In this chapter, we have shown that Equations (9.32) can be solved for the three components of the velocity vector and nine components of the stress tensor in certain simple situations. Our solutions were for a slab of infinite horizontal extent resting on a bed with a uniform slope. We first obtained a solution for a perfectly plastic material, and found that the thickness of the slab was constrained by the yield strength of the material. We then obtained solutions for a nonlinear material which are more relevant to real glaciers.

We found that azz and azx vary linearly with depth, which is probably a reasonable approximation to the situation in many real glaciers. We also found that longitudinal stresses should be extending in accumulation areas and compressive in ablation areas, although the magnitude of the longitudinal stresses is not well constrained by our simple model.

Vertical velocities vary linearly with depth for the idealized situation that we studied, and this is commonly used as a first approximation in real glaciers (e.g. Equation 6.15). Horizontal velocities decrease nonlin-early with depth, as we found in Chapter 5. In this chapter, however, we were able to investigate the effect of longitudinal stresses on the velocity profile, and found that either longitudinal extension or longitudinal compression will increase du/dz, leading to higher velocities.

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