GG 450 March 19, 2008

Stress and Strain

Elastic Constants

Deformation of solids:

We will define a set of parameters that provide models for the deformation of materials that will be used to describe seismic waves and what they tell us about the materials that they pass through.

We need two things – a set of coordinates to describe how materials can change their shape, and a set of relationships that describe how forces can cause these materials to deform, called constituative relations. You’ve likely had a similar model to work with in Structure.

Deformation of a material is called STRAIN.

We can define two types of strain, one, dilatational strain, where linear dimensions and volumes change, but not angles, and

shear strains where shapes and angles change, but, as long as the changes are relatively small, not volumes.

These two types of strain cover all types of deformation of materials, provided that the deformations are small relative to the original size/shape of the body.

dilatational strains: These components of strain are in the direction of the axis that they relate to. They represent changes in length. €

εx =∂u

∂x, ε y =

∂v

∂y, ε z =

∂w

∂z

This is a strain in the x-direction:

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ε x = du /u

Cubical dilatation, or volumetric strain:

. This is volume/volume. This strain is a change in volume.

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θ =ε x +ε y +ε z

In the figure above, the strain (u) in the x direction changes with y, but not with x. Shear strains are changes in shape, or angles.

ψ

x

y

u

Shear strains: stains along one axis change with distance in the other axis:

show CUBE

NOTE: The tangent of ψ u/y= strain in the x direction divided by y.

BUT for SMALL strains, tan ψ u/y (when measured in radians).

The assumption of small strains and similar assumptions are used ROUTINELY in seismology, which means that very large amplitude waves don't obey the same rules as very small ones. - SHOCK WAVES, and SURF for example.

STRESSES Recall Hooke’s law: F=Kx, where F is the force on a spring that deforms a distance x with a spring constant K. This works fine for springs and other linear deformations - assuming the deformation is elastic, BUT for solids, strain need not be in the direction of stress, so a simple spring constant no longer tells the whole story. Stresses are forces applied to solids per unit area. What does “elastic” mean? What other types of deformation are there?

There are normal stresses that cause dilatational strains and tangential stresses that cause shear strains. ELASTIC CONSTANTS relate stress to strain through 3-dimensional versions of Hooke's Law, our constituative relationship.

Linear strain: Young's Modulus.

A stress in the x direction, x, will result in a strain in the same direction given by:

Where E is the elastic constant called Young's Modulus. This is just a simple form of Hooke's law.

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x = E∂u

∂x= Eε x

Consider the deformation of water under a stress in the x direction. The resulting strains involve changes in all three directions as the water is pushed away, not just in the axis in which the force is directed.

We need another elastic constant that will relate how easily a body changes its other dimensions in response to a stress in the third dimension, much like Young's modulus tells us how easily its length will change.

One such elastic constant is Poisson's ratio, .

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

If =0, then the other dimensions do not change in response to stress, and volume change is maximum. If = 0.5, then the volume does not change at all. For fluids, 0.5, while for slinky 0. For most solid rocks, = 0.1-0.25.

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= - transverse strain / longitudinal strain

What if the VOLUME of the material changes when pressure is applied?

WHAT IS PRESSURE?: equal stresses in all directions).

In this case the change in volume is related to the change in pressure by the bulk modulus,

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Pf − P0 = KVf − V0

V0

Note in the above arguments that we assume that the solid is isotropic, with no changes in elastic constants with direction. This is a MAJOR assumption, but the alternative - allowing elastic constants to change with direction (anisotropy), results in difficult math still being worked out.

Shear strain: Rigidity Modulus.

We need another elastic constant to tell us how easily a body will change its shape or suffer a shear strain () under shear stress (s). The shear modulus, or rigidity modulus, G does this:

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s = Gγ

The rigidity of fluids and gasses is 0; for hard rocks, G is about 0.1-0.7 Mbar.

Other elastic constants. There are MANY elastic constants, defined by what information is needed about a material.

For analysis of an isotropic solid, only 2 elastic constants are needed. E (Young's modulus), (Poisson's ratio), and G (rigidity) are important to seismology, as are the compressibility, , and bulk modulus, : where =1/, and is the ratio of change in pressure divided by the resulting change in volume.

If we combine the relationship between stress and strain with Newton’s law of motion, ( is the density)

we arrive at a one-dimensional form of the WAVE EQUATION, a differential equations that, when solved, results in mathematical models for seismic waves. For example:€

d2q

dt2= E

d2q

dx2 (for a uniaxial compression)

€

d2q

dx2=

1

V 2

d2q

dt2

is the wave equation for a wave traveling in the x direction.

q is the displacement of the any particle along the x axis , thus

dq/dt is the particle velocity of that point in the material, and

d2q/dt2 is the particle acceleration.

V is called the propagation velocity, or the wave velocity, or phase velocity. It is a measure of how fast the wave moves through the earth.

A solution of this equation is:

which describes a wave with amplitude A traveling in the +x direction with a velocity V, and k (wave number) is the number of cycles (times 2π) per distance.

Is it really a solution to this equation? Let’s try it out…

Remember:

€

q = A sin k(Vt − x)

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d

dzsin(az) = acos(z) and

d

dzcos(z) = −asin(z)

A bit more work shows that two types of waves are possible - compressional (or p) waves generated by dilational strains, and

shear (or s) waves generated by shear strains.

Near a boundary, such as the earth’s surface, other types of waves - surface waves- are generated.

Surface waves come in two types, Rayleigh waves, with particle motions like ocean waves, and

Love waves, with only horizontal particle motion.

Rayleigh waves are a combination of compressional and shear energy, while Love waves are only shear energy.

Copressional waves are sound waves.

In terms of other elastic constants (the two velocities ARE elastic constants):

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Vp =(4 /3)μ + k

ρ

Vs =μ

ρ

where is the shear modulus, k is the bulk modulus and is the density.

Note that the compressional velocity is always greater than the shear velocity, and that the shear velocity is independent of the bulk modulus.

The shear wave equation is:

In a uniform homogeneous isotropic solid where u is the displacement.Where is the shear velocity and u is the displacement in a direction perpendicular to the direction the wave is moving in.€

∂2u

∂t 2= β 2 ∂ 2u

∂x 2 or

∂ 2u

∂x 2=

1

β 2

∂ 2u

∂t 2

Here’s a good problem from Shearer’s Intro. SEISMOLOGY, page 34, # 3.5. In this problem, the wave equation is solved to show a shear wave traveling along a 100 km long bar ( km/s). Use finite differences to write a small program to propagate a plane wave in a solid.Using distance increments of dx=1 km and time increments of dt=0.1 sec, and assuming a source at u=50 km for 0 < t < 5 sec.

Apply a stress-free boundary condition at x=0 (this is a “free surface”, like the surface of the earth), and a fixed boundary condition at x=100 km (this condition says the bar is not allowed to move at this point).

The first derivative in digital form (OF ANY WELL-BEHAVED FUNCTION!) is given by:

Don’t believe it?: Google “difference calculus”. Or try it with a sine wave.

The second derivatives can be approximated by the following:

where the subscript is a change in x and u1, u2, and u3 are time increments.

Here’s the program in Matlab:

€

∂2u

∂x2≈

Δ2u

Δx2=

ui+1 − 2ui +ui−1

Δx2 and

Δ2u

Δt 2=

u3i − 2u2i +u1i

Δt 2

€

∂u

∂x≈

Δu

Δx=

ui+1 −ui−1

2Δx and

Δu

Δt=

u3i −u1i

2Δt

% SeismicWave% WAVE EQN for bar using finite differences% problem 3.6 page 34 of Shearer Seismologybeta=4.; % shear velocitydt=0.1; % time incrimentdx=1.; % displacement incrimenttlen=5.; % time that the source is active (seconds)% initialize variables u1=zeros(1,101); % displacement at time j-1 (previous time step) u2=zeros(1,101); % displacement at time j (current time step) u3=zeros(1,101); % displacement at time j+1 (next time step)for time=[1:dt:33]; %increment time for 33 seconds for i=[2:100]; %calculate new displacement at each point for next time step rhs=beta^2*(u2(i+1)-2.*u2(i)+u2(i-1))/dx^2; % right-hand-side of wave eqn u3(i)=dt^2*rhs+2.*u2(i)-u1(i); % new displacement (u3) end % apply boundary conditions u3(1)=1*u3(2); % du/dx=0|x=1, so: u3(1)=u3(2); boundary condition at x=1: stress-free, so u3(101)=0.; % boundary condition at x=101: bar fixed, so u3(x=101)==0 if time<=tlen u3(51)=sin(3.14159*time/tlen)^2; % source-time function for plane wave% u3(1)=sin(3.14159*time/10); % another source function end for i=[1:101]; % for each x, propagate the displacement u1(i)=u2(i); u2(i)=u3(i); end plot(u2); % plot the current displacement axis([1 101 -2 2]); % set axes for plot pause % pause at each time stepend

This program displays the expected wave properties when a wave reflects from a fixed boundary and from a free surface.

Based on how density changes in the earth, how might you expect seismic velocities to change with depth?

P, Sv, Sh, Rayleigh, and Love waves are all commonly observed waves. How might these waves be observed at the earth's surface?

Think about a P wave arriving at the surface from the south at an angle of 45°. You have three seismometers, one sensitive to vertical motion, and the other two sensitive to horizontal motion in the N_S and E_W directions. What might you see on the seismometers?