Continuum mechanics

Abstract

In this course, students will learn crucial partial differential equations (conservation laws) that are applicable to any continuum including the Earth's mantle, core, atmosphere and ocean. The course will provide step-by-step introduction into the mathematical structure, physical meaning and analytical solutions of the equations. The course has a particular focus on solid Earth applications.

Objective

The goal of this course is to learn and understand few principal partial differential equations (conservation laws) that are applicable for analysing and modelling of any continuum including the Earth's mantle, core, atmosphere and ocean. By the end of the course, students should be able to write, explain and analyse the equations and apply them for simple analytical cases. Numerical solving of these equations will be discussed in the Numerical Modelling I and II course running in parallel.

Content

A provisional week-by-week schedule (subject to change) is as follows: Week 1: Density of rocks. Methods of calculation of rock density. Dependence of density on pressure, temperature and composition of rocks. Isostatic equilibrium. Poisson equation for gravity potential. Computing components of gravitational acceleration vector from gravitational potential Week 2: Definition of geological media as a continuum. Vector and scalar field variables used for representation of continuum. Methods of continuous and discrete definition of field variables. Week 3: Continuity equation. Continuity equation for incompressible fluid and its application for geodynamic problems. Week 4: Deformation and stresses. Definition of stress and strain-rate tensors. Deviatoric stresses. Mean stress as a dynamic (non-lithostatic) pressure. Orientation of stress axes. Transformations of tensors. Tensor invariants. Week 5: Viscosity and Newtonian law of viscous friction. Navie-Stokes equation of motion for viscous fluid. Week 6: Stokes equation of slow laminar flow for highly viscous incompressible fluid and its application for geodynamics. Poisson equation and its model significance. Analytical examples: Couette flow, channel flow. Week 7: Heat conduction law. Heat conservation equation and its geodynamic applications. Radioactive, viscous and adiabatic heating and their significance. Week 8: Analytical examples of solving heat conservation equations: stable geotherms, steady and non-steady temperature profiles in case of channel flow. Week 9: Solid-state creep as a major mechanism of deformation of Earth’s interior. Viscous rheology. Solid-state creep of minerals and rocks. Dislocation and diffusion creep mechanisms. Rheological equations for minerals and rocks. Week 10: Effective viscosity and its dependence on tempreature, pressure, and deformation rate. Rheological profiles across the crust and mantle. Week 11: Elastic rheology. Maxwell viscoelastic rheology. Rotation of stresses during advection. Analytical solution for stress build up. Week 12: Plastic rheology. Plastic yielding criterion. Plastic potential. Plastic flow rule. Week 13: Review GRADING will be based on oral exams.

Resources