Euler-Rodrigues and Cayley formulas for rotation of elasticity tensors

Transcription

1 arxiv:cond-mat/06765v [cond-mat.mtrl-sci] 30 Nov 006 Euler-Rodrigues and Cayley formulas for rotation of elasticity tensors A. N. Norris Mechanical and Aerospace Engineering, Rutgers University, Piscataway NJ , USA Dedicated to Michael Hayes on the occasion of his 65th birthday Abstract It is fairly well known that rotation in three dimensions can be expressed as a quadratic in a skew symmetric matrix via the Euler-Rodrigues formula. A generalized Euler-Rodrigues polynomial of degree n in a skew symmetric generating matrix is derived for the rotation matrix of tensors of order n. The Euler-Rodrigues formula for rigid body rotation is recovered by n =. A Cayley form of the n th order rotation tensor is also derived. The representations simplify if there exists some underlying symmetry, as is the case for elasticity tensors such as strain and the fourth order tensor of elastic moduli. A new formula is presented for the transformation of elastic moduli under rotation: as a -vector with a rotation matrix given by a polynomial of degree 8. The tensor rotation formulae are related to Cartan decomposition of elastic moduli and projection onto hexagonal symmetry. Introduction Rigid body rotation about an axis p, p =, is described by the well known Euler-Rodrigues formula for the rotation matrix as a quadratic in the skew symmetric matrix P, where where ǫ ijk is the third order isotropic alternating tensor. Thus, P ij = ǫ ijk p k, ( Q = exp(θp = I + sin θ P + ( cosθp, ( where θ is the angle of rotation. Euler first derived this formula although Rodrigues [] obtained the formula for the composition of successive finite rotations [, 7]. The concise form of the Euler-Rodrigues formula is basically a consequence of the property P 3 = P. (3 Hence, every term in the power series expansion of exp(θp is either P k = ( k+ P or P k+ = ( k P, and regrouping yields (. Rotation about an axis is defined by the proper orthogonal rigid body rotation matrix Q(θ,p SO(3, the special orthogonal group of matrices satisfying QQ t = Q t Q = I, (4 The summation convention on repeated subscripts is assumed in ( and most elsewhere. Q t denotes transpose and I is the identity.

2 and the special property det Q =. The generating matrix P(p so(3 where so(n denotes the space of skew symmetric matrices or tensors, usually associated with the Lie algebra of the infinitesimal (or Lie transformation group defined by rotations in SO(n. Alternatively, SO(3 is isomorphic to so(3 via the Cayley form [4] where S so(3 follows from ( as Q = (I + S (I S = (I S (I + S, (5 S = tanh( θ P = tanθ P. (6 The first identity is a simple restatement of the definition in (5 with Q = exp(θp, while the second is a more subtle relation which depends upon (3 and the skew symmetry of P, and as such is analogous to the Euler-Rodrigues formula. We are concerned with generalizing these fundamental formulas for rigid body rotation to the rotation matrices associated with Q but which act on second and higher order tensors in the same way that Q transforms a vector as v v = Q t v. If one takes the view that the rotation transforms a set of orthonormal basis vectors e,e,e 3, then the coordinates of a fixed vector v = v e + v e + v 3 e 3 relative to this basis transform according to (v, v, v 3 (v, v, v 3, where v i = Q ijv j. A tensor T of order (or rank n has elements, or components relative to a vector basis, T ij...kl. Under rotation of the basis vectors by Q, the elements are transformed by the relations T ij...kl = Q ij...klpq...rs T pq...rs, (7 where Q = Q(θ,p is a tensor of order n formed from n combinations of the underlying rotation, Q ij...kl pq...rs Q ip Q jq...q kr Q ls. (8 }{{}}{{}}{{} n n n Q is also known as the n-th Kronecker power of Q [8]. Lu and Papadopoulos [4] derived a generalized n-th order Euler-Rodrigues formula for Q as a polynomial of degree n in the tensor P, also of order n, defined by the sum of n terms Thus, where P ij...klpq...rs = P ip δ jq...δ kr δ ls + δ ip P jq...δ kr δ ls δ ip δ jq...p kr δ ls + δ ip δ jq... δ kr P ls. (9 P n (θ, x = k= n e λ kθ Q = exp(θ P = P n (θ, P, (0 n ( x λ j, with λk = k i, i =. ( λ k λ j j= n j k Lu and Papadopoulos derivation of ( is summarized in Section 3. The case n = has also been derived in quite different ways by Podio-Guidugli and Virga [0] and by Mehrabadi et al. [6]. The purpose of this paper is twofold. First, to obtain alternative forms of the generalized Euler-Rodrigues formula and of the associated Cayley form. Using general properties of skew symmetric matrices we will show that the n-th order Euler-Rodrigues formula can be expressed in ways which clearly generalize the classical n = case. The difference in approach from that of Lu and Papadopoulos [4] can be appreciated by noting that their formula ( follows from the fact that the eigenvalues of P are λ k, k = n,...,n, of eq. (. The present formulation is based on the more fundamental property that P has canonical form P = P + P n P n, (

3 where P j satisfy identities similar to (3, and are mutually orthogonal in the sense of tensor product. This allows us to express exp(θ P and tanh( θ P in forms that more clearly generalize ( and (6. Furthermore, the base tensors P j can be expressed in terms of powers of P, which imply explicit equations for Q and the related S. We will derive these equations in Section 3. The second objective is to derive related results for rotation of tensors in elasticity. The underlying symmetry of the physical tensors, strain, elastic moduli, etc., allow further simplification from the n tensors Q SO(3 n and P so(3 n, where the nominal dimensionality 3 n assumes no symmetry. Thus, Mehrabadi et al. [6] expressed the rotation tensor associated with strain, n =, in term of second order tensors in 6-dimensions, significantly reducing the number of components required. By analogy, the rotation matrix for fourth order elastic stiffness tensors is derived here as a second order tensor in -dimensions Q(θ,p = exp ( θ P SO( where P so(. This provides an alternative to existing methods for calculating elastic moduli under a change of basis, e.g. using Bond transformation matrices [3] or other methods based on representations of the moduli as elements of 6 6 symmetric matrices [6]. Viewing the elastic moduli as a -vector is the simplest approach for some purposes, such as projection onto particular symmetries [, 6]. In fact, we will see that the generalized Euler-Rodrigues formula leads to a natural method for projecting onto hexagonal symmetry defined by the axis p, as an explicit expression for the projector appears quite naturally. We begin with a summary of the main findings in Section, with general results applicable to tensors of arbitrary order described in Section 3. Further details of the proofs are given in Section 4. Applications to elasticity are described in Section 5 where the dimensional rotation of elastic moduli is derived. Finally, the relation between the generalized Euler-Rodrigues formula and the Cartan decomposition of elasticity tensors is discussed in Section 6. A note on notation: tensors of order n are denoted P, Q, etc. Vectors and matrices in 3D are normally lower and capital boldface, e.g. p, Q, while quantities in 6-dimensions are denoted e.g. Q, and in -dimensions as Q. 3 Summary of results Our principal result is this: Theorem The rotation tensor Q(θ,p = exp ( θ P has the form Q = I + ( sin kθ Pk + ( cos kθ Pk where P k, k =,,..., n are orthogonal tensors that partition P and have the same basic property as P in (3, The Cayley form for Q is P = P + P n P n, (3 (4a P i P j = P j P i = 0, i j, (4b P 3 i = P i. (4c Q = ( I + S ( I S = ( I S ( I + S, (5 where S = tan(k θ P k. (6

4 The tensors P k are uniquely defined by polynomials of P of degree n, P k = p n,k ( P, where p n,k (x = x k n (x + j j k j k, k n. (7 The polynomial representation Q = P n (θ, P has several alternative forms, three of which are as follows: where P n (θ, x = + = + ( k+ (n k! (n + k! ( sin θ k k(k! = + sin θ x ( + X + ( cosθ x ( + X [ ] n k sin kθ x + ( cos kθ x (x + j (8a j k k ( k cos θ x + sin θ x (x + l (8b l= ( X ( X ( +... X n n ( X + 3 ( + X 3 4 ( +... X n n, 4 (8c X j = ( cosθ(x + j /(j +. (9 We note that n =,, 3, 4 corresponds to rotation of vectors, second, third and fourth order tensors, respectively. All these are covered by considering n = 4 in Theorem, for which the series (8c and (8a are, respectively, ( [ ( cosθ P 8 (θ, x = + sin θ x + (x ( cosθ + + (x + 4 ( ( cosθ + (x + 9 ].3.5 [ ( cosθ (x [ ] sin θ x + ( cos θx (x + 4(x + 9(x ( cosθ x ( + = + 5!3 3.7 ( cosθ (x + 4 ( ![ sinθ x + ( cos θx ] (x + (x + 9(x ![ 3 sin 3θ x + ( cos 3θx ] (x + (x + 4(x + 6 ( cosθ (x + 9 ] 4.7 8![ 4 sin4θ x + ( cos 4θx ] (x + (x + 4(x + 9. (0 This expression includes all the Euler-Rodrigues type formulas for tensors of order n 4, on account of the characteristic polynomial equation for P of degree n +, P( P + ( P ( P + n = 0. ( Thus, the last line in (0 vanishes for n = 3, the last two for n =, and all but the first line for n =, while the terms that remain can be simplified using (. The formulas for P n, n = and 3, are P 4 (θ, x = + sin θ x ( + 3 ( cosθ(x + + ( cosθ x ( + 6 ( cosθ(x +, P 6 (θ, x = + sin θ x ( + 3 ( cosθ(x + [ + 0 ( cosθ(x + 4 ] (a

5 + ( cosθ x ( + 6 ( cosθ(x + [ + 5 ( cosθ(x + 4 ]. (b These identities apply for n and n 3, respectively, since they reduce to, e.g. the n = formula using (. The second main result is a matrix representation for simplified forms of P and Q for 4th order elasticity tensors. Although of fourth order, symmetries reduce the maximum number of elements from 3 4 to, and hence the moduli can be described by a -vector and P and Q by matrices. This reduction in matrix size is similar to the 6 6 representation of Mehrabadi et al. [6] for symmetric second order tensors. The case of third order tensors (n = 3 is not discussed here to the same degree of detail as for second and fourth order, but it could be considered in the same manner. We list here the matrices for P for n =,, 4, for which the expression ( holds for ( P, Q replaced by (P,Q, ( P, Q or ( P, Q, corresponding to rotation of vectors, symmetric second order tensors, and fourth order elasticity tensors, respectively. The m m skew symmetric generator matrix for each case has the form where P = R R t, (3 R = X, for vectors, n =, (4a ( 0 Y R =, for symmetric tensors, n =, (4b Z X Y Y 0 N N Y R = 0 Z 0 0 X 0 X, 0 N N 0 0 X 0 Z 0 0 X 0 0 X 0 0 Z X X 0 X for elasticity tensors, n = 4, (4c with 0 = and 0 p p 0 0 p 0 p 0 0 X = 0 0 p, Y = 0 0 p 3, Z = 0 0 p, N = 0 p 0. (5 p 0 0 p 0 0 p p 3 In each case, the Q-matrix is given by eq. (, and the tensors acted on by Q are m-vectors (m = 3, 6, which transform like v = Qv. The specific form of the 6- and -vectors are given in Section 5. 3 General theory for tensors 3. P s and Q s The transpose of a n-th order tensor is defined by interchanging the first and last n indices. In particular, the skew symmetry of P and the definition of P in (9 implies that P t = P. Based on

6 the definitions of Q in (8 and the properties of the fundamental rotation Q SO(3, Q of (0 satisfies d Q = P Q, (6 dθ where the product of two tensors of order n is another tensor of order n defined by contracting over n indices, (AB ij...klpq...rs = A ij...klab...cd B ab...cdpq...rs. This gives meaning to the representation Based on the skew symmetry of P it follows that Q Q t = Q t Q = I, where I ij...klpq...rs = Q = e θ P. (7 6 n {}}{ δ ip δ jq...δ kr δ ls. (8 The isomorphism between the space of n-th order tensors and a vector space of dimension 3 n implies that Q SO(3 n and P so(3 n. The derivation of the main result in Theorem is outlined next using a series of Lemmas. Details of the proof are provided below and in the Appendix. Lemma For any N, a given non-zero N N skew symmetric matrix B has m N distinct non-zero eigenvalues of the form {±ic, ±ic,...,±ic m }, c,...,c m real, and B has the representation B = c B + c B c m B m, (9a B i B j = B j B i = 0, i j, (9b B 3 i = B i. (9c Lemma is essentially Theorem. of Gallier and Xu [9], who also noted the immediate corollary e B = I + = I + ( k c B + c B c m B m k! ( c k k! B k + ck Bk ck m m Bk = e c B + e c B e cmbm (m I m ( = I + sin cj B j + ( cosc j Bj. (30 Thus, the exponential of any skew symmetric matrix has the form of a sum of Euler-Rodrigues terms. Lemma The elements of the decomposition of Lemma can be expressed in terms of the matrix B and its eigenvalues by B j = B m (B + c k, (3 c j c k c j Equation (3 may be obtained by starting with B(B + c k = k j m (c k c jc j B j, (3 j k

7 and iterating until a single B j remains. Equation (3 in combination with (30 and B j = c j BB j implies that e B can be expressed as a polynomial of degree m in B, e B = I + m We are now ready to consider exp(θ P. [ cj sin c j B + ( cosc j B ] c j 7 m (B + c k. (33 c k c j Lemma 3 The non-zero eigenvalues of the skew symmetric tensor P defined in eq. (9 are {i, i, i, i,...,ni, ni}. This follows from Zheng and Spence [3] or Lu and Papadopoulos [4], and is discussed in detail in Section 4. P therefore satisfies the characteristic equation (, and has the properties P = P + P n P n, k j (34a P i P j = P j P i = 0, i j, (34b P 3 i = P i. (34c Equation (34a is the canonical decomposition of P into orthogonal components each of which has properties like the fundamental P in (3, although the subspace associated with P i can be multidimensional. The polynomials p n,k (x of eq. (7 follow from (3, or alternatively, p n,k (x = ( k+ kx (n k!(n + k! n (x + j, k n. (35 We will examine the particular form of the P i for elasticity tensors of order n = and n = 4 in Section 5. For now we note the consequence of Lemmas and 3, e θ P = I + j k [ ] n ( k sin kθ P + ( cos kθ P k P + j. (36 j k Together with (7 this implies the first expression (8a in Theorem. The alternative identities (8b and (8c are derived in the Appendix. Lemma 4 Q may be expressed in Cayley form where the n-th order skew symmetric tensor S is j k Q = ( I + S ( I S = ( I S ( I + S, (37 S = tanh( θ P = tan(k θ P k. (38 This follows by inverting (37 and using the expression (3 for Q, S = I ( I + Q = I [ I + ( ] sin kθ Pk + ( coskθ P k, (39

8 and then applying the following identity, [ I + ( ] ak P k + ( b k P k = I + ( a k a k + P k + ( b k b k 8 a k + Pk. (40 b k Note that Lemma 4 combined with the obvious result (38 implies that, for arbitrary φ, tanh(φ P = tan(kφ P k. (4 This is a far more general statement than the simple partition of (, and in fact may be shown to be equivalent to eq. ( Equivalence with the Lu and Papadopoulos formula We now show that (36 agrees with the polynomial of Lu and Papadopoulos [4]. They derived eq. ( directly using Sylvester s interpolation formula (p. 437 of Horn and Johnson [3]. Thus, the function f(a defined by a power series for a matrix A that is diagonalizable can be expressed in terms of its distinct eigenvalues λ, λ,...λ r as f(a = p(a, where p(x is p(x = r f(λ i L i (x, L i (x = i= r j i ( x λ j λ i λ j. (4 L i (x are the Legendre interpolation polynomials and p(x is the unique polynomial of degree r with the property p(λ i = f(λ i. Equation ( follows from the explicit form of the spectrum of P. The right member of ( can be rewritten in the following form by combining the terms e ±ikθ, P n (θ, x = (n! n (x + j + ( k+ (n k! (n + k! ( n k sin kθ x coskθ x (x + j. (43 The first term on the right hand side can be written in a form which agrees with (8a by considering the partial fraction expansion of /Λ where n Λ(x = (x + k. (44 Thus, as may be checked by comparing residues, for example, Λ = = ( k+ k (n k! (n + k! x + k ( k+ (n k! (n + k! j k ( k+ x (n k! (n + k! x + k. (45 The first term in the right member is /Λ(0 = /(n!. Multiplying by Λ(x/Λ(0 implies the identity n (x + j x ( k+ n = + (x + j, (46 (n! (n k! (n + k! which combined with (43 allows us recover the form (8a. This transformation from the interpolating polynomial ( to the alternative forms in eq. (8 is rigorous but it does not capture the physical basis of the latter. We will find the identity (46 useful in Section 6. j k

9 4 Eigenvalues of P and application to second order tensors The key quantity in the polynomial representation of Q is the set of distinct non-zero eigenvalues of P, Lemma 3. We prove this by construction, starting with the case of n =. 4. Rotation of vectors, n = Let {p, q, r} form an orthonormal triad of vectors, then the eigenvalues and eigenvectors of P are as follows eigenvalue eigenvector 0 p (47 ±i v ± (iq ± r These may be checked using the properties of the third order alternating tensor. Thus, P ij q j = r i, P ij r j = q i, implying Pv ± = ±iv ±. Hence, the spectral representation of P is P = iv + v + iv v, (48 where denotes complex conjugate. It also denotes transpose if we view eq. (48 in vector/matrix format. We can also think of this as a tensorial representation in which terms such as v v stand for the Hermitian dyadic v v, however, for simplicity of notation we do not use dyadic notation further but take the view that dyadics are obvious from the context. It is interesting to note that v ± are bivectors [5]. Referring to eq. (34, we see that P = P in this case. Equation (48 allows us to evaluate powers and other functions of P. Thus, in turn, 9 P = v + v + v v, P + I = pp, P(P + I = 0, (49a (49b (49c from which the characteristic equation (3 follows. 4. Rotation of second order tensors, n = In this case the tensors P and Q are fourth order with, see eq. (9, Equation (7 implies which clearly satisfy the decomposition (34a P ijkl = P ik δ jl + δ ik P jl. (50 P = P( P + 4/3, P = P( P + /6, (5 The skew symmetric tensor S of (6 may be expressed S = ( 4 tan θ ( 3θ tan 3 6 P + tan θ 3 6 P = P + P. (5 3θ tan P 3. (53 Consider the product of the second order tensor (dyad pv ± with P. Thus, Ppv ± = ±ipv ± which together with Pv ± p = ±iv ± p yields 4 eigenvectors, two pairs with eigenvalue i and i. The dyadics v ± v ± are also eigenvectors and have eigenvalues ±i. The remaining 3 eigenvectors

10 of P are null vectors which can be identified as pp, v + v and v v +. This completes the 3 n = 9 eigenvalues and eigenvectors, and shows that the non-zero eigenvalues are {i(, i(, i, i} where the number in parenthesis indicates the multiplicity. The tensors P and P can be expressed explicitly in terms of the eigenvectors, P = ipv + pv + + iv +pv + p ipv pv iv pv p, P = iv + v + v + v + iv v v v. (54 It is straightforward to show that these satisfy the conditions (34. We note in passing that zero eigenvalues correspond to tensors of order n with rotational or transversely isotropic symmetry about the p axis. This demonstrates that there are three second order basis tensors for transversely isotropic symmetry. We will discuss this aspect in greater detail in Section 6 in the context of elasticity tensors. Both P and Q correspond to 3 n 3 n or 9 9 matrices, which we denote P and Q, with the precise form dependent on how we choose to represent second order tensors as 9-vectors. To be specific, let T ij be the components of a second order tensor (not symmetric in general, and define the 9-vector T = ( T, T, T 33, T 3, T 3, T, T 3, T 3, T t. (55 The form of P follows from the expansion (7 for small θ, or from (9, as p p 3 0 p p p 0 p 3 p 0 p p p 0 p p 0 0 p p p 3 p P = p 0 p p 3 0 p. (56 p 3 p p p 0 0 p p 0 p 3 p p 0 p p 3 0 p p 3 p 3 0 p p This can be written in block matrix form using the X, Y, Z matrices defined in eq. (5, 0 Y Y P = R R t, where R = Z 0 X. (57 Z X 0 Note that this partition of the skew symmetric matrix is not unique. 4.3 Tensors of arbitrary order n The example of second order tensors shows that the three eigenvectors {p, v +, v } of the fundamental matrix P define all 3 n = 9 eigenvectors of P. This generalizes to arbitrary n by generation of all possible n-tensors formed from the tensor outer product of {p, v +, v }. For any n, consider the product P with the n-tensor 0 v + = v + v +...v }{{ +. (58 } n Each of the n terms of P in (9 contributes +iv + with the result Pv + = ni v +. Similarly, defining v implies Pv = ni v, from which it is clear that there are no other eigenvectors with eigenvalues ±ni. Thus, the final component in the decomposition (34a is P n = i v + v + i v v. (59

11 Next consider the n n-th order tensors that are combinations of (n times v + with a single p, e.g. v = pv + v +...v }{{ +. (60 } n The action of P is Pv = (n iv, and so there are n eigenvectors with eigenvalues ±(n i which can be combined together to form the component P n, as in eq. (54 for n =. Eigenvalues (n i are obtained by combining p twice with v + (n times, but also come from n-tensors formed by a single v with (n times v +. Enumeration yields a total of n(n + eigenvectors associated with P n. By recursion, it is clear that the number of eigenvectors with eigenvalues ki equals the number of ways n elements chosen from {, 0, } sum to n k n, or equivalently, the coefficient of x k in the expansion of ( + x + x n, ( + x + x n = k= n ( n x k. (6 k The numbers ( n are trinomial coefficients (not to be confused with q-multinomial coefficients [] k with q = 3 and can be expressed [] ( n = k j=0 ( n j + k ( j + k j, (6 where ( n j = n!/(j!(n j! are binomial coefficients. The main point for the present purpose is that ( n = 3 n, (63 k k= n which is a consequence of (6 with x =. It is clear from this process that the entire set of 3 n eigenvectors has been obtained and there are no other candidates for eigenvalue of P. The precise form of the eigenvectors is unimportant because the Euler-Rodrigues type formulas (8 depend only on the set of distinct non-zero eigenvalues. However, the eigenvectors do play a role in setting up matrix representations of P for specific values of n, and in particular for tensors with symmetries, as discussed in Section 5. 5 Applications to elasticity We first demonstrate that the underlying symmetry of second and fourth order tensors in elasticity implies symmetries in the rotation tensors which reduces the number of independent elements in the latter. 5. Rotation of symmetric second and fourth order tensors We consider second and fourth order tensors such as the elastic strain ε (n = and the elastic stiffness tensor C (n = 4 with elements ε ij and C ijkl. They transform as ε ij ε ij and C ijkl C ijkl, ε ij = Q ijpqε pq C ijkl = Q ijklpqrs C pqrs, (64 where the rotation tensors follow from the general definition (8 as Q ijpq = Q ip Q jq, Q ijklpqrs = Q ijpq Q klrs. (65

12 The strain and stiffness tensors possess physical symmetries which reduce the number of independent elements that need to be considered, ε ij = ε ji, C ijkl = C jikl = C ijlk, C ijkl = C klij. (66 In short, the number of independent elements of ε is reduced from 9 to 6, and of C from 8 to. Accordingly, we may define variants of Q ijkl and Q ijklpqrs with a reduced number of elements that reflect the underlying symmetries of elasticity. Thus, the transformation rules can be expressed in the alternative forms ε ij = Q ijpq ε pq, C ijkl = Q ijklpqrs C pqrs, (67 where the elements of the symmetrized 4th and 8th order Q tensors are Q ijpq = ( Qijpq + Q ijqp, (68a Q ijklpqrs = ( Qijpq Q klrs + Q ijrs Q klpq. (68b Alternatively, they can be expressed in terms of the fundamental Q SO(3, Q ijpq = ( Qip Q jq + Q iq Q jp, (69a Q ijklpqrs = 8( Qip Q jq Q kr Q ls + Q ip Q jq Q ks Q lr + Q iq Q jp Q kr Q ls + Q iq Q jp Q ks Q lr Using the identity (8, we have + Q ir Q js Q kp Q lq + Q ir Q js Q kq Q lp + Q is Q jr Q kp Q lq + Q is Q jr Q kq Q lp. (69b Q ijpq Q klpq = ( Iijkl + I ijlk = Iijkl ( δik δ jl + δ il δ jk, (70 which is the fourth order isotropic identity tensor, i.e. the fourth order tensor with the property s = Is for all symmetric second order tensors s. Similarly, Q ijklabcd Q pqrsabcd = ( Iijpq I klrs + I ijrs I klpq Iijklpqrs. (7 This is the 8th order isotropic tensor with the property C = I C for all 4th order elasticity tensors. In summary, the symmetrized rotation tensors for strain and elastic moduli are orthogonal in the sense that Q Q t = I, and are therefore elements of SO(6 and SO(, respectively. The (symmetrized rotation tensor for strain displays the following symmetries Q ijpq = Q jipq = Q ijqp. (7 Introducing the Voigt indices, which are capital suffices taking the values,,..., 6 according to I =,, 3, 4, 5, 6 ij =,, 33, 3, 3,, (73 then (7 implies that the elements Q ijpq can be represented by Q IJ. Similarly, the elements of the 8th order tensor Q ijklpqrs can be represented as Q IJKL, which satisfy the symmetries Q IJKL = Q JIKL = Q IJLK. (74

13 The pairs of indices IJ and KL each represent independent values, suggesting the introduction of a new type of suffix which ranges from,,...,. Thus, I =,, 3, 4, 5, 6, 7, 8, 9, 0,,, 3, 4,5, 6, 7, 8,9, 0, IJ =,, 33, 3, 3,, 44, 55, 66, 4,5, 36, 34, 5,6, 4, 35, 6,56, 64, 45. (75 Hence the elements Q ijklpqrs can be represented uniquely by the ( elements Q IJ. The elements of the symmetrized rotations Q IJ and Q IJ have reduced dimensions, 6 and respectively, but do not yet represent the matrix elements of 6- and -dimensional tensors. That step is completed next, after which we can define the associated skew symmetric generating matrices and return to the generalized Euler-Rodrigues formulas for 6- and -dimensional matrices. 5. Six dimensional representation We now use the isomorphism between second order tensors in three dimensions, such as the strain ε, and six dimensional vectors according to ε ˆε with elements ˆε I, I =,,..., 6. Similarly, fourth order elasticity tensors in three dimensions are isomorphic with second order positive definite symmetric tensors in six dimensions Ĉ with elements ĉ IJ [5]. Let {ε} be the 6-vector with elements ε I, I =,,..., 6, and [C] the 6 6 Voigt matrix of elastic moduli, i.e. with elements c IJ. The associated 6-dimensional vector and tensor are ˆε = T{ε}, Ĉ = T[C]T, where T diag (,,,,,. (76 3 Explicitly, ˆε ε c c c 3 c 4 c 5 c 6 ˆε ε c c 3 c 4 c 5 c 6 ˆε = ˆε 3 ˆε 4 = ε3 ε 33, Ĉ = c 33 c 34 c 35 c 36 ε3 c 44 c 45 c. (77 46 ˆε 5 ε S Y M c 55 c 56 ˆε 6 c 66 The terms ensure that products and norms are preserved, e.g. C ijkl C ijkl = trĉt Ĉ. The six-dimensional version of the fourth order tensor Q ijkl is Q SO(6, introduced by Mehrabadi et al. [6], see also [9]. It may be defined in the same manner as Q = S[Q]S where [Q] is the matrix of Voigt elements. Thus, Q Q t = Q t Q = Î, where Î = diag(,,,,,. It can be expressed Q(p, θ = exp(θ P, and hence is given by the n = Euler-Rodrigues formula ( for the skew symmetric P(p so(6, p p p 0 p p p P = 0 0 p, (78 p 0 p 3 p p 0 p p3 p 3 0 p p 3 0 p p 0 or in terms of the block matrices defined in (5, P = R R t, R = ( 0 Y Z X. (79

14 It is useful to compare P with the 9 9 matrix for rotation of general second order tensors, eqs. (56 and (57. The reduced dimensions and the terms are a consequence of the underlying symmetry of the tensors that are being rotated. Vectors and tensors transform as ˆε ˆε and Ĉ Ĉ where ˆε = Q ˆε, Ĉ = QĈ Q t. (80 The eigenvalues and orthonormal eigenvectors of P are as follows 4 eigenvalue eigenvector dyadic 0 î 3 I 0 ˆp d 3 3 ±i û ± (pv ± + v ± p (8 ±i ˆv ± v ± v ± where v ± are defined in (47 and / 3 / p 3 p v ±, v 3 î / p p ±, 3 v ±, ˆp d p 3 p3 v ±, 3 p p3 p 3, û ± v ±,3 p v ±,3 + p 3 v ±,, ˆv ± v ±,3 v±, v ±,3. (8 0 p p p 3 v ±, + p v ±,3 v±,3 v ±, 0 p p v ±, + p v ±, v±, v ±, The right column in (8 shows the second order tensors corresponding to the eigenvectors in dyadic form. The eigenvectors are orthonormal, i.e. of unit magnitude and mutually orthogonal. The unit 6 vectors î and ˆp d correspond to the hydrostatic and deviatoric parts of pp, respectively. The double multiplicity of the zero eigenvalue means that any 6 vector of the form aˆp d + bî is a null vector of P, including ˆp 3 d + 3 î which is the six-vector for the dyad pp. The canonical decomposition of the generating matrix P so(6 is therefore P = P + P, P = iû + û + iû û, P = iˆv +ˆv + iˆv ˆv. (83 The associated fourth order tensor P, which is the symmetrized version of P in eq. (5, is P = P + P, P = ± i (pv ± + v ± p(pv ± + v ±p, P = ± ±iv ± v ± v ±v ±. (84 The representation (83 allows us to compute powers of P, and using the spectral decomposition of the identity Î = îît + ˆp dˆp t d + û +û + + û û + ˆv +ˆv + + ˆv ˆv, (85 the characteristic equation [6] follows: 5.3 dimensional representation P( P + Î( P + 4Î = 0. (86 Vectors and matrices in -dimensions are denoted by a tilde. Thus, the -dimensional vector of elastic moduli is c with elements c I, I =,,...,. Similarly, vectors corresponding to the fourth

15 5 order tensors εε and (A B + B A are ε and ṽ, that is 4 th order, n=3 nd order, n=6 vector, n= C Ĉ c εε ˆεˆε t. (87 ε (A B + B A (âˆb t + ˆbâ t ṽ The elements follow from the indexing scheme (75 and are defined by â ˆb c c ĉ ε ˆε â ˆb c c ĉ ε ˆε â c 3 c 33 ĉ 33 ε 33 ˆε 3ˆb3 3 c 4 c3 c ĉ3 5 c3 ε ε 33 ĉ3 ˆεˆε 3 (â ˆb3 + â 3ˆb ε ε 33 ˆε3ˆε (â 3ˆb + â ˆb3 c 6 c ĉ ε ε ˆεˆε c 7 c 44 ĉ 44 ε (â ˆb + â ˆb 3 ˆε â 4ˆb4 4 c 8 c 55 ĉ 55 ε 3 ˆε â 5ˆb5 5 c 9 c 66 ĉ 66 ε ˆε â 6ˆb6 6 c 0 c 4 ĉ4 ε ε 3 ˆεˆε 4 (â ˆb4 + â 4ˆb c = c = c 5 = ĉ5 c c, ε = ε ε 3 = ˆεˆε 5 36 ĉ36 ε 33 ε, ṽ = (â ˆb5 + â 5ˆb. ˆε3ˆε 6 (â 3ˆb6 + â 6ˆb3 c 3 c 34 ĉ34 ε 33 ε 3 ˆε3ˆε 4 (â 3ˆb4 + â 4ˆb3 c 4 c 5 ĉ5 c 5 c ε ε 3 ˆεˆε 5 6 ĉ6 ε ε (â ˆb5 + â 5ˆb ˆεˆε 6 (â ˆb6 + â 6ˆb c 6 c 4 ĉ4 ε ε 3 ˆε ε 4 (â ˆb4 + â 4ˆb c 7 c 35 ĉ35 ε 33 ε 3 ˆε3ˆε 5 c 8 c 6 c 9 ĉ6 c 56 c 0 ε ε ĉ56 ˆεˆε 6 (â 3ˆb5 + â 5ˆb3 ε 3 ε c 46 c ĉ46 ˆε5ˆε 6 (â ˆb6 + â 6ˆb ε 3 ε c 45 ĉ45 ˆε6ˆε 4 (â 5ˆb6 + â 6ˆb5 ε 3 ε 3 ˆε4ˆε 5 (â 6ˆb4 + â 4ˆb6 (â 4ˆb5 + â 5ˆb4 Alternatively, (88 c = T{c}, etc., (89 where {c} is the array with elements c I, and T is the the diagonal with block structure T = diag ( I I I I I I I. (90 For instance, the elastic energy density W = C ijklε ij ε kl is given by the vector inner product where ε is the -vector associated with strain ( The -dimensional rotation matrix W = εt c, (9 Vectors in -dimensions transform as c c = Q c where Q SO( satisfies Q Q t = Q t Q = Ĩ, and Ĩ is the identity, ĨIJ = δ IJ. The rotation matrix Q can be expressed Q(p, θ = e θ ep, (9

16 where P(p so( follows from the general definition (9 applied to elasticity tensors, i.e. (69b. Thus, P = T[P] T, Q = T[Q] T (93 where [P], [Q] are the array with elements P IJ, Q IJ. We take a slightly different approach, and derive the elements of P using the expansion of Q for small θ directly. Define the rotational derivative of the moduli as ċ ijkl = dc ijkl dθ. (94 θ=0 Using (80 for instance, gives Ĉ = PĈ + Ĉ P t, (95 6 from which we obtain ċ = 4c 5 p 4c 6 p 3, ċ = 4c 4 p + 4c 6 p 3, ċ 33 = 4c 34 p 4c 35 p, ċ 3 = (c 4 c 34 p c 5 p + c 36 p 3, ċ 3 = c 4 p + (c 35 c 5 p c 36 p 3, ċ = c 4 p + c 5 p + (c 6 c 6 p 3, ċ 44 = (c 4 c 34 p c 46 p + c 45 p 3, ċ 55 = c 56 p + (c 35 c 5 p c 45 p 3, ċ 66 = c 56 p + c 46 p + (c 6 c 6 p 3, ċ 4 = (c c 3 p (c 6 c 45 p + (c 5 c 46 p 3, ċ 5 = (c 6 c 45 p + (c 3 c p (c 4 c 56 p 3, ċ 36 = (c 35 c 46 p + (c 34 c 56 p + (c 3 c 3 p 3, ċ 34 = (c 33 c 3 c 44 p (c 36 + c 45 p + c 35 p 3, ċ 5 = c 6 p (c c 3 c 55 p (c 4 + c 56 p 3, ċ 6 = (c 5 + c 46 p + c 4 p (c c c 66 p 3, ċ 4 = (c c 3 c 44 p c 6 p + (c 5 + c 46 p 3, ċ 35 = (c 36 + c 45 p + (c 33 c 3 c 55 p c 34 p 3, ċ 6 = c 5 p + (c 4 + c 56 p + (c c c 66 p 3, ċ 56 = (c 66 c 55 p + (c 36 c 6 + c 45 p (c 5 c 5 + c 46 p 3, ċ 46 = (c 36 c 6 + c 45 p + (c 44 c 66 p + (c 4 c 4 + c 56 p 3, ċ 45 = (c 5 c 35 + c 46 p (c 4 c 34 + c 56 p + (c 55 c 44 p 3. (96a (96b (96c (96d (96e (96f (96g (96h (96i (96j (96k (96l (96m (96n (96o (96p (96q (96r (96s (96t (96u The elements of P can be read off from this and are given in Table. The format can be simplified to a 7 7 array of block elements, comprising combinations of the 3 3 zero matrix 0 and the 3 3 matrices defined in (5. Thus, Y Y 0 N 0 P = R R N Y t, where R = 0 Z 0 0 X 0 X. (97 0 N N 0 0 X 0 Z 0 0 X 0 0 X 0 0 Z X X 0 X

17 Equation (4 shows that the rotation matrices in three, six and dimensions can be simplified using the fundamental 3 3 matrices X,Y,Z and N. The partition of a skew symmetric matrix in this way is not unique, but it simplifies the calculation and numerical implementation. We note that these matrices satisfy several identities, XX t + YY t + ZZ t = I, 7 (98a X t X = YY t, Y t Y = ZZ t, Z t Z = XX t, (98b X t Y = YZ t, Y t Z = ZX t, Z t X = XY t, (98c N = X = Y = Z =, (98d where the norm is defined u = tru t u. We find that the eigenvalues of P are 0(5, i(3, i(3, i(3, i(3, 3i, 3i, 4i, 4i, where the number in parenthesis is the multiplicity. The associated orthonormal eigenvectors of P are eigenvalue eigenvector 6-dyadic 0 ṽ 0a îît 3 0 ṽ 0b ˆp dˆp t d 0 ṽ 3 0c (îˆpt d + ˆp dît 0 ṽ 0d (û û t + + û +û t 0 ṽ 0e (ˆv ˆv + t + ˆv +ˆv t ±i ṽ a± (îût ± + û ±ît ±i ṽ 3 b± (ˆp dû t ± + û ±ˆp t d ±i ṽ c± (ˆv ± û t + û ˆv ± t (99 ±i ṽ a± û ± û t ± ±i ṽ b± (îˆvt ± + ˆv ± î t ±i ṽ 3 c± (ˆp dˆvt ± + ˆv ±ˆp t d ±3i ṽ 3± (û ±ˆv ± t + ˆv ± û t ± ±4i ṽ 4± ˆv ±ˆv ± t Apart from the last two, which have unit multiplicity, these are not unique. For instance, different combinations of the eigenvectors with multiplicity greater than one could be used instead. We also note from their definitions in eqs. (47 and (8 that v ± = v, û ± = û, ˆv ± = ˆv. (00 Hence, the five null vectors of P in (99 are real. P can be expressed in terms of the remaining 6 eigenvectors as P = P + P + 3 P P 4, (0 where P j = ±i ± x=a,b,c The identity is Ĩ = x=a,b,c,d,eṽ 0x ṽ 0x + ( ± ṽ jx± ṽjx±, j =, ; Pj = ṽ j± ṽj±, j = 3, 4. (0 x=a,b,c (ṽx± ṽx± + ṽ x± ṽx± + ṽ3± ṽ3± + ṽ 4± ṽ4±. (03 It is straightforward to demonstrate using these expressions that P satisfies P ( P + ( P + 4 ( P + 9 ( P + 6 = 0, (04 which is the characteristic equation for n-th order P tensors, eq. (.

18 6 Projection onto TI symmetry and Cartan decomposition It has been mentioned in passing that null vectors of P and its particular realizations for elasticity tensors correspond to tensors with transversely isotropic symmetry. These are defined as tensors invariant under the action of the group SO( associated with rotation about the axis p. We conclude by making this connection more specific, and relating the theory for the rotation of tensors to the Cartan decomposition, which is introduced below. 6. Definition of the pseudo-projectors We introduce n+ pseudo-projectors M k, k = 0,,..., n, which are symmetric tensors defined by M 0 I + P + P P n, M i = P i, i =,,..., n. (05 It may be readily verified that they satisfy, for 0 k n and i n, M i M k = M k M i = M k P i = P i M k = 0, i k, (06a M 0 = M 0, M i = M i, M i P i = P i M i = P i, (06b ( M k = m n,k P, mn,k (x = ( k ( δ k0 n (x + j. (06c (n k!(n + k! The expansion of the projector M k as an even polynomial of degree n in P, i.e. of degree n in P, follows from eqs. (35 and (46 and the identity P k = k P P k. Note that the identity is I = M 0 + M M n. (07 Since the M k tensors partition the identity they can be considered as pseudo-projectors. The distinction with proper projection operators is that a projection M, by definition must satisfy M = M, which is true only for M 0. However, it is useful to consider the set of tensors M k, k = 0,,..., n together as pseudo-projectors. We next examine the associated span of these pseudo-projectors. 6. Cartan decomposition Starting with M 0 we have j=0 j k P M 0 = M 0 P = 0, (08 indicating that M 0 is the linear projection operator onto the null space of P. In order to examine the remaining n pseudo-projectors, note that the rotation can be expressed Q = M ( coskθ Mk + sin kθ P k. (09 The action of Q on an n-th order tensor T, the rotation of T, can therefore be described as Q T = T 0 + ( coskθ Tj + sin kθ R j, (0 where T 0 and T j, R j, j =,,..., n are defined by T 0 = M 0 T, T j M j T, R j P j T, j =,,..., n. (

19 9 We note that R j may be expressed R j = P j T j = j P T j, j =,,..., n. ( The projection T 0, the component of T in the null space of P, along with the n pairs T j, R j, j =,,..., n define n + Cartan subspaces [8] which are closed under rotation: Q T 0 = T 0, Q T j = cosjθ T j + sin jθ R j, Q R j = cosjθ R j sin jθ T j, j =,,..., n. (3 This is the essence of the Cartan decomposition, see [8] for further details. However, the meaning will become more apparent by example, as we consider the various cases of tensors of order n =,, and Application to elasticity tensors The pseudo-projection operators are now illustrated with particular application to tensors of low order and elasticity tensors, starting with the simplest case: rotation of vectors Vectors, n= If n =, we have T = P of eq. (, and hence M 0 = pp, M = I pp. (4 Also, T t, an arbitrary vector, and the decomposition (0 and ( is Qt = (t pp + cosθ [t (t pp] + sin θ p t. (5 Thus, we identify t 0 = M 0 t as the component of the vector parallel to the axis of rotation, t = M t as the orthogonal complement, and r = P t is the rotation of t about the axis by π/ Second order tensors, n= For n =, we have P, P given in (5 and M 0 = ( P + ( P + 4/4, M = P ( P + 4/3, M = P ( P + /. (6 The dimensions of the subspaces are T 0, 3; T,, ; R, ; T,, ; R,. For symmetric second order tensors the projectors M 0, M and M have the same form as in (6 in terms of P, and may also be expressed in terms of the fundamental vectors introduced earlier. Thus, using (83 and (85, M 0 = 4 ( P +Î( P + 4Î = îît + ˆp dˆp t d, M = û + û + +û û, M = ˆv +ˆv + + ˆv ˆv. (7 Using these and P, P from (83, it follows that the subspace associated with M 0 is two dimensional, while M, M, M, and M have dimension one. This Cartan decomposition of the ( ( ( ( 6-dimensional space Sym agrees with a different approach by Forte and Vianello [8]. Alternatively, using (7 and the dyadics in (8, it follows that M 0 = pppp + (I pp(i pp, (8a

20 0 M = (pq + qp(pq + qp + (pr + rp(pr + rp, (8b M = (qq rr(qq rr + (qr + rq(qr + rq. (8c The action of M 0 on a second order symmetric tensor is M 0 T = T 0 = T pp + T (I pp, (9 where T = T : pp and T = (trt T : pp. It is clear that T 0 is unchanged under rotation about p. Similarly, M T transforms as a vector under rotation, hence the suffix, while M T transforms as a second order tensor with elements confined to the plane orthogonal to the axis of rotation. Let p = e 3, then using (7 or the simpler (8a yields ( ˆT + ˆT / ( ˆT + ˆT / ˆT 0 = M 0 (e 3 ˆT = ˆT = ˆT 3 0. ( The projection ˆT 0 can be identified as the part of ˆT which displays hexagonal (or transversely isotropic symmetry about the axis e 3. To be more precise, if we denote the projection M 0 ˆT as ˆT Hex, then ˆT Hex is invariant under the action of the symmetry group of rotations about p. It may also be shown that of all tensors with hexagonal symmetry ˆT Hex is the closest to ˆT using a Euclidean norm for distance. Similarly, ˆT = M ˆT = ( 0, 0, 0, ˆT4, ˆT 5, 0 t, ˆT = M ˆT = ( ( ˆT ˆT, ( ˆT ˆT, 0, 0, 0, ˆT 6 t, ( and using eqs. (78 and (, ˆR = ( 0, 0, 0, ˆT 5, ˆT 4, 0 t, ˆR = ( ˆT6, It can be checked that under rotation, we have ˆT6, 0, 0, 0, ( ˆT ˆT t. ( ˆQˆT 0 = ˆT 0, ˆQˆTj = cos jθ ˆT j + sin jθ ˆR j, ˆQ ˆRj = cosjθ ˆR j sin jθ ˆT j, j =,. (3 Thus, the singleton ˆT 0 and pairs (ˆT, ˆR and (ˆT, ˆR form closed groups under the action of the rotation Elastic moduli, n=4 The 5-dimensional null space of P is equal to the set of base elasticity tensors for transverse isotropy [3, 8, 4]. The projector follows from eqs. (0 and (03 as M 0 (4! ( P + ( P + 4 ( P + 9 ( P + 6 = x=a,b,c,d,e ṽ 0x ṽ 0x. (4 This is equivalent to the 8-th order projection tensor P Hex of eq. (68c of Moakher and Norris [7], who use the base tensors suggested by Walpole []. The latter representation is useful because

21 Walpole s tensors have a nice algebraic structure, making tensor products simple. Alternatively, in the spirit of the representation of elastic moduli, (4 and (97 yield M Hex M 0 (e 3 =, M Hex = (5 The projection M 0 of (5 is identical to the matrix derived by Browaeys and Chevrot [6] for projection of elastic moduli onto hexagonal symmetry. Specifically, the 9 9 matrix M Hex is exactly M (4 of Browaeys and Chevrot (see Appendix A of [6]. Hence, M 0 of (5 provides an algorithm for projection onto hexagonal symmetry for arbitrary axis p. The subspace of T 0 is 5-dimensional, while the subspaces ( T, R, ( T, R, ( T 3, R 3 and ( T 4, R 4 have dimensions 6, 6, and, respectively. These may be evaluated for arbitrary axis of rotation p using the prescription in eqs. (06c and (. A. Appendix: A recursion Equation (33 provides an expansion in terms of basic trigonometric functions. An alternative recursive procedure is derived here. Consider the expression (33 written e B = f m (B, (A. where the function f m (B is a finite series defined by the m distinct pairs of non-zero eigenvalues of B. Suppose the set of eigenvalues is augmented by one more pair, ±ic m+, with the others unchanged. Then the sums and products in (33 change to reflect the new eigenvalues, e B = f m+ (B = I + m+ [ cj sin c j B + ( cosc j B ] c j m+ (B + c k. (A. c k c j k j The effect of the two additional eigenvalues can be isolated using the identity B + c m+ c m+ c j = + B + c j c m+, (A.3 c j to give f m+ = f m + ( m+ This defines the sequence, starting with f 0 = I. c j sin c j B + ( cosc j B m (B + c m+ k. (A.4 (c l c j c j l= l j

22 Applying this to the rotation exp(θ P, and noting that the eigenvalues of the skew symmetric tensor P correspond to c j = jθ, gives P n+ (θ, x = P n (θ, x + ( n+ n j sin jθ x + ( cos jθ x (x + k. (A.5 (l j j n+ The sum over trigonometric functions may be simplified using formulas for sin nθ and cosnθ in terms of cosθ/ powers of sin θ/ in Section.33 of Gradshteyn et al. [0]. Thus, l= l j n+ n+ sin jθ (l j j n+ l= l j cosjθ (l j j n+ l= l j = ( sin θ n+ (n +! = ( sin θ n+ (n +! cos θ, sin θ n +, (A.6 (A.7 and hence, P n+ (θ, x = P n (θ, x + ( sin θ n+ (n +! from which the expression (8b follows. ( cos θ x + (n + sin θ x n (x + k, (A.8 References [] M. Abramowitz and I. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 974. [] G. Andrews. Euler s exemplum memorabile inductionis fallacis and q-trinomial coefficients. J. Amer. Math. Soc., 3: , 990. [3] B. A. Auld. Acoustic Fields and Waves in Solids, Vol. I. Wiley Interscience, New York, 973. [4] O. Bottema and B. Roth. Theoretical Kinematics. North- Holland Publishing Company, Amsterdam, 979. [5] Ph. Boulanger and M. Hayes. Bivectors and waves in mechanics and optics. London: Chapman and Hall, 993. [6] J. T. Browaeys and S. Chevrot. Decomposition of the elastic tensor and geophysical applications. Geophys. J. Int., 59: , 004. [7] H. Cheng and K. C. Gupta. An historical note on finite rotations. J. Appl. Mech. ASME, 56:39 45, 989. [8] S. Forte and M. Vianello. Functional bases for transversely isotropic and transversely hemitropic invariants of elasticity tensors. Q. J. Mech. Appl. Math., 5:543 55, 998.

23 [9] J. Gallier and D. Xu. Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices. Int. J. Robotics and Automation, 7:, 00. [0] I. S. Gradshteyn, I. M. Ryzhik, A. Jeffrey, and D. Zwillinger. Table of Integrals, Series, and Products. Academic Press: San Diego, CA, 000. [] J. J. Gray. Olinde Rodrigues paper of 840 on transformation groups. Arch. History Exact Sci., : , 980. [] K. Helbig. Representation and approximation of elastic tensors. In E. Fjaer, R. M. Holt, and J. S. Rathore, editors, Seismic Anisotropy, pages 37 75, Tulsa, OK, 996. Society of Exploration Geophysicists. [3] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, UK, 99. [4] J. Lu and P. Papadopoulos. Representations of Kronecker powers of orthogonal tensors with applications to material symmetry. Int. J. Solids Struct., 35: , 998. [5] M. M. Mehrabadi and S. C. Cowin. Eigentensors of linear anisotropic elastic materials. Q. J. Mech. Appl. Math., 43:5 4, 990. [6] M. M. Mehrabadi, S. C. Cowin, and J. Jaric. Six-dimensional orthogonal tensor representation of the rotation about an axis in three dimensions. Int. J. Solids Struct., 3: , 995. [7] M. Moakher and A. N. Norris. The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry. J. Elasticity, 85:5 63,, 006. [8] F. D. Murnaghan. The Theory of Group Representations. The Johns Hopkins Press, Baltimore, MD, 938. [9] A. N. Norris. Optimal orientation of anisotropic solids. Q. J. Mech. Appl. Math., 59:9 53, 006. [0] P. Podio-Guidugli and E. G. Virga. Transversely isotropic elasticity tensors. Proc. R. Soc. A, A4:85 93, 987. [] O. Rodrigues. Des lois géometriques qui regissent les déplacements d un systéme solide dans l espace, et de la variation des coordonées proventant de ces déplacements considerés indépendent des causes qui peuvent les produire. J. de Math. Pures et Appliquées, 5: , 840. [] L. J. Walpole. The elastic shear moduli of a cubic crystal. J. Phys. D: Appl. Phys., 9:457 46, 986. [3] Q.-S. Zheng and A. J. M. Spencer. On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. Int. J. Engng. Sc., 3:67 635,

24 p p p 3 p p p p p3 p 0 0 p p 0 p 3 0 p 0 0 p p p p p p 0 0 p p p p 0 0 p 0 p 0 p p p 3 p p p p p p 0 p 3 p p 0 p p p p3 0 p p 3 p p p 0 p p p p 0 0 p p p p p p 0 0 p 0 p p p p p3 0 0 p 3 0 p p p 0 0 p 0 p 0 0 p p p p p 0 p 0 0 p p p p p p p 3 p p p p p 0 p 3 p 0 p p 0 p 3 p p 0 p p3 0 p 0 0 p p p 3 0 p p 3 p3 0 p p 0 p p 0 p p 0 C A Table. The matrix P(p so( for 4th order elasticity tensors represented as -vectors. The rotation Q = exp(θ P is given by the 8-th order polynomial Q = P 8 (theta, P of eq. (0. 4