# Fracture density modeling¶

The fracture density approach in PISM is based on [101] and assumes a macroscopic measure for the abundance of (partly microscale) crevasses and rifts that form in ice (shelves) and that can be transported with the ice flow as represented in a continuum ice-flow model. This approach is similar to the Continuum Damage Mechanics (CDM) (e.g. [102] and [103]) introducing a damage state variable (\(\phi\) or \(D\)) that equals zero for fully intact ice and one for fully fractured ice, that can be interpreted as a loss of all load bearing capacity.

The feedback of damage to the ice flow (creep) works within the existing constitutive framework by introducing a linear mapping between the actual physical (damaged) state of the material and an effective state that is compatible with a homogeneous, continuum representation of the creep law (Eq. 6 in [104]).

Fractures form above a critical stress threshold \(\sigma_{\text{cr}}\) in the ice (e.g. von Mises criterion, maximum stress criterion or fracture toughness from Linear Elastic Fracture Mechanics) with a fracture growth rate proportional to \(\gamma\) (Eq. 2 in [104]), that is related to the strain rate (longitudinal spreading or effective strain rate; Eq. 9 in [101]). Fracture healing is assumed to occur with a defined healing rate below a strain rate threshold (scaled with the difference to the threshold or constant; Eq. 11 in [101]).

The fracture growth constant \(\gamma\) (`fracture_density`

`.gamma`

) is ignored if
`fracture_density`

`.borstad_limit`

is set.

To enable this model, set `fracture_density`

`.enabled`

.

Parameters

Prefix: `fracture_density.`

`borstad_limit`

(no) Model fracture growth according to the constitutive law in [105] (Eq. 4), ignoring`fracture_density`

`.gamma`

.`constant_fd`

(no) Keep fracture density fields constant in time but include its softening effect.`constant_healing`

(no) Use a constant healing rate \(-\gamma_h \dot{\epsilon}_h\) independent of the local strain rate.`fd2d_scheme`

(no) Use an alternative transport scheme to reduce numerical diffusion (Eq. 10 in [104])`fracture_weighted_healing`

(no) Multiply the healing rate by \(1 - D\), i.e. assume that highly damaged ice heals slower. This mechanism can be combined with`fracture_density`

`.constant_healing`

.`gamma`

(1) fracture growth constant \(\gamma\)`gamma_h`

(0) fracture healing constant \(\gamma_{h}\)`healing_threshold`

(2e-10*1/s*) fracture healing strain rate threshold \(\dot \epsilon_{h}\)`include_grounded_ice`

(no) Model fracture density in grounded areas (e.g. along ice stream shear zones) in addition to ice shelves`initiation_threshold`

(70000*Pa*) fracture initiation stress threshold \(\sigma_{\text{cr}}\)`lefm`

(no) Use the mixed-mode fracture toughness stress criterion based on Linear Elastic Fracture Mechanics, Eqs. 8-9 in [104]`max_shear_stress`

(no) Use the maximum shear stress criterion for fracture formation (Tresca or Guest criterion in literature), which is more stringent than the default von Mises criterion, see Eq. 7 in [104]`phi0`

(0) Fracture density value used at grid points where ice velocity is prescribed. This assumes that all ice entering a shelf at`bc_mask`

locations has the same fracture density.`softening_lower_limit`

(1) Parameter controlling the strength of the feedback of damage on the ice flow. If \(1\): no feedback, if \(0\): full feedback (\(\epsilon\) in Eq. 6 in [104])

Testing

See the scripts in `example/ross/fracture`

for a way to test different damage options
and parameter values. Build a setup for the Ross Ice Shelf and let the damage field
evolve, with fracture bands reaching all the way from the inlets to the calving front.

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