12.520 Lecture notes #1
1
12.520: Geodynamics
(Continuum mechanics applied to geological problems)
Lecture 1
Handouts - Course description & reserve list
Instructor background
Student background
Scheduling
Mechanics: the study of the motion of matter a
Lecture 11: Introduction to Constitutive Equations and Elasticity
Previous lectures developed the concepts and mathematics of stress and strain. The
equations developed in these lectures such as Cauchys formula and Newtons second
law were derived from law
12.520 Lecture Notes 13
More Special Cases of Elasticity
Lecture 11 defined the constitutive equation for elasticity and explained how to reduce
the number of terms using Lam parameters. It also examined how to apply the elasticity
equations in the specia
12.520 Lecture Notes #10
Finite Strain
The "paradox" of finite strain and preferred orientation pure vs. simple shear
Suppose:
1) Anisotropy develops as mineral grains grow such that they are
preferentially oriented with the "easy slip" direction facilita
Lecture 3: Different Representations of Stress
The stress tensor can be represented in different ways to highlight particular features or
aid in solving geodynamic problems. This lecture explores how to represent the stress
tensor in terms of principle st
12.520 Lecture Notes 9
Mohrs Circle for Strain
Lecture 5 explained a simple and convenient way to find the stress on an arbitrary plane
given the stress tensor ij. The technique involved writing equations for how the shear
stress and normal stress on the
12.520 Lecture Notes #4
Assertion: most of the stress tensor in the Earth is close to "lithostatic,"
ij ~ -gd ij,
where is the average density of the overburden, g is gravitational acceleration, and d is
the depth of the point under consideration.
Conside
Lecture 5: Changing Coordinate Systems and Mohrs Circle
Lecture 2 explained that temperature is a zeroth-order tensor, force is a first-order tensor,
and stress is a second-order tensor. The order of a tensor is called its rank and is
defined by its law o
Lecture 7: Strain
Lectures 5 explored faults and brittle deformation across a failure plane. The next set of
lectures deals with ductile deformation: stretching, compressing, and twisting materials
into different shapes without breaking them. This lecture
12.520 Lecture Notes 6
Possible cause of weak faults
Preexisting fracture
Fig. 6_1
Clay low
Fig. 6_2
Pore fluid
Fig. 6_3
How to get quantitative graphs? Make assumption!
Zoback et al example.
Fig. 6_4
Assume c constant. fault = c0 .
Given , c0 , r get .
Lecture 8: Measuring Strain
The last lecture explained the basic ideas about strain and introduced the strain tensor.
This lecture explores a few different ways to measure strain and explains how strain, like
stress, can be represented graphically using M
12.520 Lecture Notes 12
Elasticity
So far:
Stress angle of repose vs accretionary wedge
Strain reaction to stress but how?
Constitutive relations
ij = ij ( kl ) ; ij = ij ( kl )
For example,
Elasticity
Isotropic
Anisotropic
Viscous flow
Isotropic
An
12.520 Problem Set 5
1) (20%) Just NE of Los Angeles, the San Andreas fault trends approximately N65W S65E. To within observational error, the displacement gradient there is observed to be
(each year):
0.15 0.24
0.00-0.15
where x1 is East, x2 is North, an
12.520 Problem Set 4
1) (90%) Consider a dry, noncohesive critical wedge, as discussed in class, as a model of
snow in front of a snow plow. We will consider 4 cases: i) = 30, b = 10; ii) =
30, b = 30; iii) = 60, b = 60; and iv) = 60, b = 30.
a) List the
12.520 Problem Set 8
1) We can deduce some interesting results about mantle viscosity and global return flow from
some simple 1-D models. Assume that plates are 100 km thick. Below the plates a low viscosity
layer extends to 600 km depth, below which the
Lecture 2: Stress
Geophysicists study phenomena such as seismicity, plate tectonics, and the slow flow of
rocks and minerals called creep. One way they study these phenomena is by
investigating the deformation and flow of Earth materials. The science of t
12.520 Problem Set 6a
1) (60%) We will often need to solve geodynamics problems where the boundaries are
not exactly aligned with the coordinate system. As an example, consider the situation
sketched below, where compressive stresses in the x direction le
12.520 Problem Set 6
1) (20%) The relationship between stress and strain for a simple isotropic elastic material
is:
ij = ekk ij + 2 eij
( and are constants, the Lame parameters, and are called moduli)
It is sometimes useful to present ij rather than ij
12.520: Problem set 1
1) Write out symbolically the relationship between the traction vector acting on a
surface, Ti, the normal vector describing the orientation of that surface, nj, and the local
stress tensor, ji, using a) vector-matrix notation, b) th
12.520 Lecture Notes 17
Stress and strain from a screw dislocation
x3
ce
rfa
Su ault
f
Fault
displacement S
x2
ion
D
sl
Di
at
oc s
i
ax
S/2
S/2
x1
Figure 17.1
Figure by MIT OCW.
Need to get traction = 0 at surface. First consider medium.
Assume:
u1 = u3 =
12.520 Lecture Notes 16
Dislocation in Elastic Halfspace Model of the Earthquake Cycle
Interseismic: Slip below depth D
xy
5 10 15
X/D
-15 -10 -5
y
Displacement
-3
0
3
X/D
Strain
Figure 16.1
Figure by MIT OCW.
Coseismic: Region above D catches up
xy
5
-15
12.520 Problem Set 2
(Problem weights are 15%, 15%, 25%, 25%, 30%, respectively)
1) In Problem set 1 you considered the stress tensor
1 1 0
ij = 1 1 0
0 0 2
dev
and found the principal stresses and directions of ij and the deviatoric stress tensor ij
i)
12.520: Problem set 3
1) (35%) As discussed in class for the Spanish Peaks example, the stress perturbation
m that results from intruding a cylindrical magma body of radius r0 aligned along the z
axis, subjected to an excess magma pressure pm, is most eas
12.163/12.463 Surface Processes and Landform Evolution
K. Whipple
BAKER RIVER: FLOW, SEDIMENT TRANSPORT, AND BANK EROSION AT A
MEANDER BEND
This lab will introduce you to some common field techniques and some general
understanding of the geomorphic proces
12.163/12.463 Surface Processes and Landscape Evolution
K. Whipple
III. Flow Around Bends: Meander Evolution
1. Introduction
Hooke (1975) [paper available] first detailed data and measurements about what
happens around meander bends how flow velocity and
Surface Processes and Landform Evolution (12.163/12.463)
Lab 4: Bedrock Channel Profile Modeling (1-D) (40 pts)
DUE: November 19
In this lab you will work with a Matlab script (detach_ftfs.m) to explore geological and
climatic controls on channel profiles
12.163/12.463 Surface Processes and Landscape Evolution
K. Whipple
II. Alluvial Channels and Their Landforms
A. Definitions and Landforms
Types of Channel: Rill, Gully (erosion limited, no floodplain, usually straight and
steep), Bedrock Channels, Mixed B
Lab 5: DEM Analysis: San Gabriel Mountains, CA
In this lab you will work with digital topographic and geologic maps with a set of DEM
analysis tools (ArcGIS and Matlab scripts) to study the topography of the San Gabriel Range in
southern California. The c
12.163/12.463 Surface Processes and Landscape Evolution
K. Whipple
Flow Mechanics: Velocity Profiles Exercise
Complete the following exercises during lab or before the next class meeting. The
exercise is intended to give you practice working with the rela