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A NONLOCAL DAMAGE MECHANICS FORMULATION FOR SIMULATINGCREEP FRACTURE IN ICE SHEETS AND GLACIERS
Ravindra Duddu1, Haim Waisman21Department of Civil and Environmental Engineering, Vanderbilt University; 400 24th Avenue South; 267 Jacobs Hall; Nashville, TN 37212
2Department of Civil Engineering and Engineering Mechanics, Columbia University; 610 Seeley W. Mudd Bldg; 500 West 120th Street, Mail Code 4709; New York, NY 10027email: [email protected], [email protected]
1. INTRODUCTION
• Creep fracture plays an important role in iceberg calving fromglaciers and in the catastrophic collapse of ice shelves.
• A better understanding of creep fracture is required to predict thesea level change and its impact on society.
• Current theoretical models based on linear elastic fracture me-chanics are too simplistic.
• Rigorous computational models can help us gain new insightsinto the relevant mechanisms behind ice sheet break up.
2. OBJECTIVES
1. Develop a three-dimensional thermo-viscoelastic constitutivemodel for polycrystalline ice
2. Devise a strain space nonlocal formulation to simulate structuralfailure due to creep fracture
3. Calibrate and validate the model using experimental data.4. Test the accuracy and consistency of numerical results using
benchmark examples and then make predictions.
3. VISCOELASTIC CONSTITUTIVE MODEL
Additive decomposition: Assuming small strains [1],
εkl = εekl + εdkl + εvkl. (1)
Stress-strain relations: The elastic, delayed elastic and viscousstrain components are given by,
σkl =E
3(1− 2ν)εeiiδkl +
E
(1 + ν)
(εekl −
1
3εeiiδkl
), (2)
εdkl = A
(3
2Kσdev
kl − εdkl), (3)
εvkl =3
2KN
(3
2σdevmn σ
devmn
)(N−1)/2
σdevkl . (4)
Temperature dependence: The relation for KN is,
KN (T ) = KN (Tm) exp(
−QR
(1T −
1Tm
)). (5)
4. NONLOCAL CONTINUUM DAMAGE FORMULATION
Effective stress concept: We define a transformation,
σij = Mijkl σkl, (6)Mijkl = 1
2 (ωikδjl + ωjkδil), ωik(δkj −Dkj) = δij . (7)
Damage rate: In a small strain Lagrangian framework,
Dij =
{fij , if max{εij} ≥ εth,0, if max{εij} < εth.
(8)
Damage evolution function: The generalized form is,
fij = B〈χ〉r(ωmnξ
(1)m ξ(1)n
)kσ[(1− γ) δij + γξ
(1)i ξ
(1)j
], (9)
χ = ασ(1) + β
√3
2σdevmn σ
devmn + (1− α− β)σkk. (10)
Tension-compression asymmetry: The different behavior of iceunder compression and tension is captured by kσ as,
kσ =
{[k1 + k2|σii|], for 0 ≤ σii ≤ 1 MPa,−[k3 + k4|σii|], for −3 MPa ≤ σii < 0.
(11)
Temperature dependence: The relation for B is,
B(T ) = B(Tm) exp(
−UR
(1T −
1Tm
)). (12)
Nonlocal damage rule: The nonlocal damage rate at Xk is [2],
Dnlij (Xk) =
∑NGP
l=1 Φ(Xk,X l)Dij(Xl)∑NGP
l=1 Φ(Xk,X l), (13)
Φ(Xk,X l) = exp
(2||Xk −X l||2
l2c
). (14)
5. MODEL CALIBRATION
(a) Ice under uniaxial tension
(b) Ice under uniaxial compression
6. MODEL VALIDATION
(d) Uniaxial loading |σ| = 0.8 MPa (e) Four point bending (f) Biaxial compressionThe above figures show the numerical results obtained for an ice slab subjected to various loading conditions. Fromfigure (d), it is evident that after 140 hours ice under uniaxial tension exhibits sudden rupture whereas no such failure isobserved under uniaxial compression, even at a later time. Figure (e) shows the good agreement between model resultsand experimental data for the four point bending test in the initial stages of creep. From figure (f), it is clear that the modelis able to capture the increase in ice strength under biaxial compression that is observed experimentally, quite well.
7. NUMERICAL RESULTS AND DISCUSSION
Figures (g)–(i) show the simulated crack paths for pure mode I creep fracture of an ice slab subjected to a uniform appliedtensile strain. The local damage model constricts the damage zone to be one element wide. As we decrease the meshsize h the damage zone width decreases and the collapse occurs early at lesser strain, which is not physical. The nonlocalintegral damage model introduces a material length scale lc, yielding thermodynamically consistent numerical results.
(g) Local vs. nonlocal damage (h) Local damage (i) Nonlocal damage, lc = 7 mFigures (j)–(m) show the simulated crack paths for mixed mode creep fracture of a square ice plate with a center cracksubjected to biaxial tension. Only 1/4th of the domain is modeled due to symmetry. It is clear from figures (j) and (k) thatthe local damage model results are mesh dependent. The nonlocal damage model alleviates the directional mesh bias offinite element calculations, yielding consistent and accurate results, given in figures (l) and (m).
(j) Local, structured (k) Local, unstructured (l) Nonlocal, structured (m) Nonlocal, unstructured
Next, we use the nonlocal damage model to simulate crevasse formation in idealized glaciers (ice slabs) consideringdepth varying ice flow velocity profiles. The set up of the simulation is given in Figure (n) and the predicted fracturepathway for v= 2.0× 10−4 m/s is shown in Figure (o). Next, we estimate the crack propagation rates by varying the flowvelocity and the bottom boundary condition (free slip/fixed), given in Figure (p). The results indicate that fractures canpropagate through the full thickness of the glacier when no hydrostatic pressure (compression) is considered.
(n) Schematic diagram (o) v= 2.0× 10−4 m/s (p) Crack length vs timeFigure (q) shows the satellite image of the loose tooth system in the Amery ice shelf in Antarctica. Figure (r) shows thenumerical results obtained for the ice shelf approximated as a square plate with two existing cracks subjected to biaxialtension. Owing to the applied boundary conditions we can see the crack forking at the tip resembling the loose toothsystem. Next, we meshed an extract of the actual geometry containing the loose tooth system and by assuming idealizedboundary conditions we performed a sample simulation shown in figure (s).
(q) Loose tooth system, Amery ice shelf (r) Idealized simulation (s) Simulation using the actual geometry
8. CONCLUSIONS
• The constitutive model calibrated using uniaxial test data gives good results under multiaxial loading.• The nonlocal damage formulation alleviates mesh dependence and yields thermodynamically consistent results.• Numerical results demonstrate the viability of the proposed formulation for studying iceberg calving and rift propagation.• In future, we shall model the influence of sea water level on crevasse formation so as to evaluate glacier stability.
9. ACKNOWLEDGEMENTS
The authors are grateful for the funding support provided by the Department of Energy under Grant #DE-SC0002137 (the ASCR SciDACISICLES initiative of DOE).
10. REFERENCES
[1] R. Duddu and H. Waisman. A temperature dependent creep damage model for polycrystalline ice. Mech. Mat., 46 : 23-41, 2012.[2] R. Duddu and H. Waisman. A nonlocal continuum damage mechanics approach to simulation of creep fracture in ice sheets. Comp.
Mech., under review.
