Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture Notes 8
Assertion: most of the stress tensor in the Earth is close to "lithostatic,"
~ gd ,
ij
ij
where is the average density of the overburden, g is gravitational acceleration, and d
is the depth of the point under consideration.
Consider the
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 7 Notes
Newtons second law
For a point mass
F ma
We can obtain F from a freebody diagram e.g., pendulum.
~
~
mg
Both F and a are vectors.
F mai
i 1, 2, 3
For a continuum, we can also construct a freebody diagram. Its easiest to do this
component
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 4 Notes
Stress Tensor (continued)
Before getting bogged down in rotation and notation, lets consider a special case:
rotation of the coordinate system by 45o about x3.
x2
x
x2
1
45
o
x1
x3, x3
In parallel, consider the tractions on a plane with it
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 2 Notes
Back to the sand box
Figure 2.1
Figure by MIT OCW.
We believe in friction
f
s n
f
12
s
11
We understand shear and normal tractions on planes oriented along axes.
In this example, nature chooses the failure plane. Failure will occur if
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 10 Notes
A house built upon sand
Theory Sand follows Amontons Law:
n (Mohr Coulomb)
Modify for pore fluid: p f
Need to express and n on any arbitrary plane develop stress tensor (symmetric)
T n
or
i
Mohr circle construction
ij n j
n T n
Applicat
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 11 Notes
Sandbox tectonics Simple theory (after Dahlen 84)
Assumptions
Coulomb failure (1 ) n
Based decollement may have different b (or b )
Material is on the verge of failure everywhere
Mohr circle construction
Inertia negligible F ma 0
4 param
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 6 Notes
The coefficients of friction, , for a wide variety of rocks, are comparable, in the range
0.6 0.9. The only rocks with low coefficients of friction are clays, which are not stable
at the high pressures and temperatures characteristic of fa
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 5 Notes
Quantities in Different Coordinate Systems
How to express quantities in different coordinate systems?
x3
x3
P
Direction Cosines
Axis
x2
23
11
x1
x1
x2
12
x2
x3
x1
x2
x3
11
21
31
12
22
32
13
23
33
x
1
Direction cosine ij is cosine of ang
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 9 Notes
Stress Rotation, after Zoback et al, 1987
The principal stress directions are observed to rotate in the vicinity of the San Andreas
fault (SAF). In the farfield (e.g., Nevada), the maximum compressive stress is oriented
at an angle to the
Applications of Continuum Mechanics to Earth, Atmospheric, and Planetary Sciences
SCI 12.005

Fall 2012
Lecture 3 Notes
Tensors
Most physical quantities that are important in continuum mechanics like temperature,
force, and stress can be represented by a tensor. Temperature can be specified by stating
a single numerical value called a scalar and is called a