ES 111 Mathematical Methods in the Earth Sciences Matrices and Deformation Lecture Outline 13 - Thurs 9th Nov 2017 Strain Ellipse and Eigenvectors One way of thinking about a matrix is that it operates on a vector - the vector ends up pointing somewhere else. In general, the vector will have been both stretched and rotated from its initial position. For instance, if our general vector A operates on the vector [x, y], we have a b x ax + by = c d y cx + dy The matrix has operated on our original vector [x, y] and produced a new vector [ax+by, cx+dy]. In particular, the x-vector [1,0] has moved to [a, c] and the y-vector [0,1] has moved to [b, d]. So a good way of thinking about a 2x2 matrix is that the first column describes what happens to an x-vector and the second column describes what happens to a y-vector. If you think about the vector as representing some pre-existing fabric in a rock, then it is obvious that the matrix is describing deformation of that pre-existing fabric. Also recall from last time that the determinant of a matrix tells us its area or volume, and so is a measure of the volumetric strain it represents. There is thus a very close link between matrix algebra and structural geology. One way of describing deformation is to use a strain ellipse. If you start out with a perfectly circular feature, after it has been deformed it will form an ellipse (see figure). In general a preexisting line will have been both stretched and rotated. However, for any strain ellipse there are two, perpendicular directions in which lines undergo stretching but no rotation (see figure). These two directions are called the principal strain axes and are a fundamental concept in structural geology. 1
Figure 1 So what? It turns out that we can use matrix algebra to find these principal strain axes. To do so we have to talk about eigenvectors and eigenvalues. Eigenvectors and eigenvalues Square matrices have so-called eigenvectors associated with them. An eigenvector is a vector which gets squeezed or stretched, but not rotated, when operated on by the matrix. The amount of squeezing or stretching (the strain) is called the eigenvalue. There is a single, fundamental equation in eigenanalysis: A x = λx where A is a square matrix, x is an eigenvector of A and λ is the associated eigenvalue. λ is a scalar, so the equation tells us that applying A to its eigenvectors does not alter their directions, but only scales their lengths. In structural geology, the eigenvectors of a deformation matrix are the principal strain axes associated with that deformation, and the eigenvalues are the associated principal strains. In general, an n by n matrix A may have up to n eigenvectors, each with an associated eigenvalue. Example The matrix 5 2 2 2 and the associated eigenvalues are has eigenvectors x 1 = 1 2 and x 2 = 2 1 λ 1 = 1 and λ 2 = 6 How do we find the eigenvectors and eigenvalues of a matrix? First we rewrite the above equation as A x λx = 0 or (A λi)x = 0 2
Since we assume x is non-zero, it can be shown that for the above equation to be true, we must have det(a λi) = 0 Note that we have moved from a matrix equation to a scalar equation (determinants are scalars). This makes things simpler. The solution to this equation turns out to be a polynomial in terms of λ that can be solved to obtain the eigenvalues of A. For an n n matrix, the polynomial has terms up to λ n and will have from 1 to n distinct roots (the eigenvalues). Example We ll use the same matrix as we had before. det(a λi) = det 5 λ 2 = 0 2 2 λ This implies that (5 + λ)(2 + λ) = 0 which is a quadratic equation in λ and gives us two roots (eigenvalues): λ 1 =-1 and λ 2 =-6. Now, how do we find the eigenvectors? Simply plug an eigenvalue into the equation Ax = λx and solve for the elements of x. The length of the vector is not determined, so we typically normalize each x to unit length. Example Carrying on with the same example, let s take λ 1 = 1 and insert it into the above equation: 5 2 2 2 x = 1 x y y which gives us two equations, identical except for a scaling factor, telling us that y = 2x. One way of writing this down as an (un-normalized) vector is x = [1, 2]. Careful - a common mistake is to reverse the order! 3
Example with repeated roots. Find the eigenvectors and eigenvalues of 1 0 0 0 2 1 0 1 2 [Answer: λ=1,1,3. For λ=3, eigenvector = [0,1,1]. For λ=1, eigenvector = [x,-1,1] i.e. it is not uniquely determined] Note that if you get repeated roots, then you don t have enough information to uniquely define some of the eigenvectors. Eigenvalues and eigenvectors have some useful properties. 1. The sum of elements along the main diagonal of A (a quantity known as the trace of A) equals the sum of the eigenvalues (repeating repeated roots): n n Trace a ii = λ i i=1 i=1 2. The determinant of A equals the serial product of the eigenvalues det Π n i=1λ i 2D Strain Back to our general matrix A, where a c b d This matrix A is the strain matrix which describes the deformation to which the medium is being subjected. Remember the way to think about what this matrix is doing is that the first column describes where the [1,0] vector ends up, and the second column describes where the [0,1] vector goes. There are three important forms of the strain matrix: 1. Rigid (counterclockwise) rotation cos θ sin θ sin θ cos θ
What are the eigenvectors of this matrix? Why? What is the determinant? [Answer λ = cos θ + cos 2 θ 1 which has no solution except for θ = 0 (i.e. no rotation). If there is rigid rotation, all vectors are rotated and so there are no eigenvectors. The determinant is 1 (rotation involves no change in shape)]. 2. Simple shear 1 γ 0 1 This represents shear parallel to the x-axis. γ is referred to as the shear strain and γ = tan φ defines φ which is called the angular shear strain. What are the eigenvectors of this matrix? What about the determinant? [Answer λ=1. There is only one eigenvector, [x,0] i.e. the x-vector. Shear parallel to the x-axis will rotate all vectors except the x-vector. Determinant is 1 (simple shear involves no change in shape)]. 3. Pure shear a 0 0 d This implies expansion (a or d > 1) or contraction (a or d < 1). What use are these matrices? They tell us how any initially oriented line will be transformed under a particular strain field. Successive application of different strain fields is obtained by simply multiplying the relevant matrices together. Note that the order of multiplication is very important - matrix multiplication is not commutative!. The result of a series of deformations D 1,D 2, D N is found by doing the matrix multiplication in reverse order: (D N ) (D N 1 ) (D 2 ) (D 1 ). Example Examine the effects of a 5 counterclockwise rotation and 100% shear parallel to the x-axis on a line with the initial orientation (1, 1). Does the order of application matter? 2 0 [Answers Rotation followed by shear gives 1/ 2 1/. Shear followed by rotation gives 2 1/ 2 0 1/ 2. Operating these two matrices on (1, 1) yields ( 2, 2) and (1/ 2, 3/ 2), respectively. So the order does 2 matter.] 5
Example What does this strain matrix do? 5 3 3 5 [Answer It stretches by a factor of 2 in the (1,1) direction and by a factor of 1/2 in the (1,-1) direction.] 6