Jackson School of Geosciences - University of Texas at Austin

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Continuum Mechanics is a new upper division/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

- Class Time: TTH 2:00 p.m.-3:30 p.m.
- Class Room: EPS 1.126
- Class Zoom: Zoom ID 921 0937 9742 - Class (password in email or on Canvas)

Mon + Wed 2-3pm in room JGB 2.104B (new!) in the Student Service Center, close to Luck Lab Coffee [Zoom ID 921 0937 9742]

Additional zoom office hours with Afzal Tue 11-noon [Zoom ID 378 6912 461]

- 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: Tensor representation and basis, dyadic product, trace, transpose, determinant, scalar product.

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

Topics: Orthogonal tensors, change of basis, spectral decomposition, polar decomposition.

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

opics: Mass and force, Traction, Action & Reaction, Cauchy’s principle

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

- Lecture: [pdf] [rec]
- Notes: Stress examples

- Lecture: [pdf] [rec]
- Notes: [Tensor Algebra II]

- Lecture: [pdf] [rec]
- Notes: Mohr circle
- Demos: [LiveScript], [pdf]

- Topics: Local analysis of deformation, right Cauchy-Green strain tensor
- Lecture:

- Topics: Other strain tensors, Cauchy-Green strain relations, Volume changes
- Lecture:

- Topics: Changes in area, linearization of kinematic quantities, infintesimal strain tensor
- Lecture:

- Topics: Motions, Material spatial fields, Material and spatial time derivatives
- Lecture:

- Topics: Rate of strain & spin tensors, Reynolds transport theorem
- Lecture:

- Topics: Integral balance laws in discrete and continuum systems, Introduction to continuum thermodynamics
- Lecture:

- Topics: Mass and momentum balance, Net working
- Lecture:

- Topics: Energy balance and entropy inequality
- Lecture:

- Topics: Mass and momentume balancs, first and second laws in Lagrangian form
- Lecture:

- Topics: Material frame indifference, galilean transformations
- Lecture:

- Topics: Isotropic functions, 4-th order tensors, material constraints
- Lecture:

- Topic: Frame indifference of ideal fluid model, Bernoulli, Vorticity equation
- Lecture:

- Topics: Newtonian stress, Navier-Stokes Equation, Stokes Equation, Kinetic energy
- Lecture:

- Topics: Solid state creep
- Lecture:

- Topics: Elastic Solid, stress response function, isotropic material
- Lecture:

- Topis: Strain enegy density, Mechanic energy inequality, common models
- Lecture:

- Topics: Linearization, Elasticity tensor, Linear elastic solids
- Lecture: