Jackson School of Geosciences - University of Texas at Austin

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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 Time: TTH 9:30 am-11:00 am
- Class Room: EPS 4.104
- Class Zoom: Zoom ID 921 0937 9742 - Class (password in email or on Canvas)

- Tue 4-5pm (Marc) [Zoom ID 921 0937 9742]
- Mon noon-1pm (Afzal) [Zoom ID 378 6912 461]
- Some light entertainment for dark hours

- Piazza - Discussion board

We will most closely follow:

- A first course in continuum mechanics, by Gonzalez and Stewart
- Continuum Mechanics in the Earth Sciences, by Newman

Other useful books are:

- Rheology of the Earth, by Ranalli
- Physics of continous matter, by Lautrup
- Introduction to continuum mechanics, by Gurtin
- Nonlinear solid mechanics, by Holzapfel

- Topics: Introduction to the class, review of vectors, index notation
- Lecture: [pdf] [rec]
- Introduction to course
- Review of vectors
- Index notation

Topics: Body and surface forces, Hydrostatic equilibrium, Isostacy

- Lecture: [pdf] [rec]
- Notes: Force Notes

Topics: Application of hydrostatic force balance

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

- Lecture: [pdf] [rec]
- Notes: Tensor Intro

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

- Lecture: [pdf] [rec]
- Notes: Cauchy stress tensor

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

- Lecture: [pdf] [rec]
- Notes: Simple stress states

Topics: Orthogonal tensors, Euler representation

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

- Lecture: [pdf] [rec]
- Notes: Change of basis

Topics: Principal stresses, constrained optimization

- Lecture: [pdf] [rec]
- Notes: Principal stresses

Topics: 2D and 3D Mohr circle

- Lecture: [pdf] [rec]
- Notes: Mohr circle

Topics: Coordinate systems, Fault normals, stress

- Lecture: [pdf] [rec]
- Notes: Fault orientation, Matlab Fault Orientation [script] [pdf]

Topics: Divergence, Gradient

- Lecture: [pdf] [rec]
- Notes: Tensor calc

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

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

- Lecture: [pdf] [rec]
- Notes: Figure of the Earth

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

- Lecture: [pdf], [rec]
- Notes: Rates, Reynolds Transport Theorem

- Lecture: [pdf], [rec]
- Notes: Balance laws

- Lecture: [pdf] [rec]
- Notes: Objectivity

- Lecture: [pdf] [rec]
- Notes: Representation

- Lecture: [pdf] [rec]
- Notes: Ideal Fluids

- Lecture: [pdf] [rec]
- Notes: Navier-Stokes

- Lecture: [pdf] [rec]
- Notes: Temperature-dependent viscosity