Download Infinitesimal strain

Geology 5640/6640 1 Feb 2017
Introduction to Seismology
Last time: The Strain Tensor
• Stress within a continuum causes deformation, or strain
1
1
eij = (¶ j ui + ¶i u j ) , and rigid-body rotation wij = (¶ j ui - ¶iu j ).
2
2
• Infinitesimal strain assumes small relative displacement
in which case the vector between two points changes by:
• The constitutive relation for seismology is Hooke’s Law:
s ij = Cijkl ekl
• The elasticity tensor has up to 21 independent terms, but
for an isotropic solid, we only need two (Lame’s
constants  and ):
Cijkl = ldijdkl + m(dikd jl + dild jk )
Read for Fri 3 Feb: S&W 29-52 (§2.1-2.3)
© A.R. Lowry 2017
The elasticity tensor can be expressed in terms of Lame’s
constants as simply:
Cijkl = ldijdkl + m(dikd jl + dild jk )
If we substitute this into our constitutive law, we can write
s ij = lekkdij + 2meij = lqdij + 2meij
Here,  is the volumetric dilatation:
The gradient operator is also sometimes called the
divergence, and is defined as:
The Equations of Motion:
Up to this point, we’ve assumed static equilibrium (i.e.,
boundary stresses on our infinitesimal cube balance out).
If they don’t balance,
we must have
motion!
Stresses
are
21, 22, 23
12 must equal
21 in equilibrium
Suppose we add a small incremental stress on +x^1 face, so
that stresses on this face are:
11 + 11, 12 + 12, 13 + 13
^ faces (& recalling F = A):
Summing the forces on the ±x
1
We can do the same for shear stress acting on the other
two faces and sum to get the total force acting in the
x1-direction:
The other force vector elements work similarly, so
¶s ji
¶s ij
Fi =
dx1dx 2 dx 3 =
dx1dx 2 dx 3
¶x j
¶x j
(by symmetry). These must be balanced by motion per
Newton’s second law:
F = ma
Here, we are interested in relating the force balance
back to displacement u, so we express
In Einstein summation notation, time derivatives are
expressed with an overdot, so we’ll also use ai = üi.
Mass m is equal to density times volume,
m = V = dx1dx2dx3,
so we can write the force balance in the x1-direction as
or
In indicial notation,
¶ 2 ui ¶s ij
r 2 =
Þ ru˙˙i = ¶ js ij
¶x j
¶t
Note that in the wave equation, acceleration and stress
vary in both space and time!
We must also consider body forces fi = Fi/V, in which case
the dynamic equations of motion are
ru˙˙i = ¶ js ij + fi
In the Earth the only significant body force is gravity:
fi = (0, 0, g)
and in practice we neglect it ( assumed negligible) for
body waves (although it is important for surface waves).
Now we have the equations in terms of stress; we’d like to
get them entirely in terms of displacement. Recall:
s ij = ldij ekk + 2meij = lqdij + 2meij
and:
1
eij = (¶ j ui + ¶i u j )
2
Substituting these back into the equations of motion (&
letting fi = 0), we have:
(For e.g. the x1-direction):
And we have
Recall also:
And we have:
Now we introduce an additional notational sleight-of-hand:
Take the derivative:
æ ¶ 2q ö
æ ö
¶ 2 æ ¶ u1 ö
2 ¶ u1
Reordering: Þ r 2 ç ÷ = ( l + m ) ç 2 ÷ + mÑ ç ÷
¶ t è ¶ x1 ø
è ¶ x1 ø
è ¶ x1 ø
If we do this for all three coordinates and sum,
We have:
This is the wave equation for dilatations only (i.e., a P-wave!)
and is more commonly written:
where:
a=
l + 2m
r
represents the propagation velocity! (Note the units:
sqrt(Pa (kg m-3)-1) = sqrt (kg m-1 s-2 kg-1 m3) = sqrt (m2/s2)
or just m/s).
If we recall moreover that
We can write in terms of displacement as:
We arrived at the P-wave equation using
by taking the derivative with respect to xi and summing over
i. We could instead take derivatives with respect to xj and
by a similar set of steps arrive at:
the S-wave equation, in which the S-wave propagation
velocity is given by
b=
m
r
Note the important implication: For the P-wave we have
dilatation, but no shear; for the S-wave we have shear,
but no dilatation!