Chapter 5 The Orientation and Stress Tensors Introduction The topic of tensors typically produces significant anxiety for students of structural geology. That is due, at least in part, to the fact that the term is studiously avoided until it is sprung on the student when introducing the topic of stress, thus conflating two difficult to grasp concepts. In our case, the previous chapters have already introduced the concept of vectors as a type of first order tensor so the X3 X′2 X′3 X2 X′1 X1 fundamental definition — an entity that can be transformed from one coordinate system to another, changing its components in predictable ways such that its fun- damental nature doesn’t change — is already understood on an intuitive level. In the case of our vector, it has the same magnitude and orientation on the page re- gardless of the orientation of the coordinate axes (Fig. 4.3). We are now fully pre- pared to tackle second order tensors, which will allow us to quantify the relations between different families of vectors. These second order tensors will take us to the very core of structural geology. We will still be doing just multiplication and addi- tions (mostly), but first, we’ll need to introduce some concepts about matrices. CHAPTER 5 ORIENTATION & STRESS TENSORS Matrices and Indicial Notation In the last chapter, we saw that the transformation of a vector could be de- scribed with three simple equations (4.4), repeated here: v1′ = a11v1 + a12v2 + a13v3 " v2′ = a21v1 + a22v2 + a23v3 (5.1) v3′ = a31v1 + a32v2 + a33v3 As was pointed out at the time, the subscripts in these equations, all of which refer to specific coordinate axes, vary in an extremely systematic and precise way. That is because these equations are an alternative way of writing down a matrix multi- plication: ⎡ ⎤ ⎛ ⎞ ⎡ ⎤ v1′ a11 a12 a13 v1 ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ " v′ = av = ⎢ v2′ ⎥ = ⎜ a21 a22 a23 ⎟ ⎢ v2 ⎥ (5.2) ⎢ v′ ⎥ ⎜ a a a ⎟ ⎢ v ⎥ ⎣ 3 ⎦ ⎝ 31 32 33 ⎠ ⎣ 3 ⎦ Note that the subscripts of the matrix a vary in exactly the same way as the sub- scripts, or indices, of aij in (5.1). Equation (5.2) works because there are the same number of columns of a as there are rows of v. For this reason, you cannot write the left side of the equation as v′ = va because v has only one column but a has three rows. That is, matrix multiplication is non-commutative: the order of the multiplication matters. This will be an extremely important insight when we come to strain in a few more chapters; strain is mathematically represented as a matrix multiplication so the order in which strain is superimposed determines the final outcome. Matrix multiplication gives us another way to write the dot product of two vectors, where we use the transpose of one of the vectors (i.e., one of the vectors is “flipped”): ⎡ v ⎤ ⎢ 1 ⎥ " u i v = uvT = ⎡ u u u ⎤ v = u v + u v + u v (5.3) ⎣ 1 2 3 ⎦⎢ 2 ⎥ 1 1 2 2 3 3 ⎢ v ⎥ ⎣ 3 ⎦ MODERN STRUCTURAL PRACTICE "88 R. W. ALLMENDINGER © 2015-16 CHAPTER 5 ORIENTATION & STRESS TENSORS You can imagine that it gets pretty tedious to keep writing out the equations in (5.1) or the right side of (5.2) and the bold text on the left side of (5.2), known as matrix notation, is one type of shorthand. There is a second type of shorthand, known as the summation convention, that is more convenient, especially when it comes to implementing these equations in a programing language. Using the sum- mation convention, we would write Equations (5.1) and (5.2) as: " vi′ = aijvj (5.4) Let’s break this down and see how it works. As we have already seen i and j can vary from 1 to 3 in value because there are three axes to our Cartesian coordinate system. There is one i one each side of the equation and it is referred to as the free suffix; that means there will be three equations and, in each, i will have a constant value. The index j is know as the dummy suffix and appears twice on the right hand side, only, of Equation (5.4). Thus, the summation occurs with respect to j in each of the three equations as shown in the following equation: 3 " vi′ = ∑ aijvj = ai1v1 + ai2v2 + ai3v3 (5.5) j=1 The “recipe” for the summation convention takes a little practice but it turns out to be quite powerful and so is worth learning. We will encounter various properties of matrices in the subsequent chapters but for right now, we’ll wrap up with three important terms: and matrix is sym- metric if there are six independent values and the three off-diagonal components above the principal diagonal are the same as the three below the principal diagonal (Fig. 5.1). That is, T12 = T21, T13 = T31, and T23 = T . If a matrix is asymmetric then there are nine 32 T T T 11 12 13 off- independent components and T12 ≠ T21, T13 ≠ T31, diagonal T21 T22 T23 and T23 ≠ T32. Finally, a matrix can be antisym- metric with three independent components where T31 T32 T33 ( ) principal the values along the principal diagonal are all zeros off- diagonal diagonal and T12 = –T21, T13 = –T31, and T23 = –T32. Figure 5.1 — Anatomy of a 3 × 3 matrix. MODERN STRUCTURAL PRACTICE "89 R. W. ALLMENDINGER © 2015-16 CHAPTER 5 ORIENTATION & STRESS TENSORS Tensors In this Chapter, when we use the term “tensors”, we are specifically referring to second order or second rank tensors. All tensors can be expressed as matrices (a 3 × 3 matrix in the case of second order tensors) but not all matrices are ten- sors. In order to be a tensor, the matrix must represent an entity or quantity that transforms like a tensor; that is, the components must change in a logical and sys- tematic way during a coordinate transformation. Note that the transformation ma- trix, a, is itself not a tensor. A common convention is that tensors are written with brackets, [ ], whereas matrices that are not tensors are written with parentheses, ( ). We used this convention, for example, in Equation (5.2). Because all tensors can be written as matrices, they can be symmetric, asymmetric, etc. and any operation that can be performed on a matrix can likewise be performed on a tensor. Tensors as Linear Vector Operators The best way to think of a second order tensor is that it relates two families of vectors. In the case of stress, discussed later in this chapter, the stress tensor re- lates the stress vector (or traction) on a plane to the orientation of the plane. In the case of strain, as we shall see in a subsequent chapter, the displacement gradient tensor (one of many tensors related to strain!) relates the position vector of a point to the displacement of the point during a deformation. If we know the stress tensor, we can calculate the stress vector on a plane of any orientation within a body. You can imagine that this is hugely important in any study of earthquakes, induced seismicity, faulting, etc. Likewise, with the displacement gradient tensor, we can cal- culate how all points within a body are displaced as a function of position during a deformation. The basic way that this relationship is written is as follows: "ui = Tijvj (5.6a) Where u and v are generic vectors and T a generic tensor that relates the two vec- tors. This equation expands following the rules of summation or tensor notation: MODERN STRUCTURAL PRACTICE "90 R. W. ALLMENDINGER © 2015-16 CHAPTER 5 ORIENTATION & STRESS TENSORS ⎡ u ⎤ ⎡ T T T ⎤⎡ v ⎤ ⎢ 1 ⎥ ⎢ 11 12 13 ⎥⎢ 1 ⎥ " ⎢ u2 ⎥ = ⎢ T21 T22 T23 ⎥⎢ v2 ⎥ (5.6b) ⎢ u ⎥ ⎢ T T T ⎥⎢ v ⎥ ⎣ 3 ⎦ ⎣ 31 32 33 ⎦⎣ 3 ⎦ u1 = T11v1 + T12v2 + T13v3 " u2 = T21v1 + T22v2 + T23v3 (5.6c) u3 = T31v1 + T32v2 + T33v3 You already know how to do this expansion because it looks a lot like Equations (5.1-5.3). All of these equations represent the multiplication of a 3 × 3 and a 3 × 1 matrix. Beyond this mathematical similarity, there are entirely different. First, in Equation (5.1) it is the same vector, v, on both sides of the equation whereas u and v are two entirely different vectors in Equation (5.6). Secondly, T is a tensor as shown by the brackets in (5.6b), where as a in Equation (5.2) is not. There is a second way to make a second order tensor out of two vectors by taking the dyad (tensor) product of those two vectors. In indicial notation, we can write: T " T = u ⊗ v = u v or Tij = uivj (5.7a) The right hand side of (5.7) does not involve any summation because there is no dummy (i.e., repeated) suffix. So T simply works out to: ⎡ u v u v u v ⎤ ⎢ 1 1 1 2 1 3 ⎥ " Tij = ⎢ u2v1 u2v2 u2v3 ⎥ (5.7b) ⎢ u v u v u v ⎥ ⎣ 3 1 3 2 3 3 ⎦ The dyad product has important applications in earthquake and faulting studies and also in constructing the orientation tensor as we shall see below. Principal Axes of a Tensor Because our second order tensor is, well, a tensor, we can rotate the axes of the coordinate system and the values of the tensor will change. It turns out that, for MODERN STRUCTURAL PRACTICE "91 R. W. ALLMENDINGER © 2015-16 CHAPTER 5 ORIENTATION & STRESS TENSORS X3 X′2 X′3 Figure 5.2 — The magnitude ellip- soid of a tensor and two different coordinate axes through the ellip- soid.