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\begin{document}

\title{A Mathematical Modeling for Stability of Crevasses}



\author[1]{Maryam Zarrinderakht}%
\author[1]{Christian Schoof}%
\affil[1]{Department of Earth Ocean and Atmospheric Sciences  University of British Columbia}%


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  \date{May 25, 2024}


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\begin{abstract}
Calving is the process of blocks of ice detaching from an ice shelf of
grounded calving cliff. Here, we focus on calving that occurs through
the propagation of fractures through a floating ice shelf on
sufficiently short time scales to allow ice rheology to be treated as
elastic. We revisit the linear elastic fracture mechanics models of
Weertman and van der Veen, which consider the propagation of cracks into
slabs of ice, driven by an applied extensional stress and by water
pressure inside the crack, due to sea water and surface melt entering
the cracks. We extend their work by considering the interaction between
multiple cracks and developing a method that allows us to compute crack
propagation in arbitrary domain geometries. We show that the simple case
of two aligned cracks, one extending from the ice surface and the other
from the base, can be considered as a two-dimensional dynamical system.
We are able to show that viable steady crack configurations (where the
ice shelf is crevassed without calving) correspond to stable fixed
points of that dynamical systems. Calving corresponds to the
annihilation of steady states under a parameter change. That can either
take the form a bifurcation that happens at specific combinations of
forcing parameters, and leads to the abrupt, dynamic propagation of the
crack across the remaining unbroken thickness of ice. Alternatively,
calving can occur because the two crack tips gradually meet as forcing
parameters change. We derive different forms of calving laws, depending
on whether crack propagation to full calving is initiated from a
previously un-cracked floating slab of ice, or from a previously cracked
configuration. For the former, we show that calving laws take the form
of a functional relationship between a water storage parameter,
extensional stress, ice thickness and fracture toughness. For the
latter, we obtain an history-dependent relationship in the form of a
steady crack evolution problem that bears abstract similarity with
plasticity models. We also discuss how these could be implemented in ice
flow models.%
\end{abstract}%



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