Recent high-resolution pan-Arctic sea ice simulations show
fracture patterns (linear kinematic features or LKFs) that are typical of
granular materials but with wider fracture angles than those observed in
high-resolution satellite images. Motivated by this, ice fracture is
investigated in a simple uni-axial loading test using two different
viscous–plastic (VP) rheologies: one with an elliptical yield curve and a
normal flow rule and one with a Coulombic yield curve and a normal flow rule
that applies only to the elliptical cap. With the standard VP rheology, it is
not possible to simulate fracture angles smaller than

Sea ice is a granular material, that is, a material that is composed of ice
floes of different sizes and shapes

The sea ice dynamics are complicated because of sharp spatial changes in
material properties associated with discontinuities (e.g., along sea ice leads
or ridges) and heterogeneity (spatially varying ice thickness and
concentration). The sea ice momentum equations are difficult to solve
numerically because of the nonlinear sea ice rheology. Since the first sea
ice dynamics model, the elastic–plastic sea ice model based on data collected
during the Arctic Ice Dynamics Joint Experiment

In spite of its success, the standard VP rheology is not undisputed.

Previously, fracture lines (LKFs) in the pack ice were explained by brittle
fracture

High-resolution sea ice models simulate LKF patterns in pack ice, where they
appear as lines of high deformation

The simulation of fractures in sea ice models has been studied in idealized
model geometries before.

In this paper, we simulate the creation of a pair of conjugate faults in an
ice floe with two different VP rheologies in an idealized experiment at a
spatial resolution of 25 m. We explore the influence of various parameters
of the rheologies and the model geometry (scale, resolution, confinement,
boundary conditions, and heterogeneous initial conditions). The remainder of
this paper is structured as follows. Section

We use the Massachusetts Institute of Technology general circulation model

The stress tensor

Any linear combination of the principal stresses consists of stress invariants. One
common set of stress invariants is the mean normal stress (

The VP rheology was originally developed to simulate ice motion on a basin
scale (e.g., Arctic Ocean, Southern Ocean)

In this study, we use two different yield curves: an elliptical yield curve

Elliptical yield curve (black) with ellipse aspect ratio

The theoretical angle of fracture

An idealized compressive test is used to investigate the modes of sea ice
fracture (Fig.

Model domain with a solid wall on the southern (red) boundary
(Dirichlet boundary conditions with

Model parameters of the reference simulation.

The model domain is a rectangle of size

We solve the nonlinear sea ice momentum equations with a Picard or fixed
point iteration with 1500 nonlinear or outer-loop iterations. Within each
nonlinear iteration, the nonlinear coefficients (drag coefficients and
viscosities) are updated and a linearized system of equations is solved with
a line successive (over-)relaxation (LSR)

On the open eastern and western boundaries, we use von Neumann boundary
conditions for velocity, thickness and concentration, and ice can escape the
domain without any restrictions:

We use simple uni-axial loading experiments to investigate the creation of
pair of conjugate faults and their intersection angle. After presenting the
results of simulations with the default parameters
(Sect.

With default parameters (Table

After a few time steps, the ice thickness decreases, particularly along the
LKFs (Fig.

Schematic of stress states and failure in principal stress space.
Black arrows show how stresses move from zero at the beginning of loading
towards the yield curve until failure. Red points show the stress states at
failure – the intersection point between the second principal axis 2 (in
red) and the elliptical yield curve – for different ellipse ratios

In this section, we test the sensitivity of the standard VP model simulation
(Sect.

Maximum shear strain rate (second strain invariant) after 10 s of integration for the default domain size and

The angle of intersection between a pair of conjugate faults does not change
with domain size and spatial resolution (Fig.

Continuing the integration to 2700 s (45 min), compared to 20 s
in the reference simulation, leads to the creation of smaller diamond-shaped
ice floes due to secondary and tertiary fracture lines
(Fig.

Sea ice thickness

In the following, we always show results after 5 s of integration because our main focus is on the initial fracture of the ice, that is, the instant when the ice breaks for the first time under compression.

Maximum shear strain rate after 5 s of integration in a reduced
size domain (

The dynamics responsible for the ice fracture and location of the fracture
(presented above) take place far away from the eastern and western boundaries
and therefore do not depend on the choice of the corresponding boundary
conditions. We now investigate the sensitivity of the results to the choice
of boundary condition at the southern boundary. To this end, we force the
fracture line to intersect the southern boundary by reducing the domain size
to

No-slip or free-slip boundary conditions have little impact on the fracture angle in the larger domain used in the control run simulation because the LKFs always only touch one boundary and end in open water (results not shown). With the free-slip boundary conditions, the stresses and strains are only different south of the diamond fracture pattern because ice can move along the southern boundary, and the second fracture cannot form.

We now explore the effect of confining pressure on the eastern and western
boundaries on the angle of fracture when using a (convex) elliptical yield
curve with a normal flow rule. To do so, we replace the open boundaries to
the east and the west with solid walls and the open water gaps with ice of
thicknesses

With an increasing lateral confinement pressure (i.e., an increasing ice
thickness

Maximum shear strain rates (left) and stress state in stress
invariant space (right) after 5 s of integration for different confinement
pressure:

So far, all initial conditions have been homogeneous in thickness and
concentration within the ice floe. In practice, sea ice (in a numerical
model but also in reality) is not homogeneous. A local weakness in the
initial ice field is likely the starting point of a crack within the ice
field

Sea ice thickness with two ice-free areas

To illustrate this behavior, we start new simulations from an initial ice
field with two areas of zero ice thickness and zero ice concentration, hence
weaker ice (Fig.

Keeping

Fracture angles as a function of ellipse aspect ratio

Maximum shear strain

In this section, we replace the elliptical yield curve with a Coulombic yield
curve

Maximum shear strain (top) and stress state in stress invariant
space (bottom) for different internal angles of friction.

The slope of the Mohr–Coulomb limbs of the Coulombic yield curve

For the Coulombic yield curve, there are two distinct regimes of failure.
When the

Our idealized experiments using the VP rheologies resolve fracture lines as
described by

We explored some experimental choices to separate their effects from those of
the rheology parameters. The fracture angles do not depend on the spatial
resolution and domain size as expected in our idealized numerical experiment
setup (Sect.

In our setup, the no-slip boundary condition has little effect on the
fracture pattern, but our results suggest that in basin-wide simulations the
choice of boundary conditions affects the fracture depending on the geometry
and stress direction. The no-slip condition appears to be unphysical. It acts
to concentrate the stress on the corners of the floe and forces the fracture
to occur at this location. This should motivate a more thorough investigation
of the boundary conditions for LKFs that form between one shoreline and
another. Similar results were obtained from analytical solutions in idealized
geometry for the Mohr–Coulomb yield curve with a double sliding deformation
law

The confining pressure (i.e., thin ice imposed on the side of the domain)
changes the distribution of stress within the domain. This results in
different deformation patterns (shear and divergence) and different fracture
angles because the yield curve is convex and uses a normal flow rule. From
this we can conclude that by surrounding our floe with open water, we get the
most acute angles from the rheology in this uni-axial compression setup. This
is not consistent with the behavior of typical granular material for which an
angle of fracture is independent of the confining pressure

In granular material, large shear resistance is linked to contact normals
between floes that oppose the shear motion and lead to dilatation

Arctic-wide simulations improve metrics of sea ice concentration, thickness,
and velocity by decreasing the value of

The fracture angle and the sea ice opening and ridging depending on the
deformation states are consistent with the theory of the yield curve analysis
developed in

With the Coulombic yield curve, the simulated fracture angle can be smaller
than for the elliptical yield curve. For

Motivated by the observation that the intersection angles in a 2 km
Arctic-wide simulation of sea ice are generally larger than in the RGPS
dataset

In our experimental configuration with uni-axial compression, fracture angles
below 30

With a modified Coulombic yield curve, the fracture angle can be decreased to
values expected from observations, but the non-differentiable corner points
of this yield curve lead to numerical (convergence) issues and, for some
values of the coefficient of internal friction

More generally, the model produces diamond-shaped fracture patterns. Later the ice floe disintegrates and several smaller floes develop. The fracturing process in the ice floe in our configuration is independent of the experiment resolution and scale but sensitive to boundary conditions (no-slip or free-slip). The fracture angle in the VP model is also sensitive to the confining pressure. This is not consistent with the notion of sea ice as a granular material. Unsurprisingly, the yield curve plays an important role in fracturing sea ice in a numerical model as it governs the deformation of the ice as a function of the applied stress.

The idealized experiment of a uni-dimensional compression is useful to
explore the effects of the yield curve because all other parameters are
controlled. Historically, the discrimination between the different yield
curves was not possible because of the scarcity of sea ice drift data. Model
comparisons to recent sea ice deformation datasets, such as from RADARSAT,
imply that we would need to increase the shear strength with the ellipse in
the standard VP rheology to match observations

If Arctic-wide sea ice simulations with a resolution of 25 m are not
feasible today because of computational cost, we can still imagine small
experiments being useful for process modeling on small scales when local and
high-resolution observations (e.g., wind, ice velocities) are available. For
example, such process modeling studies could be used to constrain the
rheology with data from the upcoming MOSAiC campaign

No datasets were used in this article. All simulation
data have been obtained with the MITgcm (

Below, we derive a relationship between the fracture angle and the internal
angle of friction for a Mohr–Coulomb yield criterion for completeness. We
consider an arbitrary piece of a 2-D medium (Fig.

Stress state in physical stress space

Mohr's circle of stress (black) with the Mohr–Coulomb yield criterion
(red) of the angle of internal friction

A yield curve can be defined in the local stress (

We can then express the fracture angle for stress states on the yield curve
envelope by placing Eq. (

From the previous equations, some implications about the elliptical yield
curve immediately follow. As shown in Fig.

Illustration of the Mohr's circle applied to the elliptical yield
curve (black ellipse) in

Mohr's circle of stress with an arbitrary yield curve (black line)
in the fracture plane reference.

Note that a yield curve in (

The shear and bulk viscosities are symmetrical about the center of the
ellipse. This implies that they are equal for divergence and convergence.
Clearly this is not physical since, for shear deformations where ice floes
continue to interact with one another (termed the quasi-static flow regime

Applying Mohr's circle to the Coulombic yield curve explains why the
non-differentiable corners in the yield curve lead to numerical problems
(Fig.

The Coulombic yield curve with an internal angle of friction of 1 (

Mohr's circle applied to the Coulombic yield curve (in black) in

DR designed the experiments, ran the simulations, and interpreted the results with the help of ML and LBT. NH contributed to the discussion on LKFs in simulations and observations. DR prepared the paper with contributions from all co-authors.

The authors declare that they have no conflict of interest.

We would like to thank Jennifer Hutchings and Harry Heorton for their helpful reviews on this paper, as well as Amélie Bouchat and Mathieu Plante for useful discussion during this work. This project was 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” (IRTG 1904 ArcTrain). The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the program “Mathematics of sea ice phenomena” when work on this paper was undertaken. This work was supported by EPSRC grant numbers EP/K032208/1 and EP/R014604/1. This work is a contribution to the Canadian Sea Ice and Snow Evolution (CanSISE) network funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Marine Environmental Observation Prediction and Response (MEOPAR) Network, and the NSERC Discovery Grants Program awarded to L. Bruno Tremblay.The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

This paper was edited by Daniel Feltham and reviewed by Harry Heorton and Jennifer Hutchings.