Summary

In this chapter, we have reviewed some elementary principles of continuum mechanics with a particular focus on those principles needed to understand much of the classical as well as the modern literature on glacier flow.

In the first part of the chapter, we defined stress and showed that if we know the stresses in one coordinate system, we can calculate them in another system rotated with respect to the first. This allowed us to calculate the direction and magnitude of the maximum and minimum normal stresses, the principal stresses. We did the calculation in two dimensions, but the extension to three dimensions is straightforward, though tedious. We found that shear stresses vanish in coordinate systems chosen with axes aligned parallel to the principal stresses.

The orientation and magnitude of the principal stresses is a property of the stress field and not of the orientation of the axes. Thus, there are certain combinations of the stresses that must be independent of the orientation of the axes: the invariants of the stress tensor. Glen's flow law for ice is based on the second of these invariants. This is logical because it is invariant, and also because the von Mises yield criterion can be expressed in terms of this invariant. Recent experimental data have validated this approach.

In the second part of the chapter, we derived the stress equilibrium or momentum balance equations.

In the third part, we defined strain and derived equations for calculating strains or strain rates in coordinate systems rotated with respect to one another. These equations are similar to those for transformation

of stress. As with stresses, we introduced the concept ofprincipal strain rates.

Finally, we showed that if a material is isotropic and incompressible, the principal axes of stress and strain rate coincide. Ice is clearly not isotropic and incompressible, but this approximation has proven to be a convenient starting point for calculations of glacier flow.

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