please clarify that if the correct equivalent stiffnesses are implemented to a homogeneous model such as that by Monti et al. (2015) or the previous model by the authors (Rosendahl and Weißgraeber, 2020a), will the homogeneous models give good prediction on the mechanical behavior of the slab?
Thank you for pointing out this ambiguity. What we mean is that we compare the model by Monti et al. against finite element analyses and also compare the present model against FE analyses. We will improve the formulation in the revised manuscript to make this more clear.
Monti et al. have not focused on stiffnesses but instead looked at stresses within the snowpack, where the agreement was not 100% but deemed satisfying by the authors. We can only speculate that they were not aware of the stiffness discrepancies.
If slabs are layered, there is no one equivalent stiffness. Instead, layered slabs always respond differently to bending or tension. When there are only small differences in the stiffness of all layers of the slab, the use of an equivalent stiffness can provide satisfactory results because the error from neglecting layering effects is small [Ber2023]. We will include this point in our revised manuscript.
References used in the response:
[Ber2023] Bergfeld, B., van Herwijnen, A., Bobillier, G., Rosendahl, P. L., Weißgraeber, P., Adam, V., ... & Schweizer, J. (2023). Temporal evolution of crack propagation characteristics in a weak snowpack layer: conditions of crack arrest and sustained propagation. Natural Hazards and Earth System Sciences, 23(1), 293315.
[Het1946] Hetényi, M., & Hetbenyi, M. I. (1946). Beams on elastic foundation: theory with applications in the fields of civil and mechanical engineering (Vol. 16). Ann Arbor, MI: University of Michigan press.
[Kre1992] Krenk, S. (1992). Energy release rate of symmetric adhesive joints. Engineering Fracture Mechanics, 43(4), 549559.
[Len2001] Lenci, S. (2001). Analysis of a crack at a weak interface. International Journal of Fracture, 108(3), 275290.
[Ste2015] Stein, N., Weißgraeber, P., & Becker, W. (2015). A model for brittle failure in adhesive lap joints of arbitrary joint configuration. Composite Structures, 133, 707718.
[Ros2020b] Rosendahl, P. L., & Weißgraeber, P. (2020). Modeling snow slab avalanches caused by weaklayer failure–Part 2: Coupled mixedmode criterion for skiertriggered anticracks. The Cryosphere, 14(1), 131145.

AC1: 'Reply on RC2', Philipp Rosendahl, 04 Feb 2023

RC3: 'Comment on tc2022140', Anonymous Referee #3, 28 Nov 2022
The Authors propose a closedform analytical model for the mechanical behavior of stratified snow covers adopting the firstorder shear deformation theory of laminated plates under cylindrical bending. The weak layer is modelled as an infinite set of smeared springs with normal and shear stiffness at the base of the plate. The problem is worth to be studied and it is of large interest for the scientific community. There are some points that must be clarified and further investigated.
 It is not clear the remind to De Saint Venant's principle (l. 246) and how it can be applied in the model proposed by the authors as the localized stresses and effects are of primary importance for the release of the avalanche.
 The authors must clarify how the concentrated load applied on the surface of the slab enters into the solution of the firstorder system of Eqn. (11).
 The authors must clarify how the size of the crack tip can be effectively measured in a real snowpack as it affects the results of the stability analysis.
 The authors confuse the energy release rate with the failure criterion. To let the crack to propagate, it is necessary that the G term equals the critical G in the mixed mode. It is worth to mention that, in fracture mechanics, Mode 1 refers to an opening mode, differently from what the authors state in their approach. This point must be addressed in order to avoid misunderstandings.

AC2: 'Reply on RC3', Philipp Rosendahl, 04 Feb 2023
Dear reviewer, thank you for your comments and suggestions, which help us to improve our paper. Please find our response to each concern and remark of the review below:
The Authors propose a closedform analytical model for the mechanical behavior of stratified snow covers adopting the firstorder shear deformation theory of laminated plates under cylindrical bending. The weak layer is modelled as an infinite set of smeared springs with normal and shear stiffness at the base of the plate. The problem is worth to be studied and it is of large interest for the scientific community. There are some points that must be clarified and further investigated.
1. It is not clear the remind to De Saint Venant's principle (l. 246) and how it can be applied in the model proposed by the authors as the localized stresses and effects are of primary importance for the release of the avalanche.
The question is well answered by Wikipedia: “SaintVenant's principle, named after Adhémar Jean Claude Barré de SaintVenant, a French elasticity theorist, may be expressed as follows: the difference between the effects of two different but statically equivalent loads becomes very small at sufficiently large distances from load.” (https://en.wikipedia.org/wiki/SaintVenant%27s_principle).
For the model this means that introducing the skier loading as distributed loads (width of the skis) or concentrated into one force (statically equivalent) has the same effect on weaklayer stresses.
We will include this clarification in the revised manuscript.
2. The authors must clarify how the concentrated load applied on the surface of the slab enters into the solution of the firstorder system of Eqn. (11).
The ODE system is obtained by using the constitutive equations and condition of equilibrium of the slabweaklayer system (Appendix A and B in the manuscript). The concentrated loads are introduced in the next step, when boundary and transmission conditions are considered (see Appendix C for details). A concentrated force introduces a discontinuity in the equilibrium of lateral forces and, hence, requires a transmission condition. We will included this clarification in the revised manuscript.
3. The authors must clarify how the size of the crack tip can be effectively measured in a real snowpack as it affects the results of the stability analysis.
In the case of the most relevant experiment for analysis of the fracture behavior of weak layers – the propagation saw test – the crack length corresponds to the critical cut length at which the PST fails.
4. The authors confuse the energy release rate with the failure criterion. To let the crack to propagate, it is necessary that the G term equals the critical G in the mixed mode. It is worth to mention that, in fracture mechanics, Mode 1 refers to an opening mode, differently from what the authors state in their approach. This point must be addressed in order to avoid misunderstandings.
The present work does not consider failure or a failure criterion but aims at providing all quantities necessary for a failure assessment. In our view, the most important ingredients are weaklayer stresses (normal and shear) and weaklayer energy release rates (collapse and sliding). Eq. (19) provides the equations for the latter. That is, Eq. (19) enables the formulation of mixedmode weaklayer failure criteria.
An example of how mode I and mode II energy release rates can be considered individually is given in Figure 16 that analyzes the effect of slope angle on the mode mixity of the energy release rates in propagation saw tests.
Note that we have proposed a mixedmode finite fracture mechanics approach for nonlayered slabs in an earlier work [Ros2020b] and we will incorporate the present model for layered slabs in this framework in the future.
When we speak of mode I anticrack propagation, we refer to the symmetric nature of the deformation field [Bro1999]. Because we consider anticracking, mode I corresponds to weaklayer crushing and collapse. Further details of this definition are given in a previous The Cryosphere publication by the authors [Ros2020b]. We will revise the manuscript to make this more clear.
References used in the response:
[Bro1999] Broberg, K. B. (1999). Cracks and fracture. Elsevier.
[Ros2020b] Rosendahl, P. L., & Weißgraeber, P. (2020). Modeling snow slab avalanches caused by weaklayer failure–Part 2: Coupled mixedmode criterion for skiertriggered anticracks. The Cryosphere, 14(1), 131145.
Philipp Weißgraeber and Philipp Laurens Rosendahl
Philipp Weißgraeber and Philipp Laurens Rosendahl
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