Autumn Semester 2020 takes place in a mixed form of online and classroom teaching.
Please read the published information on the individual courses carefully.

651-4007-00L  Continuum Mechanics

SemesterAutumn Semester 2016
LecturersT. Gerya
Periodicityyearly recurring course
Language of instructionEnglish

Catalogue data

AbstractIn 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.
ObjectiveThe 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.
ContentA provisional week-by-week schedule (subject to change) is as follows:

Week 1: 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.

Week 2: 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.

Week 3: 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.

Week 4: 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 5: 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 6: 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.

Week 7: Elasticity and plasticity
Theory: Elastic rheology. Maxwell viscoelastic rheology. Plastic rheology. Plastic yielding criterion. Plastic flow potential. Plastic flow rule.

GRADING will be based on honeworks (30%) and oral exams (70%).
Exam questions:
Lecture notesScript is available by request to
Exam questions:
LiteratureTaras Gerya Introduction to Numerical Geodynamic Modelling Cambridge University Press, 2010

Performance assessment

Performance assessment information (valid until the course unit is held again)
Performance assessment as a semester course
ECTS credits3 credits
ExaminersT. Gerya
Typeend-of-semester examination
Language of examinationEnglish
RepetitionA repetition date will be offered in the first two weeks of the semester immediately consecutive.

Learning materials

No public learning materials available.
Only public learning materials are listed.


651-4007-00 VContinuum Mechanics28s hrs
Wed/108-10NO D 11 »
13-15NO D 11 »
T. Gerya


No information on groups available.


There are no additional restrictions for the registration.

Offered in

Earth Sciences MasterGeophysics: Methods IIW+Information
Computational Science and Engineering BachelorGeophysics: Subject 1WInformation
Computational Science and Engineering MasterGeophysics: Subject 1WInformation