The Maxwell elasto-brittle (MEB) rheology uses a damage parameterization to represent the brittle fracture of sea ice without involving plastic laws
to constrain the sea ice deformations. The conventional MEB damage parameterization is based on a correction of super-critical stresses that binds
the simulated stress to the yield criterion but leads to a growth of errors in the stress field. A generalized damage parameterization is developed
to reduce this error growth and to investigate the influence of the super-critical stress correction scheme on the simulated sea ice fractures,
deformations and orientation of linear kinematic features (LKFs). A decohesive stress tensor is used to correct the super-critical stresses towards
different points on the yield curve. The sensitivity of the simulated sea ice fractures and deformations to the decohesive stress tensor is
investigated in uniaxial compression experiments. Results show that the decohesive stress tensor influences the growth of residual errors associated
with the correction of super-critical stresses, the orientation of the lines of fracture and the short-term deformation associated with the damage,
but it does not influence the long-term post-fracture sea ice deformations. We show that when ice fractures, divergence first occurs while the elastic
response is dominant, and convergence develops post-fracture in the long term when the viscous response dominates – contrary to laboratory
experiments of granular flow and satellite imagery in the Arctic. The post-fracture deformations are shown to be dissociated from the fracture
process itself, an important difference with classical viscous plastic (VP) models in which large deformations are governed by associative plastic
laws. Using the generalized damage parameterization together with a stress correction path normal to the yield curve reduces the growth of errors
sufficiently for the production of longer-term simulations, with the added benefit of bringing the simulated LKF intersection half-angles closer to
observations (from 40–50 to 35–45

Sea ice is a thin layer of solid material that insulates the polar oceans from the cold atmosphere. When sea ice fractures and a lead opens, large
heat and moisture fluxes take place between the ocean and the atmosphere, significantly affecting the polar meteorology on short timescales and the
climate system on long timescales

As sea ice models are moving to higher spatial resolutions, they become increasingly capable of resolving LKFs

The Sea Ice Rheology Experiment

In the Maxwell elasto-brittle (MEB) rheology

With the MEB rheology being relatively new, the extent to which the sea ice deformations are sensitive to the numerical and material strength parameters
has not been thoroughly tested yet. Nonetheless, the orientation of the simulated faults in uniaxial compression experiments is known to be sensitive
to the angle of internal friction and to the Poisson ratio

In this paper, we present a generalization of the damage parameterization in which a decohesive stress tensor is introduced in the stress correction scheme such that the super-critical stresses can be brought back to the yield curve following different stress correction paths in the stress invariant space. The generalization is used to reduce the growth of the residual errors associated with the stress correction and tested in uniaxial loading experiments to examine the influence of the stress correction on the simulated sea ice fracture and deformations. The sensitivity of the simulated fracture angles to the decohesive stress tensor is also investigated to find the stress correction paths that present the added benefit of bringing the simulated fracture angles closer to observations.

This paper is organized as follows. In Sect.

Default model parameters.

The simulations are run using the MEB model implemented on an Eulerian, finite difference Arakawa C grid in the McGill SIM5

The prognostic equations for the mean ice thickness

The MEB model differs from classical sea ice models in that it represents the brittle character of sea ice using a damage parameter to represent the effect of local fracture on the large-scale sea ice material properties. The sea ice deformations in the MEB model thus occur post-fracture, rather than simultaneously as in most sea ice models using granular or plastic flow laws, and the formation of LKFs follows from the propagation of damage in space over short timescales during the fracture process.

In the MEB rheology, the ice behaves as a visco-elastic material with a fast elastic response to forcing and a slower viscous response that acts over a
longer timescale. The governing equation for this visco-elastic material can be written as

The relative importance of the elastic and viscous components (first and second terms on the left-hand side in Eq.

Damage (or fracture) occurs when the internal stress state exceeds the Mohr–Coulomb failure criterion,

The prognostic equation for the damage parameter

When the ice fractures, the damage factor

This stress correction scheme corresponds to scaling all the individual stress components by the factor

We propose a generalized damage parameterization where the super-critical stresses are corrected back to the yield curve along a line oriented at any
angle

We define the damage factor in the generalized damage parameterization in terms of the shear stress invariant only as

The damage factor can then be written in terms of the super-critical stress state invariants (

In this manner, the correction of super-critical stresses can follow any path in the stress invariant space provided that the damage increases when
ice fractures (

Note that using a stress correction path other than the standard path to the origin means that the corrected normal stress differs from the scaled
super-critical stress

Note that the decohesive stress tensor used in this parameterization has a similar role as the decohesive strain rates used in the elastic–decohesive
model

The error

Given that the uncorrected stress is close to the yield criterion (i.e.

Note that the error amplification ratio

Idealized domain for uniaxial compression simulations, with a solid boundary (Dirichlet conditions,

We test the numerical and material behaviour of the MEB model and the generalized damage parameterization in uniaxial compression
experiments. Uniaxial experiments are designed to present conditions similar to those in laboratory experiments and have been used with MEB

Note that all simulations are performed without including heterogeneity in order to clearly identify the model performance (both numerics and physics), unless specified otherwise. This allows us to quantify the growth of residual numerical errors in a problem with full symmetry and their impact on the simulated LKF orientation and post-fracture sea ice deformations.

The MEB model is implemented in the McGill Sea Ice Model Version 5 (McGill SIM5) using an Eulerian, second-order finite difference numerical scheme

We monitor the influence of the residual errors on the model solution in the simulations using a normalized domain-integrated asymmetry factor
(

Note that the field asymmetry measures the degradation of the originally fully symmetric problem as numerical errors are integrated and includes the
physical response to the integrated errors. This is in contrast with the residual error amplification ratio

We quantify the development of fractures in the experiments using the damage activity

This parameter is analogous to the damage rate in

The angles between conjugate LKFs in the Arctic are often discussed in relation with the orientation of the smaller-scale brittle fractures observed
in the laboratory under uniaxial compression loads

Here, we define the fracture angle

In the Roscoe theory

If

In our experiment, the fracture angle is calculated graphically for each individual simulation. We define the uncertainty as

In the control simulation, a pair of conjugate LKFs first appear when the surface forcing

Scatter plots of local stress invariants (

The deformation along the fully developed LKFs in our experiment is mostly shear and convergent (i.e. ridging, Fig.

The asymmetries in the solution are very small at the beginning of the simulation (

The generalized damage parameterization reduces the growth of residual errors, with decreasing asymmetry factor and maximum error amplification ratio

Sensitivity of the LKF orientation

Results show that the LKF orientation is sensitive to the decohesive stress tensor, with a decreasing angle

Time evolution of the mean normal

The correction path angle

Sensitivity of the LKF orientation (

Repeating the experiment using different angles of internal friction (

Time evolution of

Decreasing the angle of internal friction reduces the shear strength of sea ice for a given normal stress, such that the fracture develops earlier in
the simulation (i.e. under smaller surface forcing, Fig.

Sensitivity of the LKF orientation (

The orientation of LKFs is not sensitive to the Poisson ratio when the generalized stress correction scheme is used with a fixed stress correction
path angle

Time evolution of the mean normal strain rate invariant integrated over the ice cover (

The results presented above show that the generalized stress correction scheme reduces the growth of the residual error associated with the damage
parameterization. Despite the improvement, some asymmetries are still present in the simulations (

Overall, the use of a decohesive stress tensor yields smaller simulated LKF angles, without significantly impacting the material deformations. Using a
large correction path angle

The simulation results show that in the MEB model, the damage develops at short timescales during which the elastic component of the rheology is
important, while most of the deformations occur post-fracture over a longer timescale in the heavily damaged ice. This is in contrast with plastic
models, in which a flow rule simultaneously dictates both the LKF development and the relative amount of shear and normal deformations occurring along
the LKFs. The decoupling between the development of damage and the post-fracture deformations in the MEB model explains that the type of deformations
in the LKFs remains similar

The viscous dissipation timescale (

Note that the results presented above were presented using a single space and time resolution and ice sample aspect ratio and without using
heterogeneity. While the exact localization of the LKFs in the simulations is affected by these parameters, the overall physics and sensitivity to the
damage parameterization are robust to these changes. For instance, repeating the experiment by doubling the space resolution or the width of the ice
sample does not change the LKF position and orientation (not shown). On the other hand, adding heterogeneity changes the LKF development by forming
irregular sliding planes instead of the linear diamond shapes (Fig.

We propose a generalized stress correction scheme for the damage parameterization to reduce the growth of residual errors in the MEB sea ice model
documented in

Our results show that in the MEB rheology, most of the deformations occur post-fracture in heavily damaged ice, where the viscous term is dominant. This causes a predominance of convergence (ridging) in the LKFs, contrary to laboratory experiments of granular materials and satellite observations of sea ice. The use of a decohesive stress tensor influences the LKF orientation in the sea ice cover but does not influence the type of deformation rates (convergence and shear) or the simulated dilatancy. Future work will involve the modification of the non-linear relationship between the Maxwell viscosity and the damage. We also show that the sensitivity of the LKF orientation to the Poisson ratio, seen when using the standard damage parameterization, disappears when using the generalized stress correction scheme with a fixed stress correction path. This suggests that in the MEB model the stress concentration and fracture propagation are governed by the stress correction rather than by the relaxation of the mechanical properties associated with the damage.

Based on our results, using the generalized damage parameterization with a stress correction path normal to the yield curve reduces the growth of
residual errors and allows longer-term simulations with post-fracture deformations. Using this stress correction path also reduces the orientation of
LKFs by

Our sea ice model code is available upon request.

Our model outputs are available upon request.

MP coded the model, ran all the simulations, analyzed results and led the writing of the manuscript. BT participated in regular discussions during the course of the work and edited the manuscript.

The contact author has declared that neither they nor their co-author have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is a contribution to the research program of Québec-Océan and to the ArcTrain International Training Program. We thank the three anonymous reviewers for their useful comments and suggestions during the open discussion process. We also thank Amélie Bouchat, Damien Ringeisen, Martin Losch and Jean-François Lemieux for useful discussions during the implementation of the MEB model and the generalized stress correction.

We are grateful to the Fonds de recherche du Québec – Nature et technologies (FRQNT) for financial support to Mathieu Plante during the course of this work as well as to the Natural Science and Engineering and Research Council (NSERC) Discovery Program and the Environment and Climate Change Canada Grant & Contribution for grants awarded to Bruno Tremblay.

This paper was edited by Yevgeny Aksenov and reviewed by three anonymous referees.