Unlike the prevalent maximum strain formalism commonly used by the sea ice community when modelling wave–ice interactions for example,, we develop a breakup criterion from the framework of fracture mechanics, based on the consideration that in solid, brittle materials, fracture happens to minimise the internal energy associated with deformation . In this framework, the total energy to be minimised is the sum of the elastic energy Eel associated with the deformation (in our case, bending) of the material, and of the fracture energy Efr associated with the creation of new surfaces around a crack. This energy decomposition is consistent with mode I fracturing, which in the case of our model translates to vertical fractures due to in-plane tensile stress.

The elastic energy density (per unit length in the transverse horizontal direction, and per unit thickness) stored in a material that is elastic, isotropic, and homogeneous, and stretched only in the longitudinal direction, is 7Wel=D2hκ2(x) with 8κ(x)=d2wdx2 the local linear floe curvature due to the deformation. Equation () stems from integrating the density of elastic energy (per unit volume) along the axis normal to the neutral plane of the plate (in our case, the z-direction), and takes into account stretching or compression in the directions of the plane (in our case, simply the x-direction). By integrating Eq. () along the floe, we obtain the surface energy density, 9Eel=D2h∫0Lκ2(x)dx.

The fracture energy density Efr relates to the energy required to create a new surface. In the case where the only admissible fractures vertically break the ice through its entire thickness, we simplify the formulation from  as 10Efr=NfrG with Nfr the number of fractures, and G the energy release rate. Again, note that this energy is expressed per unit surface normal to the x-direction.

To determine whether a floe breaks, we compare two energy states: that of the unbroken, deformed floe, and a hypothetical state in which this floe has fractured into several fragments. In the former state, the elastic energy, noted Eel0, is that of the deformed floe, as defined in Eq. (). In the latter state, the total elastic energy, noted Eels, is the sum of the elastic energies of the individual newly broken floes. If the broken state is – from an energy standpoint – favourable, it should replace the unbroken state. Formally, we look for the finite set of the fracture locations, xfr={xj∣xj∈(0,L)}. This set has size Nfr, the number of fractures, dividing the original floe into Nfr+1 fragments. It should minimise the free energy F defined as 11F(xfr)=Efr(Nfr)+Eels(xfr)-Eel0 with the additional constraint that for breakup to occur, we must have F<0. In other words, a floe breaks if the elastic energy released by the breakup exceeds the energetic cost of the breakup. If no such set can be found, we conclude that the current deformation of the floe is not sufficient to fracture it. The post-fracture elastic energy expands to 12Eels(xfr)=D2h∑j=0Nfr∫xjxj+1κj2(x)dx with x0=0 and xNfr+1=L. The curvatures κj are obtained from solving Eq. (6) individually for every (at this stage, still hypothetical) fragments.

Equation () has an explicit dependency to the number of fractures allowed to happen for a given quasi-static state, that is, at a given time. It suggests that the size of xfr should be a dimension to the minimisation problem. In practice, when considering travelling waves, floes of reasonable size, and the succession of such quasi-static states, at most one single fracture is admissible at a given timestep, which greatly diminishes the numerical cost of the procedure. In what follows, we will thus use Nfr=1, xfr={x1}, and we will have 13Eels(x1)=Eel<(x1)+Eel>(x1) with Eel<(x1),Eel>(x1) the elastic energies of the left (x<x1) and right (x>x1) fragments obtained from that single fracture, while the fracture energy reduces to Efr=G. Hence, we look for xfr=argminx1∈(0,L)(Eel<+Eel>),(14a)F(xfr)<0.(14a)

For given ice and wave conditions, fracture search can thus be conducted in a completely deterministic manner. In practice, we proceed by sampling Eels(x1) regularly on (0,L), ensuring the sampling rate is sufficient to capture its oscillations. We find the set of arguments of the peaks (the local maxima) of this discretised Eels, which we augment with the bounds {0,L} of the domain, to obtain an ordered sequence of at least two coordinates bounding, two by two, local minima of Eels. We conduct local minimisation between the bounds using Brent's method . Finally, the smallest of these minima is validated against Eq. (14b). If this condition is verified, its argument is the fracture location xfr. These steps are summarised in Fig. , and a fracture search is illustrated in Fig. . Note that in this case, the asymmetry of the total energy profile comes from differences in wave phase at the edges of the floe, and wave attenuation by the ice cover, discussed in Sect. .

To allow future comparisons, we additionally implement a conventional strain criterion for fracture. Under that formulation, the floe is allowed to fracture if the bending strain ε locally exceeds a prescribed critical strain εcr, that is if 15∃x∣|ε(x)|>εcr with 16ε(x)=-h2d2wdx2 the maximum (when taking the absolute value) bending strain, here defined as evaluated at the top of the floe.

Typically, if Eq. () holds anywhere, it holds on continuous intervals along the floe. We illustrate this in Fig. . A second criterion must then be chosen to constrain the fracture. Herein, we arbitrarily choose to consider the global strain extremum (single fracture). We thus have xfr=argmaxx∈(0,L)|ε(x)|,(17a)|ε(xfr)|>εcr.(17b) This criterion can be straightforwardly extended to the case of multiple fracture by considering all the local extrema exceeding the critical value.

Our two fracture parametrisations rely on two different parameters: the energy release rate G (energy criterion) or the critical strain εcr (strain criterion). These parameters have to be measured or estimated from sea ice samples. Sea ice properties can vary greatly based on its history and environmental conditions, such as temperature, brine fraction, or past loading rate. Nevertheless, G and εcr are physical quantities that can be measured, and are not completely free parameters. For a detailed compilation of sea ice property measurements, we refer the reader to .

Additionally, ice strength depends on the direction of the applied stress. We are here solely interested in failure from bending (mode I, or opening mode; see ), which is compatible with Griffith's model of fracture as well as with wave action. In particular, in the plane strain approximation, εcr can be related to the flexural strength σf so that εcr=1-ν2Yσf, and G to the fracture toughness K1c so that G=1-ν2YK1c2. compile values of K1c in the range 75 to 150 kPam12. measured values down to 26 kPam12 for floating samples in the lab, a reduction that could be attributed to temperature or size effects . Reported values of εcr are typically in the range of 10-5 to 10-4 , even though larger value (on the order of 1×10-3) have been reported for lab-grown, saline ice . Sea ice is subject to fatigue, and repeated cyclic loading was shown to lower its apparent flexural strength .

For a prescribe angular frequency ω, we derive wavenumbers k from the dispersion relations 18ω2g=ktanh⁡(kH) in the open-water parts of the domain, and 19ω2g=(Dρwgk4+1-ω2gd)ktanh⁡(k(H-d)) in the ice-covered parts. We use the single symbol k for brevity, and the appropriate dispersion relation should be understood from context. In the right-hand side of Eq. (), the term Dρwgk4 corresponds to the elastic response of the ice cover, while the term ω2gd corresponds to its mass-loading response. Whether the former has a significant contribution to the dispersion relation when the ice is heavily fragmented is debated . As it can easily be turned off, we include it here for completeness.

We note that this relation dispersion can be derived by considering the bending of a plate conforming to a fluid surface excited by harmonic waves. This can therefore be seen as a soft coupling of the fluid to the plate. The dispersion relation of plate excited by harmonic waves, without a fluid foundation, would otherwise be ω2=Dρihk4.

For any given floe in the domain, a linear monochromatic wave can be parametrised with two complex variables, amplitude a^ and wavenumber k^. The modulus a=|a^| denotes the amplitude of the wave at the left edge of the floe, while the argument ϕ=Anga^ denotes the phase of that wave mode at the left edge of the floe. The real part of the wave number, k=Rek, describes wave propagation while its imaginary part α=Imk^ describes the spatial rate of attenuation in the direction of propagation. Following , we implement a parametrisation with attenuation linear in ice thickness and quadratic in wavenumber, so that 20α=14hk2. Other parametric attenuations can easily be added to the current framework; either directly to SWIIFT's codebase

A parametrisation derived from was added to a later version.

A linear polychromatic plane wave can be defined by superposition. The wave state is then 21η(x)=∑jIm[a^jexp⁡(k^jx)] where the subscript j denotes spectral modes. The modal amplitudes can be derived from any spectral density S(ω), using the relationship 2212aj2=S(ωj)Δωj where Δωj is the width of the angular frequency bin corresponding to the amplitude aj.

To allow for the advection of a developing sea into the ice-covered domain, we apply a semi-Gaussian kernel to the sea state. We implement this modification to avoid the non-realistic situation of a fully developed sea appearing under a potentially kilometre-wide MIZ. Therefore, Eq. () is modified into 23η(x)=∑jIm[a^jexp⁡(k^jx)]Kj(x) with 24Kj(x)=exp⁡(-(x-μj)22σj2)x≥μj,1x≤μj. To each wave mode, we associate a coordinate μj. The wave is considered fully developed in the half-plane left of that coordinate. The parameters σj control the width of the transition between a fully developed wave, and a near-rest state (as the wave envelop of a given mode is reduced to about 1 % of its maximum at x=μj+3σj).

Let τ be a model timestep. Each wave mode in Eq. () propagates at a phase speed 25cj=ωjkj. Between time tn and time tn+1, the phase ϕj of mode j increases by -ωjτ. The limit μj of the fully developed wave advances of a distance cjτ. Therefore, we iterate in time by updating the values of our a^j and μj by these quantities. A new quasi-static wave profile η(x) can then be computed across the domain.

Once the sea surface has been computed, the resulting sea ice deflection w(x) is computed for all individual floes, which are scanned for possible fractures. For a given floe, if μj>L∀j (that is, the wave acting on the floe is fully grown), the deflection can be determined analytically, as developed in Appendix . Otherwise, we obtain a solution to Eq. (6) with a numerical solver . In turn, the deflection is used to compute the curvature. Depending on the chosen fracture mechanism (energy-based or strain-based), the floes are considered for breakup, as described in Sect.  and .

If using the energy criterion, we evaluate the post-fracture total energy along the discretised floe. We use a peak detection algorithm to separate intervals of convex free energy (as can be identified in Fig. a), onto which Eq. () is minimised. If the global minimum among these local minima satisfies Eq. (14b), fracture occurs; these steps are summarised in Fig. . If using the strain criterion, we evaluate the bending strain along the discretised floe. Again, a peak detection algorithm is run on -ε2(x) to detect convex intervals, and we conduct local minimisation on these, which is equivalent to maximising |ε(x)|. If the global minimum satisfies Eq. (17b), fracture occurs.

The input necessary for both parametrisations is thus the floe curvature. In Sect.  and , we merely suggest a simple mechanical model to infer this curvature from wave forcing. Other 1D models outputting the curvature field, or actual curvature measurements, can be substituted without having to alter the fracture formalism presented in Sect. . However, for the fracture parametrisation to be sensible, it is necessary that the mechanical model can be stepped forward in small time increments. Thus, we choose here not to rely on harmonic solutions to the bending problem, such as in , as these rely on the hypothesis that a steady state has been reached in the whole fluid domain. We do so at the cost of neglecting floe bending inertia and relaxing constraints on the fluid itself. In particular, wave scattering induced by different boundary conditions imposed on the fluid when transitioning between open water and ice-covered water regions is not accounted for. A comparison between the two types of solution can be found in Appendix . We find minor differences in terms of floe curvature (impacting the strain parametrisation) and resulting elastic energy (impacting the energy parametrisation). We compute the ratio of elastic energy (Eel, defined in Eq. ()), derived from the scattering model, to that same energy derived from SWIIFT, for the case of a polychromatic forcing. The elastic energy is, generally (77.5 % of the ensemble considered), overestimated by SWIIFT; but the distribution of these ratios being skewed, the two models yield, on average, a similar value (mean of the ensemble considered: 1.02, geometric mean: 0.69). Additionally, we do not find these ratios to depend on model parameters such as ice thickness or floe length. We thus conclude that even though differences exist between the two solutions, they are less meaningful than random fluctuations of the wave state.

Care must be taken when selecting a model timestep, τ. The theoretical upper limit for crack propagation in an elastic, isotropic, and homogeneous material is set by the speed of Rayleigh waves, cR=Y2ρi(1+ν). Using Y=3.8GPa and ν=0.33, values of the Young's modulus and Poisson's ratio estimated in situ for sea ice , this speed is on the order of cR≈1250ms-1. As we consider cracks to instantaneously fracture floes through their thickness, we must have τ>hcR. For 1 m thick ice, it corresponds to τ>0.8ms. Fractures propagate faster in stiffer ice, and increasing the Young's modulus (or reducing the thickness) would lower this bound, which must be considered on a case-by-case basis.

However, we also want to keep the timestep small enough that we can detect fractures as soon as it is possible for them to occur, as delaying the onset of a fracture may affect the length of the resulting floes. Therefore, we aim to keep the ratio of the progression of the wave front to the wave amplitude small. In the monochromatic case, with phase speed c, it translates to having τca<r⇔τ<rakω, with r<1. Setting r=15 ensures sufficient convergence. An analogous relationship can be derived for polychromatic cases, by substituting the amplitude by the significant wave height HS of the spectrum, and the phase speed c by the maximum phase speed of within the sampled spectrum, so that τ<rHSmax⁡jcj.

We show numerical results in Fig.  (colour-coded lines), which we compare to experimental data from (circle and square markers). We obtain results similar to those presented in Sect. . The critical amplitude profiles do not depend on the mechanical properties of the simulated material, up to a multiplicative constant, that increases with flexural rigidity (or, equivalently, flexural length).

This fact extends to the other variables shown in Fig. . Within the ranges of mechanical parameters explored, which are in agreement with the values and internal variability estimated by , the order of magnitude of the critical amplitude and its decreasing tendency with increasing kLD agree between the simulations and the laboratory experiments. However, and as expected, this agreement is only qualitative. Indeed, we recall that in the experimental setting, fracture was only obtained with nonlinear waves. It is likely the reason why the critical amplitudes measured by varies as k-1, not as k-2 as we find numerically. The differences between experimental and numerical critical amplitudes are therefore deepened at small wavenumbers.

We also note that the way we define the relaxation length Lκ in our linear waves simulations differs from the definition of , who used the full width at half-maximum (hereinafter, FWHM) of pre-fracture curvature in their nonlinear experiments. On Fig. , we thus also represent the experimental critical amplitudes as a function of LDFWHM (pink squares), which horizontally shifts the experimental points, in an attempt to correct the discrepancy between their nonlinear waves forcing and our linear model.

We do not adopt their definition, as it would be incompatible with our region 4 results, where fracture does not happen around curvature peaks or troughs, while experimental fractures always happened in the vicinity of a deflection antinode. If we were to apply it to regions 1 to 3, we would obtain something very similar to the FWHM of sin⁡(kx), that is 2π3k=λ3. Because of their nonlinear wave forcing, measured FWHMs that varied like λ12. Bending was thus concentrated in a smaller fraction of their forcing wavelengths, and LDFWHM can be seen as an alternative, rescaled dimensionless wavenumber. Using this definition, we can improve the overlap between experimental and numerical results, with experimental points falling in regions 1 to 3, where both model end experiments show the critical curvature depends on the forcing wavelength, precluding a constant strain threshold.

So far, we focused on the harmonic number n=3. However, for large enough kLD, the response of the model depends on n, and therefore on the geometry of the domain. In particular, for even n, fracture in region 4 happens systematically in the middle of the floe. Due to the symmetry property of the forcing, this means our model predicts fracture with κcr=0m-1, inconsistent with bending fracture but reminiscent of shearing or tearing fracture. The fundamental configuration, with n=1, is a particular case. The maxima of deflection, curvature, and free energy happen at x=L2 independently of kLD. We illustrate this difference in Fig.  by presenting the relationships between the nondimensionalised wavenumber and two dimensionless quantities, for different harmonic wavenumbers. We choose these two quantities because they exhibit the remarkable property of depending on the dimensionless wavenumber, but not on the individual variations of the mechanical parameters.

The first of these quantities, shown in Fig. a, is the relaxation length Lκ (as shown in Fig. d for the fixed parameters experiment), normalised by the augmented flexural length 2LD. We retrieve, independently of the harmonic number, the limit lim⁡k→0Lκ=2LD already stated in Sect. . Behaviours depending on the harmonic number emerge from kLD≈0.4. If all curves show a downward trend in what corresponds to kLD in regions 2 and 3, this trend is more pronounced for smaller numbers, in particular n∈{1,2}. The second striking difference, is that between even and odd harmonic numbers. There is a discontinuity at the transition from region 3 to region 4 for even numbers, with an upward jump preceding a sustained downward trend. This transition is continuous for n=3. There is no region 4 behaviour for n=1, as in this configuration, fracture always happens at xfrL=12.

The second dimensionless number, shown in Fig. b, can be built as the product of two distinct dimensionless quantities: critical curvature multiplied by thickness (that is, twice the critical strain), and critical curvature multiplied by relaxation length. Notably, both these quantities do depend on thickness and Young's modulus, without showing the ordering critical amplitude does in Fig. . However, their product, κcr2hLκ, only depends on kLD. This quantity was also derived by , who interpreted it as a constant independent of the wave forcing. We do not replicate this result outside of regions 1 and 2, that is, the wavenumber band where neither critical curvature nor relaxation length vary. As in Fig. a, the different curves are indistinguishable within these two regions (that is, at small kLD), and a discontinuity exists for even harmonic numbers between regions 3 and 4, as for these, the critical curvature drops to 0 m-1 in region 4.

Finally, it can be seen that the upper bound of the range of kLD that sees fracture happens depends on n. It first increases with n but peaks for n=3, and then decreases. This is despite keeping the same acrk≤0.5 criterion on the wave slope. The lower bound, however, does not change and keeps the value kLD=0.1167. The differences between the different harmonics in region 4 are explained by the loss of deflection extrema near the boundaries as kLD increases, which have dissimilar effects on the deflection profile in this region for different n. For n>2, there exists a single kLD for which the slope of the deflection at the edges of the plate, (dwdx)x={0,L}, cancels. It converges exponentially towards 1, so that noting it (kLD)0, we have log⁡|1-(kLD)0|∼-n. The cancellation of the slope conveys the transition from a deflection profile with n antinodes to one with n-2 antinodes. When kLD keeps increasing, the higher n, the higher the variety of behaviours shown by w. However, for large enough kLD, w(x)→2anπ for odd n, and w(x)→2an(-2xL+1) for even n. These are rigid motions, independent of the forcing, corresponding respectively to heave (translation) or pitch (rotation).