Research Papers

The Path-Independent M Integral Implies the Creep Closure of Englacial and Subglacial Channels

[+] Author and Article Information
Colin R. Meyer

John A. Paulson School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138
e-mail: colinrmeyer@gmail.com

John W. Hutchinson

Fellow ASME
John A. Paulson School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138
email: hutchinson@husm.harvard.edu

James R. Rice

Fellow ASME
John A. Paulson School of Engineering and Applied Sciences,
Harvard University,
Cambridge, MA 02138;
Department of Earth and Planetary Sciences,
Harvard University,
Cambridge, MA 02138
e-mail: rice@seas.harvard.edu

1Corresponding author.

Manuscript received September 7, 2016; final manuscript received September 21, 2016; published online October 18, 2016. Editor: Yonggang Huang.

J. Appl. Mech 84(1), 011006 (Oct 18, 2016) (9 pages) Paper No: JAM-16-1433; doi: 10.1115/1.4034828 History: Received September 07, 2016; Revised September 21, 2016

Drainage channels are essential components of englacial and subglacial hydrologic systems. Here, we use the M integral, a path-independent integral of the equations of continuum mechanics for a class of media, to unify descriptions of creep closure under a variety of stress states surrounding drainage channels. The advantage of this approach is that the M integral around the hydrologic channels is identical to same integral evaluated in the far field. In this way, the creep closure on the channel wall can be determined as a function of the far-field loading, e.g., involving antiplane shear as well as overburden pressure. We start by analyzing the axisymmetric case and show that the Nye solution for the creep closure of the channels is implied by the path independence of the M integral. We then examine the effects of superimposing antiplane shear. We show that the creep closure of the channels acts as a perturbation in the far field, which we explore analytically and numerically. In this way, the creep closure of channels can be succinctly written in terms of the path-independent M integral, and understanding the variation with applied shear is useful for glacial hydrology models.

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Grahic Jump Location
Fig. 1

Schematic of the path-independent M integral around a void. The path of integration is Γ and the vector normal to this integration path is ni. The down glacier component is x which points out of the page, i.e., antiplane, the in-plane coordinates are y (across glacier) and z (depth).

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Fig. 2

Nye problem setup and boundary conditions. Inward-pointing arrows indicate the creep closure of the ice (Ur), and the eddy structures in the channel highlight the turbulence (turbulent pipe flow simulation by Feldmann and Wagner [20]; photo credit: Noel Fitzpatrick).

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Fig. 3

Schematic for an idealized ice stream shear margin with a Röthlisberger [2] subglacial channel. Surface ice velocity increases away from the margin and is nearly stagnant in the ridge. Adapted from Ref. [29].

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Fig. 4

Numerical evaluation of M at the edge of the channel Minner and in the far-field Mouter at R = 500. The scalings show that for S ≪ 1, M ∼ S1∕n and for S ≫ 1, M ∼ S(n+1)∕n. Numerical artifacts lead to negative values for Minner when S is small; therefore, these simulations are omitted but yield absolute values close to Mouter.

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Fig. 5

Numerical simulations showing the nondimensional channel creep closure velocity |Vr| at the edge of the channel R = 1. For S ≪ 1, the antiplane straining is negligible and the creep closure velocity approaches the nondimensional Nye solution. When the antiplane motions are dominant, i.e., S ≫ 1, the creep closure velocity increases as S(n−1)/n.




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