T June 6, 2015 9:08 pm A F Tensors R a n n o n D c a B r R e b e c 1 n 2 Vm “type” notation and Voigt or Mandel matrices 3 4 This section defines a special “Vn ” tensor classification notation. Any tensor of type 5 m 6 n n n 7 Vm will be seen to have m components. Furthermore, any tensor of type Vm is also of 8 1 n 9 type VN where N= m . This latter identification helps us to convert indicial formulas 10 11 involving 3u 3 tensor matrices into matrix formulas involving 9u 1 arrays. Similarly, 12 13 indicial formulas involving 3 3 3 third-order tensor matrices may be alternatively cast 14 u u 15 in terms of 3 9 , 9 3 or even 27 1 arrays, depending on what is most convenient for 16 u u u 17 18 actual computations using linear algebra libraries in modern computing environments. 19 20 th st 21 Scalars are often called 0 -order tensors. Vectors are sometimes called 1 -order ten- 22 th n * 23 sors. In general, an n order engineering tensor has 3 components, and we say that 24 n 2 25 these tensors are of type V3 . For example, stress is a second-order tensor, so it has 3 26 27 (nine) components. Stress is symmetric, so these components are not all independent, but 28 29 that doesn’t change the fact that there are still nine components. 30 31 32 When solving a problem for which all tensors have symmetry with respect to some 2D 33 34 35 plane embedded in 3D space, it is conventional to set up the basis so that the third basis 36 37 vector points perpendicular to that plane. Doing this permits the 3D problem to be reduced 38 39 to a 2D problem where vectors of interest now have only 2 nonzero (in-plane) components 40 41 and second-order tensors are characterized by their (in-plane) 2 2 matrices, along with 42 u TENSOR ALGEBRA TENSOR 43 their lone 33 component. For example, biaxial stretching of a thin plate is nominally a 2D 44 45 46 problem in which you only need to track the in-plane components of stress and strain. The 47 48 out-of-plane 33 components can be analyzed separately, and all other components are 49 50 zero. In a case like this, the pertinent tensors still have 3u 3 component matrices and 51 52 hence nine components — we are merely setting up a naturally aligned basis that allows 53 54 55 some of those components to be consistently zero and hence ignorable. 56 57 When working in two dimensions, an nth -order engineering tensor has 2n compo- 58 59 60 nents. Similarly, when working in an m-dimensional linear manifold (which is the higher 61 th n 62 dimensional version of a plane), an n order engineering tensor has m components, and 63 n n 64 we say it is of type Vm . The tensor’s Vm type is highly pertinent to converting tensor 65 66 expressions into matrix forms for actual calculations, so we now provide a summary of 67 68 69 terminology and conventions for matrix versions of tensor equations. 70 71 72 * The base numeral 3 is used because of our emphasis on three-dimensional engineering contexts. ABCDEFGHIJKLMNOPQRSTUVWXYZ 334 Copyright is reserved. No part of this document may be reproduced without permission of the author. June 6, 2015 9:08 pm D Tensors R A R e b e c F c a B r T a n n o n 1 2 ALERT: The remainder of this section is rather abstract and can be skipped. It explains 3 4 various ways to convert higher-order tensor operations into matrix forms needed to per- 5 form actual computations. Different matrix conventions are motivated by applicable con- 6 straints (such as symmetry) that might apply to tensors in the operations. Skip now to 7 You have been warned! 8 page 462 if you are a newcomer to tensor analysis. 9 10 11 A second-order tensor of type V2 is also a first-order tensor of type V1 , where 12 m N 13 N= m 2 . For example, an ordinary second-order engineering tensor (type V2 ) is also a 14 3 15 1 16 first-order vector in a 9-dimensional space (type V9 ). Each has a total of 9 independent 17 18 components, regardless of whether they are collected in the form of a 3u 3 array or 9u 1 19 20 array. Each of these nine components is associated with a basis tensor. With respect to the 21 22 T e e lab basis, for example, the matrix version of = Tij might be written as 23 ˜ ˜i ˜j 24 25 T T T 26 11 12 13 1 0 0 0 1 0 0 0 1 27 T21 T22 T23 = T11 0 0 0 +T12 0 0 0 + T13 0 0 0 28 0 0 0 0 0 0 0 0 0 29 T31 T32 T33 30 31 0 0 0 0 0 0 0 0 0 32 + T +T + T 33 21 1 0 0 22 0 1 0 23 0 0 1 34 0 0 0 0 0 0 0 0 0 35 36 0 0 0 0 0 0 0 0 0 37 + T +T + T (12.106) 38 31 0 0 0 32 0 0 0 33 0 0 0 39 1 0 0 0 1 0 0 0 1 40 41 In this case, the implied summation is expanded explicitly so that the 9 tensor components 42 43 show up in the order ALGEBRA TENSOR 44 45 T T T T T T T T T (12.107) 46 >11 12 13 21 22 23 31 32 33 @ 47 48 These components are paired with corresponding basis tensors, 49 50 e e e e e e e e e e e e e e e e e e , (12.108) 51 ˜1 ˜1 ˜1 ˜2 ˜1 ˜3 ˜2 ˜1 ˜2 ˜2 ˜2 ˜3 ˜3 ˜1 ˜3 ˜2 ˜3 ˜3 52 53 which have lab component matrices 54 55 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 56 0 0 0 ,,,,,,,,.0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 (12.109) 57 58 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 59 60 As with any basis, the ordering can always be changed. We could, for example, identify 61 62 the 9 tensor components instead as 63 64 >T11 T22 T33 T23 T31 T12 T32 T13 T21 @ (12.110) 65 66 67 with associated basis tensors having component matrices given by 68 69 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 70 0 0 0 ,,,,,,,,.0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 (12.111) 71 72 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 ABCDEFGHIJKLMNOPQRSTUVWXYZ 335 Copyright is reserved. No part of this document may be reproduced without permission of the author. T June 6, 2015 9:08 pm A F Tensors R a n n o n D c a B r R e b e c 1 This simple reordering of the basis tensors is a rudimentary form of a basis change for ten- 2 3 sor space. Now let’s do a non-trivial basis change. * 4 5 6 Just as you can speak of planes embedded in ordinary 3D space, you can also limit 7 your attention to subspaces or linear manifolds contained within 9D tensor space. The set 8 9 of all symmetric second-order engineering tensors, for example, is closed under tensor 10 11 addition and scalar multiplication. By this we mean that any linear combination of sym- 12 13 metric tensors will be itself a symmetric tensor. Any symmetric engineering tensor (which 14 has, at most, six independent components) can be therefore regarded as a six -dimensional 15 16 vector belonging to a six-dimensional plane embedded in general nine-dimensional tensor 17 18 space. Since symmetric engineering tensors belong to a linear manifold (a 6D plane), we 19 1 20 say that they are of type V6 . Thus, symmetric second-order engineering tensors are simul- 21 taneously of type V2 , V1 , and V1 . It’s a matter of how you want to interpret them for 22 3 9 6 23 your problem, especially for matrix computations. A symmetric engineering tensor T still 24 ˜ 25 has 9 components, but 3 constraints (T12 = T21 , T23 = T32 , and T31 = T13 ) exist 26 among those components so that it has only 9–3=6 independent components. The goal of 27 28 this chapter is to motivate a tensor basis change that is aligned with this type of symmetry. 29 30 31 32 An ordinary basis change to motivate the abstract one. This section dis- 33 cusses a simple 3D basis change from the lab basis to a basis that is aligned with a ramp.