Xx

We give the name principal stresses to the remaining normal stresses, and the axes in this coordinate system are called the principal axes of stress. Similarly, if the coordinate system is oriented such that shear strain rates vanish, the remaining strain rates are called the principal strain rates and the axes are the principal axes of strain rate.

Equation (2.14) shows that the octahedral shear stress is the root-mean-square of the principal stress deviators. When the coordinate axes are aligned parallel to the principal stresses, the octahedral shear stress is the resolved shear stress on the octahedral plane, a plane that intersects the three axes at points equidistant from the origin (Figure 2.7). Hence the name: octahedral shear stress.

The flow law

The most commonly used flow law for ice is Glen's flow law, named after John W. Glen upon whose experiments it is based (Glen, 1955). We will normally write Glen's flow law in the form he originally used:

where B is a viscosity parameter that increases as the ice becomes stiffer, and n is an empirically determined constant. Most studies have found that n « 3. At very low stresses, however, there is some evidence that n ^ 1. An alternative form of the flow law that is commonly used is:

Here, B is normally given in MPa a1/n, while A is in MPa-n a-1 or kPa-n s-1. If the octahedral shear stress and strain rate are used, the numerical values of B and A must be adjusted accordingly, but the units stay the same.

Octahedral plane

Figure 2.7. A plane that intersects the x-, y-, and z-axes at points equidistant from the origin, in this case a unit distance, is called the octahedral plane. If similar planes are drawn involving the negative directions along the axes, the solid figure formed is a regular octahedron.

Octahedral plane

Figure 2.7. A plane that intersects the x-, y-, and z-axes at points equidistant from the origin, in this case a unit distance, is called the octahedral plane. If similar planes are drawn involving the negative directions along the axes, the solid figure formed is a regular octahedron.

Both forms of the flow law have their advantages, and as A = (1 /B)n it is easy to convert between the two forms as long as n is known. The form ee = Aaen resembles conventional constitutive relations in rheol-ogy, and is also easier to generalize if greater precision is needed in situations involving complicated stress configurations (Glen, 1958). For example, some materials, when subjected to a shear stress, swell or contract perpendicular to the plane of shear. In other words, deformation occurs in directions in which the stress is zero. Such rheologies require an extra term in the flow law, and this is more readily accommodated with a flow law of the form ee = Aaen. So far, however, the forms presented in Equations (2.16) and (2.17) seem adequate to represent phenomena observed in studies of ice deformation, both in the laboratory and on glaciers, so the additional term is not needed.

The form ee = (ae/B)n is similar to that used in fluid mechanics with the viscosity, n, defined by:

Here t is the shear stress. Thus B, like n,isa ratio of stress to strain rate. An increase in B results in a decrease in strain rate. Scientists interested in geomorphological applications of glaciological principles are more likely to be familiar with principles of fluid mechanics than with those of rheology, so the form ee = (ae/B)n is used throughout this book.

In Chapter 9, we will show that if the principle axes of stress and strain rate coincide, as is normally the case, the flow law can be written as:

'J B" i where i and J can represent x or y or z. Eliminating ae from Equations (2.15) and (2.18) yields:

Equation (2.18) re-emphasizes a fundamental tenet of Glen's flow law mentioned above: namely that the strain rate in a given direction is a function not only of the stress in that direction, but also of all of the other stresses acting on the medium. Equation (2.19) shows that we can express this concept in terms of strain rates, which are generally easier to measure than stresses.

In the next several chapters we will be dealing with situations in which it is feasible to assume that one stress so dominates all of the others that the others can be neglected. However, the reader should be aware of the implications of this assumption.

Chapter 3

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