In this chapter we will derive general equations for calculating the force per unit area, or traction, on a plane that is not parallel to the coordinate axes, and then use these equations to determine the orientation of the plane on which tractions are a maximum. We will see how this leads to the concept of the invariant of a tensor, and show that this provides the fundamental basis for Glen's flow law. Then we derive the stress equilibrium equations.

In the second half of the chapter we derive expressions for strain rates in terms of velocity derivatives, and develop some relations based on these expressions and some other basic equations. This will set the stage for calculating stresses and velocities in a very simple ice sheet, consisting of a slab of ice of uniform thickness on a uniform slope (Chapter 10) and for investigating some more realistic problems (Chapter 12).

Stress

Although we have been referring to stresses and strain rates throughout the last few chapters, we will now enter into a much more detailed discussion, involving the tensor properties of these quantities. The reader may find it helpful, therefore, to review the section on stresses and strain rates in Chapter 2.

General equations for transformation of stress in two dimensions

Consider a domain in a slab of material of unit thickness (measured normal to the page) as shown in Figure 9.1.