Purdue University · Web viewThe relation between stress and strain allows us to relate the applied forces to the resulting deformation in much the same way that the extension of

Purdue University

EAS 557Introduction to Seismology

Robert L. Nowack

Lecture 5

Now we begin developing some of the basic concepts of continuum mechanics that we will need to study the propagation of elastic waves in the Earth. It is these elastic waves that are recorded on seismographs. The information they contain provides us with much of what we know about both earthquake sources and the structure of the Earth’s interior. So, it is important for us to understand the fundamentals if we are to properly interpret the information brought to us about the Earth’s interior by the elastic wave types we see on seismograms.

The concepts of stress and strain are fundamental to describing the force balance and geometrical description of continuous deformable bodies (like the Earth). The relation between stress and strain allows us to relate the applied forces to the resulting deformation in much the same way that the extension of a spring is related to the force pulling on it.

The particular kind of relation between stress and strain that we shall derive is called linear elasticity because the resulting strain is linearly proportional to the applied stress. So, to a high degree of precision, the Earth can be modeled as a linear elastic system over the short time span of seismic waves and, using observations of elastic waves, we can derive the elastic properties within the Earth.

Analysis of Strain

We shall consider a fixed set of axes and express all vectors and “tensors” in terms of their components with respect to these axes. First, several definitions:

A Deformable body = a body that changes shape under the action of forces (internal (like gravity) or external forces)

A Rigid (nondeformable) body = a body that undergoes only rigid motions (i.e., translations and rotations).

Deformation = strain = the change of relative position of points within the body (distortion stretch, etc.).

The assumptions on a continuum are

1) continuity of deformation (i.e., no tears or dislocations within body)

2) single valued deformation (i.e., 1-1 correspondence in strained and unstrained configuration

Consider a material in which a particle initially having position X0

moves to some other point X1

. We will write

X1 ( X0 ,t ) = X0 + u

where u is the position and time dependent displacement.

The distance between particles P and Q in the initial state is d X0

. After deformation, P is

displaced by u to P' and Q is displaced to Q'. Note that we define u in terms of X

0 [a

Lagrangian description - a function of where it comes from], or in terms of X1

[a Eularian description – a function of where the particle is]. For infinitesimal deformation of linear elasticity, it is simpler to use the Lagrangian description.

The separation between the displaced particles P' and Q' is d X

1. This can be written

from the figure

d X1 = { X0 + d X 0 + u ( X0 + d X 0 ) }(1)

− { X0 + u ( X0 ) }

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Now we want to express d X1 in terms of d X

0 and y ( X 0 , t ) . Use a Taylor expansion

of u ( X0 + d X0 )

u ( X0 + d X0 ) = u ( X 0 ) + ∂u∂ x1

|X0 dx1 + ∂u∂ x2

|X0 dx2 + ∂u∂ x3

|X0 dx3 + O | d X0| 2

where d X0 = ( dx1 , dx2 , dx3 ) .

In the vicinity of P at X0

, we can find the changes in length of any elementary line,

segment d X0

to first order as

u i ( X0 + d X0 ) = ui ( X0 ) + ∑j=1

3 ∂u i∂ x j

dx j + O ( dx j2 )

where the summation convention over repeated indices can be used to suppress the summation sign. In vector notation,

u ( X0 + d X0 ) = u ( X 0 ) + ( ∇ u )⋅d X0(2)

where ( ∇ u ) is the outer product,

∇ u =∂ u i∂ x j

= [ ∂ u1

∂ x1

∂ u1

∂ x2

∂ u1

∂ x3

∂ u2

∂ x1

∂ u2

∂ x2

∂ u2

∂ x3

∂ u3

∂ x1

∂ u3

∂ x2

∂ u3

∂ x3

]where ui = u i(X

0 ) are the components of the u ( X0 , t ) displacement vector in the figure above. This includes 9 partial derivatives and is a rank 2 tensor. A linearized version of equation (1), in component form, can then be written as,

dx i1 = dxi

0 +∂ ui∂ x j

dx j0

(sum on j) (3)

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In order to give a physically meaningful representation of the local deformation, we

separate ( ∇ u ) into its symmetric and antisymmetric parts. Thus,

[∂u1

∂ x1

∂ u1

∂ x2

∂ u1

∂ x3

∂u2

∂ x1

∂ u2

∂ x2

∂ u2

∂ x3

∂u3

∂ x1

∂ u3

∂ x2

∂ u3

∂ x3

] = [∂u1

∂ x1

12 [ ∂u1

∂ x2+

∂u2

∂ x1 ] 12 [ ∂u1

∂ x3+

∂u3

∂ x1 ]12 [ ∂u1

∂ x2+

∂ u2

∂ x1 ] ∂ u2

∂ x2

12 [ ∂u2

∂ x3+

∂u3

∂ x2 ]12 [ ∂u1

∂ x3+

∂ u3

∂ x1 ] 12 [ ∂u2

∂ x3+

∂u3

∂ x2 ] ∂u3

∂ x3

]− [− 0 1

2 [ ∂ u2

∂ x1−

∂u1

∂ x2 ] 12 [ ∂u3

∂ x1−

∂u1

∂ x3 ]12 [ ∂u2

∂ x1−

∂u1

∂ x2 ] 0 12 [ ∂u3

∂ x2−

∂u2

∂ x3 ]− 1

2 [ ∂ u3

∂ x1−

∂ u1

∂ x3 ] − 12 [ ∂u3

∂ x2−

∂u2

∂ x3 ] 0]

In an abridged notation, let

e ij =12 [ ∂ ui

∂ x j+

∂u j∂ x i ] = 1

2 [ ui , j + u j , i ]

This is the infinitesimal strain tensor which is symmetric (in the infinitesimal case, there is no distinction between Langrangian and Euclerian descriptions), and

ωij =12 [ ∂u j

∂ x i−

∂ ui∂ xj ]

This is the rotation tensor which is antisymmetric.

The above equation can then be written

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[ ∂ui∂ x j ] = eij − ωij

Thus, the infinitesimal strain tensor has six independent components, e11 , e22 , e33 and e12 , e13 , e23 , and the rotation tensor has three independent components, ω12 , ω13 , ω23 . We will now give a physical interpretation of these components.

The Physical Interpretation of Strain and Rotation

Consider first in 2-D

A) Tensional strains

Assume only u1 exists (i.e. u2 = 0). Then, from equation (3)

dx i1 = dxi

0 +∂ ui∂ x j

dx j0

(sum on j)

for the only non-zero component i = 1,

dx11 = dx1

0 +∂ u1

∂ x1dx1

0 = dx10 ( 1 +

∂ u1

∂ x1)

Since e11 =

∂u1

∂ x1 , the relative extension is

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dx11 − dx1

0

dx10 = e11 =

∂u1

∂ x1

Thus, e11 measures the relative extension along the x1 direction. In a similar fashion, e22 and e33 measure the relative extensions in the x2 and x3 directions. The sign convention used here is,

e11 > 0 for extensione11 < 0 for contraction

Also, note that strains are dimensionless.

B) Shear Strains

The off diagonal terms, e12 , e23 , e13 , are usually called “shear strains” because they measure the local shearing. First, we will look at simple shear

From the figure and equation (3), then

dx i1 = dxi

0 +∂ ui∂ x j

dx j0 = dx i

0 +∂ ui∂ x1

dx10 +

∂u i∂ x2

dx20

Since dx 20 = 0 ,

dx i1 = dxi

0 +∂ ui∂ x1

dx10

Then,

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dx11 = dx1

0 +∂ u1

∂ x1dx1

0 = ( 1 + e11 ) dx10

and,

dx21 = 0 +

∂ u2

∂ x1dx1

0 =∂ u2

∂ x1dx1

0

Now, from the figure

tan γ1 =

∂ u2

∂ x1

( 1 +∂u1

∂ x1 )~

∂u2

∂ x1

For small angles, γ1 ~ tan γ 1 (for small shear strains). Thus,

γ1 ~∂ u2

∂ x1

and, from the equation for strain

e12 = 12 ( ∂ u1

∂ x2+

∂ u2

∂ x1) = 1

2γ 1

Thus, for simple strain, e12 is just half the shearing angle.

The general shear strain case in 2-D is shown below.

Expressing the displacements of the points Q, R, and S in terms of the displacement and its first derivatives at the point P, we can approximate

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γ1 ~ tan γ 1 ~∂ u2

∂ x1

γ2 ~ tan γ 2 ~∂ u1

∂ x2

and

e12 = 12 ( ∂ u1

∂ x2+

∂ u2

∂ x1) = 1

2 ( γ1 + γ2 )

is the average shear angle change between line segments PQ and PS. Note that the

approximations to use γ1 and γ2 require that

∂ui∂ x j

<< 1

In fact, we require that all the strain components be infinitesimal. This is satisfied in the Earth where maximum strains associated with elastic waves are on the order of 10-4 – 10-5. The other shear strains e13 and e23 are similarly defined with respect to the different coordinate axes.

C) Rotation

In 2-D, the rotation tensor is associated with the angle of rotation of the diagonals. For example, in 2-D

ω12 = 12 ( ∂u2

∂ x1−

∂u1

∂ x2) = 1

2 ( γ1 − γ 2 )

where ω12 > 0 indicates rotation in the counterclockwise direction. Note that rotation does not necessarily imply shear or deformation. It is simply a local solid rotation by an angle related to ω12 . Referring to the diagonal line segments from P to R and P' to R' in the figures above, simple shear has a non-zero rotation of the block. This is true for a general shear as well. Pure

shear has γ1 = γ2 resulting in no block rotation since then ω12 = 0 . In 3-D, we can similarly

define e13 , ω13 and e23 , ω23 with respect to each pair of coordinate axes.

Dilatation

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An important variable we have to define is the relative change of volume (or area in 2-D) due to the deformation. Consider again the 2-D example. If the initial area of the 2-D block is S0 = dx1

0 dx20, the area of the strained block is approximately

S1 ~ ( 1 + e11 + e22 ) dx10 dx2

0

We then define the dilatation θ to be

θ =S1 − S0

S0= e11 + e22 =

∂ u1

∂ x1+

∂u2

∂ x2=

∂ ui∂ x i

= ∇⋅u

where ∇⋅u is the divergence of the displacement field u .

By a similar argument, in 3-D

θ = ΔVV

= e11 + e22 + e33 =∂ ui∂ xi

= ∇⋅u

The dilatation (or dilation) is the relative increase of volume due to deformation and equals the sum of the diagonal elements of the strain tensor. Thus,

θ = ∇ ⋅u = e11 + e22 + e33 = Trace e

where ∇⋅u is the divergence of the displacement field.

As we shall see, the dilatation moves through the body at the P velocity.

Rotation

Another important variable useful in dealing with elastic waves is the rotation vector defined by

Ω = ω23 x̂1 + ω31 x̂2 + ω12 y x̂3 = 12

∇ × u

That is, the rotation vector is the curl of the displacement field.

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In 3-D, the rotation vector Ω describes the rotation of a material element surrounding the point P. The rotation and corresponding shear strain propagates at the velocity of shear waves (S waves) in an elastic material.

An example of the static horizontal displacement field measured geodetically after the 1927 Tango, Japan earthquake is shown in Figure 1. This shows the decay of fault parallel displacement with distance perpendicular to the fault.

Figure 1 (from Stein and Wysession, 2003)

In the last decade, high precision GPS measurements can now attain accuracies of 10 mm in displacement and relative ground velocities of several mm/year.

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Synthetic aperture radar interferometry (InSAR) uses high resolution radar mapping from spacecraft or aircraft to map the surface. By taking phase differences between images taken before and after an earthquake, very high resolution interferometric images of ground displacement can be obtained as shown in Figure 2. The top figure is a measured interferometric image of ground displacement and the bottom is from a computer model of ground displacement resulting from slip on the fault.

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Figure 2 (from Stein and Wysession, 2003)

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