Grain size-sensitive creep and its relationship to grain size-insensitive attenuation in ice-I

Caswell, Tess E.1, Reid F. Cooper1, and David L. Goldsby2

1Brown University, Providence, RI, USA 2 University of Pennsylvania, Philadelphia, PA

Motivation:The geodynamics of icy moons in the Outer Solar System hinge upon their “thermal budget.” The primary source of energy in these bodies is the dissipation of tidal energy (attenuation) within the ice shell.

Recent work [1] demonstrates several unexpected features of the attenuation response of polycrystalline ice creeping under conditions appropriate to icy satellites:

What are the physics of low-frequency attenuation in polycrystalline ice?

Background: Hart’s model describes the constant-microstructure creep compliance of a material in terms of a distribution of microstructural barriers to deformation characterized by the state variable σ*.

The form of the Lambda Law arises from the distribution of “strengths” of obstacles to dislocation glide (e.g. the distribution of subgrain sizes in crept, single crystal halite at right). [2]

The “Lambda Law,” a mathematical form of Hart’s model. [3]σ* = Hardness Parameterε* = Strain Rate Parameter

Methodology: Stress-reduction experiments probe the constant-microstructure creep compliance of polycrystalline ice at steady-state creep under conditions relevant to icy satellite geodynamics.

(1) (2) (3)

stra

in

structural change begins

}st

ress

TransientSteadystate

Iso-structural state

New steady state

Thermocouples

Results & Discussion: Polycrystalline ice displays a constant-microstructure creep compliance consistent with Hart’s Model as enhanced by Grain Boundary Sliding, but the response is independent of the presence of subgrains.

We gratefully acknowledge assistance from Dr. Christine McCarthy, Dr. David Prior and Dr. Rachel Obbard for their mentorship and assistance in performing cryogenic microstructural analyses for this research. This material is based upon work supported by the NSF Graduate Research Fellowship. Any opinion, �ndings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily re�ect the views of the National Science Foundation.

Log

Stre

ss (M

Pa)

Log Strain Rate (s-1)

1.74

1.54

1.34

1.14

0.94

-3-4-5-6-7-8-9-10

Ni 270T = 600°Cεtot = 0.013

No GBS~10% GBS

~20% GBS

~30% GBS

[1] McCarthy, C. (2009) Ph.D. Dissertation, Brown University; [2] Stone, D.S. et al. (2004) Similarity and Scaling in creep and load relaxation of single-crystal halite (NaCl), JGR; [3] Alexopoulos, P. et al. (1982)Experimental investigation of nonelastic deformation emphasizing transient phenomena by using a state variable approach, Me-chanical Testing for Deformation Model Development, ASTM STP; [4] Korhonen, M.A. et al. (1985) A State Variable Approach to Transient Creep Deformation in Stress Reduction Experiments; [5] Goldsby, D.L. and D.L. Kohlstedt (2001) Superplastic deformation of ice; Experimental observations, JGR; [6] Gifkins, R.C. (1976) Grain-Boundary Sliding and its Accommodation During Creep and Superplasticity, Met. Trans. A; [7] Crossman, F.W. and M.F. Ashby. (1975) The non-uniform �ow of polycrystals by grain-boundary sliding accommodated by power-law creep, Acta Metallurgica. [8] Barr, A.Barr, A. C., & Showman, A. P. (2009). Heat transfer in Europa’s icy shell. Europa, edited by RT Pappalardo, WB McKinnon, and K. Khurana, 405-430.

Our experiments were performed in both the dislocation creep and Grain Boundary Sliding regimes [5].

Our constant-microstructure data conform to the Lambda Law, including the e�ect of Grain Boundary Sliding - Consistent with a distribution of obstacle strengths - Independent of steady state creep regime

In the GBS regime, deformed microstructures do not display the stress-dependent subgrain size distribution observed in materials undergoing dislocation creep. - The distribution of obstacles in GBS may instead be on the grain boundary -Indicates that the grain boundary structure evolves with deviatoric stress over a �nite strain

The grain size-independent scaling of attenuation data for polycrystalline ice may be the result of stress-dependent grain boundary structure.

A material that deforms by Grain Boundary Sliding (e.g. polycrystalline Nickel, left) experiences an enhanced strain rate at low stresses due to GBS-induced stress concentrations (right). [3, 4, 7]

The data follow the form of the Andrade Model when normalized by stress-e�ected subgrain size, but are more attenuating than predicted by either the Andrade or Maxwell Models.

Over 3 orders of magnitude in grain size, the response is grain size-insensitive.

MR23C-4380

The behavior is similar to that of deformed single crystals, also normalized by predicted subgrain size.

We seek a microstructural feature that reconciles the Andrade Model-like behavior, which implies di�usional rheology, with the observed grain size insensitivity.

Why are icy satellites pervasively fractured?

Why are plumes of water erupting from Enceladus’s south pole?

How do Earth’s ice sheets respond to ocean tides?

Elastic recovery+

rearrangement of free (mobile)

dislocations

Instantaneous Stress Reduction

Re�ects response of microstructure established in (1) to new stress.

time

(1)(3)

(3)

(3)

(3) (1)(3)

(3)

(3)

(3)

log(

σ)

log(ε)

(�ow

law

)

Steady state@ σ1

Separate stress drops

A series of stress drops from steady state (e.g., left) maps out the response of a single microstructure in stress-strain rate space (schematically, below). [4]

Basal Slip

GBS

Dislocation Creep

Di�.

log(

σ) (M

Pa)

T (K)15010050 200 250

0

2

-2

Our experiments were conducted in dislocation creep and the GBS-limited basal slip regime, appropriate to icy satellite interiors (red box in Def. Mech. map, above, for d = 1 mm, [8]), utilizing a dead-weight creep apparatus at Brown University (schematically, above right).

Dislocation Creep Microstructure:- Well-developed subgrain boundaries with regular spacing- Irregular grain boundaries

GBS Microstructure:- No subgrain boundaries (con�rmed by EBSD)- Numerous four-grain junctions

Steady state@ σ2 < σ1

Sam

ple

Cera

mic

Pis

ton

Steady State: Dislocation Creep, Coarse Grains

Steady State: GBS, Fine Grains

In this model, GBD obstacles are analogous to subgrain boundaries; their stress-dependent size distribution gives rise to Lambda Law behavior.

GBD “pinning points” emit lattice dislocations, which contribute to deformation.

Persp

ectiv

e vie

w o

f

gra

in b

oundar

y

Grain Boundary Dislocations (GBD’s)

ε σn

Adapted from [6]

Cross section through grain