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

## 2.2 Tensors

-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

### Week 4:  2/5 & 2/7 (Lai, Ch. 4, 182-187, Jaeger and Cook, 9-17; 86-94)

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, 73-80)

3.6 Conservation of momentum

-Divergence Theorem

-Momentum equation

-Application to the isostatic equilibrium of Earth’s crust

# Week 6:  2/19 & 2/21 (Lai, Ch. 2, p. 79-91)

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

-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

# Week 17:  5/13 & 5/15

Class Project Presentations