The standard viscous–plastic (VP) sea ice model with an elliptical yield curve and a normal flow rule has at least two issues. First, it does not simulate fracture angles below 30

Sea ice plays a significant role in the energy budget of the climate system and therefore has a strong influence on future climate projections. Sea ice dynamics are located primarily along narrow lines of deformation, called linear kinematic features (LKFs), where floes slide along and grind against each other. LKFs can form in divergence, creating stretches of open water or leads, or in convergence, creating piles of ice or ridges. LKFs in the Arctic sea ice cover influence the Earth system in many ways: heat and moisture exchange take place primarily over open water

LKFs are ubiquitous features of granular media, and sea ice is often described as such a granular material

Different

The viscous–plastic rheology is an appropriate continuum rheology for modeling sea ice as a granular material because it includes (1) a yield condition for plastic deformation and (2) a flow rule that allows the representation of the divergent and convergent motion along shear lines, that is, the dilatancy observed in granular media. Continuum plastic flow models with normal or non-normal flow rules are often used in other scientific fields to model granular geo-materials

LKFs have been studied in satellite observations

New models have been designed to represent sea ice fracture, for example, brittle models with a damage parameter that keeps the memory of previous fracture

The orientation of LKFs is a well-studied subject in the field of engineering and granular materials (LKFs are called shear bands in this field). Two classical solutions coexist and set two limit angles for the orientation of fractures: the Coulomb angle (static behavior) and the Roscoe angle (dynamic behavior). The Coulomb angle of fracture

The fracture angles with the standard VP rheology cannot be smaller than 30

The flow rule has the advantage that it can be observed with remote sensing methods, in contrast to observing stress which requires in situ measurements. The ratio of shear to divergence along the shear bands or LKFs allows the inference of the dilatancy angle of granular material. Observations of sea ice drift in the Arctic show that most of the deformation takes place in shear with some divergence

In this paper, we investigate the effects of a non-normal flow rule on fracture angles. We use the non-normal flow rule as a means of separating the state of stress (at failure) and the post-fracture deformation. To this end, we study the non-normal flow rule in the context of the standard VP rheological model using a similar shape for the plastic potential (i.e., an ellipse) because (1) the ellipse is widely used in the community and (2) its behavior is well documented (compared to other models), providing a solid basis for comparison. For these two reasons, we use the elliptical yield curve despite the fact that it is not the most appropriate yield curve to model sea ice as a granular material. This paper provides a new generalized theoretical framework for any viscous–plastic material with normal or non-normal flow rules. Following

The paper is structured as follows. Section

We consider sea ice a 2D viscous–plastic material. The ice velocities are calculated from the sea ice momentum equations:

One of the state variables in the model is the maximum compressive strength

The yield curve represents the stress states for which sea ice deforms plastically while enclosing the stress states for which sea ice slowly deforms viscously. We express the yield curve as a function of the stresses

Schematic yield curve

The plastic potential determines the direction of deformation for stress states on the yield curve. The flow rule represents the direction of deformation in the grid cell. The orientation of the flow rule in the coordinate system (

Using Eqs. (

After deriving these constitutive equations, we assume that the stress and strain rate tensors are symmetric; that is,

An ideal plastic model, with the stresses independent of the strain rates, has a singularity because the non-linear viscosities tend to infinity as the strain rates tend to zero.

We now build a rheology with an elliptical yield curve and a non-normal flow rule; that is, we use a plastic potential

Following

Elliptical yield curve with a non-normal flow rule, a yield curve ellipse aspect ratio

In this section, we generalize the theory linking the rheological model and the fracture angles in a simple uni-axial compressive test

Link between fracture angle and yield curve:

Figure

We calculate the fracture angles for the elliptical yield curve with a non-normal flow rule in uni-axial compression along the

Trajectory of maximum normal stress (red arrow) in a uni-axial loading test experiment in a material with two different elliptical yield curves (blue) and plastic potentials (dashed orange and dash-dotted teal). The orange and teal arrows show the flow rule normal to the plastic potential of the same color for the same stress state. For

In the following, we use the normalized stress invariants

The slope of the tangent at

Following

Model domain with a solid wall at

The non-linear momentum equation, Eq. (4), is integrated using a Picard solver with 15 000 non-linear (or outer-loop) iterations

The intersection angles between the LKFs are measured manually with the Measure Tool from the GNU Image Manipulation Program (GIMP; version 2.8.16;

We study the evolution of the fracture angle

Diamond-shaped fracture pattern in the shear deformation field

Figure

Asymmetric secondary fracture lines appear, in contrast to the normal flow rule simulation. We attribute the asymmetry and presence of secondary fractures to the lack of full numerical convergence associated with the violation of Drucker's principle, or the non-normality of the flow rule (the ratio of divergence to shear strain rate differs from that of the shear to normal stress). For instance, the

The width and activity of the LKFs is also affected by the flow rule. With

The fracture angle changes as the plastic potential changes. The angles are wider with

We now present results from four sets of simulations with fixed yield curve ellipse ratios at

Figure

The colored, dashed lines in Fig.

For completeness, Fig.

The idealized experiments using the elliptical yield curve with a non-normal flow rule confirm that the type of deformation and the fracture angle are intimately linked with the shape of the plastic potential. We observe that, irrespective of the plastic potential elliptical aspect ratio, a yield curve ellipse ratio of

Understanding the link between the rheology and fracture angle is necessary for choosing or designing a rheology that is capable of reproducing the observed intersection angles between pairs of LKFs and consequently the emerging anisotropy. An independent plastic potential may resolve several inconsistencies of the standard elliptical yield curve with a normal flow rule

In the standard VP model with an elliptical yield curve and normal flow rule, adding shear strength increases the fracture angle, in contradiction to granular matter theory

Because of the elliptical shape of the yield curve, the angle of fracture in the standard VP model changes with confining pressure

In the standard VP model with a normal flow rule, the divergence and convergence are set by the ellipse ratio of the yield curve and thus by the relative amounts of compressive and shear stress. The plastic potential ellipse ratio

The fracture angles in the standard VP models are larger than observed. Using a non-normal flow rule allows us to change the fracture angle in uni-axial compression to values below 30

We discuss the elliptical yield curve here because it is the most commonly used one and its behavior is better documented than any other model in use in the community. This provides a known reference for studying the use of non-associated flow rules. Our goal is to provide a reference for the future development of viscous–plastic rheologies with non-normal flow rules rather than suggest a new VP rheology. Alternatives to the elliptic yield curve have been used before, for instance, the Mohr–Coulomb, the Coulombic yield curve, or the teardrop yield curves (Fig.

Alternative yield curves and flow rules: the Mohr–Coulomb yield curve with shear (non-normal) flow rule

Non-normal flow rules can be combined with the Mohr–Coulomb family of yield curves. For a Mohr–Coulomb yield curve with a double-sliding law

The Coulombic yield curve uses the two straight limbs from the Mohr–Coulomb yield curve and an elliptical cap of the standard VP rheology for large compressive stresses

The teardrop yield curve with a normal flow rule

As the main disadvantage of a non-normal flow rule, we found that it leads to slower convergence of the numerical solver. Solving the momentum equation accurately requires more solver iterations, and failure to converge is more frequent than for standard normal-flow-rule rheologies. In our simulations, this numerical issue manifests itself by the presence of multiple and asymmetrical fracture lines despite the fact that our experiment geometry and forcing are exactly symmetrical. This asymmetry is not expected and is not found with normal flow rules. The fracture lines with a normal flow rule are symmetrical and come in pairs

The following criteria should be considered when building a new rheology: the spatial and temporal scaling of sea ice deformation should agree with observations

Although high-spatial-resolution observations from satellites are available from optical instruments (e.g., from the Landsat or Sentinel programs), higher temporal resolutions of sea ice deformation and flow size distributions are still unavailable. The new Sentinel constellation and in situ observations from the field program the Multidisciplinary drifting Observatory for the Study of Arctic Climate (MOSAiC) may bridge this gap. There is also a knowledge gap in the interplay between yield stresses and the post-fracture deformation in a 2D granular material such as sea ice. This interplay is likely different than for the well-studied case of a solid, homogeneous, 3D block of ice

The flow rule, which dictates the post-fracture deformation, has a fundamental effect on the orientation of fractures lines in a viscous–plastic (VP) sea ice model. To test this, we added an elliptical plastic potential (allowing for a non-normal flow rule) to the standard VP rheology with an elliptical yield curve, therefore modifying the flow rule without changing the yielding stress state. We tested this new rheology with numerical experiments in uni-axial compression using the standard VP model of

Designing a rheology for high-resolution simulations requires information about sea ice fracture angles and sea ice strength in a wide range of stress conditions (i.e., compression with or without confinement, pure shear, tension), yet this is unavailable at high temporal and spatial resolution. The observations of MOSAiC

The sea ice rheology used in this paper is implemented in the sea ice package of the MITgcm, Mid 2021 release (

No datasets were used in this article. All simulation data have been obtained with the model cited in the “Code availability” section below. The idealized model configuration is described in detail in Sect. 3.

DR designed the rheology and implemented the code changes with ML. DR ran the experiments. DR and BT designed the theory linking sea ice rheologies and granular matter theory. DR wrote the manuscript with contributions from BT and ML.

The authors declare that they have no conflict of interest.

The authors would like to thank Véronique Dansereau and Harry Heorton for their review and numerous comments that helped improve this paper. The authors also thank Jennifer Hutchings and Yevgeny Aksenov for their comments and involvement as editors. The authors are thankful to Mathieu Plante for his comments on this paper, as well as to Stephanie Deboeuf and Guillaume Ovarlez for discussions on granular materials rheologies.

This project has been supported by the Deutsche Forschungsgemeinschaft (DFG) through the International Research Training Group “Processes and impacts of climate change in the North Atlantic Ocean and the Canadian Arctic” (grant no. IRTG 1904 ArcTrain). This work is a contribution to the Natural Sciences and Engineering Research Council Discovery Grant awarded to L. Bruno Tremblay.The article processing charges for this open-access publication were covered by the University of Bremen.

This paper was edited by Yevgeny Aksenov and Jennifer Hutchings and reviewed by Véronique Dansereau and Harry Heorton.