Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Physical constants relevant to ice
- Derived SI units and conversion factors
- 1 Why study glaciers?
- 2 Some basic concepts
- 3 Mass balance
- 4 Flow and fracture of a crystalline material
- 5 The velocity field in a glacier
- 6 Temperature distribution in polar ice sheets
- 7 The coupling between a glacier and its bed
- 8 Water flow in and under glaciers: geomorphic implications
- 9 Stress and deformation
- 10 Stress and velocity distribution in an idealized glacier
- 11 Numerical modeling
- 12 Applications of stress and deformation principles to classical problems
- 13 Finite strain and the origin of foliation
- 14 Response of glaciers to changes in mass balance
- Appendix: Problems
- References
- Index
9 - Stress and deformation
Published online by Cambridge University Press: 24 November 2009
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- Physical constants relevant to ice
- Derived SI units and conversion factors
- 1 Why study glaciers?
- 2 Some basic concepts
- 3 Mass balance
- 4 Flow and fracture of a crystalline material
- 5 The velocity field in a glacier
- 6 Temperature distribution in polar ice sheets
- 7 The coupling between a glacier and its bed
- 8 Water flow in and under glaciers: geomorphic implications
- 9 Stress and deformation
- 10 Stress and velocity distribution in an idealized glacier
- 11 Numerical modeling
- 12 Applications of stress and deformation principles to classical problems
- 13 Finite strain and the origin of foliation
- 14 Response of glaciers to changes in mass balance
- Appendix: Problems
- References
- Index
Summary
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.
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- Principles of Glacier Mechanics , pp. 252 - 270Publisher: Cambridge University PressPrint publication year: 2005