EART 118 SeismotectonicsLecture 3

Second of two mathematical lectures in this class that will develop the basic equations for P and S waves. CONCEPTS: Constitutive Law Isotropic Material Elastodynamic Equation P and S Waves Nature of Waves Supplementary Reading (Optional, for more details/rigor) Lay and Wallace, Modern Global Seismology, Ch. 2 Stein and Wysession, An Introduction to Seismology, Earthquakes and Earth Structure, Ch. 2

F(xs,t) U(xr,t)

A force, associated with some form of energy release at position xs, in or on a body will produce a motion at position xr, somehow. If the body is perfectly rigid, the motion will be an instantaneous translation, rotation or both. Use F=ma. If the body is elastic, we still use F=ma, but the action of the force spreads through the body causing internal deformations that vary in space and time. Do describe these we need general representations of internal forces and deformations. We use stress and strain.

U(xr,t) is vector ground displacement at position xr, defined in some coordinate system, as a function of time. This is what we can record.

Basic Problem:

KEY Result: SPATIAL GRADIENTS OF STRESS terms are balanced by INERTIAL terms. This is how F=ma manifests in a continuum.

Last time – idea of distributed body stresses leads to stress tensor. Balance of forces on any sub-element of the body gives an equation of motion.

Key result: all infinitesimal deformations can be represented by this strain tensor. The diagonal terms change lengths of lines, the off-diagonal terms change angles between orthogonal lines. Strains all involve first spatial derivatives of the displacement field. The equation of motion involves temporal derivatives of the displacement field and stresses. So, if we have a relationship between stress terms and strain terms, we can use that to write the equation of motion with all terms involving displacements. That is appealing because displacements can be recorded.

Deformations involve changes in displacement vectors between adjacent points in the body (space and time dependent).

OK, so now we have a general formulation of internal forces distributed through a medium, for which any choice of coordinate system gives us a specific stress tensor that describes the local state of stress (along with specific rules for how the stress tensor changes if we change the coordinate system). That gives us a local expression for F=ma; the elastic equation of motion, which specifies accelerations in terms of specific spatial gradients of the stress tensor. Then we have a general formulation of how internal deformations are expressed as linear spatial derivatives of displacements of points in the medium (for infinitesimal strain theory). We would like to relate stress and strain to change the equation of motion to an equation involving just spatial and temporal derivatives of displacement. So, what is the relationship between stress and strain?

Stress and strain tensors are general representations of internal forces and deformation within an elastic body. They are independent representations, but must be related (F=ma connects forces to accelerations, which are second time derivatives of displacements, and strains are spatial derivatives of displacements. It would be ‘nice’ to have a first-principle’s theory for a relationship between stress and strain terms, but until very recently we have relied on experimental observation of such behavior for rocks.

When compressed, all rock materials exhibit an interval of linear relationship between stress and strain that is totally recoverable (elastic). stress = constant x strain For relatively low pressure (< 400 Mpa, T < 600°C), rocks will fracture as stress Increases to some limit.

ε Strain (δL/L)

Constitutive Law: ε = f(σ)

I – crack closure

II – linear elasticity

III - dilatancy

IV - microcracking

σ Stress

V - failure (fracture)

VI – frictional sliding

For an experiment under fixed pressure and temperature of the rock.

F

A (area)

σ = F/A

L

δL

Constitutive Equation – Empirical relationship between stress and strain

SOME

^

SOME

^

SOME

^

Now, we can combine the generalized Hookes’s Law with the equation of motion and the relationships between strain and displacement to combine everything into a single equation involving only constants and displacement.

Three equations: Elastodynamic Equations for Linear Elastic, Isotropic, Homogeneous Medium

Vector calculus is very useful at this point.

Helmholtz Theorem: ANY continuous field can be represented in this form with Potentials (other fields) satisfying specific criteria.

This is a wave equation for the P wave – φ is the space-time P wavefield

This is a wave equation for the S waves – ϕ is the space-time S wavefield

OK, let’s take a breath. All this algebra has given a profound result that we want to take stock of. The general mathematical representation of motions everywhere in the medium satisfying the elastodynamic equations, U(x,t), is given by simple spatial derivative operations on two space-time functions, φ(x,t) and ψ(x,t), which are themselves solutions of the three-dimensional wave equation. φ(x,t) is the P wavefield ψ(x,t) is the S wavefield Physical displacements of the P wave are calculated by taking the gradient of φ(x,t). Physical displacements of the S wave are calculated by taking the curl of ψ(x,t). Because the wavefields satisfy the wave equation, P and S wave motions have basic behavior of waves – this makes the final solution very straightforward: we just need to understand properties of waves.

x1

Thin, elastic rod; all motion in the x1 direction.

dx1

σ11 + dσ11/dx1 x dx1

dx1

σ11

Mass = density x volume = ρ dA dx1

Equation of motion : sum forces = Mass x acceleration

ρ dx1dA x d2u1/dt2 = (σ11 + dσ11/dx1 x dx1) dA – σ11 dA

ρ d2u1/dt2 = dσ11/dx1 Spatial gradient of stress balanced by inertial terms

Need constitutive law, measure in lab: σ11 = E ε11, E= Young’s modulus

U1(x1,t)

σ11 = E ε11 = E du1/dx1

ρ d2u1/dt2 = dσ11/dx1 = E d2u1/d2x1 : now have equation of motion all in terms of u1

Let c = (E/ρ)1/2

d2u1/dt2 = c2 d2u1/d2x1 We finally arrive at the one-dimensional wave equation. The implication is that the displacements in the rod will spread along it as a wave, obeying the wave equation.

Note – this one-dimensional treatment is not physically correct; for a finite colume rod, there is more than 1 strain term produced by σ11, and the full 3D solution before is needed for rigorous solution, and it would give eise to two waves in the rod (P and S waves), as a general solution to elastic waves in a solid. But, we see the parallel development easily for this simple example. The same holds for the string problem, next.

OK, let’s quickly derive a simple 1D wave equation from a standard physics problem: motion of a string. Tension along/within the string is constant, τ

The sum of the forces in the y-direction equals the mass times acceleration in the y-direction. We use linear density, ρ, such That ρdx is mass.

NOTE: The spatial imbalance in forces gives accelerations. The wave equation always has two spatial derivatives and two temporal derivatives – this controls the nature of solutions.

The wave equation is a hyperbolic partial differential equation, with two space and two time derivatives of the solution weighted by a relative weighting factor which is the inverse square of the wave velocity. This is a classic PDE, and applied mathematicians have figured out all properties of the solutions for different choice of coordinate system (Cartesian, spherical, cylindrical). The most important point is that the solutions (waves) are disturbances with a given shape (functional form) that retains that exact shape as space and/or time vary. The connection between the space and time domains that keep the waveshape fixed is given by the wave velocity (distance/time). Essentially, whatever shape the P wave or S wave has generated by the source will translate through the medium as a function of time without distortion. This enables analysis of seismic sources using wave motions recorded thousands of miles away!

For our string; if we put an initial arbitrary shape on the string, that shape will move through space as a function of time, with a velocity controlled by the physical parameters of the medium.

Here, f is ANY function, of an argument involving space (x) and time (t) with a weighting factor of wave velocity (v), in the form of x ± vt. The chain rule for two derivatives with respect to x and two derivatives with respect to t ensure the wave equation will be satisfied.

α is the P velocity, and it depends on the two Lamé parameters λ, µ, or on incompressibility K, µ, along with density, ρ.

β is the S velocity, and it depends on only µ, and ρ. For a Poisson medium which has equal Lamé parameters, α = 1.732 β.