Book contents
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
4 - Stress and strain
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Acknowledgements
- Introduction
- 1 The continuity equation
- 2 Density and gravity
- 3 Numerical solutions of partial differential equations
- 4 Stress and strain
- 5 The momentum equation
- 6 Viscous rheology of rocks
- 7 Numerical solutions of the momentum and continuity equations
- 8 The advection equation and marker-in-cell method
- 9 The heat conservation equation
- 10 Numerical solution of the heat conservation equation
- 11 2D thermomechanical code structure
- 12 Elasticity and plasticity
- 13 2D implementation of visco-elasto-plastic rheology
- 14 The multigrid method
- 15 Programming of 3D problems
- 16 Numerical benchmarks
- 17 Design of 2D numerical geodynamic models
- Epilogue: outlook
- Appendix: MATLAB program examples
- References
- Index
Summary
Theory: Deformation and stresses. Definition of stress, strain and strain-rate tensors. Deviatoric stresses. Mean stress as a dynamic (non-lithostatic) pressure. Symmetry of stress tensor. Stress and strain rate invariants.
Exercises: Computing the strain rate tensor components in 2D from the material velocity fields.
Stress
Tensors are field variables which characterise the internal state of a continuum and are, perhaps, the most difficult quantities to intuitively understand. Indeed, at least three of them have to be used in the following and these are the stress, strain and strain rate tensors.
Stress is the internal distribution and intensity of force acting at any point within a continuum in response to various internal and external loads applied to the continuum. Stress is defined as a force per unit area and we can easily ‘apprehend’ its effect by pressing two fingers against each other – equal force is applied from both sides and therefore nothing moves, but we have a feeling of pressure between the fingers, which is a sign of the presence of stress. This stress is directly proportional to the applied force – the stronger we press the stronger the feeling is. On the other hand, the stress is inversely proportional to the contact surface between the fingers – if we press one finger with the nail of the other the feeling is much stronger because the same force is applied to a much smaller area.
- Type
- Chapter
- Information
- Introduction to Numerical Geodynamic Modelling , pp. 51 - 60Publisher: Cambridge University PressPrint publication year: 2009