GEO 325C/398C Continuum Mechanics

Jackson School of Geosciences - University of Texas at Austin

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Course Description

Earth Science Examples

Continuum Mechanics is a new upper division undergraduate/beginning graduate course focused on developing the foundations of solid and fluid mechanics and thermodynamics of continua, and their application to geophysical and geological models such as mantle convection, ice sheet flow, and geophysical fluid dynamics. The emphasis is on a rigorous quantitative three-dimensional description, beginning with tensor analysis, forces, and stresses, and kinematics, motion, and strain; moving to the basic balance (i.e. conservation) laws of mass, momentum, and energy as well as constitutive relations for fluid and solid media; and ending with the application of these laws and relations to the governing equations of oceanography, seismology, glaciology, and geodynamics.

Previous versions of this course: Fall 2021, Fall 2022, Fall 2023, Fall 2024, Fall 2025

Class basics

Office hours

Additional course websites:

Relevant textbooks

We will most closely follow:

Other useful books are:

Topic I: Force and Torque

Lecture 1 (Aug 25): Vector review and index notation

Lecture 2 (Aug 27): Linear momentum and force

Topics: Newton’s laws, Body and surface forces, Hydrostatic equilibrium, Isostacy

Lecture 3 (Sep 1): Angular momentum and torque

Topics: Angular momentum, torque, moment, tipping

Topic II: Tensors and Stress

Lecture 4 (Sep 3): Tensor algebra

Topics: Tensor representation and basis, dyadic product, trace, transpose.

Lecture 5 (Sep 8): Cauchy stress tensor

Topics: Traction, Action & Reaction, Cauchy’s principle

Lecture 6 (Sep 10): Prinipal stresses

Topics: Normal and shear stress, principal stresses

Lecture 7 (Sep 15): Rotations

Topics: Orthogonal tensors, Euler representation, Fault normals

Lecture 8 (Sep 17): Change in basis and spectral decomposition

Topics: Change in basis tensor, eigen problem

Lecture 9 (Sep 22): Mohr circle

Topics: Mohr circle, maximum shear stress, failure

Topic III: Tensor Calculus

Lecture 10 (Sep 24): Divergence and gradient

Topics: Gradient, divergence, Laplacian, Poisson’s equation for gravity

Lecture 11 (Sep 29): Integral theorems

Topics: Curl, Divergence and Stokes theorems, Derivatives of tensor functions

Lecture 12 (Oct 1): Equilibrium Equations

Topics: Equilibrum equations, symmetry of stress tensor, hydrostatic shapes, Figure of the Earth

Midterm 1 (Oct 6): Topics I and II

Topic IV: Kinematics and Strain

Lecture 13 (Oct 8): Deformation Map and Gradient

Topics: Deformation map and gradient; change of material lines, volumes and areas

Lecture 14 (Oct 13): Cauchy-Green Strain Tensor

Lecture 15 (Oct 15): Interpretation of the Strain Tensor

Lecture 16 (Oct 20): Infinitesimal Strain Tensor

Lecture 17 (Oct 22): Motions and Rates

Lecture 18 (Oct 27): Reynolds Transport Theorem

Topic VI: Balance Laws and Constitutive Theory

Lecture 19 (Oct 29): Eulerian balance laws

Lecture 20 (Nov 3): Energy balance - heating

Lecture 21 (Nov 5): Energy balance - working

Lecture 22 (Nov 10): Constitutive laws

Midterm 2 (Nov 13): Topics III and IV

Topic V: Applications

Lecture 23 (Nov 17): Navier Stokes

Lecture 24 (Nov 19): Boundary layers

Thanksgivings break (Nov 24-28)

Lecture 25 (Nov 1): Static problems

Lecture 26 (Dec 3): Elastic waves

Final Exam (Dec 14): 8:00-10:00 Comprehensive

So Long, and Thanks for All the Fish