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Minor edits in fracture doc
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talbrecht committed Sep 18, 2021
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Fracture density modeling
-------------------------

The fracture density (Continuum Damage Mechanics) approach in PISM based on
:cite:`AlbrechtLevermann2012` 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. :cite:`Lemaitre1996Damage` and
The fracture density approach in PISM is based on :cite:`AlbrechtLevermann2012`
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. :cite:`Lemaitre1996Damage` and
:cite:`Borstad2013Creep`) introducing a damage state variable (`\phi` or `D`) that
equals zero for fully intact ice and one for fully fractured ice. This can be interpreted
as a loss of all load bearing capacity.
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 :cite:`AlbrechtLevermann2014softening`).

Fractures form above a critical stress threshold `\sigma_{\text{cr}}` in the ice (von
Mises or maximum stress criterion or fracture toughness from Linear Elastic Fracture
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
:cite:`AlbrechtLevermann2014softening`), that is related to the strain rate (longitudinal
spreading or effective strain rate; Eq. 9 in :cite:`AlbrechtLevermann2012`). Fracture
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