We develop a phase-field model of brittle fracture to model fracture in sea ice floes. Phase fields allow for a variational formulation of fracture by using an energy functional that combines a linear elastic energy with a term modeling the energetic cost of fracture. We study the fracture strength of ice floes with stochastic thickness variations under boundary forcings or displacements. Our approach models refrozen cracks or other linear ice impurities with stochastic models for thickness profiles. We find that the orientation of thickness variations is an important factor for the strength of ice floes, and we study the distribution of critical stresses leading to fracture. Potential applications to discrete element method (DEM) simulations and field data from the ICEX 2018 campaign are discussed.

The fracture of sea ice at intermediate scales,

Modeling intermediate-scale fractures is challenging, and most smaller-scale observations are limited to the lab scale on the order of

Previous studies have connected fracture angles and intermediate-scale models

There are multiple candidates for fracture models in sea ice with various strengths and drawbacks.

Phase-field models of fracture

In this work, we investigate floe-scale fracture with phase-field models. We simulate fracture under boundary force or displacement conditions for a distribution of stochastic ice thickness fields, which model refrozen ice cracks or ice ridges. Our experiments show that the critical forcing at which fracture occurs depends on the geometry of the thickness anomalies and their relation to the forcing to which the floe is subjected. Additionally, we discuss measurement data from the ICEX 2018 expedition in fracture simulations and prospects for incorporating physical fracture in discrete element models.

The outline of the paper is as follows. In Sect.

In this section, we look at how phase-field models of brittle fracture
can be used to model kilometer-scale fractures. We define a
phase-field formulation of brittle fracture with linear elasticity in
Sect.

We describe ice floe deformation using a linear constitutive relation.
The floe geometry is modeled by a two-dimensional domain

We use a phase-field model of quasi-static brittle fracture developed
by

Determining a crack set

Combined with other constitutive relations,
phase-field models have also been used to capture different modes of
fracture

Following

The algorithm to solve Eq. (

Staggered minimization for phase-field model of fracture.

Solve Eq. (6) for

Solve Eq. (7) for

We introduce the ice thickness

We use reference Lamé parameter values from available sea ice
measurements, namely, Young's modulus

For the fracture toughness we choose

Our implementation is based on the open-source finite-element software
FEniCS

In this section, we use the phase-field formulation described in Sect.

For the following experiments, we use a 1

Our stochastic model is similar to
a model of random linear features, also called discrete
fracture networks (DFNs)

Diagram of the mollifying function

Here,

Random
realizations of this stochastic model can be seen in the top row in Fig.

In our experiments, we assume the following boundary conditions on the
right side

In all experiments below, we use

Illustration of experimental setup. The floe is fixed along the right edge, stress-free along the top and bottom edges, and is subject to normal stress or displacement along the left edge. The blue color indicates the floe thickness, where lighter regions are thinner.

We first study fractures that occur as a result of a prescribed tensile
displacement. In these numerical experiments, we augment the displacement boundary
conditions (10) with the following condition on the
left boundary:

A staggered algorithm or any method that computes critical points of Eq. (

In separate calculations, we have performed compression experiments by enforcing a positive normal displacement in Eq. (

Top row: random realizations of ice thickness; brighter
features correspond to thinner ice. Bottom row: corresponding
fractures under tensile stress, where the phase field

Next, we consider a follow-up experiment on the ice thickness realizations with shear displacements. Different from Eq. (

The same as Fig.

In our next set of experiments, we combine the boundary conditions (10) with a
normal pulling force condition on the left edge

Histogram of critical stresses leading to vertical fracture using 1000 random samples of linear weakness fields.

Using this approach, we study the distribution of critical stresses for the various
ice thickness fields. In Fig.

Each dot in the scatterplot in panel

In Fig.

Next, we study if the angle between the force
direction and the linear features of thinner ice affects the floe
strength. We define the average absolute value of the sine of the
angles of the linear thinner ice regions as

To further study which weaknesses are the most important for floe
fracture, we compare floes with all

Additionally, we study the critical stress for ice floes with a single vertical feature of thinner ice that spans the entire floe and goes through the middle.
In Fig.

Height

Here, we discuss the potential use of phase-field models for individual floe fractures for incorporation into large-scale DEMs and for analyzing measurements from the 2018 ICEX field campaign. The following two subsections begin by outlining the applications, followed by discussing technical challenges and potential paths forward.

Time-evolving DEM simulations of interacting ice floes are likely to be more accurate if they include realistic intermediate-scale fractures. Ideally, such fracture calculations should not drastically increase the computational cost of the time steps in a DEM. This creates critical hurdles to implementing fracture models for floes. Below we discuss potential paths forward in order to incorporate a physics-based fracture model, such as the phase-field model proposed here, into a DEM.

Existing DEM implementations stimulate sea ice as tightly bounded particles

Computational cost prohibits running the phase-field fracture algorithm in each time step and for each floe. In a typical DEM, each floe element requires estimating stresses due to collisions, which, at most, amounts to a partial differential equation (PDE) solve.
Running the staggered phase-field solver on each element would require many PDE solves for each floe, making the straightforward use of the phase-field fracture model as part of a DEM infeasible.
Nevertheless, we imagine two possible approaches to combining phase-field fracture models with a DEM: (1) speeding up the computation using a fracture dictionary generated by a large number of individual ice floe experiments, potentially interpolating between fracture experiments using machine learning, and (2) by detecting the instances when a phase-field fracture computation is crucial in order to decide whether and how a floe should fracture. In Sect.

A particular challenge would be to model leads across multiple floes, which can be observed in data. Such a behavior could for instance be achieved if one floe's fracture increases the stresses in the neighboring floe, which, as a result, also fractures. This likely requires a fracture model where fractures are initiated through stress localization and occur in the lead direction.
In Sect.

Motivated by the high-frequency displacement data available from the ICEX 2018 expedition, we are interested in whether such data can be used to identify fractures when or even before they occur. The ICEX expedition ran from 8 to 21 March 2018 at Ice Camp Skate in Beaufort Sea, roughly 230 km north of Prudhoe Bay, Alaska.
Researchers spread 24 reflectors on the ice within 1 km of the camp. The observation area contained both first-year and multi-year ice. A high-precision robotic system measured the location of the reflectors roughly every 1–3 min. Measurements were taken both before and after a crack appeared within the observation area. For a more detailed description of the data and its acquisition, we refer the reader to

There are several challenges to using such data to reproduce and estimate the time and shape of the physical crack. First,
using position time series data in a quasi-static fracture model as proposed in this paper is not straightforward. Reflector positions need to be converted to displacements, and we do not have information on existing large-scale background stresses or displacements. Second, key data are missing. The experiments on stressed ice floes in Sect.

Our code implementation is available at

Data reproducing the floe fracture experiments in this study can be generated by running the code at

All authors contributed to the model formulation and the conceptualization of the experiments. HD developed the model code and performed the simulations. All authors contributed to the analysis of the numerical experiments. HD and GS prepared the paper with contributions from all co-authors.

The contact author has declared that none of the authors has any competing interests.

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

We appreciate many helpful discussions with Gilles Francfort, Blaise Bourdin, Georgy Manucharyan, Brandon Montemuro, and Matthew Parno.

This research has been supported by the Office of Naval Research (grant no. N00014-19-1-2421).

This paper was edited by Yevgeny Aksenov and reviewed by Damien Ringeisen and two anonymous referees.