Introduction to Seismology Geology 5640/6640 Introduction to Seismology 25 Jan 2017 Last time: “Review” of Vectors • A normal is the unit vector orthogonal to a plane or surface at a point • The dot product is a measure of how parallel two vectors are: (in indicial notation) • The cross product is a vector whose length is a measure of how orthogonal two vectors are, with direction perpendicular to the two vectors: ||c|| = ||ab|| = ||a|| ||b|| sin Read for Fri 27 Jan: S&W 29-52 (§2.1-2.3) © A.R. Lowry 2017

“Review” of Derivatives: Since calculus is a pre-req for the course, I assume you’ve seen derivatives and integrals before, so I will only review a few basics and some notation we will use. Derivatives are used for finding minima and maxima of functions as well as describing rates of change.

Consider an example function: f(x,y) = x2 + 2y2 Then we have the following: partial derivatives: indicial notation: If a function has directional elements, e.g., uz, then indicial notation for a partial derivative in the y direction would be: Another notation we’ll see often is the gradient operator:

Complex Numbers: A complex number can be written z = x + iy where x = Re{z} is the real part of z, and y = Im{z} is the imaginary part of z. We can graphically represent z in either Cartesian or polar (r, ) coordinates as a vector quantity: y = r sin x = r cos

Substituting polar coordinates into z = x + iy we have: z = r cos + i r sin = r (cos + i sin) From Euler’s Identity, ei = cos + i sin So we can write: z = r ei The complex conjugate of a complex number z, which we denote z*, is: z* = x – iy = r cos – i r sin = r cos(–) + i r sin(–) = r e–i Thus the squared modulus (magnitude) of z is: |z|2 = zz* = (x + iy)(x – iy) = (x2 + y2) = rei re–i = r2

Grasping this opens up a seamy underbelly of the math world. ** Note also that: ei = cos + i sin& e–i = cos – i sin  cos= (ei+ e–i)/2 & sin= –i(ei– e–i)/2

Introduction to Tensors Motivation: Physical laws must be independent of any coordinate system used to mathematically describe them. Tensors are a compact representation of spatial properties that can easily be transformed from expression in one coordinate system to another. Tensors are very useful for: Stress, Strain & Elasticity (our application) Deformation via other rheologies… Hydrodynamics Electromagnetism General relativity And many other applications...

In 3D, a point is represented by three numbers (or coordinates) Example: In 3D, a point is represented by three numbers (or coordinates) as a vector. Tensors are a generalization of scalars and vectors. The following table describes rank (or order) of some commonly-used tensors: Order n (or rank) Number of Components (3n) Name (& example) 30 = 1 Scalar (temperature) 1 31 = 3 Vector (displacement) 2 32 = 9 2nd order tensor (stress) … n 3n (The elasticity tensor, Cijkl, has n = 4  81 components!)

Stress: Consider a body acted on by external and/or internal (body) forces. We represent these forces on arbitrary planes within the body as stress (force/unit area). This has application in earthquake physics, as earthquakes represent stress/strain release by slip on planar surfaces. Imagine a plane or surface dissecting the medium: (Note: Here, “Imaginary” ≠ complex-valued! Better “arbitrary”?)

Assume the body is in static equilibrium (the forces balance such that the body doesn’t move.) Now, “remove” the right half and consider the equivalent set of forces acting on the plane that equate to the effect of the forces we removed (i.e., that are needed to maintain the equilibrium)…

Now, divide up the plane into infinitesimal elements with area A, normal , and force ΔF: The stress vector (“traction”) is defined as: acting on element A at point P with normal .

Note that we’re talking about forces on an elastic medium which is a continuum. Thus any force acting on the body is transmitted through every point within the body. Also, the orientation of our imaginary plane was completely arbitrary, so we can choose it to be parallel to the yz-plane for any given x.

We denote the stress components acting on the plane with normal (i.e. a plane on which x = constant) as:

ij More generally we can denote: index i denotes index j denotes direction of normal to the plane acted on by the force index j denotes direction of the force Hence σxx is a normal stress and σxy and σxz are stresses in the plane (called “shear stresses”). We can do the same with the y direction: σyx, σyy, σyz act on the plane parallel to xz-plane. For the z-direction, σzx, σzy, σzz act on the plane parallel to the xy-plane. Thus we can use all nine σij to represent the internal force distribution at point P for any plane passing through P.

We combine our nine components of stress defined in these coordinate planes to represent the stress on any surface through the medium. The 9 σij define the state of stress at point P. The full stress tensor is given by : Recall that strain is the deformation that results from a given stress. A common PhD exam question is to derive stress and strain & elucidate their difference, particularly in relation to your specific discipline…