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

Class room

Office hours

Additional course websites:

Relevant textbooks

We will most closely follow:

Other useful books are:

Topic I: Review

Lecture 1 (Aug 26): Vector review and index notation

Lecture 2 (Aug 28): Linear momentum and force

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

Lecture 3 (Sep 2): Angular momentum and torque

Topics: Angular momentum, torque, moment, tipping

Topic II: Tensors and Stress

Lecture 4 (Sep 4): Tensor algebra

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

Lecture 5 (Sep 9): Cauchy stress tensor

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

Lecture 6 (Sep 11): Rotations

Topics: Orthogonal tensors, Euler representation, Fault normals

Lecture 7 (Sep 16): Change in basis and spectral decomposition

Topics: Change in basis tensor, eigen problem, spectral decomposition, invariants

Lecture 8 (Sep 18): Prinipal stresses

Topics: Normal and shear stress, principal stresses

Lecture 9 (Sep 23): Mohr circle

Topics: Mohr circle, maximum shear stress, failure

Lecture 10 (Sep 25): Divergence and gradient

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

Lecture 11 (Sep 30): Integral theorems

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

Lecture 12 (Oct 2): Equilibrium Equations

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

Midterm Exam (Oct 7):

Topic III: Kinematics and Strain

Lecture 13 (Oct 9): Deformation Map and Gradient

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

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

Lecture 15 (Oct 16): Interpretation of the Strain Tensor

Lecture 16 (Oct 21): Infinitesimal Strain Tensor

Lecture 17 (Oct 23): Motion and Material Derivative

Lecture 18 (Oct 28): Rates of deformation & Reynolds Transport Theorem

Topic IV: Balance Laws and Constitutive Theory

Lecture 19 (Oct 30): Eulerian balance laws

Lecture 20 (Nov 4): Continuum Thermodynamics

Lecture 21 (Nov 6): Lagrangian balance laws

Lecture 22 (Nov 11): Constitutive laws I - Objectivity and Galilean transformations

Lecture 23 (Nov 13): Constitutive laws II - Representation theorem and constraints

Topic V: Applications

Lecture 24 (Nov 18): Fluids I - Navier Stokes

Lecture 25 (Nov 20): Fluids II - Creeping Flows

Thanksgivings break (Nov 24-28)

Lecture 26 (Dec 2): Power-law creep

Lecture 27 (Dec 4)

Final Exam (Dec 9)

So Long, and Thanks for All the Fish