GG711:  Continuum Mechanics in Geophysics

Fall 2002

Instructor:  Garrett Ito

 

General Information

Time:  9:00 a.m.-10:15 a.m., Tuesday and Thursday

Place:  Post 613

Homework: There will be approximately 7 homework assignments.  Homework is due at the beginning of class on the due date. You have 3 "grace days" for late homework. That is, you can turn in a total of three homework assignments a day late, or one assignment three days late. Assignments turned in late without "grace credit" will be penalized by 20% per day.

Reading:  Textbook assignments and occasional research papers

Exams:  One midterm and a final exam

Grading:  40% homework; 20% midterm; 20% final; 20% class discussion

Required Text:  

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

Recommended Texts:

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

•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 and Lecture Outline

Week 1

Tue 8/27                        I. Introduction to Continuum Mechanics

II. Tensors and Vectors

1. Scalars

2. Vectors, definition and indicial notation

2.1 vector addition

2.2 scalar/dot product

2.3 vector/cross product

2.4 tensor/dyadic product

3. Tensors, definitions, examples

Thu 8/29                            3.1 tensor components                  Read Lai, Ch. 1 (Malvern, Ch. 1)

3.2 rotation tensor

3.3 coordinate transformation of vectors (e.g. 2B12.1)

3.4 coordinate rotation of tensors (e.g. 2B13.1)

3.5 tensor arithmetic

3.6 symmetric and asymmetric tensors

 

Week 2

Tue 9/3                        4. Matrix concepts                  Read Lai, Ch. 2 (Malvern Ch. 2)

4.1 inverse of a Matrix

4.2 eigenvalues and eigenvalues (e.g. 2B17.3, 2B17.4)

4.3 invariants of a 3 x 3 matrix

Thu 9/5                        5. Vector calculus- Div, Grad, Curl                  (e.g. 2B13.2; 2CX.x)

5.1 scalars

5.2 vectors

5.3 tensors

 

Week 3

Tue 9/10                        III. Stress                        HW 1 due

1. Body and Surface forces

2. Cauchy stress tensor

3. Traction on plane of arbitrary orientation in terms of stress tensor components

4. Principal axes of stress tensor

4.1 General case in 3D

 

Thu 9/12                        4.2 Rotation and principal axes of stress tensor in 2D

5. Mohr’s circle

6. Application of Mohr’s circle to fracture mechanics

6.1 Mohr-Coulomb failure criterion

6.2 Angle of fracture plane at minimum differential stress

6.3 Pre-existing crack of arbitrary orientation

 

Week 4

Tue 9/17    no class (Garrett out of town)                  Read Lai, Ch. 4 (Malvern, Ch. 3)

Thu 9/19    no class (Garrett out of town)   

 

Week 5

Tue 9/24                      6.4 Effects of pore pressure     (ex. 4.9) HW 2a due

6.5 Failure angles and coefficient of internal friction

6.6 Yield stress envelopes

7. Conservation of momentum

7.1 Divergence Theorem

7.2 Momentum equation

Thu 9/26                        9. Non-cohesive critical coulomb wedge                        Read/discuss Dahlen [1984]                       

 

Week 6

Tue 10/1                      7.3 Application to the Earth’s crust                  Turcotte & Schubert (Ch. 2)

7.4 Deviatoric stress tensor

 

Thu 10/1    IV. Strain and Kinematics:   

1. Material versus spatial reference frames (ex. 3.2.1; 3.3.1)

2. Material time derivative

3. Infinitesmal strain tensor

 

Week 7

Tue 10/8                      3.1 diagonal components                  Read Lai, Ch. 3 p. 79-149

3.2 off-diagonal components (Malvern Ch. 4)  HW 2b due

3.3 pure rotation w/o deformation (ex. 3.8.1)

 

Thu 10/10                        3.4 pure and simple shear

3.5 analogies to stress tensor

 

Week 8     

Tue 10/15                    4. Rate of deformation                   (ex. 3.15.1)

4.1 Velocity gradient tensor

4.2 Rate of deformation and spin tensor

 

Thu 10/17                        5. Finite Strain                        (ex. 3.20.2)

5.1 Polar decomposition of the deformation gradient tensor F = UR = RV

5.2 Lagrangian formulation:  FTF = UTU = C

5.3 Lagrangian strain tensor:  E*that on this is to the review of his 10

                        5.4 Eulerian formulation:  FFT = VTV = B                        (ex. 3.24.1; 3.26.1))

5.5 Eulerian strain tensor:  e*

                        5.6 Changes in area and volume                                   

 

Week 9

Tue 10/22                        5.7 “Finite strain and rotation from fault-slip data”

                        Read/discuss Cladouhos and Allmendiger   

Thu 10/24                  Review          HW 3 due

 

Week 10

Tue 10/29                  Midterm             

                                         

 

Thu 10/31                        V. Elasticity   Read Lai, 114-121

1. Compatibility equations

2. Constitutive relations

Week 11

Tue 11/5                        Election Day Holiday

Thu 11/7                      3. Formulation of Elastic BV problem                  Read Lai, 217-251

3.1 General 3D and BC’s

3.2 Displacement formulation and Navier’s Equation (seismology)

Week 12

Tue 11/12                    4. 2D problems                   Read Lai, 254-266

4.1 Plane Strain

4.2 Plane Stress

Thu 11/14                    4.3 Airy Stress Function    HW 4 due; Read Lai, 269-291

                        4.4 Solution to a pressurize cylinder

Week 13

Tue 11/19                    4.4 Thin plate theory and plate bending                  Read T & S Ch 3.

Thu 11/21                    4.5 Plate bending applications   

 

Week 14

Tue 11/26                        VI. Fluid mechanics

1. Conservation of mass

2. Constitutive law

3. Governing Equations- Navier-Stokes

Thu 11/28                  Happy Thanksgiving 

 

Week 14

Tue 12.3                        4. Stream functions                        Read Lai, Ch 6. 348-380

5. Applications HW 5 due

5.1 Post-glacial rebound

5.2 Corner flow

Thu 12/5                         6. Laminar Flow:  Poiselle, Couette, Stokes flow    

 

Week 15

Tue 12/10  No class (AGU Meeting)

Thu 12/12                        7. Porous Flow

 

Week 16

Tue 12/17                  Review                  HW6 due

Thu 12/19                  Final Exam