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Continuum Mechanics
Last Updated: 2026-02-05 15:36:08
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 points of view. Continuity equation in Eulerian and Lagrangian forms and their derivation. Advective transport term. Continuity equation for an incompressible fluid. 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. Exercise: Computing density, thermal expansion and compressibility from an equation of state. 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. Exercise: Analysing strain rate tensor for solid body rotation. Weeks 7,8: The momentum equation Theory: Momentum equation. 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: 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. Week 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. Exercise: steady temperature profile in case of channel flow. Weeks 12,13: Elasticity and plasticity Theory: Elastic rheology. Maxwell viscoelastic rheology. Plastic rheology. Plastic yielding criterion. Plastic flow potential. Plastic flow rule. Week 14: Coupled fluid-solid systems Theory: Fluid percolation processes. Darcy law and its derivation from simple principles. Permeability and its dependence on porosity. Mass and momentum conservation equations for modelling coupled fluid-solid systems. GRADING will be based on homeworks (30%) and oral exams (70%). Exam questions: http://www.erdw.ethz.ch/people/geophysics/tgerya/EXAM_QUESTIONS
Resources
Lecture Notes
Script and exam questions are available by request [email protected]
Literature
Taras Gerya Introduction to Numerical Geodynamic Modelling 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
The lecturers will communicate the exact lesson times of ONLINE courses.
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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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