Computational Modeling of Fracture, Large Deformations, and Transport Mechanics in Multiphysics ProblemsImplementing calving laws based on linear elastic fracture mechanics (LEFM) into CISM
Ravindra Duddu Department of Civil and Environmental Engineering
Department of Earth and Environmental Sciences Vanderbilt University
Email: ravindra.duddu@vanderbilt.edu
CESM Annual Workshop, Boulder, CO June 19, 2018
This work was supported by NSF PLR #1341428 grant from the Antarctic Glaciology program
Antarctic ice sheet bedrock elevation map (Source: BEDMAP 2 British Antarctic Survey)
Conceptualization of ice sheet flow and fracture (Courtesy: Prof. Helen Fricker, UC San Diego)
Antarctic Ice Sheet Fracture and Sea Level Rise
• The Antarctic ice sheet is comprised of 27 million km3 of ice, with a sea level equivalent of ~ 58 m [Fretwell et al., 2013]
• Iceberg calving from Antarctic glaciers and ice shelves due to fracturing is a dominant mode of ice mass loss [Rignot et al., 2010; Liu et al., 2015]
• Can hydrofracture of crevasses cause 1– 3 m of sea level rise within the 21st and 22nd centuries? [DeConto and Pollard, 2016]
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Rifts – large horizontal fractures (Larsen C ice shelf, Antarctic Peninsula)
Crevasses – deep vertical fractures (Ross ice shelf, West Antarctica)
Technical Challenges with Ice Sheet/Shelf Fracture Simulation
Objective: To understand crevasse propagation and its relation to viscous ice flow and deformation and parametrize calving laws in ice sheet models
Technical challenges: • Fluid mechanics (Stokes flow) coupled with solid mechanics (fracture) • Vast separation of time scales of viscous flow and brittle fracture • Vast separation of spatial scales of ice sheet and individual crevasses
Introduction
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Land terminating glacier (fast flow)
Damage field in the domain at time (a) t = 0; (b) t = 2 weeks; (c) t = 2 years
**Resolved with a 2 m mesh size
t = 0
Full Stokes Modeling Combined with LEFM Introduction
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Schematic of the combination of ISSM and LEFM [Yu et al., 2017]
(a) Initial condition; (b) crevasses propagate; (c) crevasses advect downstream; (d) crevasse grow
Resolved using 5 m mesh size near crevasses.
A mesh moving method is used to update the mesh as crevasses grows
Yu, H., Rignot, E., Morlighem, M. and Seroussi, H. The Cryosphere, 11, 1283 – 1296, 2017
Ice shelf fracture parameterization in an ice sheet model Introduction
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Schematic of vertically integrated continuum damage model [Sun et al., 2017]
The ice sheet divided in 3 layers:
(1) damaged by surface crevasses (2) intact (3) damaged by basal crevasses
Implemented in BISICLES
Resolved using 0.5 km near the grounding line and 4 km mesh size elsewhere.
Sun, S., Cornford, S. L., Moore, J. C., Gladstone, R., and Zhao, L. The Cryosphere, 11, 2543 – 2554, 2017
Scheme to Incorporate the LEFM Calving Law Models and Methods
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Velocity Stress
Hydraulic pressure
Crevasse depth
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1. Vertically integrated 2D (plan view) model for ice shelves and ice streams
2. Solves for velocity
3. Nonlocal stress balance
(ice shelves) basal drag Horizontal velocity is vertically
invariant (plug flow)
τD: driving stress
Winkelman, R., Martin, M.A., Haseloff, M., Albrecht, T., Beuler, E., Khroulev, C., and Leverman, A. The Cryosphere, 5, 715 – 726, 2011
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Idealization of Crevasse Propagation in LEFM Models and Methods
Water filled crevasses in a marine terminating glacier with free basal slip or floating boundary condition [Jimenez and Duddu, 2018]
Jimenez and Duddu, Journal of Glaciology, Accepted, 2018
Central through crack in a finite width plate – basal crevasse in a grounded free slip glacier
Double edge crack in a finite width plate – surface crevasse in a grounded free slip glacier
Single edge crack in a finite width plate – surface crevasse in a floating ice shelf
Linear Elastic Fracture Mechanics Model
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Models and Methods
Net stress on the crack walls is obtained by adding hydraulic pressure
Longitudinal Cauchy stress as a function of depth (obtained from SSA model)
Stress intensity factor using the weight function method [Bueckner, 1970; Rice, 1972; Petroski and Achenback, 1978]
Depth of crevasses is evaluated using a iterative (bi-section) algorithm
= − ( − )
net z = + ( − − )
I = 0
I(s) = IC I(b) = IC Jimenez and Duddu, Journal of Glaciology, Accepted, 2018
Feedback Between Fracture and Flow
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Vertically integrated effective viscosity defined as
Damage is allowed only in a small region at mid-length
Jimenez and Duddu, Journal of Geophysical Research, under review, 2018
= 1 − +
−1/3 eq
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Superposition Principle for Water-filled Fractures Models and Methods
Stress intensity factor can be obtained by integrating the far field stress on the crack surface only in a linear elastic medium
Stress at the crack tip is mainly dependent on the nonlinear viscous ice rheology [Jimenez et al., 2017]
Superposition principle is not valid for nonlinear viscous media
Jimenez and Duddu, Journal of Geophysical Research, under review, 2018
Conclusion
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1. Is computationally tractable than full Stokes and damage mechanics
2. Avoids explicit meshing or description of crevasses
3. Accounts for the feedback between crevasses and flow
4. Incorporate hydrofracture in crevasses based on superposition principle
Parametrizing ice shelf fracture in CISM Models and Methods
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1. How to combine depth integrated approximation models with LEFM? ([Sun et al., 2017] used zero stress model)
2. How to avoid explicit meshing or description of crevasses? ([Sun et al., 2017] defined crevasse depth ratio as depth averaged damage)
3. How to account for the feedback between crevasses and flow? ([Sun et al., 2017] modified ice viscosity based on local damage)
4. How to incorporate hydrofracture in crevasses? ([Sun et al., 2017] employed superposition to add water pressure)
Implementing calving laws based on linear elastic fracture mechanics (LEFM) into CISM
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