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Continuum Mechanics
Last Updated: 2026-02-05 16:02:06
Abstract
In this course, students 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: Weeks 1,2: The continuity equation Theory: Definition of a geological media as a continuum. Field variables used for the representation of a continuum.Methods for definition of the field variables. Eulerian and Lagrangian reference frames. Continuity equation in Eulerian and Lagrangian forms. Derivation of Eulerian continuity equation from simple principles. Advective transport term. Incompressible continuity equation. Exercise: Computing the divergence of velocity field. Weeks 3,4: Density and gravity Theory: Density of rocks and minerals. Thermal expansion and compressibility. Dependence of density on pressure and temperature. Equations of state. Poisson equation for gravitational potential and its derivation from simple principles. Exercises: Computing density, thermal expansion and compressibility from an equation of state. Derivation of gravitational acceleration and its divergence from gravitational potential. Weeks 5,6: Stress and strain Theory: Deformation and stresses. Definition of stress, strain and strain-rate tensors. Deviatoric stresses. Mean stress as a dynamic (nonlithostatic) pressure. Stress and strain rate invariants. Exercises: Analysing strain rate tensor for solid body rotation. Computing stress invariants Weeks 7,8: The momentum equation Theory: Momentum equation and its derivation from simple principles. Viscosity and Newtonian law of viscous friction. Navier-–Stokes equation for the motion of a viscous fluid. Stokes equation of slow laminar flow of highly viscous incompressible fluid and its application to geodynamics. Simplification of the Stokes equation in case of constant viscosity and its relation to the Poisson equation. Exercises: Deriving momentum equation. Computing velocity for magma flow in a channel. Week 9: Viscous rheology of rocks Theory: Solid-state creep of minerals and rocks as themajor mechanism of deformation of the Earth’s interior. Dislocation and diffusion creep mechanisms. Rheological equations for minerals and rocks. Effective viscosity and its dependence on temperature, pressure and strain rate. Formulation of the effective viscosity from empirical flow laws. Exercise: Deriving viscous rheological equations for computing effective viscosities from empirical flow laws. Weeks 10,11: The heat conservation equation Theory: Fourier’s law of heat conduction. Heat conservation equation and its derivation. Radioactive, viscous and adiabatic heating and their relative importance. Heat conservation equation for the case of a constant thermal conductivity and its relation to the Poisson equation. Exercises: Computing of heat fluxes. Deriving equation for steady state temperature profile in a magmatic channel. Week 12,13: Elasticity and plasticity Theory: Elastic rheology. Maxwell viscoelastic rheology. Plastic rheology. Plastic yielding criterion. Plastic flow potential. Plastic flow rule. Exercise: compute viscoelastic stress evolution. Week 14: Fluid flow in deforming porous media. Darcy equation for fluid percolation. Derivation of Darcy equation from Stokes equation for channel flow. Dependence of permeability on porosity and grain size. Coupled hydro-mechanical momentum and continuity equations for solid matrix and percolating fluid. Fluid and solid Lagrangian reference frames. GRADING will be based on honeworks (1/3) and oral exam (2/3).
Resources
Lecture Notes
Script and Exam questions are available by [email protected]
Literature
Taras Gerya Introduction to Numerical Geodynamic Modelling. Second Edition. Cambridge University Press, 2019
General Information
- Language
- English
- Levels
- BSC , MSC
- Frequency
- Yearly recurring
Examination
- Type
- graded semester performance
Course Components
| Type | Title | Time & Place | Hours |
|---|---|---|---|
| lecture | Continuum Mechanics |
|
2 h weekly |
Offered In
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Geophysics (Recommended combinations: Subject 1 + Subject 2 Subject 1 + Subject 3 Subject 2 + Subject 3 Subject 3 + Subject 4 Subject 5 + Subject 6 + Subject 8 Subject 4 + Subject 5 Subject 7 + Subject 8)
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Geophysics (Recommended combinations: Subject 2 + Subject 5 + Subject 6 + Subject 7 Subject 2 + Subject 4 + Subject 5 + Subject 6 + Subject 8 Subject 2 + Subject 5 + Subject 6 + (Subject 1 or Subject 3))
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