**GG681: Continuum Mechanics in Geophysics**

**Fall 2004**

**Instructor: Garrett Ito**

**General Information**

**Time: ** 10:30 a.m.-11:45 a.m., Tuesday and Thursday

**Place: ** Post 706

**General Format:** Physics is best learned by solving problems
and discussing ideas and approaches with colleagues. This course will therefore emphasize
student-led discussions of problems dealing with basic concepts of stress,
deformation, elasticity, and fluid dynamics as applied to problems in solid
Earth science. Class projects will allow
students to apply what they have learned to a topic related to their own
interests.

**Course Objectives:**

·Teach basic concepts of
continuum mechanics for applications in Earth science and beyond

·Provide familiarity with many classic and
ongoing problems and applications in geodynamics

·Improve skills in independent learning,
critical and quantitative reasoning, and communication

**Homework:** We will do and discuss at least one homework
assignment per week. Homework should be
done by the day we discuss the assignment and will be turned in for grading the
following week.

**Grading: **40% homework; 30% class discussion; 30% Class
Project

**Required Textbook: **

•*Introduction to Continuum Mechanics*, Third Edition, by* *W. M. Lai, D. Rubin, and E. Krempl

**(strongly) Recommended
Textbook:**

•*Geodynamics*, by D. L. Turcotte and G. Schubert

**Others:**

•*Introduction to the Mechanics of a Continuous Medium*, by L. Malvern

•*Introduction to Fluid Dynamics*, G. Batchelor

•*Incompressible Flow*, R. Panton

•*Elasticity, by* J. R. Barber

•*Elasticity, Tensor, Dyadic, and Engineering Approaches
by*
P. C. Chou and N. J. Pagano

**Working Syllabus**

__Week 1: 1/15 & 1/17 (Lai, Ch. 1 & 2)__

**1. Introduction to
Continuum Mechanics**

** 2. ****Tensors and Vectors**

2.1
Indicial Notation: definitions and applications

-Definition

-How to extract tensor components

-Tensor arithmetic

-Some properties, “trace”, “Identity
tensor”, and “inverse”

-Orthogonal tensors and coordinate
transformations

-Symmetric and Antisymmetric Tensors

__Week 2: 1/22
& 1/24__

-Eigenvalues and Eigenvectors

-Principal values and principal directions

-Principal scalar invariants

2.3 Tensor Calculus (e.g.,
Div, Grad, Curl, etc)

__Week 3: 1/29 & 1/31 (Lai,
Ch. 4, 173-181)__

**3. Stress**

3.1 Body and Surface
forces

3.2 Stress vector

3.3 Cauchy stress tensor
and its components

3.4 Principal axes of
stress tensor

-General case in 3D

-Rotation and principal axes of stress tensor in 2D

3.5 Mohr’s circle and
applications to lithospheric mechanics

-Mohr-Coulomb failure criterion

-Failure angles and coefficient of internal friction

-Yield stress envelopes

__Week 5: 2/12 & 2/14 ( Lai p. 187-191; Turcotte
& Schubert, 73-8__

3.6 Conservation of
momentum

-Divergence Theorem

-Momentum equation

-Application to the isostatic equilibrium of
Earth’s crust

* ***4. Strain and Kinematics**

4.1 Material versus
spatial reference frames

4.2 Material time
derivative

__Week 7: 2/26 & 2/28 ( Lai, Ch. 2, p.
92-112)__

4.3 Infinitesmal
strain tensor

diagonal/off-diagonal components

-pure rotation without deformation, pure and simple shear

-analogies to stress tensor

4.4 Rate of deformation

-Velocity gradient tensor

-Rate of deformation and spin tensor

__Week 8: 3/4 & 3/6__

4.5 Finite Strain

**·****Class project
introductions**

__Week 9: 3/11 & 3/13 (Lai, Ch. 5)__

**5. Elasticity**

5.1 Compatibility equations

5.2 Constitutive relations and Lamé
constants

5.3 Formulation of Elastic BV problem

-General 3D and BC’s

- Displacement formulation and Navier’s Equation (seismology)

__Week 10: 3/18 & 3/20__

5.4 2D problems

-Plane Strain

-Plane Stress

__Week 11: 3/25 & 3/27 __

* *5.5 Airy Stress Function and applications

__Week 12: 4/1 & 4/3 ( Turcotte
& Schubert Ch 3.)__

5.6 Thin plate theory and plate bending* *

__Week 13: 4/8 & 4/10 ( Lai Ch. 6)__

**6. Fluid mechanics **

6.1 Conservation of mass

6.2
Constitutive law

6.3
Governing Equations-Navier-Stokes

__Week 14: 4/15 & 4/17__

6.4. Stream functions and applications

__Week 15: 4/22 & 4/24 ( Turcotte & Schubert
Ch 6)__

6.5 Laminar Flow:
Poiselle, Couette,
Stokes flow

__Week 16: 4/29 & 4/24__

** Class Project Presentations**