Time lapse photography Glaciers Dislocations Bernal-Fowler rule Generation of defect structures Crystal structures Crystal structures Ice Basal glide Strain rate for glide on basal systems Critical resolved shear stress Non-basal glide Non-basal glide Diffusional flow Plastic Deformation Primary creep Secondary creep Tertiary creep Deformation maps Grain growth and grain size reduction Anisotropic flow Index
Anisotropic flow law for ice
The prolonged deformation of polycrystalline aggregates in glaciers leads to the development of a bulk cystallographic fabric. As a polycrystalline aggregate accumulates strain, the individual crystals rotate by dislocation glide into rotationally stable end orientations. The c-axes tend to rotate towards the principal compressive direction and away from the principal tensile direction. Recrystallisation also contributes to the development of the bulk fabric as grains that are poorly oriented for basal glide inhibit those that are more appropriately oriented. The geometric restriction on the movement of dislocations in appropriately oriented grains produces dislocation tangles and a site for the initiation of dynamic recrystallisation (Wilson & Zhang 1994,Wilson 1986). The new grains that are produced by this process tend to form with basal planes at 45° to the compressive direction (see fig. 1. 4. 1). Thus, the poorly oriented grains tend to be replaced by grains with the maximum resolved shear stress on their basal planes. The alignment of basal planes to more appropriate orientations for glide leads to strain softening and enhanced flow. Pimienta et al. (1987) report that if an aggregate with a single c-axis maximum is deformed by uniaxial or biaxial compression, such that the maximum shear stress is resolved onto the basal planes, then the strain rate is enhanced by 10 times and if the load is applied co-axially with the single c-axis maximum, then the strain rate is enhanced by 0.1.
 
Figure 1. 4. 1
Figure 1. 4. 1 Polycrystalline aggregate showing: Undeformed shape of grains and slip-plane traces and initial random two-dimensional orientation of slip-plane normal (c-axes) with respect to the specimen orientation. Deformed by 29% shortening, the grains in the aggregate show elongation parallel to the extension direction and the c-axes are concentrated in a bi-symmetrical pattern about the shortening axis. This 2D FLAC model is taken from Wilson & Zhang (1996).
When a fabric develops in an ice aggregate, Glen's Law breaks down as the fabric produces a strong mechanical anisotropy and the strain rate is no longer independent of the orientation of the applied stress. A geometric term based on the mean orientation of c-axes in the polycrystalline aggregate must be introduced to equation 1 (Glen's Law), so that the strain rate varies with the orientation of any applied stress. To simplify the approach it is common to assume that single crystals deform by basal glide only (Azuma 1994, Azuma & Goto-Azuma 1996). This method is followed in the example of the implementation of an anisotropic model presented by Marmo & Wilson (1999).

For the case when glide is assumed to occur on basal systems only, the resolved shear stress on the basal plane of any crystal under uniaxial compression or tension, can be related by a geometric function referred to as the Schmid Factor, Schmid Factor (Azuma & Higashi 1985):

Equation 11----------------------------------------equation (11)

where Angle is the angle between the c-axis and the unique stress axis. The Schmid factor for an aggregate as a whole, S, is given by the mean distribution of c-axes relative to the unique stress axis:

Equation 12----------------------------------------equation (12)

When the c-axes are randomly oriented Schmid Factor for Pure Shear, for pure shear and Schmid Factor for Simple Shear for simple shear, and when a fabric is completely developed such that all the c-axes are inclined at 45° to the compressive stress axis, Another Schmid Factor. Azuma 1995 showed that the strain on an individual grain was related to the net strain of the aggregate by the ratio:

Equation 13----------------------------------------equation (13)

where Strain is the strain of an individual grain and the macroscopic strain of the aggregate is given by Definition of Strain. Azuma (1995) also demonstrated by in situ experimental observation, that the strain varies with S approximately to the fourth power:

Equation 14----------------------------------------equation (14)

The Schmid factor is related to the strain rate by assuming that the stress on any one grain, , is inversely proportional to the Schmid factor for the grain, and proportional to the Schmid factor for the aggregate as a whole:

Equation 15----------------------------------------equation (15)

where Sigma is the uniaxial stress on the aggregate. The shear stress resolved onto the base of a single grain, Shear Stress, is given by:

Equation 16----------------------------------------equation (16)

By dividing both sides of equation 13 by time, the strain rate of the aggregate becomes:

Equation 17----------------------------------------equation (17)

By substituting Glen's Law and equation 16 into equation 17, a law describing flow of anisotropic ice under uniaxial deformation is arrived at:

Equation 18----------------------------------------equation (18)

The geometric factor introduced to Glen's Law acts as a scalar quality on the flow parameter A Flow Parameter. As n=3 for polycrystalline ice, the strain rate varies with the fourth power of the Schmid factor. A fully developed fabric Another Schmid Factor therefore enhances the strain rate by ~5 times under uniaxial compression, compared to an isotropic aggregate Another Schmid Factor.

 
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Created: August 23, 1999
Last modified: March 15, 2004
Authorised by: Head, School of Earth Sciences
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