V “Type” Notation and Voigt Or Mandel Matrices

Home , Voigt notation

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.

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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. 34 35 This discussion serves as a familiar precursor to what we will then do analogously as a rel- 36 37 atively unfamiliar change of tensor basis. 38 39 Consider the set of ordinary 3D x -position vectors that are constrained to lie within a 40 ˜ 41 “tilted ramp” defined by the equation x = x . This set is of type V1 because each of the 42 2 3 3

TENSOR ALGEBRA TENSOR 1 43 vectors still has three components, ^x1x2 x3 ` , but it is also of type V2 because the com- 44 45 ponents are subjected to one constraint (namely x2 and x3 must be equal). Any linear 46 combination of 3D vectors lying in this 2D plane will itself lie in the plane. Accordingly, 47 48 the 2D tilted ramp is a linear manifold embedded within 3D space. 49 50 1 1 51 By saying that this set of vectors is both V3 and V2 , we are implying that it is possible 52 53 to set up a new orthonormal basis for which all vectors in the ramp have at most two non- 54 zero components, while vectors perpendicular to the ramp have only one nonzero compo- 55 56 nent. Let’s get such a basis. The conventional laboratory expansion of a vector is 57 58 59 x 60 1 1 0 0 61 x = x e +x e + x e , which has matrix form x = x 0 +x 1 + x 0 . (12.112) ˜ 1 ˜1 2 ˜2 3 ˜3 2 1 2 3 62 0 0 1 63 x3 64 65 66 Knowing that we are interested in breaking up general vectors in to parts lying in the tilted 67 ramp or perpendicular to this ramp, the above expansion could be written instead as 68 69 70 * As the discussion proceeds, keep in mind that we will be changing the basis used for tensors with- 71 out changing any underlying ordinary 3D “laboratory” basis vectors ^e e e ` . 72 ˜1 ˜2 ˜3 ABCDEFGHIJKLMNOPQRSTUVWXYZ

336 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 x + x x – x x e §2 3 · e e §3 2 · e e 2 = x1 1 +------2 + 3 + ------3 – 2 , 3 ˜ ˜ ©2 ¹ ˜ ˜ ©2 ¹ ˜ ˜ 4 x 5 1 1 x + x 0 x – x 0 6 §2 3 · §3 2 · which has matrix form x = x3 0 +------1 + ------1– . (12.113) 7 2 ©2 ¹ ©2 ¹ 0 1 1 8 x3 9 10 By defining 11 12 13 x2 + x3 x3 – x2 14 x 2 := §------· , x 3 := §------· , 15 ©2 ¹ > @ ©2 ¹ 16 b e b e e b e e 17 1 = 1 , 2 = 2 + 3 , and3 = 3 – 2 , (12.114) 18 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ 19 the new expansion may be written a bit more compactly as 20 21 x = x b +x b + x b (12.115) 22 1 1 2 2 >3 @ 3 23 ˜ ˜ ˜ ˜ 24 If the vector happens to fall on the tilted ramp where x = x , the last term would drop 25 2 3 out to give x = x b + x b , making the vector effectively 2D even though it still has 3 26 1 1 >2 @ 2 27 ˜ ˜ ˜ 28 components in the lab basis. 29 30 The expansion in Eq. (12.115) decomposes an arbitrary vector into a part contained 31 32 within the ramp (where x2=x3 ) and a part perpendicular to it. The associated basis, 33 b b b ^12 3 ` is orthogonal — but not orthonormal . It is appealing because it is aligned 34 ˜ ˜ ˜ 35 with the ramp. For convenience of calculations, a rational engineer would normalize the 36 37 basis to obtain yet another expansion: 38 x e e e 39 = x1 ˆ +x2 ˆ + x ˆ , (12.116) 40 ˜ ˜1 ˜2 3 ˜3 41 42 where 43 ALGEBRA TENSOR 44 x := 2 x = ------2 x + x , x := 2 x = ------2 x – x , 45 2 2 2 2 3 3 >3 @ 2 3 2 46 b b e e b e e 47 1 2 2 + 3 3 3 – 2 48 eˆ := ------˜ = e ,eˆ := ------˜ = ------˜ ˜ , andeˆ := ------˜ = ------˜ ˜ . (12.117) 1 b 1 2 b 3 b 49 ˜ 1 ˜ ˜ 2 2 ˜ 3 2 50 ˜ ˜ ˜ 51 The act of dividing the last two basis vectors, b and b , by their own 2 magnitudes has 52 ˜ 2 ˜ 3 53 required multiplying the corresponding components by 2 so that the linear combination 54 55 of components times basis vectors is unchanged. This new expansion has the matrix form 56 57 58 59 x 60 1 1 0 0 61 x = x 0 +2x + 2x 1– . (12.118) 62 2 1 2 1 >3 @ 0 1 63 x3 1 64 ------65 2 2 66 67 This form merely uses a new orthonormal basis that is aligned with the tilted ramp. Now 68 69 let’s go through a similar process for tensor space, where the analog of the 2D tilted ramp 70 will be the 6D linear manifold (hyperplane) of symmetric tensors and the analog of the 71 72 line perpendicular to the tilted ramp will be the 3D linear manifold of skew tensors. ABCDEFGHIJKLMNOPQRSTUVWXYZ

337 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 Voigt and Mandel representations of second-order tensors. When consid- 2 3 ering symmetric tensors embedded within 9D tensor space, your calculations will often 4 simplify if you switch away from the conventional laboratory e e basis used in 9D tensor 5 ˜i ˜j 6 space to a different basis that is “aligned” with symmetric tensors. For example, instead of 7 8 e e e e e e using 1 2 as one of your basis tensors, you might use 1 2 + 2 1 , which is symmetric. 9 ˜ ˜ ˜ ˜ ˜ ˜ 10 All of the other basis tensors would need to be redefined as well in order to switch to a 11 12 basis that is aligned with symmetric tensors. 13 14 To see how this change-of-basis might be developed, note that the component expan- 15 16 sion for the general (not necessarily symmetric) tensor in Eq. (12.106) can be written 17 18 equivalently as 19 20 21 F11 F12 F13 1 0 0 0 0 0 0 0 0 22 =F + F + F 23 F21 F22 F23 11 0 0 0 22 0 1 0 33 0 0 0 24 F F F 0 0 0 0 0 0 0 0 1 25 31 32 33 26 0 0 0 0 0 1 0 1 0 27 F23 + F32 F31 + F13 F12 + F21 28 + §------· + §------· + §------· ©2 ¹ 0 0 1 ©2 ¹ 0 0 0 ©2 ¹ 1 0 0 29 0 1 0 1 0 0 0 0 0 30 31 0 0 0 0 0 1 0 1–0 32 F32 – F23 F13 – F31 F21 – F12 33 + §------· + §------· + §------· (12.119) ©2 ¹ 0 0 1– ©2 ¹ 0 0 0 ©2 ¹ 1 0 0 34 0 1 0 1–0 0 0 0 0 35 36 37 This equation is analogous to Eq. (12.113). Analogous to Eq. (12.114), we now define 38 39 40 F = ---1 F + F andF = ---1 F – F , (12.120) 41 ij 2 ij ji >ij @ 2 ij ji 42 TENSOR ALGEBRA TENSOR 43 Then Eq. (12.119) may be written more compactly as 44 45 46 47 F11 F12 F13 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 48 F F F =F 0 0 0 + F 0 1 0 + F 0 0 0 + F 0 0 1 + F 0 0 0 + F 1 0 0 49 21 22 23 11 22 33 23 31 12 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 50 F31 F32 F33 51 52 0 0 0 0 0 1 0 1–0 53 F F F (12.121) 54 + >32 @ 0 0 1– + >13 @ 0 0 0 + >21 @ 1 0 0 55 0 1 0 1–0 0 0 0 0 56 57 58 The nine matrices on the right side of this equation can be regarded as an alternative basis 59 60 for 9D tensor space. The first six of these basis tensors are symmetric, and the last three 61 62 are skew. This basis is still capable of describing arbitrary non-symmetric tensors. If >F @ 63 happens to be symmetric, then F = F , F = F , and F = F , and the above 64 23 32 31 13 12 21 65 expansion reduces to only the first six terms since the three F ij would all be zero. Thus, 66 > @ 67 if you are dealing exclusively with symmetric tensors, then you only need the first six basis 68 tensors. The components with respect to these particular basis tensors are called Voigt 69 70 components . A disadvantage of the Voigt system is that only the first three basis tensors 71 72 implied by Eq. (12.121) are unit tensors. The remaining ones have magnitude of 2 .

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338 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 The non-orthonormality of the Voigt basis prevents you from treating the Voigt array 2 3 as if it were an ordinary vector (and thus precludes efficient use of vector functions built 4 5 into modern computing environments). For example, the magnitude of a tensor is not 6 found by the magnitude of its associated Voigt array. As was done in the analogous ramp- 7 8 aligned-vector basis example, this inconvenience is easily remedied by simply normaliz- 9 10 ing the Voigt basis: each of the non-unit basis tensors is divided by its own magnitude, 11 2 . As in the ramp analogy, dividing the basis tensors by 2 requires multiplying the 12 13 coefficient by 2 to avoid changing the result of the summation. Specifically, defining 14 15 2 F = ------2 F + F andF = ------F – F , (12.122) 16 ij 2 ij ji ij 2 ij ji 17 18 19 Eq. (12.121) may be written alternatively as 20 21 ^ ^ ^ ^ ^ ^ 22 F11 F12 F13 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 23 F F F =F + F + F + F + F + F 24 21 22 23 11 0 0 0 22 0 1 0 33 0 0 0 23 0 0 1 31 0 0 0 12 1 0 0 25 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 F31 F32 F33 26 ^ ^ ^ 27 0 0 0 0 0 1 0 1–0 28 +F + F + F Alert: the “hats” normalize! (12.123) 29 32 0 0 1– 13 0 0 0 21 1 0 0 30 0 1 0 1–0 0 0 0 0 31 32 33 The very important “hat” on the matrices denotes normalization ( i.e. , division of the 34 35 matrix by its own magnitude). This normalization of the Voigt basis produces the Mandel 36 basis . The corresponding Mandel components are the same as Voigt components 37 38 except that the components corresponding to off-diagonal positions within the original 39 40 3u 3 matrix are multiplied by 2 in the Mandel 9u 1 array. For example, the fourth 41 Voigt component is F , whereas the fourth Mandel component is F23 = 2F 23 . 42 23 43 ALGEBRA TENSOR 44 The change of basis to either the Voigt or Mandel system clearly shows that the set of 45 all symmetric tensors needs only six basis tensors, making symmetric tensors of type V1 . 46 6 1 47 Skew tensors are of type V3 , and we have defined the three skew-tensor components to 48 49 coincide with “axial vector” components defined later in Eq. (13.29). 50 51 In engineering mechanics, Voigt arrays for symmetric tensors most commonly arise in 52 53 stress-strain constitutive models, where phrases like “strain-like” or “stress-like” are used 54 for what should be properly termed as covariant and contravariant , respectively. For sym- 55 56 metric tensors, these are defined as follows: 57 58 59 Covariant “strain-like” Voigt array for a symmetric tensor H : 60 ˜ 61 >HVHV HV HV HV HV @= > H H H 2H 2H 2H @ (12.124a) 62 1 2 3 4 5 6 11 22 33 23 31 12 63 These are paired with contravariant basis tensors having lab components given by 64 65 G1 G2 G3 G4 G5 G6 = 66 >V @ >V @ >V @ >V @ >V @ >V @ 67 ˜ ˜ ˜ ˜ ˜ ˜ 68 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 69 0 0 0 , 0 1 0 , 0 0 0 , 0 0 1 , 0 0 0 , 1 0 0 . (12.124b) 70 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 71 ------72 2 2 2 ABCDEFGHIJKLMNOPQRSTUVWXYZ

339 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 2 9 I 3 The expansion, = e e , may be then written as = VG . (12.124c) H Hij i j H HI V 4 ˜ ˜ ˜ ˜ I¦ ˜ 5 1= ˜ 6 7 Contravariant “stress-like” Voigt array for a symmetric tensor V : 8 ˜ 9 1 2 3 4 5 6 10 >VVVV VV VV VV VV @= > V 11 V22 V33 V23 V31 V12 @ (12.125a) 11 12 These are paired with covariant basis tensors having lab components given by 13 GV GV GV GV GV GV = 14 >1 @ >2 @ >3 @ >4 @ >5 @ >6 @ 15 ˜ ˜ ˜ ˜ ˜ ˜ 16 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 17 0 0 0 , 0 1 0 , 0 0 0 , 0 0 1 , 0 0 0 , 1 0 0 (12.125b) 18 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 19 20 21 9 e e IGV 22 The expansion, V = Vij i j , may be then written as V = VV I . (12.125c) 23 ˜ ˜ ˜ ˜ I¦ ˜ 24 1= 25 26 Consistent with standard notation for non-orthonormal basis systems, covariant indices 27 28 are subscripts (for which the mnemonic “co-go-below” is helpful). The contravariant indi- 29 ces are superscripts. Such conventions are needed only when the basis is not orthonormal. 30 31 For orthonormal bases, like the Mandel system, all indices are subscripts. Thus, 32 33 T 34 Mandel array for a symmetric tensor : 35 ˜ 36 >T1T2 T3 T4 T5 T6 @ = >T11 T22 T33 2T23 2T31 2T12 @ (12.126a) 37 38 These are paired with orthonormal basis tensors having lab components given by 39 40 >e @ >e @ >e @ >e @ >e @ >e @ = 41 ˜1 ˜2 ˜3 ˜4 ˜5 ˜6 42

TENSOR ALGEBRA TENSOR 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 43 , , , , , (12.126b) 44 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 . 45 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 46 ------47 2 2 2 48 9 49 T e e T e 50 The expansion, = Tij i j , may be then written as = TI I . (12.126c) 51 ˜ ˜ ˜ ˜ I¦1= ˜ 52 53 54 Advantages of the Mandel system become apparent when you start doing calculations. For 55 A 56 example, the magnitude of a tensor, , is defined in terms of conventional components by 57 ˜ 58 A A:A = = Aij Aij 59 ˜ ˜ ˜ 60 2 2 2 2 2 2 2 2 2 61 = A11 ++++++++A22 A33 A23 A31 A12 A32 A13 A21 (12.127) 62 63 If the tensor is symmetric, the off-diagonal terms combine to give 64 65 66 A = A2 +++A2 A2 2 A2 +2 A2 + 2 A2 . (12.128) 67 ˜ 11 22 33 23 31 12 68 ˜ 69 When computed using contravariant (stress-like) Voigt components, this operation is 70 71 A = A1 2 +++ A2 2 A3 2 2 A4 2 +2 A5 2 + 2 A6 2 (12.129) 72 ˜ V V V V V V ABCDEFGHIJKLMNOPQRSTUVWXYZ

340 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 When computed using covariant (strain-like) components, the same operation is 2 3 1 1 1 4 A V 2 V 2 V 2 --- V 2 --- V 2 --- V 2 (12.130) 5 = A1 + A2 + A3 +2 A4 +2 A5 + 2 A6 . 6 ˜ 7 When computed using Mandel components, on the other hand, the magnitude of the tensor 8 9 identically equals the magnitude of its Mandel vector array: 10 11 12 A 2 2 2 2 2 2 = A1 +A2 +A3 +A4 +A5 + A6 (12.131) 13 ˜ 14 15 Such intoxicating simplicity is lost when using a Voigt system, which requires jolting fac- 16 17 tors or divisors of 2. 18 19 Some moderately recent discourses on Mandel components and their associated basis 20 21 can be found in Refs. [48, 94, 57, 67], but publications that use the 2 in component 22 23 arrays usually don’t give a name to the approach (and often don’t even explain that the 24 components are the result of a tensor basis change). We have chosen the name “Mandel” 25 26 after one of the earliest researchers that we found to extensively use the factor of 2 ( cf . 27 28 ___). 29 30 31 Voigt and Mandel representations of fourth-order tensors. Now that we 32 2 33 have established that a second-order (type V3 ) tensor can be regarded as a nine-dimen- 34 sional (type V1 ) vector with a corresponding 9 1 component array, it follows that 35 9 u 4 2 36 fourth-order (type V3 ) tensors can be viewed as (type V9 ) second-order tensors with 37 38 respect to a 9D space. Consequently, they can be manipulated in computations by using a 39 9 9 matrix, with the indices ranging from 1 to 9 corresponding to index pairs selected 40 u 41 above for 9D Voigt or Mandel representations of second-order tensors: 11, 22, 33, 23, 31, 42 43 12, 32, 13, 21. If you limit attention to fourth-order tensors that are minor symmetric ALGEBRA TENSOR 44 (which means A =A = A ), then you would be well-advised to abandon the 45 ijkl jikl ijlk 46 standard lab dyad basis used in Eq. (12.38) in favor of the alternative basis tensors listed in 47 48 Eq. (12.119). When using that basis, the last three columns and last three rows of a 9u 9 49 matrix for a minor-symmetric fourth-order tensor will contain all zeros . In other words, 50 51 you will be dealing only with the upper 6u 6 part of the matrix. Consequently, minor- 52 2 2 53 symmetric fourth-order tensors are of type V6 and they have at most 6 , or 36, nonzero 54 components. The 6 6 Mandel component matrix is the same as the 6 6 Voigt matrix 55 u u 56 except that the last three rows and columns are multiplied by 2 (making the lower right 57 58 3u 3 submatrix ultimately multiplied by 2). The Mandel matrix will be symmetric if the 59 original fourth-order tensor is major symmetric. 60 61 62 Being orthonormalized, the Mandel basis tends to produce more intuitive component 63 64 matrices than the Voigt system. For example, the fourth-order identity tensor for symmet- 65 ric matrices (called Psym later in this book) maps to the 6u 6 identity matrix under the 66 ˜ 67 Mandel system (but not˜ with the Voigt system). Also, the concept of an eigenvector, found 68 69 using standard numerical eigensolvers, applies to Mandel representations but not Voigt. A 70 standard eigenvector of a Voigt matrix is physically meaningless. The Mandel system is 71 72 relatively intuitive because it uses an orthonormal basis for tensors. As explained in ABCDEFGHIJKLMNOPQRSTUVWXYZ

341 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 Ref. [14], Voigt component formulas typically contain surprising factors or divisors of 2 2 3 or 4 because the underlying Voigt basis is not orthonormal. According to parlance of non- 4 5 orthonormal basis (cf. [14]), Voigt component arrays have covariant , contravariant , 6 and (for higher-order tensors) mixed forms. 7 8 9 Voigt and Mandel notation for 2 nd -order tensors. An ability to change how 10 11 you regard the “Vn type” of a second-order tensor is useful in materials mechanics. For 12 m 13 14 example, in plasticity, the trial elastic stress rate is found by assuming that a material is 15 16 behaving elastically under strain-controlled loading. If it is found that this assumption 17 18 would move the stress into a “forbidden” region that violates the yield condition, then 19 20 plastic flow must be occurring. The set of admissible elastic stresses is defined by a yield 21 22 1 23 function such that f V 0 . When V is regarded as a vector of type V6 , then f V = 0 ˜ ˜ ˜ 24 x x 2 25 defines a yield surface in 6D space. For example, just as the equation x –R 0= 26 ˜ ˜ 27 defines a sphere of radius R in ordinary 3D space, the equation V:V –R2 0= would 28 ˜ ˜ 29 define a hypersphere in 6D stress space. When the trial assumption of elastic behavior is 30 31 found to move the stress into inadmissible stress states (i.e., those for which f 0 ), 32 V˜ ! 33 ˜ 34 then the equations governing classical nonhardening plasticity can be used to show that 35 36 the actual stress rate is obtained by projecting the trial elastic stress rate onto the yield sur- 37 38 face [16]. The projection operation in plasticity is similar in structure to the projection 39 40 shown in Fig. 11.4 except that the vector dot product is replaced by the tensor inner prod- 41 42

TENSOR ALGEBRA TENSOR : B 43 uct ( ). The outward “normal” that defines the target plane is the gradient of the yield 44 ˜ 45 function (i.e., Bij = wf e wV ij ). Statements like this implicitly regard tensors as vectors 46 1 47 (higher dimension vectors of type V6 ). This determination of the yield surface normal 48 49 direction is perfectly analogous to finding the normal to an ordinary 3D surface by taking 50 51 the spatial gradient of its defining level-set function f x . 52 53 ˜ 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72

ABCDEFGHIJKLMNOPQRSTUVWXYZ

342 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 Examples of Voigt and Mandel notation for 3 rd -order tensors. 2 st nd 3 This section revisits two previously analyzed transformations relating 1 -order and 2 - 4 rd rd 5 order tensors, each characterized by a 3 -order tensor. Conventionally, a 3 order tensor 6 has a 3 3 3 component array. We can consider this instead as a 3 9 matrix when the 7 u u u 8 transformation input is a second-order tensor (which has 9 components). We can consider 9 10 the 3u 3u 3 component array as a 9u 3 matrix when the input is a vector. As illustrated 11 in these examples, the matrix may be reduced to 3 6 or 6 3 if the tensor is symmetric. 12 u u 13 14 15 EXAMPLE 1 : Transformation of a second-order tensor to a vector V2 V1 . 16 3 o 3 17 Consider a relationship of the form v = A x c . This way of writing the formula 18 ˜ ˜ ˜ 19 emphasizes that the vector v depends linearly on˜ the vector c , but it also depends linearly 20 A ˜ ˜ 21 on the tensor . Recalling the operator Yijk defined in Eq. (12.97), we may also write 22 ˜ 23 A x c = Y:A , where Y = G c (12.132) 24 ˜ ˜ ˜ ˜ ijk ij k 25 ˜ ˜ ˜ 26 The third-order tensor Y is introduced to emphasize that the vector v = A x c depends 27 ˜ ˜ ˜ ˜ 28 linearly on the tensor A˜ , allowing the operation to be written as v = Y:A . This is the 29 ˜ ˜ ˜ ˜ 30 standard symbolic (direct) notation for a vector-valued linear transformation of a tensor. 31 We desire now to express this formula in standard matrix form (where the motivation 32 33 might be, for example, to exploit computational linear algebra libraries that are available 34 35 in modern computing environments). We also aim to find a matrix formulation that is 36 A computationally convenient for the special case that is symmetric, which allows Yijk to 37 ˜ 38 be replaced without loss in generality by the formula in Eq. (12.98), 39 40 Y = ---1 c + c . (12.133) 41 i jk 2 Gij k Gik j 42 43 Wrapping the last two indices ( jk ) in parentheses indicates imposed symmetry with respect ALGEBRA TENSOR 44 to this index pair. For now, we will presume that A is potentially nonsymmetric. Then: 45 ˜ 46 v A c 47 Explicit form of = x : v1 = A11 c1 +A12 c2 + A13 c3 (12.134a) 48 ˜ ˜ ˜ 49 v2 = A21 c1 +A22 c2 + A23 c3 (12.134b) 50 51 v3 = A31 c1 +A32 c2 + A33 c3 (12.134c) 52 53 If this is interpreted as a linear operator for which A is the input, 54 ˜ 55 ˜ 56 A11 57 58 A22 59 A 60 33 61 v1 c1 00 00 c2 0 c3 0 A23 62 v A c 63 Matrix form of = x : v2 = 0 c2 0 c3 00 00 c1 A31 . (12.135) ˜ ˜ ˜ 64 v 0 0 c 0 c 0 c 0 0 A 65 3 3 1 2 12 66 A32 67 68 A13 69 A 70 21 71 72 This is a special case of a more generic statement: ABCDEFGHIJKLMNOPQRSTUVWXYZ

343 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 2 A11 3 A 4 22 5 A33 6 7 v1 Y111 Y122 Y133 Y123 Y131 Y112 Y132 Y113 Y121 A23 8 Matrix form of v = Y:A : v = Y Y Y Y Y Y Y Y Y A (12.136) 9 ˜ ˜ ˜ 2 211 222 233 223 231 212 232 213 221 31 10 ˜ ˜ v3 Y311 Y322 Y333 Y323 Y331 Y312 Y332 Y313 Y321 A12 11 12 A32 13 A 14 13 15 A21 16 17 1 1 18 The 3u 9 matrix in Eq. (12.135) is the V3 u V9 component representation of the third- 19 order tensor Y for which v = A x c is re-written in the form v = Y:A . Comparing the 20 ˜ ˜ ˜ ˜ ˜ ˜ ˜ 21 corresponding matrix equations shows, for example, that Y113 = c3 , which is, of course, 22 23 the same as what is implied from the indicial formula Yijk = Gij ck derived in Eq. (12.97). 24 25 By expansion, you can confirm that Eq. (12.136) may be written alternatively as 26 27 A 28 11 29 A22 30 31 A33 32 v Y Y Y 2Y 2Y 2Y 2Y 2Y 2Y A 33 1 111 122 133 1 23 1 31 1 12 1> 32 @ 1> 13 @ 1> 21 @ 23 34 v2 = Y211 Y222 Y233 2Y2 23 2Y2 31 2Y2 12 2Y2 32 2Y2 13 2Y2 21 A 31 (12.137) 35 > @ > @ > @ v Y Y Y 2Y 2Y 2Y 2Y 2Y 2Y A 36 3 311 322 333 3 23 3 31 3 12 3> 32 @ 3> 13 @ 3> 21 @ 12 37 A 38 >32 @ 39 A 13 40 > @ A 41 >21 @ 42 TENSOR ALGEBRA TENSOR 43 where 44 45 1 1 Y = --- Y + Y and A = --- A + A (12.138a) 46 i jk 2 ijk ikj ij 2 ij ji 47 48 ---1 ---1 Yi jk = 2 Yijk – Yikj and A ij = 2 Aij – Aji (12.138b) 49 > @ > @ 50 Applying Eq. (12.137) to the special case of Eq. (12.135) gives: 51 52 53 A11 54 A 55 22 56 A33 57 v c 0 0 0 c c 0 c –c A 58 1 1 3 2 3 2 23 59 v A c New matrix form of = x : v2 = 0 c2 0 c3 0 c1 0 0 c1 A 31 . (12.139) 60 ˜ ˜ ˜ 61 v3 0 0 c3 c2 c1 0 c2 –c1 0 A 12 62 A 63 >32 @ 64 A 65 >13 @ 66 A 21 67 > @ 68 If the tensor A is symmetric, the last three components of its 9u 1 array in this for- 69 ˜ 70 mula are zero (i.e.,˜ A = 0 ), which allows these explicit formulas to be written alterna- 71 >ij @ 72 tively in reduced matrix form as

ABCDEFGHIJKLMNOPQRSTUVWXYZ

344 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 contravariant Voigt array 2 A11 A11 3 A A 4 v c 0 0 0 c c 22 Y Y Y 2Y 2Y 2Y 22 5 1 1 2 3 A 111 122 133 1 23 1 31 1 12 A v =0 c 0 c 0 c 33 = Y Y Y 2Y 2Y 2Y 33 , (12.140a) 6 2 2 3 1 211 222 233 2 23 2 31 2 12 7 A A v 0 0 c c c 0 23 Y Y Y 2Y 2Y 2Y 23 8 3 3 2 1 A 311 322 333 3 23 3 31 3 12 A 9 31 31 10 A 12 A 12 11 12 13 covariant Voigt matrix 14 15 We refer to each 3u 6 matrix in this formula as the covariant Voigt representation of 16 Y A 17 the third-order tensor , applicable to the special case that is symmetric. An alternative 18 representation moves the˜ factors of 2 to a different location in˜ such a manner that the result 19 20 of the matrix multiplication is unchanged: 21 22 23 covariant Voigt array A11 A11 24 c2 c3 25 c1 0 0 0 ------A A v 2 2 22 Y Y Y Y Y Y 22 26 1 111 122 133 1 23 1 31 1 12 27 c c A33 A33 v =0 c 0 -----3 0 -----1 = Y Y Y Y Y Y , (12.140b) 28 2 2 2 2 2A 211 222 233 2 23 2 31 2 12 2A 29 v 23 Y Y Y Y Y Y 23 3 311 322 333 3 23 3 31 3 12 30 c2 c1 2A 2A 31 31 31 0 0 c3 ------0 2 2 2A 2A 32 12 12 33 34 35 contravariant Voigt matrix 36 37 We refer to each 3u 6 matrix in this formula as the contravariant Voigt representa- 38 tion of the third-order tensor Y , again applicable to the special case that A is symmetric. 39 ˜ ˜ 40 ˜ ˜ 41 With Voigt notation, each matrix multiplication must be between one covariant and 42 43 one contravariant representation (i.e., the covariant factor must include the metric multi- ALGEBRA TENSOR 44 plier of 2 and the contravariant factor must omit it). The metric arises because the Voigt 45 46 basis is not normalized. The Mandel representation for second (and higher) order tensors 47 48 eliminates the confusing metrics by normalizing the basis for second-order tensors, so 49 there is no need to distinguish between contravariant and covariant components. In partic- 50 51 ular, the Mandel revision of Eq. (12.137) is 52 53 54 A11 55 Mandel array 56 A22 57 A 58 33 59 Y Y Y Y Y Y Y Y Y A23 60 v1 111 122 133 123 131 112 132 113 121 61 v = Y Y Y Y Y Y Y Y Y A31 (12.141) 62 2 211 222 233 223 231 212 232 213 221 63 A v3 Y Y Y Y Y Y Y Y Y 12 64 311 322 333 323 331 312 332 313 321 A 65 32 66 A 67 13 68 A 69 Mandel matrix 21 70 71 72 where ABCDEFGHIJKLMNOPQRSTUVWXYZ

345 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 ------2 ------2 2 Yi jk =2Yi jk = Yijk + Yikj and A ij =2A ij = Aij + Aji (12.142a) 3 2 2 4 2 2 5 Y =2Y = ------Y – Y and A =2A = ------A – A (12.142b) 6 ijk i> jk @ 2 ijk ikj ij >ij @ 2 ij ji 7 8 By avoiding the confusion of two different component types (contravariant and covariant), 9 10 the Mandel system is much easier to work with. To create a Mandel matrix, simply multi- 11 ply the Voigt contravariant components by 2 . In the case of symmetry, the size of the 12 13 matrices may be reduced as previously discussed for Voigt notation. In this example, 14 15 therefore, 16 17 Mandel matrix: 18 c2 c3 19 c1 0 0 0 ------20 Y Y Y Y Y Y 2 2 21 111 122 133 123 131 112 22 c c Y Y Y Y Y Y = 3 1 (12.143) 23 211 222 233 223 231 212 0 c2 0 ------0 ------24 2 2 25 Y311 Y322 Y333 Y323 Y331 Y312 26 c2 c1 27 0 0 c3 ------0 28 2 2 29 30 31 To obtain Voigt contravariant components, just divide the Mandel components by 2 . 32 33 The reason for forming a matrix representation in the first place is typically to facilitate 34 efficient computations — the Mandel system serves this purpose far better than Voigt 35 36 components. Not only does the Mandel system treat off-diagonal components in the same 37 38 way (no worries about whether or not to use a factor of 2), it also allows meaningful use of 39 standard linear algebra packages such as singular-value decomposition. 40 41 Equations (12.140a) and (12.140b) are different matrix-notation formulations for the 42 TENSOR ALGEBRA TENSOR 43 single direct-notation formula, 44 45 46 v =A x c = Y:A (12.144a) 47 ˜ ˜ ˜ ˜ ˜ 48 49 The conventional indicial form for this equation is 50 51 v = Y A , (12.144b) 52 i ijk jk 53 54 where each index ranges from 1 to 3 and repeated indices are implicitly summed. 55 56 Recalling the Voigt components and basis tensors defined in Eqs. (12.124b) and 57 58 (12.125b), the indicial-notation versions of Eqs. (12.140a) and (12.140b) are respectively 59 J 60 Voigt indicial form for a contravariant input: v = Y V A (12.144c) 61 i iJ V 62 J V 63 Voigt indicial form for a covariant input: vi = YiV AJ (12.144d) 64 65 The index i ranges from 1 to 3, while the repeated index J is implicitly summed from 1 to 66 67 6. The label “V” is just indicating that these are Voigt components (it isn’t a summed 68 69 index). Recalling the normalized basis and associated component definitions used in the 70 Mandel system (where there is no need to distinguish between contravariant and covariant 71 72 components), the Mandel indicial formula corresponding to Eq. (12.141) is simply

ABCDEFGHIJKLMNOPQRSTUVWXYZ

346 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 Mandel indicial form: v = Y A (12.144b) 2 i iJ J 3 4 As with the Voigt system, the free index i varies from 1 to 4, while the repeated index J 5 is summed from 1 to 9. Unlike the Voigt system, the Mandel system uses an orthonormal 6 7 basis and hence doesn’t need distinction between covariant and contravariant components. 8 9 10 11 EXAMPLE 2 : Transformation of a vector to a tensor 1 2 . 12 V3 o V3 13 14 Regarding c as a known constant vector, it was proved on page 331 that an operation 15 B ---1 cu ˜ uc u 16 = 2 + may be written as a linear transformation of . Namely, 17 ˜ ˜ ˜ ˜ ˜ ˜ 18 B Z u ---1 19 = x , which means Bij = Zijk uk with Zijk = 2 Gkj ci + Gki cj (12.145) 20 ˜ ˜ ˜ 21 B Z u u 22 The input to the operation = x is a vector , while the output is the symmetric ten- 1 23 B --- cu uc ˜ ˜ ˜ ˜ sor = 2 + . This symmetry implies that Zijk = Z ij k . Accordingly, we should 24 ˜ ˜ ˜ ˜ ˜ 25 represent Z as a 6u 3 matrix. The Voigt representations of this matrix are 26 ˜ 27 ˜ 28 Z111 Z112 Z311 c1 0 0 29 30 Z221 Z222 Z322 0 c2 0 31 Z Z Z 0 0 c 32 Covariant Voigt: 331 332 333 = 3 (12.146) 33 2Z 2Z 2Z 0 c c 23 1 23 2 23 3 3 2 34 2Z 2Z 2Z c 0 c 35 31 1 31 2 31 3 2 1 36 2Z 2Z 2Z c c 0 37 12 1 12 2 12 3 3 1 38 39 c 0 0 40 1 41 Z Z Z 0 c2 0 42 111 112 311 0 0 c ALGEBRA TENSOR 43 Z221 Z222 Z322 3 44 Z Z Z c3 c2 45 Contravariant Voigt: 331 332 333 = 0 ------(12.147) 46 2 2 Z 23 1 Z 23 2 Z 23 3 47 c c 48 Z Z Z 2 1 31 1 31 2 31 3 ----- 0 ----- 49 2 2 Z Z Z 50 12 1 12 2 12 3 c c 51 -----3 -----1 0 52 2 2 53 54 The Mandel representation is constructed by multiplying the contravariant Voigt matrix by 55 56 a factor 2 at the locations corresponding to off-diagonal ` ij ` indices: 57 58 59 c1 0 0 60 Z Z Z 0 c 0 61 111 112 311 2 62 0 0 c Z221 Z222 Z322 3 63 c c 64 Z331 Z332 Z333 3 2 Mandel: = 0 ------(12.148) 65 Z Z Z 2 2 66 231 232 233 67 c2 c1 Z311 Z312 Z313 ------0 ------68 2 2 69 Z121 Z122 Z123 70 c3 c1 71 ------0 72 2 2 ABCDEFGHIJKLMNOPQRSTUVWXYZ

347 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 6u 1 representation of Z is the transpose of the 1u 6 component matrix in 2 ˜ 3 Eq. (12.143) for the tensor Y of˜ the previous example! This observation exploited in 4 ˜ 5 applications of small-deformation mechanics (see page 1043). 6 B Z u 7 In the covariant Voigt representation, matrix and indicial forms of = x are 8 ˜ ˜ ˜ 9 covariant Voigt array 10 conventional lab array 11 12 B Z Z Z c 0 0 13 11 111 112 311 1 14 B22 Z221 Z222 Z322 0 c2 0 15 u1 u1 16 B Z Z Z 0 0 c 33 =331 332 333 u = 3 u (12.149a) 17 2B 2Z 2Z 2Z 2 0 c c 2 18 23 23 1 23 2 23 3 u 3 2 u 19 3 3 2B 31 2Z 31 1 2Z 31 2 2Z 31 3 c2 0 c1 20 2B 2Z 2Z 2Z c c 0 21 12 12 1 12 2 12 3 3 1 22 23 24 25 covariant Voigt matrix 26 V V j (12.149b) 27 and BI = ZI j u , respectively. 28 I 29 Here, is a Voigt free index ranging from 1 to 6, while j is a regular laboratory index 30 implicitly summed from 1 to 3. The laboratory basis is orthonormal, so the notation uj 31 32 means the same thing as uj , but superscripting is used here to conform to standard syntax 33 34 that applies when any part of an indicial formula (in this case the Voigt part) does not cor- 35 respond to an orthonormal basis. 36 37 B Z u 38 In the contravariant Voigt representation, matrix and indicial forms of = x are 39 ˜ ˜ ˜ contravariant Voigt array 40 conventional lab array 41 42 c1 0 0 TENSOR ALGEBRA TENSOR 43 B Z Z Z 0 c 0 44 11 111 112 311 2 45 0 0 c B22 Z221 Z222 Z322 3 46 u1 u1 B Z Z Z c3 c2 47 33 =331 332 333 = 0 ------(12.150a) 48 u2 2 2 u2 B 23 Z 23 1 Z 23 2 Z 23 3 49 u c c u 50 3 2 1 3 B 31 Z 31 1 Z 31 2 Z 31 3 ----- 0 ----- 51 2 2 B Z Z Z 52 12 12 1 12 2 12 3 c c 53 -----3 -----1 0 54 2 2 55 56 57 contravariant Voigt matrix 58 I I j 59 and BV = ZV uj , respectively. (12.150b) 60 61 62 63 64 65 66 67 68 69 70 71 72

ABCDEFGHIJKLMNOPQRSTUVWXYZ

348 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 In the Mandel representation, these forms of the operation B = Z x u are 2 ˜ ˜ ˜ 3 ˜ 4 conventional lab array 5 Mandel array 6 7 c 0 0 8 1 9 B Z Z Z 0 c2 0 10 11 111 112 311 0 0 c 11 B22 Z221 Z222 Z322 3 12 u1 c c u1 13 B33 Z331 Z332 Z333 3 2 = = 0 ------(12.151a) 14 u2 2 2 u2 B23 Z231 Z232 Z233 15 u u 16 3 c2 c1 3 B31 Z311 Z312 Z313 ------0 ------17 2 2 18 B12 Z121 Z122 Z123 19 c c ------3 ------1 0 20 2 2 21 22 23 24 Mandel matrix 25 26 and BI = ZI j uj , respectively. (12.151b) 27 28 29 All of these matrix formulations are equivalent. Hence, it is up to your personal prefer- 30 ences which one to use. The Voigt system is more common in the literature, but (as previ- 31 32 ously mentioned) it is becoming increasingly seen as an unwise choice if you wish to 33 34 apply conventional linear-algebra matrix operations like singular-value decomposition, as 35 these make no sense for contravariant or covariant components of any tensor (Voigt or oth- 36 37 erwise). The Mandel system provides components with respect to an orthonormal tensor 38 39 basis and thus has no confusing distinction between contravariant and covariant compo- 40 nents. The Mandel system allows meaningful application of standard linear algebra 41 42 decompositions that would be otherwise inappropriate when applied to Voigt matrices. 43 ALGEBRA TENSOR 44 45 46 47 EXAMPLE 3 : Transformation of a tensor to a tensor 2 2 . 48 V3 o V3 49 50 Consider a tensor-to-tensor function such that 51 52 53 input = ordinary engineering second-order tensor X (12.152a) 54 ˜ 55 ˜ 56 output = ordinary engineering second-order tensor Y (12.152b) 57 ˜ 58 ˜ 59 According to the representation theorem, there exists a local tangent fourth-order tensor 60 61 M that is independent of dX (but might depend on X ) such that dY = M:dX . If the 62 ˜ ˜ X ˜ ˜ ˜ ˜ 63 fourth-order˜ tensor happens to be independent of , then the operation is linear˜ so that 64 ˜ 65 66 Y = M:X Y = M X (12.153) 67 ˜ ˜ ˜ ij ijkl kl 68 ˜ ˜ ˜ 69 70 An assumption of linearity isn’t really needed for the ensuing discussion, but it is adopted 71 72 to keep the equations simpler ( i.e., the same concepts apply to tangent tensors). ABCDEFGHIJKLMNOPQRSTUVWXYZ

349 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 A very simple 9D matrix form of the above equation is 2 3 4 Y M M M M M M M M M X 5 11 1111 1122 1133 1123 1131 1112 1132 1113 1121 11 6 Y22 M2211 M2222 M2233 M2223 M2231 M2212 M2232 M2213 M2221 X22 7 8 Y33 M3311 M3322 M3333 M3323 M3331 M3312 M3332 M3313 M3321 X33 9 Y M M M M M M M M M X 10 23 2311 2322 2333 2323 2331 2312 2332 2313 2321 23 11 Y31 = M3111 M3122 M3133 M3123 M3131 M3112 M3132 M3113 M3121 X31 (12.154) 12 13 Y12 M1211 M1222 M1233 M1223 M1231 M1212 M1232 M1213 M1221 X12 14 Y M M M M M M M M M X 15 32 3211 3222 3233 3223 3231 3212 3232 3213 3221 32 16 Y13 M1311 M1322 M1333 M1323 M1331 M1312 M1332 M1313 M1321 X13 17 18 Y21 M2111 M2122 M2133 M2123 M2131 M2112 M2132 M2113 M2121 X21 19 20 21 Though this seems compact, it admits no significant simplification in the special (and very 22 common) situations for which input and/or output is symmetric. When the input is always 23 24 symmetric, the fourth-order tensor may be presumed without loss to have right minor 25 26 symmetry (meaning that Mijkl = Mijlk ). When the output is symmetric, the fourth-order 27 tensor has left minor symmetry (meaning M = M ). As in the preceding exam- 28 ijkl jikl 29 ples, the following 9D alternative representation naturally exploits symmetries: 30 31 32 Contravariant Voigt array Covariant Voigt array 33 (has no factors of 2) (has factors of2) 34 35 36 37 Contravariant Voigt matrix 38 (has no factors of 2) Y M M M M M M M M M X 39 11 11 11 11 22 11 33 11 23 11 31 11 12 11 >32 @ 11 >13 @ 11 >21 @ 11 40 Y M M M M M M M M M X 41 22 22 11 22 22 22 33 22 23 22 31 22 12 22 >32 @ 22 >13 @ 22 >21 @ 22 Y M M M M M M M M M X 42 33 33 11 33 22 33 33 33 23 33 31 33 12 33 >32 @ 33 >13 @ 33 >21 @ 33 TENSOR ALGEBRA TENSOR 43 Y M M M M M M M M M 2X 44 23 23 11 23 22 23 33 23 23 23 31 23 12 23 >32 @ 23 >13 @ 23 >21 @ 23 Y = M M M M M M M M M 2X (12.155) 45 31 31 11 31 22 31 33 31 23 31 31 31 12 31 >32 @ 31 >13 @ 31 >21 @ 31 46 Y M M M M M M M M M 2X 47 12 12 11 12 22 12 33 12 23 12 31 12 12 12 >32 @ 12 >13 @ 12 >21 @ 12 Y M M M M M M M M M 2X 48 >32 @ >32 @ 11 >32 @ 22 >32 @ 33 >32 @ 23 >32 @ 31 >32 @ 12 >32 @ >32 @ >32 @ >13 @ >32 @ >21 @ >32 @ 49 Y M M M M M M M M M 2X 50 >13 @ >13 @ 11 >13 @ 22 >13 @ 33 >13 @ 23 >13 @ 31 >13 @ 12 >13 @ >32 @ >13 @ >13 @ >13 @ >21 @ >13 @ Y M M M M M M M M M 2X 51 >21 @ >21 @ 11 >21 @ 22 >21 @ 33 >21 @ 23 >21 @ 31 >21 @ 12 >21 @ >32 @ >21 @ >13 @ >21 @ >21 @ >21 @ 52 53 54 where 55 56 1 1 X = --- X + X , Y = --- Y + Y (12.156a) 57 ij 2 ij ji ij 2 ij ji 58 59 1 1 X = --- X – X , Y = --- Y – Y (12.156b) 60 >ij @ 2 ij ji >ij @ 2 ij ji 61 62 1 M = --- M +M +M + M (12.156c) 63 ij kl 4 ijkl jikl ijlk jilk 64 1 65 M = --- M + M – M – M (12.156d) 66 ij >kl @ 4 ijkl jikl ijlk jilk 67 1 68 M = --- M – M + M – M (12.156e) 69 >ij @ kl 4 ijkl jikl ijlk jilk 70 1 71 M = --- M – M – M + M (12.156f) 72 >ij @ >kl @ 4 ijkl jikl ijlk jilk

ABCDEFGHIJKLMNOPQRSTUVWXYZ

350 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 This Voigt representation has a major disadvantage: the tensor basis that is paired with the 2 3 Voigt components is not orthonormal, making the tensor arrays different on the two sides 4 5 of Eq. (12.155) — the one on the right-hand-side of the equation has confusing factors of 6 2 (which are actually metrics) that are not present on the left side. Similar to what was 7 8 done in the previous examples, the Mandel representation normalizes the Voigt basis to 9 10 give the following more consistent structure: 11 12 Mandel form! 13 14 M M M M M M M M M 15 1111 1122 1133 1123 1131 1112 1132 1113 1121 Y11 X11 16 M M M M M M M M M 2211 2222 2233 2223 2231 2212 2232 2213 2221 17 Y22 X22 18 M3311 M3322 M3333 M3323 M3331 M3312 M M M 19 Y33 3332 3313 3321 X33 20 Y M2311 M2322 M2333 M2323 M2331 M2312 M M M X 21 23 2332 2313 2321 23 22 Y31 = M3111 M3122 M3133 M3123 M3131 M3112 M M M X31 23 3132 3113 3121 (12.157) 24 Y12 M M M M M M M M M X12 25 1211 1222 1233 1223 1231 1212 1232 1213 1221 26 Y M M M M M M M M M X 27 32 3211 3222 3233 3223 3231 3212 3232 3213 3221 32 28 Y X 13 M M M M M M M M M 13 29 1311 1322 1333 1323 1331 1312 1332 1313 1321 Y X 30 21 M M M M M M M M M 21 31 2111 2122 2133 2123 2131 2112 2132 2113 2121 32 33 The 9 9 matrix in this formula is the same as the contravariant matrix in Eq. (12.155) 34 u 35 except that the last six rows and last six columns (i.e., the ones that are associated with off- 36 37 diagonals) are multiplied by 2 . Only the shaded parts are required in cases that the input 38 and output are always symmetric tensors. Unlike Eq. (12.155), which needed factors of 2 39 40 in one of the 9u 1 vector arrays (but not in the other one), the arrays for both X and Y 41 ˜ ˜ 42 are constructed in an identical way: the last six components are the same as the contravari- 43 ant Voigt components except multiplied by 2 . Specifically, ALGEBRA TENSOR 44 45 X = 2X Y = 2Y (12.158a) 46 ij ij , ij ij 47 48 X = 2X , Y = 2Y (12.158b) ij >ij @ ij >ij @ 49 50 M = 2M M = 2M M = 2 M (12.158c) ijkl ij kl , ij kl ij kl ij kl ij kl 51 52 M = 2Mij kl , M = 2 M ij kl (12.158d) 53 ijkl > @ ijkl > @ 54 M = 2M , M = 2 M (12.158e) 55 ij kl >ij @ kl ijkl >ij @ kl 56 M = 2 M (12.158f) 57 ij kl >ij @ >kl @ 58 59 The Mandel representation may seem strange at first with all of its factors of 2 or 2 , but 60 61 it is actually much easier to work with than a conventional Voigt representation for the fol- 62 lowing reasons: 63 64 65 • The non-orthonormal covariant basis that is paired with the 9u 1 Voigt 66 67 contra variant component array is the set of tensors with traditional 3u 3 component 68 matrices given by 69 70 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1–0 71 0 0 0 , 0 1 0 , 0 0 0 , 0 0 1 , 0 0 0 , 1 0 0 , 0 0 1– , 0 0 0 , 1 0 0 See Exercise 12.3. 72 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1–0 0 0 0 0 ABCDEFGHIJKLMNOPQRSTUVWXYZ

351 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 The non-orthonormal contravariant basis that is paired with the 9 1 Voigt co variant 2 u 3 array is the set of tensors with traditional 3 3 component matrices given by 4 u 5 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 0 1 0 0 1 1 0 1–0 ------6 0 0 0 , 0 1 0 , 0 0 0 , 2 0 0 1 , 2 0 0 0 , 2 1 0 0 , 2 0 0 1– , 2 0 0 0 , 2 1 0 0 See Exercise 12.4. 7 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1–0 0 0 0 0 8 9 The simple orthonormal basis that is paired with the 9u 1 Mandel component array 10 11 is the set of tensors with traditional 3u 3 component matrices given by 12 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1–0 13 1 1 1 1 1 1 0 0 0 , 0 1 0 , 0 0 0 , ------0 0 1 , ------0 0 0 , ------1 0 0 , ------0 0 1– , ------0 0 0 , ------1 0 0 2 2 2 2 2 2 14 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1–0 0 0 0 0 15 16 Equation 12.154 also has an orthonormal basis. Namely, it is given by 17 18 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 19 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 20 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 21 This basis, however lacks the advantage shared by both Voigt and Mandel bases of 22 23 splitting both the input and output into symmetric and skew parts. For both the 24 25 Mandel and Voigt system, the 9u 9 transformation matrix will have its last three 26 X 27 columns zero if the transformation depends only on sym ; the first six columns will 28 ˜ 29 be zero if the transformation depends only on skw X . The last three rows of either the 30 ˜ 31 Voigt or Mandel matrix will be zero if the output is symmetric; the first six rows will 32 33 be zero if the output is skew. Thus, in these special and very common cases, the 34 effective size of the transformation matrix is significantly reduced. 35 36 • Unlike the Voigt system, which has a factor of 2 in the 9 1 covariant (strain-like) 37 u 38 array but no factor of 2 in the contravariant (stress-like) array, the 9 1 39 u 40 representations are the same for any tensor when using the Mandel system because 41 the Mandel system uses an orthonormal basis . The factors of 2 in the Voigt system 42 TENSOR ALGEBRA TENSOR 43 are actually metrics (resulting from the inner product of the covariant basis tensors 44 45 with themselves). If the Voigt representation is adjusted so that both of the 9u 1 46 arrays are of the same type (both covariant or both contravariant), the corresponding 47 48 9 9 transformation matrix would be of “mixed” form. 49 u 50 • The Mandel system has the appealing property that major symmetry of the fourth- 51 52 order tensor (Mijkl = Mklij ) implies symmetry of the Mandel matrix MIJ = MJI , 53 54 whereas the mixed Voigt transformation matrices would not be symmetric. 55 56 • The identity operation, Y = X , has a transformation matrix that is simply the 9u 9 57 ˜ ˜ 58 identity matrix when using the Mandel system. Such is not the case for the Voigt 59 60 system (see Exercise 12.7). 61 62 • The ordinary eigenvalues and eigenvectors of the Mandel 9u 9 representation are 63 64 meaningful. Such is not the case for the 9u 9 Voigt representation, which requires a 65 generalized eigenproblem involving the basis metrics. See Ref. [14]. 66 67 • The Mandel matrix for the inverse of a fourth-order tensor is found by inverting the 68 69 Mandel matrix for the fourth-order tensor. With the Voigt system, on the other hand, 70 inverting the contra variant Voigt matrix will give you the covariant matrix for the 71 72 inverse (converting this to the contravariant form requires multiplying the last three

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352 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 rows and last three columns by a factor of 2, which is again a needlessly cumbersome 2 3 consequence of the Voigt basis not being orthonormal. 4 5 6 7 Exercise 12.3 Confirm that the ordinary fully populated 3u 3 component matrix, 8 9 Y Y Y 10 11 12 13 11 Y21 Y22 Y23 , is indeed obtained by multiplying the nine contravariant components, 12 13 Y31 Y32 Y33 14 Y Y Y Y Y Y Y Y Y times the corresponding nine covariant basis tensors 15 ^ 11 22 33 23 31 12 >32 @ >13 @ >21 @ ` 16 having component matrices respectively given by 17 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1–0 18 0 0 0 , 0 1 0 , 0 0 0 , 0 0 1 , 0 0 0 , 1 0 0 , 0 0 1– , 0 0 0 , 1 0 0 19 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1–0 0 0 0 0 20 21 Hint: Multiply the first component times the first basis matrix plus the next component times the next basis 22 23 matrix, etc. 24 25 26 27 28 29 Exercise 12.4 Confirm that the ordinary fully populated 3u 3 ordinary component matrix, 30 31 Y11 Y12 Y13 32 33 Y21 Y22 Y23 , is indeed obtained by multiplying the nine covariant Voigt components, 34 Y Y Y 35 31 32 33 36 ^Y11 Y22 Y33 2Y 23 2Y 31 2Y 12 2Y 32 2Y 13 2Y 21 ` times the corresponding nine contravariant 37 > @ > @ > @ 38 Voigt basis tensors having component matrices respectively given by 39 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1–0 ---1 ---1 ---1 ---1 ---1 ---1 40 0 0 0 , 0 1 0 , 0 0 0 , 2 0 0 1 , 2 0 0 0 , 2 1 0 0 , 2 0 0 1– , 2 0 0 0 , 2 1 0 0 41 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1–0 0 0 0 0 42 43 Hint: Same procedure as the previous problem. ALGEBRA TENSOR 44 45 46 47 48 49 Exercise 12.5 The operation v = A x c is said to be bilinear because it is linear with respect to both c ˜ ˜ ˜ ˜ 50 and A . As such, there exists a fourth-order tensor M (independent of both c and A ) such that 51 ˜ ˜ ˜ ˜ 52 v = A M c , which may be written in direct notation as v = A:M x c . Find the Mandel component i mn mniq q ˜ ˜ ˜ ˜ 53 matrix for M in the case that A is a general, potentially nonsymmetric˜ tensor. Find the simplest Mandel 54 ˜ ˜ component ˜matrix for M in the case that A is known to be always a skew tensor. 55 ˜ ˜ 56 ˜ ˜ 57 Hint: In the first case, explain why the answer is the general fourth-order identity (see previous exercises for 58 the corresponding Mandel matrix). In the second case, you must impose, without loss in generality , a certain 59 60 symmetry. Reasoning is similar to the symmetry imposed in Eq. (12.133). 61 62 63 64 65 66 67 68 69 70 71 72 ABCDEFGHIJKLMNOPQRSTUVWXYZ

353 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 Computer algorithm for evaluating 9x9 Mandel matrices. To compute the 2 3 9u 9 Mandel matrix of a fourth-order tensor M , 4 ˜ 5 6 STEP 1.Find the conventional 3u 3 u 3 u 3 representation, denoted Mijkl in indicial 7 8 notation. 9 10 STEP 2.Define a 9-member array whose elements are not numerals but instead are the 11 12 3u 3 component matrices for the Mandel basis. 13 14 15 STEP 3.Loop over I and J each ranging from 1 to 9: 16 th 17 (i) Define >U @ to be the 3u 3 component matrix of the I 18 Ith 19 Mandel basis tensor ( i.e., the element from step 2) 20 th 21 (ii) Define >V @ to be the 3u 3 component matrix of the J 22 Mandel basis tensor ( i.e., the Jth element from step 2). 23 24 (iii) Set M equal to U M V , where the indices i , j , k , 25 IJ ij ijkl kl 26 and l are each implicitly summed from 1 to 3. 27 28 The last two steps of this algorithm are implemented in Mathematica as 29 30 31 32 33 34 35 36 37 38 39 40 41 42 TENSOR ALGEBRA TENSOR 43 44 45 46 47 48 49 50 51 52 EXAMPLE: suppose we wish to find the Mandel array for the fourth-order tensor defined 53 54 by M = abC where the right-hand side of this equation is formed from dyadic multipli- 55 ˜ ˜ ˜ ˜ 56 cation˜ of vectors and a tensor having laboratory components given by 57 58 59 2 8 4 5–0 60 61 ^a ` = 0 ,^b ` = 3 ,and >C @ = 0 1 7 . (12.159) 62 ˜ ˜ ˜ 63 5– 2 2 3 0 64 65 66 The conventional indicial formula for the desired fourth-order tensor is 67 68 69 Mijkl = aibjCkl (12.160) 70 71 72 To get the `16` component of the Mandel matrix, first define

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354 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 1 0 0 2 st 3 >U @ = 0 0 0 = 1 Mandel basis matrix, and (12.161a) 4 0 0 0 5 6 0 1 0 7 1 th 8 >V @ = ------1 0 0 = 6 Mandel basis matrix. (12.161b) 2 9 0 0 0 10 11 Then only two of the 81 terms in the implied summation U M V survive: 12 ij ijkl kl 13 14 M16 =Uij Mijkl Vkl =U11 M1112 V12 + U11 M1121V21 = a1b1C12 e 2 + a1b1C21 e 2 15 16 = 2 8 5– e 2 + 2 8 0 e 2 –= 40 2 (12.162) 17 18 This tedious hand calculation verifies the circled `16` component in the following applica- 19 tion of the previously defined automated Mathematica function: 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 Exercise 12.6 Using the components defined in Eq. (12.159), find the 9 9 Mandel matrix for u ALGEBRA TENSOR 43 M = abA in which A = sym C . Provide at least one “by hand” verification calculation like what was 44 ˜ ˜ ˜ ˜ ˜ ˜ 45 done˜ in Eq. (12.162). Then also find the Mandel Matrices for T = aAb and Y = Aba . 46 ˜ ˜ ˜ ˜ ˜ ˜ ˜ ˜ 47 48 Hint: This is like the example. The symmetry Aij = Aji implies a new symmetry Mijkl = Mijlk , which results 49 50 in the Mandel matrix having its last three columns all zeros! What symmetries are evident for the other parts 51 of this question? 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 ABCDEFGHIJKLMNOPQRSTUVWXYZ

355 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 2 Exercise 12.7 Show at least two hand calculations (then use the computer) for the following tasks: 3 4 (a) Find the 9u 9 Mandel and contravariant Voigt matrices for the identity operation, Y = X . ˜ ˜ 5 ˜ ˜ (b) Find the 9u 9 Mandel and contravariant Voigt matrices for the symmetry operation, Y = sym X . 6 ˜ ˜ 7 ˜ ˜ (c) Find the 9u 9 Mandel and contravariant Voigt matrices for the skew operation, Y = skw X . 8 ˜ ˜ 9 (d) Let A and B be known symmetric tensors. For the following operations, find the 9u 9 Mandel matrix 10 ˜ ˜ 11 for the linear operator M such that Y = M:X : 12 ˜ ˜ ˜ ˜ 13 ˜ ˜ ˜ ˜ 14 i. Y = A B:X 15 ˜ ˜ ˜ ˜ 16 ii. Y = A x X x B 17 ˜ ˜ ˜ ˜ A B 18 For each, express the results in terms of the Mandel components of and , written as A1A2 } A9 ˜ ˜ 19 and B B B , as defined in Eq. (12.126a). 20 12 } 9 21 (e) What revisions of the operation in part ( d)ii would give a similar operation that produces both a major 22 and minor symmetric fourth-order tensor? Resist peeking at the hint! 23 24 25 Hint: (a) Mandel’s will be the simple identity matrix. (b) and (c) will be a “piece” of the identity matrix (if 26 using the Mandel system) — this result nicely shows these to be projection operators! For part (d)i., note 27 B B:X B: X that symmetry of implies that = sym = B1X1 +} + B6X6 (i.e., there is no involvement of X7 , 28 ˜ ˜ ˜ ˜ ˜ 29 X , or X , and hence the last three columns of the 9 9 matrix will be zero; ultimately, your answer will 30 8 9 u 31 look like a higher-dimensional form of the component matrix for an ordinary vector-vector dyad ab . 32 ˜ ˜ 33 34 35 36 For part (d)ii., you need to first prove that Mijkl = Aik Bjl . Then work out the matrix painstakingly for at 37 38 least two components to show the required hand calculations. For example, the Mandel M14 component is 39 40 2 2 M =2M =------M + M = ------A B + A B . This result now needs to be written in 41 1123 11 23 2 1123 1132 2 12 13 13 12 42 TENSOR ALGEBRA TENSOR A B 43 terms of the single-index Mandel components for and . By virtue of symmetry, A12 =A12 = A6 e 2 and 44 ˜ ˜ 45 B =B = B 2 . The other components are similar. The final answer for the M component is there- 46 13 13 5 e 14 47 1 48 fore ------A B + A B . Once hand calculations like these have been given for two components, use the com- 2 2 6 5 5 6 49 50 puter algorithm to get the rest. (e) The answer is to replace both the input and the output by their symmetric 51 parts to get Y = sym A x sym X x B . Your job is to confirm that this does indeed produce a 9x9 matrix that is 52 ˜ ˜ ˜ ˜ 53 nonzero only in the upper-left 6x6 submatrix! 54 55 56 57 58 This concludes our overview of how to convert higher-order tensor expressions and 59 60 operations to matrix forms. Such transformations are needed for actual computations 61 62 (especially when built-in linear algebra computer packages are to be used). Throughout 63 the remainder of this book, where the discussions are concept-focused, and direct nota- 64 65 tions and traditional indicial formulas (where indices range from 1 to 3) are typically suffi- 66 67 cient, but even a passing understanding of what was covered in this section can 68 69 significantly enhance your understanding of the upcoming survey of operations. Also, 70 some of these operations are summarized in the Mandel system, which was why we had to 71 72 introduce it here despite its abstract nature.

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356 Copyright is reserved. No part of this document may be reproduced without permission of the author.