# Fracture density modeling¶

The fracture density approach in PISM is based on [92] 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. [93] and [94]) 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 [95]).

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 [95]), that is related to the strain rate (longitudinal spreading or effective strain rate; Eq. 9 in [92]). 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 [92]).

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.

1. borstad_limit (no) Model fracture growth according to the constitutive law in [96] (Eq. 4), ignoring fracture_density­.gamma.

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

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

4. fd2d_scheme (no) Use an alternative transport scheme to reduce numerical diffusion (Eq. 10 in [95])

5. 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.

6. gamma (1) fracture growth constant $$\gamma$$

7. gamma_h (0) fracture healing constant $$\gamma_{h}$$

8. healing_threshold (2e-10 1/s) fracture healing strain rate threshold $$\dot \epsilon_{h}$$

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

10. initiation_threshold (70000 Pa) fracture initiation stress threshold $$\sigma_{\text{cr}}$$

11. lefm (no) Use the mixed-mode fracture toughness stress criterion based on Linear Elastic Fracture Mechanics, Eqs. 8-9 in [95]

12. 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 [95]

13. 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.

14. 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 [95])

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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