GEO 325C/398C Continuum Mechanics

Jackson School of Geosciences - University of Texas at Austin

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Course Description

Continuum Mechanics is a new upper division undergraduate/beginning graduate course focussed 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 particularization of these laws and relations to the governing equations of oceanography, seismology, glaciology, and geodynamics.

Previous versions of this course: Fall 2021, Fall 2022

Class room

Office hours (TBD)

Additional course websites:

Relevant textbooks

We will most closely follow:

Other useful books are:

Topic I: Tensors and Stress

Lecture 1 (Aug 22): Introduction and Review

Lecture 2 (Aug 24): Continuum Mass and Force Concepts

Topics: Body and surface forces, Hydrostatic equilibrium, Isostacy

Lecture 3 (Aug 29): Isostacy and index notation

Topics: Application of hydrostatic force balance

Lecture 4 (Aug 31): Introduction to Tensors

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

Lecture 5: Cauchy stress tensor

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

Lecture 6: Stress examples

Topics: Normal and shear stress, Projections, simpe stress states

Lecture 7: Rotations

Topics: Orthogonal tensors, Euler representation

Lecture 8: Change in basis and spectral decomposition

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

Lecture 9: Extremal stress values

Topics: Principal stresses, constrained optimization

Lecture 10: Mohr circle and failure

Topics: 2D and 3D Mohr circle

Lecture 11: Computing fault stresses

Topics: Coordinate systems, Fault normals, stress

Lecture 12: Gradient and Divergence

Topics: Divergence, Gradient

Lecture 13: Integral theorems

Topics: Curl, Laplacian Divergence and Stokes theorems, Poisson’s equation for gravity

Lecture 14: Equilibrium Equations

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

Topic II: Kinematics and Strain

Lecture 15: Deformation Map and Gradient

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

Lecture 16: Analysis of local deformation

Lecture 17: Cauchy-Green Strain Tensor

Lecture 18: Infinitesimal strain

Lecture 19: Motion and Material Derivative

Lecture 20: Rates of deformation & Reynolds Transport Theorem

Topic III: Balance Laws and Constitutive Theory

Lecture 21: Local Eulerian balance laws

Lecture 22: Energy balance

Lecture 23: Constitutive Theory: Objectivity

Lecture 24: Constitutive Theory: Representation Theorem

Topic IV: Applications

Lecture 25: Ideal fluids

Lecture 26: Newtonian fluids

Lecture 27: Stokes flow

Lecture 28: Power-law creep

So Long, and Thanks for All the Fish