The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry

Abstract

The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.

Full text

arXiv:cond-mat/0608311v1 [cond-mat.mtrl-sci] 14 Aug 2006
The closest elastic tensor of arbitrary symmetry
to an elasticity tensor of lower symmetry
Maher Moakher
∗
Andrew N. Norris
†
February 4, 2008
Abstract
The closest tensors of higher symmetry classes are d erived in explicit f orm for
a given elasticity tensor of arbitrary symmetry. The mathematical problem is to
minimize the elastic length or distance between the given tensor and the closest
elasticity tensor of the specified symmetry. Solutions are presented for three distance
functions, with particular attention to the Riemannian and log-Euclidean distances.
These yield solutions that are invariant under inversion, i.e., the same whether
elastic stiffness or compliance are considered. The Frobeniu s distance function,
which corresponds to common notions of Euclidean length, is not invariant although
it is simple to apply us ing p rojection operators. A complete description of the
Euclidean projection method is presented. The three metrics are considered at a
level of d etail far greater than heretofore, as we develop the general framework
to best fit a given set of moduli onto higher elastic symmetries. The procedures
for finding the closest elasticity tensor are illustrated by application to a set of 21
moduli with no underlying symmetry.
1 Introd uction
We address the question of finding the elastic moduli with a given material symmetry
closest to an arbitrary set of elastic constants. There are several reasons for reducing a
set of elastic constants in this way. One might desire to fit a data set to the a priori known
symmetry of the material. Alternatively, a seismic simulation might be best understood in
terms of a model of the earth as a layered transversely isotropic medium, even though core
samples indicate local anisotropy of a lower symmetry. More commonly, one might simply
want t o reduce the model complexity by decreasing the number of elastic parameters. In
each case a distance function measuring the difference between sets of elastic moduli is
necessary to define an appropriate closest set. The most natural metric is the Euclidean
∗
Laboratory for Mathematical and Numerical Modeling in Engineering Science, National Engineering
School at Tunis, ENIT-LAMSIN, B.P. 37, 1002 Tunis Belv´ed`ere, Tunisia, maher.moakher@enit.rnu.tn
†
Rutgers University, Department of Mechanical and Aerospace Engineering, 98 Brett Road, Piscat-
away, NJ 08854-8058, norris@rutgers.edu
1

Moakher and Norris February 4, 2008 2
norm in which the “length” kCk of a set of moduli
1
C
ijkl
is kCk = (C
ijkl
C
ijkl
)
1/2
. The Eu-
clidean distance function is, however, not invariant under inversion of the elasticity t ensor
- one obtains a different result using compliance as compared with stiffness. Alternative
distance functions have been proposed which do not have this failing. In particular, the
Riemannian distance function of Moakher [1] and the log-Euclidean metric of Arsigny et
al. [2] are invaria nt under inversion, and as such are preferred general elastic norms.
The problem of simplifying elastic moduli by increasing the elastic symmetry was
apparently first considered by Gazis et al. [3]. They provided a general framework for
defining and deriving the Euclidean projection, i.e., the closest elasticity using the Eu-
clidean norm. Several examples were given, including the closest isotropic material, which
agrees with the isotropic approximant derived by Fedorov [4]. Fedorov obtained isotropic
moduli using a different criterion: the isotropic material that best approximates the elas-
tic wave velocities of the given moduli by minimizing the difference in the orientation
averaged acoustical tensors. It may be shown [5] that the generalization of Fedorov’s
criterion to other symmetries is satisfied by the Euclidean projection of the stiffness ten-
sor onto the elastic symmetry considered. The Euclidean projection is also equivalent to
operating on the given elasticity tensor with the elements of the transformation group of
the symmetry in question [6]. This approach has been used by Fr an¸cois et al. [7] to find
the closest moduli of trig onal and other symmetries for a set of ultrasonically measured
stiffnesses. The equivalence of the projection and group transformation averaging will
be discussed in Section 5. The Euclidean projection approach has received attention in
the geophysical community from modelers interested in fitting rock data to particular
elastic symmetries [8 , 9, 10, 11]. The work of Helbig [9] is particularly comprehensive.
He considers the problem both in terms of the 6×6 matrix notation [12, 13] and in t erms
of 21-dimensional vectors representing the elastic moduli. The latter a pproa ch has been
developed further by Browaeys and Chevrot [14] who describ e the projection operators
for different elastic symmetries in the 21-dimensional viewpoint. Dellinger [15] presents
an algorithm for finding the closest transversely isotropic medium by searching over all
orientations of the symmetry axis. Dellinger e t al. [16] proposed a distinct a pproach to
the problem of finding closest elasticity, based on the idea of generalized rotat io n of the
21-dimensional elasticity vector.
Although the Euclidean projection is commonly used in applications, it suffers f rom
the fundamental drawback alluded to earlier, i.e., it is not invariant under inversion of
the elasticity tensor. The projection fo und using the elastic stiffness is different from that
obtained using the compliance tensor. This fundamental inconsistency arises from the
dual physical properties of the elasticity tensor/matrix and its inverse. The projection of
one is clearly not the same as the inverse of the projection of the other, although both
projections by definition possess the same elastic symmetry. While there are circumstances
in which the Euclidean projection is preferred, e.g., for the generalized Fedorov problem
of finding the best acoustical approximant of a given symmetry [5], there is a clear need
for a consistent t echnique to define a “closest” elastic material of a given symmetry.
The solution to this quandary is to use an elastic distance function that is invariant
under inversion. Several have been proposed of which we focus on two, the Riemannian
distance function due to Moa kher [1] and the log-Euclidean length of Arsigny et al. [2],
described in Section 3. Moakher [1] describes how the Riemannian distance function gives
a consistent method for averaging elasticity tensors. The only other application so far of
these invariant distance functions to elasticity is by Norris [17] who discusses the closest
1
Lower case Latin s uffices take on the values 1, 2, and 3, and the summation convention on repeated
indices is assumed.

Moakher and Norris February 4, 2008 3
triclinic 21
monoclinic 13
tetragonal 7 orthotropic 9 trigonal 7
tetragonal 6 trigonal 6
cubic 3
hexagonal 5
isotropic 2
Figure 1: The sequence of increasing elastic sym metries, from triclinic to isotropic.
The numb er of independent elastic constants are listed. The dashed boxes
for trigonal 6 and tetragonal 6 indicate that these are obtained from the
lower symmetries by r otation, and d o n ot represent new elastic symmetries.
The true symmetries are the eight in solid boxes [18].
isotropic elasticity.
The outline of the paper is as follows. The three distance functions are introduced in
Section 3 after a brief review of notation in Section 2. Section 4 defines the different types
of elastic symmetry using the algebraic tensor decomposition of Walpole [19]. Euclidean
projection is presented in Section 5, with results for particular elastic symmetries summa-
rized in Appendix A. Before discussing the closest tensors using the logarithmic norms,
methods to evaluate the exponential and other functions of elasticity tensors are first de-
scribed in Section 6. Section 7 discusses the closest tensors using the logarithmic norms,
with particular attention given to isotropy and cubic symmetry as the ta rget symmetries.
Numerical examples are given in Section 8, and final conclusions in Section 9.
2 Notation
Elasticity tensors relate stress T and infinitesimal strain E linearly according to
T = CE, E = ST, (1)
where C and S denote the fourth-order stiffness and compliance tensors, respectively.
They satisfy C S = S C = I, the identity. Although we are concerned primarily with
fourth-order tensors in 3-dimensional space, calculation and presentation are sometimes
better performed using second-order tensors in 6-dimensions. Accordingly we define

Moakher and Norris February 4, 2008 4
S(d, r) as the space of symmetric tensors of order r in d−dimensions. Elasticity ten-
sors, denoted by Ela ⊂ S(3, 4), are positive definite, i.e., A ∈ Ela if hB, ABi > 0 for
all nonzero B ∈ S(3, 2). Components are defined relative to the basis triad {e
1
, e
2
, e
3
};
thus, a = a
j
e
j
, A = A
ij
e
i
⊗ e
j
, and A = A
ijkl
e
i
⊗ e
j
⊗ e
k
⊗ e
l
, where the summation
convention is assumed over 1, 2, 3 for lower case subscripts. Symmetry of second-order
(r = 2) tensors implies A
ij
= A
ji
, while for A ∈ S(3, 4) the elements satisfy
A
ijkl
= A
jikl
= A
ijlk
, A
ijkl
= A
klij
. (2)
The first pair of identities reflects the symmetry of the stress and the strain, while the last
one is a consequence o f the assumed existence of a strain energy function, and consequently
elasticity tensors have at most 21 independent components.
Throughout the paper lower case Latin, upper case Latin and “ghostscript” indicate
respectively 3−dimensional vectors and tensors of order 2 and 4; e.g., vector b, A ∈
S(3, 2), A ∈ S(3, 4). The basis vectors are assumed orthonormal, e
i
· e
j
= δ
ij
, so that
products of tensors are defined by summation over pairs of indices: (AB)
ij
= A
ik
B
kj
and
( A B)
ijkl
= A
ijpq
B
pqkl
. The inner product for tensors is defined as
hu, vi = tr(uv), (3)
where tr A = A
ii
, and for elasticity tensors, tr A = A
ijij
. The norm of a tensor is
kuk ≡ hu, ui
1/2
. (4)
We ta ke advantage of the well known isomorphisms between S(3, 2) and S(6 , 1), and
between S(3, 4) and S(6, 2). Thus, fourth-order elasticity tensors in 3 dimensions are
equivalent to second-order symmetric tensor of 6 dimensions [20], with properties BA ↔
b
B
ˆ
a, A B ↔
b
A
b
B, and hA, Bi = h
b
A,
b
Bi. Vectors and second-order tensors in six dimension
are distinguished by a hat, e.g., vector
ˆ
a,
b
A ∈ S(6, 2). Components are defined relative to
the orthonormal sextet {
ˆ
e
I
, I = 1, 2, . . . , 6},
ˆ
e
I
·
ˆ
e
J
= δ
IJ
, by
ˆ
a = a
I
ˆ
e
I
,
b
A =
b
A
IJ
ˆ
e
I
⊗
ˆ
e
J
,
with the summation convention over 1, 2, . . . , 6 f or capital subscripts. Also, tr
b
A =
b
A
II
.
The connection between S(6, 1) and S(3, 2) is made concrete by relating the basis vectors:
6 − vector
ˆ
e
1
ˆ
e
4
ˆ
e
2
ˆ
e
5
ˆ
e
3
ˆ
e
6
↔
dyadic
e
1
⊗ e
1
1
√
2
(e
2
⊗ e
3
+ e
3
⊗ e
2
)
e
2
⊗ e
2
1
√
2
(e
3
⊗ e
1
+ e
1
⊗ e
3
)
e
3
⊗ e
3
1
√
2
(e
1
⊗ e
2
+ e
2
⊗ e
1
)
(5)
This implies a unique
b
A ∈ S(6, 2) for each A ∈ S(3, 4), and vice versa. Let C ∈ Ela be
the tensor of elastic stiffness, usually defined by the Voigt notation: C
ijkl
≡ c
IJ
, where I
or J = 1, 2, 3, 4 , 5, 6 correspond to ij or kl = 11, 22, 33, 23, 13, 12, respectively, and c
IJ
are

Moakher and Norris February 4, 2008 5
the elastic moduli in the Voigt notation [21], i.e.,


















c
11
c
12
c
13
c
14
c
15
c
16
c
22
c
23
c
24
c
25
c
26
c
33
c
34
c
35
c
36
c
44
c
45
c
46
S Y M c
55
c
56
c
66


















. (6)
The isomorphism implies that the associated matrix
b
C ∈ S(6, 2 ) is positive definite and
has elements
b
C ≡



















c
11
c
12
c
13
2
1
2
c
14
2
1
2
c
15
2
1
2
c
16
c
12
c
22
c
23
2
1
2
c
24
2
1
2
c
25
2
1
2
c
26
c
13
c
23
c
33
2
1
2
c
34
2
1
2
c
35
2
1
2
c
36
2
1
2
c
14
2
1
2
c
24
2
1
2
c
34
2c
44
2c
45
2c
46
2
1
2
c
15
2
1
2
c
25
2
1
2
c
35
2c
45
2c
55
2c
56
2
1
2
c
16
2
1
2
c
26
2
1
2
c
36
2c
46
2c
56
2c
66



















. (7)
The spectral decomposition of elasticity tensors can be expressed in both the fourth-
order and second-order notations [12, 20],
C =
6
X
I=1
Λ
I
N
I
⊗ N
I
↔
b
C =
6
X
I=1
Λ
I
ˆ
n
I
⊗
ˆ
n
I
, (8)
where Λ
I
and
ˆ
n
I
, I = 1, 2, . . . , 6 are the eigenvalues and eigenvectors of the matrix
b
C, N
I
are t he associated dyadics, with h
ˆ
n
I
,
ˆ
n
J
i = hN
I
, N
J
i = δ
IJ
. Also, Λ
I
> 0 by virtue of the
positive definite nature of the strain energy.
The elastic moduli C can also be expressed as a vector with 21 elements [14],
X = (x
1
, x
2
, x
3
, x
4
, x
5
, x
6
, x
7
, x
8
, x
9
, x
10
, x
11
, x
12
, x
13
, x
14
, x
15
, x
16
, x
17
, x
18
, x
19
, x
20
, x
21
)
t
=

c
11
, c
22
, c
33
,
√
2c
23
,
√
2c
13
,
√
2c
12
, 2c
44
, 2c
55
, 2c
66
, 2c
14
, 2c
25
, 2c
36
,
2c
34
, 2c
15
, 2c
26
, 2c
24
, 2c
35
, 2c
16
, 2
√
2c
56
, 2
√
2c
46
, 2
√
2c
45

t
=

ˆc
11
, ˆc
22
, ˆc
33
,
√
2ˆc
23
,
√
2ˆc
13
,
√
2ˆc
12
, ˆc
44
, ˆc
55
, ˆc
66
,
√
2ˆc
14
,
√
2ˆc
25
,
√
2ˆc
36
,
√
2ˆc
34
,
√
2ˆc
15
,
√
2ˆc
26
,
√
2ˆc
24
,
√
2ˆc
35
,
√
2ˆc
16
,
√
2ˆc
56
,
√
2ˆc
46
,
√
2ˆc
45

t
.
(9)

Moakher and Norris February 4, 2008 6
The
√
2’s ensure that the inner product preserves the norm of the elastic moduli, whether
it is a tensor, matrix or vector. Thus, the norm of an elasticity tensor can be expressed
in a various ways depending o n how C is represented,
kCk
2
≡ hC, Ci = C
ijkl
C
ijkl
= bc
IJ
bc
IJ
= Λ
I
Λ
I
= kXk
2
. (10)
3 Elastic distance functions
We consider three metrics for Ela: the Euclidean or Frobenius metric d
F
, the log-
Euclidean norm d
L
[2] and the Riemannian metric d
R
[1]. They are defined for any
pair of elasticity tensors as
d
F
( C
1
, C
2
) = kC
1
− C
2
k, (11a)
d
L
( C
1
, C
2
) = kLog( C
1
) − Lo g( C
2
)k, (11b)
d
R
( C
1
, C
2
) =



Log( C
−1/2
1
C
2
C
−1/2
1
)



. (11c)
The Riemannian distance d
R
is related to the expo nential map induced by the scalar
product on the tangent space to Ela at C [1, 22]. The log-Euclidean metric d
L
can be
motivated by the following definition of tensor multiplication [2] :
C
1
⊙ C
2
≡ exp (Log( C
1
) + Log( C
2
)) . (12)
The ⊙ product preserves symmetry and positive definiteness, unlike normal multiplica-
tion. The three metrics have the usual at tr ibutes of a distance function d:
1. non-negative, d( C
1
, C
2
) ≥ 0 with equality iff C
1
= C
2
,
2. symmetric in the arguments, d( C
1
, C
2
) = d( C
2
, C
1
),
3. invariant under a change of basis, d( C
′
1
, C
′
2
) = d( C
1
, C
2
) for all proper orthogonal
coordinate transformations {e
1
, e
2
, e
3
} → {e
′
1
, e
′
2
, e
′
3
}, and
4. it satisfies the triangle inequality, d( C
1
, C
3
) ≤ d( C
1
, C
2
) + d( C
2
, C
3
).
The Riemannian and log-Euclidean distances possess the additional properties that
they are invariant under inversion and (positive) scalar multiplication
d
L,R
( C
−1
1
, C
−1
2
) = d
L,R
( C
1
, C
2
), and d
L,R
(a C, a C) = d
L,R
( C
1
, C
2
), a > 0. (13)
The following inequality between the lo garithmic distance functions is a consequence of
the metric increasing property of the exponential [23]
d
L
( C
1
, C
2
) ≤ d
R
( C
1
, C
2
). (14)
We also have (see [23] for the inequality)
d
L
( C
b
1
, C
b
2
) = |b|d
L
( C
1
, C
2
) , b ∈ R (15a)
d
R
( C
b
1
, C
b
2
) ≤ b d
R
( C
1
, C
2
) , b ∈ [0, 1], (15b)

Moakher and Norris February 4, 2008 7
and the Riemannian distance is invariant under congruent transformations, i.e.,
d
R
( T C
1
T
t
, T C
2
T
t
) = d
R
( C
1
, C
2
), ∀T invertible. (16)
Distance functions satisfying (13) are called bi-invariant, a property that makes the
Riemannian and log-Euclidean distances consistent metrics for elasticity tensors. The
Frobenius norm, not being invariant under inversion, gives different results depending
on whether the stiffness or compliance is considered. Other metric functions may be
considered. For instance, the Kullback-Leibler metric [1] is invariant under inversion and
congruent transformation but does not satisfy the triangle inequality, which we take here
as prerequisite for consideration as a distance function.
Each distance function has a geometrical interpretation. For instance, t he midpoint
between C
1
and C
2
is defined as the unique C
3
such that d( C
1
, C
3
) = d( C
3
, C
2
) =
1
2
d( C
1
, C
2
). Using d
F
, the midpoint is the vector halfway between the 21−vectors X
1
and
X
2
as defined by (9), i.e., C
3
=
1
2
( C
1
+ C
2
). The midpoint using d
L
is C
3
= ( C
1
⊙ C
2
)
1/2
.
The midpoint with d
R
is [24] C
3
= C
1
( C
−1
1
C
2
)
1/2
which can be expressed in several other
ways, see [1]. More generally, the midpoint is the value of the geodesic C(t) at t =
1
2
,
where
C(t) =





(1 − t) C
1
+ t C
2
, Frobenius,
exp ((1 −t) Log( C
1
) + t Log( C
2
)) , log −Euclidean,
C
1
exp

t Log( C
−1
1
C
2
)

, Riemannian,
0 ≤ t ≤ 1. (17)
4 Elasticity tensors of the different symmetry classes
Elasticity tensors for the different symmetry classes are described in this section. The
choice of representation of Ela is important for discussing the closest approximants, par-
ticularly for the bi-invariant metrics. It should preferably be independent of the coordinate
system, although at the same time, practical application is normally in terms of the Voigt
notation, so the format should not deviate too far from t his. We begin with a review of
alternative representatio ns for elastic moduli.
4.1 Representation of elasticity tensors
There are many representations of Ela in addition to the standard Voigt notation of eq.
(6). The 21−dimensional vector format [14] is useful for some applications, including the
Euclidean proj ection, as we show later. However, these two representations depend upon
the co ordinate system. Among the coor dinate-f r ee forms of Ela that can be identified in
the literature we distinguish (i) spectral decomposition, (ii) algebraic decomposition, (iii)
groups and reflection symmetries, (iv) harmonic decomposition, (v) integrity bases.
The first, (i), spectral decomposition dates back to Kelvin [25]. The idea is simple:
the elasticity tensor operates on the six-dimensional space of symmetric second-order
tensors, and therefore has a six-dimensional spectral form. The associated eigenvalues
are called the Kelvin moduli [12, 13]. Recent related developments are due to Rychlewski
[12] and Mehrabadi and Cowin [20, 13], who independently rediscovered and extended
Kelvin’s approach, see also [26, 27]. The associated six-dimensional tensor representation
will be used here for practical implementation. (ii) Tensor functions operating on Ela
can be greatly simplified using the irreducible tensor algebra proposed by Walp ole [19],

Moakher and Norris February 4, 2008 8
and independently by Kunin [28]. This procedure provides the most efficient means of
representing elastic tensors of a given symmetry class, especially those of high symmetry.
We adopt this method to develop most of the results here. (iii) The group of rot ations
associated with elastic symmetry provides an irreducible representation [29]. There are
various related ways of considering elasticity tensors in terms of rotational group properties
of tensors, e.g., ba sed on Cartan decomposition [30], complex vectors and tensors [31, 32],
and subgroups of O(3) [33, 34]. These ideas are closely related to definitions of elastic
symmetry in terms of a single symmetry element: reflection about a plane. The necessary
algebra and the relationship to the more conventional crystallographic symmetry elements
are described in detail by Cowin and Mehrabadi [35 ]. Comparison between the rotation-
based approach and that using symmetry planes is provided by Chadwick et al. [18].
(iv) Backus [36] proposed a representation of Ela in terms of harmonic tensors. These
are based on an isomorphism between the space of homogeneous harmonic polynomials of
degree q and the space of totally symmetric tensors of order q. There has been considerable
interest in Backus’s representation [37, 38, 6, 39]. For instance, Baerheim and Helbig
[40] provide an orthonor mal decomposition of Ela in terms of harmonic tensors. (v)
Elasticity decomposition via integrity bases has been studied considerably [41, 42, 4 3,
44, 45]. An integrity basis is a set of polynomials, each invariant under the group of
symmetry transformations, such that any po lynomial function invariant under the group
is expressible as a polynomial in elements of the integrity basis. For instance, Tu [42] used
an integrity basis to construct five hierarchies of orthono rmal tensor bases which span the
space of elastic constants of all crystal systems. Any elastic tensor of order four possessing
certain symmetry may be decomposed into a sum of tensors of increasing symmetry. The
resulting decomposition has considerable similarity to the decomposition generated in
Section 5 using Walpo le’s irreducible elements with Euclidean projection. Among other
methods for decomposing elasticity tensors, we mention the scheme of Elata and Rubin
[46] who use a set of six vectors related to a regular icosahedron to form a basis for Ela.
This basis naturally splits a tensor into deviatoric and non-deviatoric parts.
We use Walpole’s [19] alg ebraic represent ation as it provides a consistent and straight-
forward means to define projections of f ourth-order tensors onto the given elastic sym-
metry. This representation is better for our purposes than the spectral decomposition,
since it is independent of the elastic moduli (i.e., no distributors [13]) and depends only
on the crystallographic orientation. It uses the notion of basis tensors, similar to but not
the same as for a vector space, which makes it very suitable for Euclidean projection. We
work mainly with fourth-o r der tensors directly, although the associated six-dimensional
matrix notation is also provided. The lat t er is simpler for purposes of computation, e.g.,
many of the matrix operations are easily implemented using MATLAB. We begin with
the highest elastic symmetry.
4.2 Isotropic system
A general isotro pic fourth-order tensor is given by
A = a J + b K, (18)
where J and K are the two linearly independent symmetric tensors defined by
J =
1
3
I ⊗ I , K = I − J . (19)

Moakher and Norris February 4, 2008 9
The compo nent forms follow from I
ijkl
=
1
2
(δ
ik
δ
jl
+ δ
il
δ
jk
) and J
ijkl
=
1
3
δ
ij
δ
kl
. Note that
the tensors J and K sum to the identity,
I = J + K, (20)
and they satisfy the multiplication table
J
2
= J, K
2
= K, JK = KJ = O. (21)
The Euclidean lengths are kJk = 1, kKk =
√
5. Tensors J and K form, respectively, one
and five-dimensional Kelvin subspaces [13].
An important result (see [3, 4]) is that the closest isotropic elasticity tensor in the
Euclidean sense is
C
iso
= 3κ J + 2µ K , (22)
where
9κ = 3hC, Ji = ˆc
11
+ ˆc
22
+ ˆc
33
+ 2ˆc
12
+ 2ˆc
13
+ 2ˆc
23
,
30µ = 3 hC, Ki = 2(ˆc
11
+ ˆc
22
+ ˆc
33
− ˆc
12
− ˆc
23
− ˆc
31
) + 3 (ˆc
44
+ ˆc
55
+ ˆc
66
) .
(23)
Gazis e t al. [3] obtained (22) using methods similar to those we will generalize in Section 5.
Fedorov actually f ound κ and µ using an apparently different approach - the minimization
of the mean square difference in the acoustical or Christoffel matrices for the original a nd
isotropic systems. However, it can be shown [5] that the generalization of Fedorov’s
approach to other symmetries is identical to the Euclidean minimization. The moduli
(23) will be derived later in the context of the general theory for Euclidean projection.
4.3 Cubic system
Let a, b and c be three mutually orthogonal unit vectors that describe the three crystal-
lographic directions of a cubic medium. We introduce the second-order tensors
U =
1
√
2
(a ⊗ c + c ⊗a), V =
1
√
2
(b ⊗ c + c ⊗ b), W =
1
√
2
(a ⊗ b + b ⊗a),
X =
1
√
2
(c ⊗ c − a ⊗a), Y =
1
√
2
(b ⊗ b − c ⊗ c), Z =
1
√
2
(a ⊗ a − b ⊗b),
(24)
and fourth-order tensors L and M defined by them,
L = U ⊗U + V ⊗ V + W ⊗ W, M =
2
3
(X ⊗ X + Y ⊗Y + Z ⊗Z). (25)
The tensors J, L and M sum up to the identity tensor, and L and M partition K, i.e.,
I = J + L + M, K = L + M . (26)
The multiplication table for cubic tensors is
J
2
= J, L
2
= L, M
2
= M, JL = LJ = JM = MJ = LM = ML = O, (27)
and the Euclidean lengths are kLk =
√
3, kMk =
√
2.
A general symmetric fourth-order tensor for a cubic media is given by a linear combi-
nation of the three linearly independent tensors J, L and M
A = a J + b L + c M. (28)
The tensors J, L and M form, respectively, one, three and two-dimensional Kelvin sub-
spaces [13].

Moakher and Norris February 4, 2008 10
4.4 Transversely isotropic system
Assume that the unit vector c characterizes the preferred direction of a transversely
isotropic medium. Let P and Q be the second-order tensors
P = c ⊗ c, Q =
1
√
2
(I − c ⊗ c), (29)
and define the six linearly independent fourth- order elementary tensors
E
1
= P ⊗ P, E
2
= Q ⊗ Q, E
3
= P ⊗ Q, E
4
= Q ⊗ P,
F = W ⊗ W + Z ⊗ Z, G = U ⊗ U + V ⊗V.
(30)
Both E
1
and E
2
are symmetric but the tensors E
3
and E
4
are not, although they are
the transpose of one another and so their sum is a symmetric tensor. These asymmetric
tensors are introduced because the set {E
1
, E
2
, E
3
+ E
4
} is not closed under tensor
multiplication. The multiplication table for the six elementary tensors is
E
1
E
2
E
3
E
4
F G
E
1
E
1
O E
3
O O O
E
2
O E
2
O E
4
O O
E
3
O E
3
O E
1
O O
E
4
E
4
O E
2
O O O
F O O O O F O
G O O O O O G
(31)
The tensors E
1
, E
2
, F and G sum up to the identity,
I = E
1
+ E
2
+ F + G. (32)
The Euclidean lengths are kE
1
k = kE
2
k = 1, kE
3
+ E
4
k = kFk = kGk =
√
2.
A general symmetric fourth-order tensor of transversely isotropic symmetry is given
by the 5-parameter linear combination
A = a E
1
+ b E
2
+ c( E
3
+ E
4
) + f F + g G. (33)
Note that the base tensors do not correspond to the Kelvin modes, although some can
be identified as such, e.g., G is a two-dimensional Kelvin subspace [13]. The three bases
tensors {E
1
, E
2
, E
3
+ E
4
} together define a two-dimensional subspace.
4.5 Tetragonal system
Nine base tensors are required: {E
1
, E
2
, E
3
, E
4
, F
1
, F
2
, F
3
, F
4
, G}, where the new
tensors are a symmetric pair F
1
, F
2
, and a pair that are mutual transposes, F
3
, F
4
,
F
1
= W ⊗W, F
2
= Z ⊗ Z, F
3
= W ⊗Z, F
4
= Z ⊗ W . (34)
In comparison with the transversely isotropic system, F is replaced by the symmetric pair
of tensors, i.e., F = F
1
+ F
2
, a nd the decomposition of the identity is
I = E
1
+ E
2
+ F
1
+ F
2
+ G. (35)

Moakher and Norris February 4, 2008 11
The multiplication table of the nine elementary tensors is
E
1
E
2
E
3
E
4
F
1
F
2
F
3
F
4
G
E
1
E
1
O E
3
O O O O O O
E
2
O E
2
O E
4
O O O O O
E
3
O E
3
O E
1
O O O O O
E
4
E
4
O E
2
O O O O O O
F
1
O O O O F
1
O F
3
O O
F
2
O O O O O F
2
O F
4
O
F
3
O O O O O F
3
O F
1
O
F
4
O O O O F
4
O F
2
O O
G O O O O O O O O G
(36)
Under multiplication the E
i
’s and F
i
’s decouple fr om one another and from G. Further-
more, the algebra of the F
i
’s is similar to that of the E
i
’s. The Euclidean lengths o f the
new tensors are kF
1
k = kF
2
k = 1, kF
3
+ F
4
k =
√
2.
A general symmetric fourth-order tensor of tetragonal symmetry is given by the
6−parameter combination
A = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p F
1
+ q F
2
+ r( F
3
+ F
4
) + g G. (37)
Tetrag onal symmetry is often represented by 6 rather than 7 independent parameters.
Fedorov [4] pointed out that transformation by rotation about the c axis by angle θ yields
zero for the transformed coefficient r if (see eq. (9.7) in [4])
tan 4θ = r/(q − p) . (38)
The transformation (38) depends on knowledge of the coefficients. However, it is assumed
here that the only properties of the tetragonal symmetry available a priori are the orthog-
onal crystal axes a, b and c. The reduction from 7 to 6 parameters can be achieved after
the effective tetragonal material is found, but the axes of t he final 6-parameter material
depend upon the initial moduli. Hence, it is important to retain the seven-parameter
representatio n (37).
4.6 Trigonal system
Trig onal symmetry is characterized by three planes of reflection symmetry with normals
coplanar and at 120
◦
to one another. This symmetry class is also known as hexagonal [19]
but we refer to it as trigonal since hexagonal symmetry is often used synonymously with
transverse isotropy. It is advantageous to choose non-orthogonal basis vectors a
′
, b
′
, c
with a
′
and b
′
normals to two of the planes and c perpendicular to them (the pair a
′
and b
′
may also be represented in terms of the or t honor mal vectors via, for instance,
a
′
= a, b
′
= −
1
2
a +
√
3
2
b). In order t o describe symmetric fourth- order tensors of trigonal
symmetry, fo llowing [19] we introduce the second-order tensors
S =
q
2
3
(a
′
⊗ b
′
+ b
′
⊗ a
′
+ a
′
⊗ a
′
), T =
q
2
3
(a
′
⊗ b
′
+ b
′
⊗ a
′
+ b
′
⊗ b
′
), (39a)
U
′
=
1
√
2
(a
′
⊗ c + c ⊗ a
′
), V
′
= −
1
√
2
(b
′
⊗ c + c ⊗ b
′
), (39b)

Moakher and Norris February 4, 2008 12
the two symmetric f ourth-order t ensors
R
1
=
4
3
(S ⊗ S + T ⊗ T −
1
2
S ⊗ T −
1
2
T ⊗ S) , (40a)
R
2
=
4
3
(U
′
⊗ U
′
+ V
′
⊗ V
′
−
1
2
U
′
⊗ V
′
−
1
2
V
′
⊗ U
′
) , (40b)
and two pairs of mutually t ransp ose tensors:
R
3
=
4
3
(S ⊗ U
′
+ T ⊗V
′
−
1
2
T ⊗ U
′
−
1
2
S ⊗ V
′
) , (41a)
R
4
=
4
3
(U
′
⊗ S + V
′
⊗ T −
1
2
U
′
⊗ T −
1
2
V
′
⊗ S) , (41b)
and
R
5
=
2
√
3
(S ⊗ V
′
− T ⊗ U
′
) , R
6
=
2
√
3
(V
′
⊗ S −U
′
⊗ T) . (41c)
To make the R
i
’s closed under multiplication, we need to further introduce two skew-
symmetric tensors
R
7
=
2
√
3
(T ⊗ S − S ⊗T) , R
8
=
2
√
3
(U
′
⊗ V
′
− V
′
⊗ U
′
) . (42)
Thus, E
i
R
j
= R
j
E
i
= O for i = 1, 2, . . . , 4 and j = 1, 2, . . . , 8, and the multiplication
table is
2
R
1
R
2
R
3
R
4
R
5
R
6
R
7
R
8
R
1
R
1
O R
3
O R
5
O R
7
O
R
2
O R
2
O R
4
O R
6
O R
8
R
3
O R
3
O R
1
O R
7
O R
5
R
4
R
4
O R
2
O R
8
O R
6
O
R
5
O R
5
O −R
7
O R
1
O −R
3
R
6
R
6
O −R
8
O R
2
O −R
4
O
R
7
R
7
O −R
5
O R
3
O −R
1
O
R
8
O R
8
O −R
6
O R
4
O −R
2
(43)
The algebra of the R
i
’s is equivalent to that of 2 × 2 complex matrices [19].
We note that R
1
= F and R
2
= G, and hence the decompo sition for the identity
tensor is
I = E
1
+ E
2
+ R
1
+ R
2
. (44)
The Euclidean lengths of the new tensors are kR
1
k = kR
2
k =
√
2, kR
3
+ R
4
k = kR
5
+
R
6
k = 2.
A general symmetric fourth-order tensor of trigonal symmetry is given by the linear
combination
A = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p R
1
+ q R
2
+ r( R
3
+ R
4
) + s( R
5
+ R
6
). (45)
4.7 Rhombic, Monoclinic and Triclinic systems
The algebra for orthorhombic (equivalently orthotropic) symmetry is the same as that of
3 × 3 real matrices plus three real numbers. For monoclinic symmetry it is the same as
the algebra of pairs of 4 ×4 and 2 ×2 real matrices. The lowest symmetry, triclinic or no
symmetry, has algebra the same as that of 6 ×6 real matrices. These low symmetries are
probably better defined in terms of their group properties, which also serves as efficient
means for Euclidean projection, as described in Section 5.
2
The multiplication table in [19] contains some typographical errors; specifically the elements (3,8),
(4,5), (5,8), (6,3), (8,4) and (8 ,6) in the original need to be multiplied by −1.

Moakher and Norris February 4, 2008 13
4.8 Six-dimensional representation
An equivalent 6-dimensional matrix format is described for the various elementary tensors.
For each symmetry class the 6-dimensional tensors possess the same algebraic properties
as the f ourth-order t ensors, with the same multiplication tables and Euclidean lengths.
The 6-dimensional vectors {
ˆ
a,
ˆ
b,
ˆ
c} represent the second-order tensors {a ⊗ a, b ⊗b,
c ⊗ c} that corresp ond to the a xes {a, b, c}, where according to eq. (5),
a = a
1
e
1
+ a
2
e
2
+ a
3
e
3
⇒
ˆ
a =

a
2
1
, a
2
2
, a
2
3
,
√
2a
2
a
3
,
√
2a
3
a
1
,
√
2a
1
a
2

t
, etc. (46)
The associated 6 × 6 base matrices are given next.
4.8.1 Isotropic system
The six-dimensional analogs of I, J and K are
b
I, the 6 × 6 identity matrix, and
b
J,
b
K,
where
b
J =
1
3
(
ˆ
e
1
+
ˆ
e
2
+
ˆ
e
3
)(
ˆ
e
1
+
ˆ
e
2
+
ˆ
e
3
)
t
,
b
K =
b
I −
b
J . (47)
4.8.2 Cubic system
The analogs of L and M are
b
L =
b
I −
ˆ
a
ˆ
a
t
−
ˆ
b
ˆ
b
t
−
ˆ
c
ˆ
c
t
,
c
M =
b
K −
b
L. (48)
4.8.3 Transversely isotropic system
Define the 6−vectors
ˆ
p =
ˆ
c,
ˆ
q =
1
√
2
(
ˆ
a +
ˆ
b),
ˆ
z =
1
√
2
(
ˆ
a −
ˆ
b),
ˆ
w =

√
2a
1
b
1
,
√
2a
2
b
2
, a
√
2
3
b
3
, (a
2
b
3
+ a
3
b
2
), (a
3
b
1
+ a
1
b
3
), (a
1
b
2
+ a
2
b
1
)

t
,
(49)
then the 6-dimensional matrices for the tensors E
1
, E
2
, E
3
, E
4
, F and G are
b
E
1
=
ˆ
p
ˆ
p
t
,
b
E
2
=
ˆ
q
ˆ
q
t
,
b
E
3
=
ˆ
p
ˆ
q
t
,
b
E
4
=
ˆ
q
ˆ
p
t
,
b
F =
ˆ
w
ˆ
w
t
+
ˆ
z
ˆ
z
t
,
b
G =
b
L −
ˆ
w
ˆ
w
t
.
(50)
4.8.4 Tetragonal system
The six-dimensional representation is as for TI, with the addition
b
F
1
=
ˆ
w
ˆ
w
t
,
b
F
2
=
ˆ
z
ˆ
z
t
b
F
3
=
ˆ
w
ˆ
z
t
,
b
F
4
=
ˆ
z
ˆ
w
t
. (51)
The trigonal system can be treated in the same manner using equations similar to ( 46)
for
ˆ
a and ( 49) for
ˆ
w, but the details are omitted for brevity.

Moakher and Norris February 4, 2008 14
5 Euclidean projection
Euclidean projection is an essential ingredient for determining the closest tensors in bot h
the Euclidean and the log-Euclidean metrics. It is defined in abstract terms in the next
subsection, with the remainder of the section devoted to explicit procedures for the pro-
jection.
5.1 Definition of the projection
We wish to find the tensor C
sym
of a specific symmetry class which minimizes the Eu-
clidean distance kC − C
sym
k o f an elasticity tensor of arbitrary symmetry, C, from the
particular symmetry. The solution is a Euclidean decomposition
C = C
sym
+ C
⊥sym
, (52)
where C
sym
possesses the symmetries appropriate to the symmetry class considered. The
complement, or residue [3], is orthogonal to C
sym
,
hC
sym
, C
⊥sym
i = 0 , (53)
and hence
kCk
2
= kC
sym
k
2
+ kC
⊥sym
k
2
, (54a)
kC − C
sym
k
2
= kC
⊥sym
k
2
. (54b)
We illustrate the recursive nature of the Euclidean projection for the special case of
isotropy. Any elasticity may be partitioned into orthogonal components a s
C = C
iso
+ C
⊥iso
= C
iso
+ C
cub/iso
+ C
tet/cub
+ C
ort/tet
+ C
mon/ort
+ C
⊥mon
, (55)
where
C
sym B/A
≡ C
sym B
− C
sym A
, (56)
with sym A of higher symmetry than sym B. Equation (55) partitions C using the path
from no symmetry to isotropy via cubic symmetry, see Figure 1. An alternative decom-
position via hexagonal, gives
C = C
iso
+ C
hex/iso
+ C
tet/hex
+ C
ort/tet
+ C
mon/ort
+ C
⊥mon
. (57)
The different paths in Figure 1 were identified by Gazis et al. [3] and discussed more
recently in greater detail by Chadwick et al. [18]. The idea of sequent ia l decomposition
in the Euclidean no rm is not new, e.g., [42, 14], but the explicit projection and complement
have not been represented previously.
Details of the recursive Euclidean partition of C are given in the Apendix. For in-
stance, eq. (A.30) represents kCk
2
using three different routes fro m triclinic to isotropic.
In particular, the length of the isotropic projection is given by
kC
iso
k
2
= 9κ
2
+ 20µ
2
, (58)
where κ and µ are defined in (23), and the distance of C from isotropy is given by
kC
⊥iso
k
2
= kCk
2
− 9κ
2
− 20µ
2
. (59)

Moakher and Norris February 4, 2008 15
It appears that Fedorov [4] was the first to show that the distance from isotropy is mini-
mized if the isotropic tensor is as given in (22).
The remainder of this Section details three alternative projection methods using basis
tensors, using symmetry planes or groups, and using 21-dimensional vectors. The pro-
jection operators are valid for all symmetries, but the method based on symmetry planes
becomes more complicated for the higher symmetries. Conversely, the projection operator
using basis tensors becomes simpler at the higher symmetries. The 21-dimensional vector
procedure is straightforward but is coo r dinate dependent.
5.2 Projection using a tensor basis
We assume that the set of tensors {V
i
∈ Ela} form a linearly independent basis for the
symmetry sym in the sense that any elasticity tensor of that symmetry may be expressed
uniquely in terms of N linearly independent tensors V
1
, V
2
, . . . V
N
, where 2 ≤ N ≤ 13
is the dimension of the space for the material symmetry. We have seen the explicit form
of several sets of basis tensors in Section 4 , e.g., N = 2 for isotropic elasticity. N = 13
corresponds to monoclinic, which is t he lowest symmetry apart from triclinic (technically
N = 21) which is no symmetry. The N elements of the orthogonal basis are assumed to
be independent o f the elasticity itself, i.e., they do not require “elasticity distributors”,
which are necessary for the spectral decomposition of Ela [13]. They depend only on
the choice of the planes and/or axes which define the gro up G of symmetry preserving
transformations (see below). The precise form of the basis tensors is irrelevant, all that
is required is that they be linearly independent in the sense of elements of a vector space,
and consequently any tensor with the desired symmetry can be expressed
C
sym
=
N
X
i=1
a
i
V
i
. (60)
Minimizing d
2
F
( C, C
sym
) with respect to the coefficients a
i
implies
∂
∂a
i
kC − C
sym
k
2
= 2hC, V
i
i − 2hC
sym
, V
i
i = 0. (61)
Hence,
N
X
j=1
a
j
hV
j
, V
i
i = hC, V
i
i. (62)
Let D be the N × N symmetric matrix with elements
D
ij
≡ hV
i
, V
j
i. (63)
This is invertible by virtue of the linear independence of the basis tensors, and so
a
i
=
N
X
j=1
D
−1
ij
hV
j
, Ci. (64)

Moakher and Norris February 4, 2008 16
Noting t hat C
⊥sym
= C − C
sym
, we have
hC
⊥sym
, C
sym
i = hC −
N
X
i=1
a
i
V
i
,
N
X
j=1
a
j
V
j
i
=
N
X
j=1
a
j
hC, V
j
i −
N
X
i=1
N
X
j=1
a
i
a
j
hV
i
, V
j
i = 0. (65)
Hence, the partition (60) satisfies the fundamental projection property (52). It is a simple
exercise to show that C
sym
of eq. ( 60) minimizes the Euclidean distance kC − C
sym
k iff
the co efficients a r e given by eq. (64).
The Euclidean projection can also be expressed in terms of tensors of order eight,
P
sym
, such that
C
sym
= P
sym
C, ⇔ C
sym
ijkl
= P
sym
ijklmnrs
C
mnrs
. (66)
The projector can be expressed in terms of dyadics of basis tensors,
P
sym
=
N
X
i,j=1
D
−1
ij
V
i
⊗ V
j
. (67)
We now apply these ideas to particular symmetries, focusing only on the higher sym-
metries since the lower ones (monoclinic, orthorhombic) are relatively trivial and can be
handled using the projector of eq. (71).
5.3 Projection operators for particular symmetries
The projector for the symmetries described in Section 4 follow naturally from the vectorial
formulation. In particular, we note that the D is diagonal in each case, and
P
iso
= J ⊗ J +
1
5
K ⊗ K, (68a)
P
cub
= J ⊗ J +
1
3
L ⊗ L +
1
2
M ⊗ M, (68b)
P
hex
= E
1
⊗ E
1
+ E
2
⊗ E
2
+
1
2
( E
3
+ E
4
) ⊗ ( E
3
+ E
4
)
+
1
2
F ⊗ F +
1
2
G ⊗ G, (68c)
P
tet
= E
1
⊗ E
1
+ E
2
⊗ E
2
+
1
2
( E
3
+ E
4
) ⊗ ( E
3
+ E
4
)
+ F
1
⊗ F
1
+ F
2
⊗ F
2
+
1
2
( F
3
+ F
4
) ⊗ ( F
3
+ F
4
) +
1
2
G ⊗ G, (68d)
P
trig
= E
1
⊗ E
1
+ E
2
⊗ E
2
+
1
2
( E
3
+ E
4
) ⊗ ( E
3
+ E
4
) +
1
2
R
1
⊗ R
1
+
1
2
R
2
⊗ R
2
+
1
4
( R
3
+ R
4
) ⊗ ( R
3
+ R
4
) +
1
4
( R
5
+ R
6
) ⊗ ( R
5
+ R
6
) . (68e)
Expressions can be g iven for the lower symmetries, but a s discussed, the number of elemen-
tary tensors involved is much larger. The form of the projectors in (68) are independent
of the crystal axes, and provide a co ordinate-free scheme for Euclidean projection. They
can be easily programmed using the 6-dimensional matrix representation.
If sym A > sym B, we may convert P
symB
into a form that yields C
sym B
and the
complement C
sym B/A
of (56). A general procedure for achieving this uses the fact that

Moakher and Norris February 4, 2008 17
basis tensors for a given symmetry are not unique, a nd accordingly, alternate for ms of
P
sym
may be found. The decomposition
P
sym B
= P
sym A
+ P
sym B/A
, (69)
can be found by a Gram-Schmid type of process. As an example, let sym A= iso and sym
B= cub, and consider the tensor basis for cubic symmetry {V
1
, V
2
, V
3
} = {J, K, L −
a M}, where a is a free parameter. Requiring that D is diagonal implies a = 3/2, and
consequently we obtain
P
cub
= J ⊗ J +
1
5
K ⊗ K +
1
30
(2 L −3 M) ⊗ (2 L − 3 M) = P
iso
+ P
cub/iso
. (70)
The analogous partition for isotropy and transverse isotropy is described in Appendix A,
with P
hex/iso
given in eq. (A.24).
5.4 Projection using the transformation group
Let G be the group of transformations for the material symmetry. G ⊂ SO(3), the
space of 3-dimensional special (or proper) orthogonal tra nsformations. Under the change
of basis associated with Q ∈ SO(3), an elasticity tensor transforms C → C
′
where
C
′
ijkl
= Q
ip
Q
jq
Q
kr
Q
ls
C
pqrs
. The projection is
C
sym
ijkl
=
1
n
n
X
m=1
Q
(m)
ip
Q
(m)
jq
Q
(m)
kr
Q
(m)
ls
C
pqrs
, (71)
where n is the order of the group G, and Q
(m)
, are elements of the group. The order
of the groups a r e as follows (see Table 6 of [35]): triclinic 1, monoclinic 2, trigonal 6,
orthorhombic 4, tetragonal 8, transverse isotropy ∞+2, and isotropy ∞
3
. The proj ection
defined by (71) was discussed by, among others, Gazis et al. [3], and has been applied to
ultrasonic data by Fran¸cois et al. [47].
The six-dimensional version of the projection (71) is
b
C
sym
=
1
n
n
X
m=1
b
Q
(m)
b
C
b
Q
(m)
t
, (72)
where
b
Q
(m)
∈ SO(6) is the orthog onal second-order tensor corresponding to Q
(m)
, satis-
fying
b
Q
b
Q
t
=
b
Q
t
b
Q =
b
I. A necessary and sufficient condition that
b
C
sym
has the correct
symmetry is that
b
Q
(j)
b
C
sym
b
Q
(j)
t
, corresponding to the transformation Q
(j)
, is also a mem-
ber of the symmetry class. Thus, for j = 1, 2, . . . , n,
b
Q
(j)
b
C
sym
b
Q
(j)
t
=
1
n
n
X
i=1
b
Q
(j)
b
Q
(i)
b
C
b
Q
(i)
t
b
Q
(j)
t
=
1
n
n
X
k=1
b
Q
(k)
b
C
b
Q
(k)
t
. (73)
The latter identity follows from
b
Q
(j)
b
Q
(i)
=
b
Q
(k)
and the fact that the
b
Q
(i)
form a group of
order n. This proves that C
sym
defined in eq. (71) is indeed an element of the symmetry
class.
The projection fo r mula (71) or (72) may be effected in several ways, e.g., by referring
to the extensive lists of the 6 × 6 matrices
b
Q
(i)
provided by Cowin and Mehrabadi [35].

Moakher and Norris February 4, 2008 18
We note that each Q
(i)
can be decompo sed into a product of reflection operators [3 5].
The fundamental operator is R(n) ≡ I − 2n ⊗ n, where the unit vector n is normal to
the plane. Chadwick e t al. [18] provide a complete description of these sets and groups.
It is also possible to represent the group of transformations in terms of elements each
corresponding t o a single rotation [35, 18 ]. The equivalence between reflection operator
and rotations follows from the identity Q(e, π) = −R(e), where Q(n, θ) ∈ SO(3) defines
rotation about n by angle θ.
We illustrate this general procedure with an example.
5.4.1 Example: Projection onto monoclinic symmetry
There is only one plane of symmetry [18], with normal c, say, and n = 2 g r oup elements
G = {I, −R(c)}. Using (71) we have
C
mon
ijkl
=
1
2
C
ijkl
+
1
2
R
ip
(c)R
jq
(c)R
kr
(c)R
ls
(c)C
pqrs
. (74)
For instance, let c = e
3
, and using the explicit fo r m of
b
R(e
3
) from eq. (54) o f [35] (where
it is called
b
R
(3)
),
b
R(e
3
) = diag(1, 1, 1, −1, −1, 1) , (75)
then (72) gives
b
C
mon
=
1
2
b
C +
1
2
b
R(e
3
)
b
C
b
R(e
3
) =


















ˆc
11
ˆc
12
ˆc
13
0 0 ˆc
16
ˆc
22
ˆc
23
0 0 ˆc
26
ˆc
33
0 0 ˆc
36
ˆc
44
ˆc
45
0
S Y M ˆc
55
0
ˆc
66


















. (76)
The example illustrates that the group projection method is practical if the order n
of the group G is not large. This is the case for monoclinic, orthorhombic and perhaps
trigonal systems, but even for these symmetries it is simpler to implement the algorithm
by machine. It is not practical for isotropy and transverse isotropy, each with an infinity
of symmetry planes. Application of (71) for cubic isotropy is unwieldy although has been
performed [47]. The explicit orthogonal basis metho d of ( 60) and (64) provides a simpler
procedure for the higher symmetries.
5.5 Projections using 21-dimensional vectors
The projection is simplest for this representation o f Ela b ecause the elastic moduli are
explicitly represented as a vector. The operator is a 21 × 21 matr ix P
sym
, a nd the closest
elasticity is
X
sym
= P
sym
X, (77)

Moakher and Norris February 4, 2008 19
where X is defined in (9). The var io us projection matrices can be read off from the
formulas of Appendix A. Thus, the projector for cubic symmetry is
P
cub
=


p
cub
0
9×12
0
12×9
0
12×12


, p
cub
=














1
3
1
3
1
3
0 0 0 0 0 0
1
3
1
3
1
3
0 0 0 0 0 0
1
3
1
3
1
3
0 0 0 0 0 0
0 0 0
1
3
1
3
1
3
0 0 0
0 0 0
1
3
1
3
1
3
0 0 0
0 0 0
1
3
1
3
1
3
0 0 0
0 0 0 0 0 0
1
3
1
3
1
3
0 0 0 0 0 0
1
3
1
3
1
3
0 0 0 0 0 0
1
3
1
3
1
3














. (78)
Browaeys and Chevrot [14] derived P
sym
for isotropy, transverse isotropy (hexagonal sym-
metry), t etragonal, orthorhombic a nd monoclinic symmetry. They f ound that in all cases
except monoclinic that the non-zero part of P
sym
reduces to a 9 × 9 matrix p
sym
, as in
(78). However, it should be noted that t his is not true fo r the tetragonal projector derived
here, see eq. (A.6), for reasons discussed in Section 4.5. Nor it is true for the trigonal
projector, see eq. (A.10) .
6 Exponential, logarithm and s qu are root of sym-
metric fourth-order tens ors
Unlike the Euclidean projection for the Frobenius norm, finding minimizers for the loga-
rithmic and Riemannian distance functions involves evaluation of analytic functions with
symmetric fourth-order tensor arguments. In this section we derive the necessary ex-
pressions with emphasis on exponential, logarithm and square root f unctions that will be
used to obtain the closest elastic tensors in section 7. For this purpose, let h denote the
analytic function given by its power series
h(x) =
∞
X
m=0
a
m
x
m
.
The higher symmetries are simpler, and are considered first.
6.1 Isotropic system
Using (20) and (21), for A defined in (18) we have
h( A) =
∞
X
m=0
a
m
(a J + b K)
m
= a
0
I +
∞
X
m=1
a
m
(a
m
J + b
m
K)
= a
0
I +
∞
X
m=0
a
m
(a
m
J + b
m
K) − a
0
( J + K) = h(a) J + h(b) K. (79)
It follows that
exp A = e
a
J + e
b
K. (80)

Moakher and Norris February 4, 2008 20
When a 6= 0 and b 6= 0, the symmetric tensor A is invertible and its inverse is
A
−1
=
1
a
J +
1
b
K. (81)
The tensor A is positive definite if a > 0 and b > 0, with square root given by
A
1/2
=
√
a J +
√
b K, (82)
and logarithm
log A = ln a J + ln b K. (83)
6.2 Cubic system
Using (26)
1
and (2 7), f or A defined in (28) we have
h( A) =
∞
X
m=0
a
0
(a J + b L + c M)
m
= a
0
I +
∞
X
m=1
a
m
(a
m
J + b
m
L + c
m
M)
= a
0
I +
∞
X
m=0
a
m
(a
m
J + b
m
L + c
m
M) − a
0
( J + L + M) = h(a) J + h(b) L + h(c) M.
(84)
In particular,
exp( A) = e
a
J + e
b
L + e
c
M. (85)
When a 6= 0, b 6= 0 and c 6= 0, the symmetric tensor A is invertible a nd its inverse is
given by
A
−1
=
1
a
J +
1
b
L +
1
c
M. (86)
If a > 0, b > 0 and c > 0 then A is positive definite. Its square root is
A
1/2
=
√
a J +
√
b L +
√
c M, (87)
and its lo garithm is given by
Log A = ln a J + ln b L + ln c M. (88)
6.3 Transversely isotropic system
Using (32) and (31), for A defined in (33) we have
h( A) =
∞
X
m=0
a
m
[a E
1
+ b E
2
+ c( E
3
+ E
4
) + f F + g G]
m
= a
0
I +
∞
X
m=1
a
m
[(a E
1
+ b E
2
+ c( E
3
+ E
4
))
m
+ f
m
F + g
m
G]
=
∞
X
m=0
a
m
(a E
1
+ b E
2
+ c( E
3
+ E
4
))
m
+ h(f) F + h(g) G − a
0
( F + G)
= h(0)( E
1
+ E
2
− I) + h(a E
1
+ b E
2
+ c( E
3
+ E
4
)) + h(f) F + h(g) G. ( 89)

Moakher and Norris February 4, 2008 21
Contrary to the two previous classes of material symmetries, the upper left 4 ×4 multipli-
cation table for the elementary tensors of a transversely isotropic medium is not diagonal.
This fact makes the algebra of transversely isotropic tensors a bit more difficult. Fortu-
nately, Walpole [19] showed that the algebra for E
1
, E
2
E
3
and E
4
is equivalent to the
algebra of 2 × 2 matrices. The details are described in Appendix B.
For the sake of simplicity of notation, we use the symbol a to denote the quadruple
(a, b, c, d) and introduce the f unctions
α(a) =
1
2
(a + b),
β(a) =
1
2
(a − b),
γ(a) =
1
2
p
(a − b)
2
+ 4(c
2
+ d
2
),
δ(a) =
√
ab − c
2
− d
2
(if ab − c
2
− d
2
≥ 0).
(90)
The variable d is not needed in this subsection, so we t ake d = 0. We used it to keep the
number of auxiliary functions to a minimum.
The procedure is to diagonalize the comp ound tensor argument in (89),
B ≡ a E
1
+ b E
2
+ c( E
3
+ E
4
) = (α(a) + γ(a)) E
+
+ (α(a) − γ(a)) E
−
, (91)
where
E
±
=
1
2
( E
1
+ E
2
) ±
1
2
[cos ψ(a)( E
1
− E
2
) + sin ψ(a)( E
3
+ E
4
)], (92)
with ψ(a) defined by



cos ψ(a) =
β(a)
γ(a)
, sin ψ(a) =
c
γ(a)
if γ(a) > 0,
ψ(a) = 0, if γ(a) = 0.
The derivation is apparent from the results below, and explained in detail in Appendix B.
The central feature of this decomposition is that E
±
satisfy E
±
E
±
= E
±
, E
+
E
−
= O,
E
+
+ E
−
= E
1
+ E
2
, and consequently any function, h, of the tensor B may be expressed
h( B) = h(α(a) + γ(a)) E
+
+ h(α( a) − γ(a)) E
−
+ h(0)( I − E
1
− E
2
) , (93)
and hence
h( A) = h(α(a) + γ(a)) E
+
+ h(α( a) − γ(a)) E
−
+ h(f) F + h(g) G . (94)
Converting back to the E
i
’s gives
h( A) = h
+
(a)( E
1
+ E
2
) + h
−
(a) [β(a)( E
1
− E
2
) + c( E
3
+ E
4
)] + h(f) F + h(g) G , (9 5)
where
h
+
(a) =
h(α(a) + γ(a)) + h(α( a) − γ(a))
2
,
and
h
−
(a) =



h(α(a) + γ(a)) − h(α(a) − γ(a))
2γ(a)
, if γ(a) > 0,
h
′
(α(a)), if γ(a) = 0.

Moakher and Norris February 4, 2008 22
The results of Appendix B and above imply
exp( A) = e
1
(a) E
1
+ e
2
(a) E
2
+ e
3
(a)( E
3
+ E
4
) + e
f
F + e
g
G, (96)
where
e
1
(a) = exp(α(a))[cosh(γ(a)) + β(a) sinhc(γ(a))],
e
2
(a) = exp(α(a))[cosh(γ(a)) − β(a) sinhc(γ(a))],
e
3
(a) = c exp(α(a)) sinhc(γ(a)),
(97)
and sinhc(·) is the hyperbolic sine cardinal function defined by
sinhc(x) =



1 if x = 0 ,
sinh(x)
x
otherwise.
(98)
Note that sinhc(x) is continuous at 0.
When ab − c
2
6= 0, f 6= 0 and g 6= 0, the symmetric tensor A is invertible and
A
−1
=
1
ab − c
2
[b E
1
+ a E
2
− c( E
3
+ E
4
)] +
1
f
F +
1
g
G. (99)
A is positive definite if a > 0, b > 0, ab − c
2
> 0, f > 0 and g > 0, with square root
A
1/2
=
1
p
2(α(a) + δ(a))
[(a + δ(a)) E
1
+ (b + δ(a)) E
2
+ c( E
3
+ E
4
)] +
p
f F +
√
g G,
(100)
and logarithm
Log A = l
1
(a) E
1
+ l
2
(a) E
2
+ l
3
(a)( E
3
+ E
4
) + ln f F + ln g G, (101)
where
l
1
(a) = ln δ(a) + β(a)ℓ(a),
l
2
(a) = ln δ(a) − β(a)ℓ(a),
l
3
(a) = c ℓ(a),
(102)
with the function ℓ(·) defined by
ℓ(a) =







1
α(a)
if γ(a) = 0,
1
2γ(a)
ln
α(a) + γ(a)
α(a) − γ(a)
otherwise.
(103)
Note that for γ(a) we have
1
2γ(a)
ln
α(a)+γ(a)
α(a)−γ(a)
=
1
γ(a)
tanh
−1
γ(a)
α(a)
and that the limit of this
expression is 1/α(a) as γ(a) goes to zero.

Moakher and Norris February 4, 2008 23
6.4 Tetragonal system
Using the decomposition (35) and the multiplication table (36), an analysis similar to
that used for the transversely isotropic system implies that for a tetragonal tensor A as
defined in (37),
h( A) = h
+
(a)( E
1
+ E
2
) + h
−
(a) [β(a)( E
1
− E
2
) + c( E
3
+ E
4
)]
+ h
+
(p)( F
1
+ F
2
) + h
−
(p) [β(p)( F
1
− F
2
) + r( F
3
+ F
4
)] + h(g) G . (104)
As before, a = (a, b, c, d) (with d = 0) and p = (p, q, r, s) (with s = 0 ) .
Thus,
exp( A) = e
1
(a) E
1
+ e
2
(a) E
2
+ e
3
(a)( E
3
+ E
4
)
+ e
1
(p) F
1
+ e
2
(p) F
2
+ e
3
(p)( F
3
+ F
4
) + e
g
G. (105)
The tensor A is invertible if ab − c
2
6= 0, pq − r
2
6= 0 and g 6= 0, and
A
−1
=
1
ab − c
2
[b E
1
+ a E
2
− c( E
3
+ E
4
)] +
1
pq − r
2
[q F
1
+ p F
2
− r( F
3
+ F
4
)] +
1
g
G.
(106)
A is positive definite if a > 0, b > 0, ab − c
2
> 0, p > 0, q > 0, pq − r
2
> 0 and g > 0,
with square root
A
1/2
=
1
p
2(α(a) + δ( a))
[(a + δ(a)) E
1
+ (b + δ(a)) E
2
+ c( E
3
+ E
4
)]
+
1
p
2(α(p) + δ(p))
[(p + δ(p)) F
1
+ (q + δ(p)) F
2
+ r( F
3
+ F
4
)] +
√
g G, (107)
and logarithm
Log A = l
1
(a) E
1
+ l
2
(a) E
2
+ l
3
(a)( E
3
+ E
4
)
+ l
1
(p) F
1
+ l
2
(p) F
2
+ l
3
(p)( F
3
+ F
4
) + ln g G. (108)
6.5 Trigonal system
The decomposition (44) and the multiplication t able (43) yield for A defined in (45)
h( A) = h
+
(a)( E
1
+ E
2
) + h
−
(a) [β(a)( E
1
− E
2
) + c( E
3
+ E
4
)]
+ h
+
(p)( R
1
+ R
2
) + h
−
(p) [β(p)( R
1
− R
2
) + r( R
3
+ R
4
) + s( R
5
+ R
6
)] ,
(109)
where, as befor e, a = (a, b, c, d) (with d = 0) and p = (p, q, r, s) (here s need not be zero).
Using these results we have
exp( A) = e
1
(a) E
1
+ e
2
(a) E
2
+ e
3
(a)( E
3
+ E
4
)
+ e
1
(p) R
1
+ e
2
(p) R
2
+ e
3
(p)( R
3
+ R
4
) + e
4
(p)( R
5
+ R
6
), (110)

Moakher and Norris February 4, 2008 24
where e
i
(·), i = 1, 3 are as defined in (97) and e
4
(p) = s exp(α(p)) sinhc(γ(p)). The tensor
A is invertible if ab − c
2
6= 0 and pq − r
2
− s
2
6= 0, with
A
−1
=
1
ab − c
2
[b E
1
+ a E
2
− c( E
3
+ E
4
)]
+
1
pq − r
2
− s
2
[q R
1
+ p R
2
− r( R
3
+ R
4
) −s( R
5
+ R
6
)] . (111)
A is p ositive definite if a > 0, b > 0, ab −c
2
> 0, p > 0, q > 0 and pq −r
2
−s
2
> 0, with
square root
A
1/2
=
1
p
2(α(a) + δ(a))
[(a + δ(a)) E
1
+ (b + δ(a)) E
2
+ c( E
3
+ E
4
)]
+
1
p
2(α(p) + δ(p))
[(p + δ(p)) R
1
+ (q + δ(p)) R
2
+ r( R
3
+ R
4
) + s( R
5
+ R
6
)] ,
(112)
and logarithm
Log A = l
1
(a) E
1
+ l
2
(a) E
2
+ l
3
(a)( E
3
+ E
4
)
+ l
1
(p) R
1
+ l
2
(p) R
2
+ l
3
(p)( R
3
+ R
4
) + l
4
(p)( R
5
+ R
6
), (113)
where l
i
(·), i = 1, 3 a r e as defined in (102) and l
4
(p) = sℓ(p).
6.6 Rhombic, monoclinic and triclinic systems
There are no practical analytical results for the exponential, logarithm and square root
for these low symmetries. The purely numerical route is recommended.
7 The closest tensors using logarithmic norms
We now consider the problem of finding the closest elasticity tensors using the Riemannian
and the logarithmic distance functions. The solutions use the machinery developed in t he
previous Section for evaluating functions of tensors. We begin with the higher symmetries.
7.1 The closest isotropic tensor using the Riemannian norm
We want to find t he closest isotropic tensor C
iso
R
= 3κ
R
J + 2µ
R
K to a given elasticity
tensor C, i.e., find κ
R
> 0 and µ
R
> 0 such t hat the Riemannian distance d
R
( C, C
iso
R
) is
minimized, where d
R
is defined in eq. (11c). To find the optimality conditions, we note
that minimizing the Riemannian distance is equivalent to minimizing the square of this
distance, and r ecall the following result [24, Prop. 2.1]
d
dt
tr

Log
2
X(t)

= 2 tr

Log X(t)X
−1
(t)
d
dt
X(t)

.
Here X(t) is a real matrix-valued function of the real variable t such that, for all t in
its domain, X(t) is an invertible matrix which does not have eigenvalues on the closed

Moakher and Norris February 4, 2008 25
negative real line. This r esult applies to fourth-order elasticity tensors via the isomorphism
(6) and (7 ) .
Using this general result implies two conditions,
∂ d
2
R
∂κ
R
( C, C
iso
R
) =
2
κ
R
tr

Log( C
−1
C
iso
R
) J

, (114a)
∂ d
2
R
∂µ
R
( C, C
iso
R
) =
2
µ
R
tr

Log( C
−1
C
iso
R
) K

. (114b)
Hence, the optimality conditions for C
iso
R
are
tr

Log( C
−1
C
iso
R
) J

= 0, tr

Log( C
−1
C
iso
R
) K

= 0. (115 )
In view o f the partition (20), addition o f these two equations yields
tr

Log( C
−1
C
iso
R
)

= 0,
which can be rewritten as
3
(see Appendix B)
det( C
−1
C
iso
R
) = 1,
or, equiva lently,
det C = det C
iso
R
. (116)
This is the counterpart of the fact that, for the Euclidean distance, the closest isotropic
tensor has the same trace as the given elasticity tensor.
7.1.1 The closest isotropic tensor to a given cubic tensor
For C = a J + b L + c M, the conditions (115a) and (116), after some algebra, reduce to
ln
3κ
R
a
= 0, ab
3
c
2
= 3κ
R
(2µ
R
)
5
, (117)
which can readily be solved to yield
3κ
R
= a, 2µ
R
= (b
3
c
2
)
1/5
.
7.1.2 The closest isotropic tensor to a given transversely isotropic tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + f F + g G, the conditions (115a) and (116) yield
3 ln δ(x) −ℓ(x)(β(x) − 2
√
2z) = 0 , (ab − c
2
)f
2
g
2
= 3κ
R
(2µ
R
)
5
, (118)
where x = (x, y, z, 0) with

x z
z y

= Σ
t
Λ

A C
C B

ΛΣ,

A C
C B

= Σ

a c
c b

Σ
t
, (119)
3
Here and throughout, the determinant of a fourth-order elasticity tensor C is det C =
Q
6
I=1
Λ
I
,
where the Λ
I
’s are the eigenvalues of C. From (8 ) we have det C = det
b
C, where
b
C is the associated
six-dimensional second-order tensor.

Moakher and Norris February 4, 2008 26
where
Σ =
1
√
3

1
√
2
−
√
2 1

, Λ = diag

1
√
3κ
R
,
1
√
2µ
R

. (120)
Since Σ is an orthogonal transformation, it follows that AB −C
2
= ab−c
2
, A+B = a+b,
xy − z
2
= (ab − c
2
)/(6κ
R
µ
R
) and x + y = (a + b)/(6κ
R
µ
R
).
Equation (118)
2
can be used to eliminate κ
R
in equation (118)
1
and hence we obtain a
single nonlinear equation for a single unknown. An appropriate choice for this unknown
is the positive variable ξ defined by the ratio
ξ
2
=
3κ
R
2µ
R
=
1 + ν
R
1 − 2ν
R
,
where ν
R
is the Poisson’s ratio of the closest isotropic tensor with respect to the Rieman-
nian distance. Then (118) yields the following equation for ξ
(
¯
A −
¯
Bξ
2
) ln
¯
A +
¯
Bξ
2
+ R( ξ)
¯
A +
¯
Bξ
2
− R(ξ)
− R(ξ) ln
ξ
4/3
¯
A
¯
B −
¯
C
2
= 0, (121)
where
R(ξ) =
q
(
¯
A −
¯
Bξ
2
)
2
+ 4
¯
C
2
ξ
2
,
and
¯
A = A/(det C)
1/6
,
¯
B = B/(det C)
1/6
,
¯
C = C/(det C)
1/6
. Note here that for this
type of symmetry det C = (ab − c
2
)f
2
g
2
.
The isotropic moduli therefore depend upon three unidimensional combinations of the
transversely isotropic moduli,
¯
A,
¯
B and
¯
C. Once the solution ξ of eq. (121) is found, the
isotropic moduli are given by
3κ
R
= (det C)
1/6
ξ
5/3
, 2µ
R
= (det C)
1/6
ξ
−1/3
. (122)
7.1.3 The closest isotropic tensor to a given tetragonal tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p F
1
+ q F
2
+ r( F
3
+ F
4
) + g G, the conditions (115a)
and (116) yield
3 ln δ(x) −ℓ(x)(β(x) − 2
√
2z) = 0 , (ab − c
2
)(pq − r
2
)g
2
= 3κ
R
(2µ
R
)
5
, (123)
where again x = (x, y, z, 0) is defined by (119). The solution is obtained in similar
manner to that for transverse isotropy, thus, κ
R
and µ
R
are given by (122) but in this
case det C = (ab − c
2
)(pq − r
2
)g
2
.
7.1.4 The closest isotropic tensor to a given trigonal tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p R
1
+ q R
2
+ r( R
3
+ R
4
) + s( R
5
+ R
6
), the conditions
(115a) and (116) yield
3 ln δ(x) −ℓ(x)(β(x) − 2
√
2z) = 0 ,
1
2
(ab − c
2
)(pq − r
2
− s
2
)
2
= 3κ
R
(2µ
R
)
5
, (124)
with x = (x, y, z, 0) of eq. (119). The isotropic elastic mo duli are obtained in similar
fashion as for of transverse isotropy, i.e., κ
R
and µ
R
are given by (122) but now det C =
1
2
(ab − c
2
)(pq − r
2
− s
2
)
2
.

Moakher and Norris February 4, 2008 27
7.2 The closest cubic tensor using the Riemannian norm
The optimality conditions that define the closest cubic tensor C
cub
R
= 3κ
R
J + 2µ
R
L +
2η
R
M to a given elasticity tensor C of lower symmetry, are obtained by minimizing the
Riemannian distance d
R
( C, C
cub
R
):
∂ d
2
R
∂κ
R
( C, C
cub
R
) =
2
κ
R
tr

Log( C
−1
C
cub
R
) J

= 0, (125a)
∂ d
2
R
∂µ
R
( C, C
cub
R
) =
2
µ
R
tr

Log( C
−1
C
cub
R
) L

= 0, (125b)
∂ d
2
R
∂η
R
( C, C
cub
R
) =
2
η
R
tr

Log( C
−1
C
cub
R
) M

= 0. (125c)
As κ
R
> 0, µ
R
> 0 and η
R
> 0, these conditions are equivalent to
tr

Log( C
−1
C
cub
R
) V

= 0, V = J, L, M. (126)
In view of the partition (26)
1
, combining these three equations yields, in the same manner
as for (127),
det C = det C
cub
R
= 3κ
R
(2µ
R
)
3
(2η
R
)
2
. (127)
If C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p F
1
+ q F
2
+ r( F
3
+ F
4
) + g G, i.e., is of t etragonal
symmetry, then equation (126a) reduces to
3 ln δ(
˜
x) − ℓ(
˜
x)(β(
˜
x) − 2
√
2˜z) = 0, (128)
where
˜
x = (˜x, ˜y, ˜z, 0) with

˜x ˜z
˜z ˜y

= Σ
t

˜
A
˜
C
˜
C
˜
B

Σ = Σ
t
˜
ΛΣ

a c
c b

Σ
t
˜
ΛΣ, (129)
where Σ is as defined in (120) and
˜
Λ = diag

1
√
3κ
R
,
1
√
2η
R

.
Again, since Σ is an orthogonal transformation, it follows that
˜
A
˜
B −
˜
C
2
= ab − c
2
,
˜
A +
˜
B = a + b, ˜x˜y − ˜z
2
= (ab − c
2
)/(6κ
R
η
R
), and ˜x + ˜y = (a + b)/(6κ
R
η
R
). Equation
(127) becomes
(ab − c
2
)(pq − r
2
)g
2
= 3κ
R
(2µ
R
)
3
(2η)
2
. (130)
Let ζ and σ be the nondimensional positive variables defined by
ζ
2
=
3κ
R
2η
R
, σ
2
=
µ
R
η
R
,
then (128) and (126b) yield the fo llowing coupled two equations for ζ and σ
(
˜
A −
˜
Bζ
2
) ln
˜
A +
˜
Bζ
2
+ Q
1
(ζ)
˜
A +
˜
Bζ
2
− Q
1
(ζ)
− Q
1
(ζ) ln
ζ
4/3
σ
−2
(det C)
1/3
ab − c
2
= 0, (131a)
(p − qσ
2
) ln
p + qσ
2
+ Q
2
(σ)
p + qσ
2
− Q
2
(σ)
− Q
2
(σ) ln
ζ
−2
σ
−1
(ab − c
2
)
g
2
= 0. (131b)

Moakher and Norris February 4, 2008 28
where
Q
1
(ζ) =
q
(
˜
A −
˜
Bζ
2
)
2
+ 4
˜
C
2
ζ
2
, Q
2
(σ) =
p
(p − qσ)
2
+ 4r
2
σ
2
.
The three moduli of the closest cubic elasticity tensor are then given by
3κ
R
= ζ
5/3
σ
−1
(det C)
1/6
, 2µ
R
= ζ
−1/3
σ(det C)
1/6
, 2η
R
= ζ
−1/3
σ
−1
(det C)
1/6
.
7.3 The closest transversely isotropic tensor using the Rieman-
nian norm
We want to find the closest, in the Riemannian metric, transversely isotropic tensor
C
hex
R
= a E
1
+ b E
2
+ c( E
3
+ E
4
) + f F + g G to a given elasticity tensor C of lower
symmetry, i.e., find a > 0, b > 0 f > 0, g > 0 and c with ab − c
2
> 0 such that the
Riemannian distance d
R
( C, C
hex
R
) is minimized.
The optimality conditions a re
∂ d
2
R
∂a
( C, C
hex
R
) =
2
ab − c
2
tr

Log( C
−1
C
hex
R
)(b E
1
− c E
4
)

= 0, (132a)
∂ d
2
R
∂b
( C, C
hex
R
) =
2
ab − c
2
tr

Log( C
−1
C
hex
R
)(a E
2
− c E
3
)

= 0, (132b)
∂ d
2
R
∂c
( C, C
hex
R
) =
2
ab − c
2
tr

Log( C
−1
C
hex
R
)(−c E
1
− c E
2
+ b E
3
+ a E
4
)

= 0, (132c)
∂ d
2
R
∂f
( C, C
hex
R
) =
2
f
tr

Log( C
−1
C
hex
R
) F

= 0, (132d)
∂ d
2
R
∂g
( C, C
hex
R
) =
2
g
tr

Log( C
−1
C
hex
R
) G

= 0. (132e)
As a > 0, b > 0, ab −c
2
> 0, f > 0 and g > 0, these five conditions are equivalent to
tr

Log( C
−1
C
hex
R
) V

= 0, V = b E
1
− c E
4
, a E
2
− c E
3
, E
1
+ E
2
, F, G. (133)
Combining the conditions for E
1
+ E
2
, F and G, a nd partition (32) implies the constraint
det C = det C
hex
R
.
7.4 The closest tensors using the log-Euclidean norm
The closest elasticity tensors according to the log-Euclidean metric follow by minimizing
d
2
L
( C, C
sym
). The stationarity condition implies that (Log C − Log C
sym
) is orthogonal
(in the Euclidean sense) to the symmetry class, and hence
Log C
sym
= P
sym
Log C, (134)
where the Euclidean projector is defined in (67). We may therefore write
C
sym
= exp (P
sym
Log C) . (1 35)

Moakher and Norris February 4, 2008 29
This may be evaluated for the particular symmetries using t he explicit expressions f or
P
sym
in (68), combined with the formulas for the logarithm and exponential of elasticity
tensors in Section 6. We note in particular the Euclidean property,
d
2
L
( C, C
sym A
) = d
2
L
( C, C
sym B
) + d
2
L
( C
sym B
, C
sym A
) , sym A ≥ sym B. (136)
We note the following identity, which is a consequence of the requirement that (Log C−
Log C
sym
) is orthogonal to each of the basis tensors,
tr[ V
i
Log C] = tr[ V
i
Log C
sym
]. (137)
In the remainder of this Section we apply this to the particular cases of isotropy a nd cubic
isotropy, for which we derive explicit formulas for the closest moduli.
7.4.1 The closest isotropic tensor using the log-Euclidean norm
There are two basis tensors and therefore the general conditions (137) become, using (83)
with C
iso
L
= 3κ
L
J + 2µ
L
K,
ln 3κ
L
= tr[ J Log C], ln 2µ
L
=
1
5
tr[ K Log C]. (138)
Hence, we obtain explicit formulas
3κ
L
= exp (tr[ J Log C]) , 2µ
L
= exp

1
5
tr[ K Log C]

. (139)
Using I = J + K and eq. (B.14), it follows that
3κ
L
(2µ
L
)
5
= det C. (140)
7.4.2 The closest isotropic tensor to a given cubic tensor
For C = a J + b L + c M, we use (88) and K = L + M to simplify the conditions (138).
After some algebra we find [17]
3κ
L
= a, 2µ
L
= (b
3
c
2
)
1/5
.
These moduli coincide with those obtained f or the Riemannian norm. This is a conse-
quence of the fact that when two tensors C
1
and C
2
commute under multiplication we
have Log( C
1
C
−1
2
) = Log C
1
− Log C
2
.
7.4.3 The closest isotropic tensor to a given transversely isotropic tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + f F + g G, we use eqs. (10 1), (138)
1
and (A.22)
1
to
get
ln 3κ
L
=
1
2
ln(ab − c
2
) +
b − a + 4
√
2c
12γ(a)
ln
a + b + 2γ(a)
a + b − 2γ(a)
. (141)
Thus,
3κ
L
=
√
ab − c
2
a + b +
p
(a − b)
2
+ 4c
2
a + b −
p
(a − b)
2
+ 4c
2
!
b−a+4
√
2c
6
√
(a−b)
2
+4c
2
, (142)

Moakher and Norris February 4, 2008 30
and the shear modulus then follows from (140) as
2µ
L
=

1
3κ
L
(ab − c
2
)f
2
g
2

1/5
. (143)
Note that the equation (141) for κ
L
can be cast in the form of eq. (122)
1
, where ξ satisfies
an equation similar to (121),
(
¯
A −
¯
B) ln
¯
A +
¯
B + R(1)
¯
A +
¯
B − R(1)
− R(1) ln
ξ
10/3
¯
A
¯
B −
¯
C
2
= 0. (144)
7.4.4 The closest isotropic tensor to a given tetragonal tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p F
1
+ q F
2
+ r( F
3
+ F
4
) + g G, we again use eqs.
(101), (138)
1
and the identity (A.22 )
1
to derive (142) for κ
L
. The shear modulus is
2µ
L
=

1
3κ
L
(ab − c
2
)(pq − r
2
)g
2

1/5
. (145)
7.4.5 The closest isotropic tensor to a given trigonal tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p R
1
+ q R
2
+ r( R
3
+ R
4
) + s( R
5
+ R
6
), t he condition
(142) for κ
L
is again recovered, while µ
L
follows from
2µ
L
=

1
3κ
L
(ab − c
2
)(pq − r
2
− s
2
)
2

1/5
. (146)
7.4.6 The closest cubic tensor using the log-Euclidean norm
Proceeding in the same way as for the isotropic case, in this case with three basis tensors,
C
cub
L
= 3κ
L
J + 2µ
L
L + 2η
L
M, we find that t he closest cubic tensor has explicit solution
3κ
L
= exp (tr[ J Log C]) , 2µ
L
= exp

1
3
tr[ L Lo g C]

, 2η
L
= exp

1
2
tr[ M Log C]

.
(147)
The moduli satisfy the same determinant constraint as for the Riemannian norm, in this
case
3κ
L
(2µ
L
)
3
(2η
L
)
2
= det C. (148)
7.4.7 The closest cubic tensor to a given tetragonal tensor
For C = a E
1
+ b E
2
+ c( E
3
+ E
4
) + p F
1
+ q F
2
+ r( F
3
+ F
4
) + g G, the bulk modulus κ
L
is again given by (142). We use
L = F
1
+ G, (149)
to obtain
2µ
L
= (pq − r
2
)
1/6
g
2/3
p + q +
p
(p − q)
2
+ 4r
2
p + q −
p
(p − q)
2
+ 4r
2
!
p−q
2
√
(p−q)
2
+4r
2
. (150)

Moakher and Norris February 4, 2008 31
The shear modulus η
L
follows from
2η
L
=

(ab − c
2
)(pq − r
2
)g
2
3κ
L
(2µ
L
)
3

1/2
. (151)
7.4.8 The closest t ransversely isotropic tensor using the log-Euclidean norm
In this case there are five basis tensors, C
hex
L
= a E
1
+ b E
2
+ c( E
3
+ E
4
) + f F + g G.
Using eqs. (101) and (137), we have
f = exp

1
2
tr[ F Log C]

, g = exp

1
2
tr[ G Lo g C]

, (152)
and the r emaining three moduli follow from the identities
l
1
(a) = tr[ E
1
Log C], l
2
(a) = tr[ E
2
Log C], l
3
(a) =
1
2
tr[( E
3
+ E
4
) Log C]. (153)
Thus,







a = δ
cosh(θ + φ cosh θ)
cosh θ
, b = δ
cosh(θ − φ cosh θ)
cosh θ
, c = δ
sinh(φ cosh θ)
cosh θ
, if φ 6= 0,
a = δe
ψ
, b = δe
−ψ
, c = 0, if φ = 0,
(154)
where
φ =
1
2
tr[( E
3
+ E
4
) Log C], ψ =
1
2
tr[( E
1
− E
2
) Log C], (155a)
δ = exp

1
2
tr[( E
1
+ E
2
) Log C]

, θ = log
"
ψ
φ
+

ψ
2
φ
2
+ 1

1/2
#
. (155b)
8 Application and numerical e xamples
We illustrate the methods developed ab ove by considering an example of a general elas-
ticity tensor, with no assumed symmetry. This type of data raises a problem typically
encountered, i.e., find the optimal orientation of the symmetry axes in addition to finding
the closest elastic tensors for a given symmetry axes or planes. We first present the data,
and the isotropic approximations, and then consider the question of orientation.
8.1 Example
A complete set of 21 elastic constants were determined ultrasonically by Fran¸cois et al .
[7]. The reader is referred to their paper for details of the measurement technique. The
raw moduli are
C =








243 136 135 22 52 −17
136 239 137 −28 11 16
135 137 233 29 −49 3
22 −28 29 133 −10 −4
52 11 −49 −10 119 −2
−17 16 3 −4 −2 130








(GPa). (156)

Moakher and Norris February 4, 2008 32
We first search for t he presence of symmetry planes, which are an indicator of underlying
symmetry. Following Cowin and Mehrabadi [35], define A and B by
A
ij
= C
ijkk
, B
ij
= C
ikjk
. (157)
If A and B have no common eigenvectors then there are no planes of reflection symmetry
and the material has no effective elastic symmetry [48]. We find that the two sets of eigen-
vectors are not coincident, and the smallest angle between any pair from the two sets of
eigenvectors is 16
◦
. Materials with symmetry higher than orthorhombic have five or fewer
distinct eigenvalues Λ
I
[20]. For the given moduli, we find Λ
I
= 47, 79, 244, 285, 312, 512,
consistent with the material having no symmetry plane.
We next consider isotropic approximations to the 21 moduli. The Euclidean and log-
Euclidean approximations follow from eqs. (68a) and (139), respectively. The numerical
procedure for finding the closest, in the Riemannian norm, isotropic tensor C
iso
R
= 3κ
R
J+
2µ
R
K is as follows. Define the function
F (3κ
R
, 2µ
R
) = tr[Log( C
−1
(3κ
R
J + 2µ
R
K)) J].
By the equal determinant rule (116), we have 3κ
R
(2µ
R
)
5
= det C, and therefore we need
only solve the equation for one of κ
R
or µ
R
. In practice, we solve
F (3κ
R
, (
det C
3κ
R
)
1/5
) = 0,
using Newton’s method.
The isotropic approximations are
κ µ
Euclidean, C 170.11 96.87
Euclidean, S 169.33 55.81
log −Euclidean
169.84 75.91
Riemannian 169.69 75.92
(158)
The Euclidean projection is obviously not invariant under inversion. Another way to see
this is to consider the pro duct
b
S
iso
b
C
iso
≈ 1.00
b
I + 0.74
b
K, (159)
where
b
C
iso
and
b
S
iso
are the isotropic projections for the stiffness
b
C and compliance
b
S =
b
C
−1
, respectively. The product (159) is an isotropic tensor, as expected, but not the
identity. The analogous product
b
S
sym
b
C
sym
of the Euclidean stiffness a nd compliance
projections becomes closer to the identity as the symmetry is reduced, although not
uniformly. As one measure, we note that the parameter det(
b
S
sym
b
C
sym
), which is unity for
the full triclinic matrices, takes the values 4.2, 4.6 , 6.6, 4.5, 8 .1 , 5.3, 15.8 for mon, ort,
trig, tet, hex, cub and iso, respectively.
8.2 Orientation effects
This set of moduli present the general problem of finding the optimal orientation of the
symmetry axes, fo r given symmetries. This can be done by a “brute force” approach of

Moakher and Norris February 4, 2008 33
searching over all po ssible orientations of the basis vectors. In practice this is achieved
using Euler angles (θ
1
, θ
2
, θ
3
) to transform from {e
1
, e
2
, e
3
} → {e
′
1
, e
′
2
, e
′
3
} by first rotating
about the e
3
axis by θ
1
, then about the intermediate e
′′
1
axis by θ
2
, a nd finally about t he
e
′
3
axis by θ
3
. The moduli transform as C → C
′
, and for each triple (θ
1
, θ
2
, θ
3
) the
closest elasticity tensors C
sym
of different symmetries are found. Numerically, we fix the
symmetries by reference to the original basis, with {a, b, c} = {e
1
, e
2
, e
3
}.
Let d be any of the three distance functions, and define
ρ
sym
(θ
1
, θ
2
, θ
3
) =
d
2
( C
sym
, C
iso
)
d
2
( C, C
iso
)
. (160)
Thus, 0 < ρ
sym
≤ 1, with equality only if the rotated C
′
has symmetry sym. The
preliminary analysis above indicates the absence o f any symmetry so we expect ρ
sym
to
be less than unity. We also define
ρ
∗
sym
= max
θ
1
,θ
2
,θ
3
ρ
sym
(θ
1
, θ
2
, θ
3
). (161)
Orientations at which ρ
sym
= ρ
∗
sym
are candidates f or symmetry axes that best approximate
the mo duli.
d
F
, C
ρ
cub
ρ
hex
ρ
tet
ρ
ort
ρ
trig
ρ
mon
cub 0.91 0.60 0.91 0.92 0.61 0.94
hex 0.85 0.60 0.91 0.87 0.62 0.93
tet 0.01 0.45 0.92 0.88 0.47 0.94
ort
0.03 0.54 0.90 0.94 0.58 0.95
trig 0.12 0.20 0.21 0.20 0.95 0.22
mon
0.02 0.07 0.35 0.81 0.08 0.98
Table 1: ρ
sym
calculated using the Euclidean projection of the stiffness C. The diago-
nal elements in bold are the values of ρ
∗
sym
for the symmetries indicated in the
left column. The other numbers in each row are the values of ρ
sym
evaluated
at the same orientation as ρ
∗
sym
.
Ta bles 1 and 2 list the results using the Euclidean projection for the stiffness and compli-
ance, respectively. Table 3 gives the results for the log -Euclidean distance. The results in
Ta bles 1-3 were obtained using 60 × 60 × 60 discretized Euler angles. In general, larger
values give an indication of the proximity of the symmetry. These numbers taken to gether
suggests that the material is not well approximated by hex, but that cub and certainly tet
are reasonable candidates for approximating symmetries. The analogo us log-Euclidean
computations, given in Table 3, reinforce this view. Also, ρ
∗
cub
= 0.95 for the Riemannian
norm, using the method below. We focus on the cubic approximation for the remainder
of this Section.

Moakher and Norris February 4, 2008 34
d
F
, S ρ
cub
ρ
hex
ρ
tet
ρ
ort
ρ
trig
ρ
mon
cub 0.82 0.54 0.83 0.97 0.54 0.99
hex
0.18 0.71 0.90 0.83 0.72 0.97
tet 0.8 2 0.35 0.96 0.97 0.35 0.98
ort
0.82 0.35 0.96 0.97 0.35 0.98
trig 0.21 0.32 0.33 0.42 0.84 0.43
mon 0.82 0.54 0.83 0.97 0.54 0.99
Table 2: ρ
sym
calculated using the Euclidean projection of the compliance C
−1
. The
elements are determined in the same manner as in Table 1.
d
L
ρ
cub
ρ
hex
ρ
tet
ρ
ort
ρ
trig
ρ
mon
cub 0.92 0.68 0.94 0.96 0.69 0.96
hex 0.87 0.69 0.93 0.92 0.70 0.96
tet 0.11 0.47 0.95 0.71 0.48 0.96
ort
0.09 0.20 0.37 0.96 0.21 0.98
trig 0.26 0.43 0.43 0.44 0.94 0.47
mon 0.91 0.60 0.92 0.96 0.62 0.99
Table 3: ρ
sym
calculated using the log-Euclidean distance function, determined in the
same manner as in Table 1. In particular the diagonal elements are ρ
∗
sym
.
8.3 Cubic approximations
We note some properties of the optimally oriented cubic approximations. First, the de-
composition (70) for the Euclidean cubic projection has an interesting implication. Noting
that
2 L −3 M = 5 J + 2 K − 5 (a ⊗ a ⊗ a ⊗a + b ⊗ b ⊗ b ⊗ b + c ⊗c ⊗ c ⊗ c) , (162)
and using (58), we may write the cubic length
kC
cub
k
2
= 9κ
2
+ 20µ
2
+
5
6
(3κ + 4µ − c
aa
− c
bb
− c
cc
)
2
, (163)
where κ and µ are Fedorov’s isotropic moduli, and c
aa
= hC, a⊗a⊗a⊗ai = C
ijkl
a
i
a
j
a
k
a
l
,
etc. Consider kC
cub
k as a function of the orientation of the cube axes. Since κ and µ are
isotropic invariants, and c
aa
> 0, c
bb
> 0, c
cc
> 0 on account of the positive definite nature
of C, it follows that the largest length occurs when c
aa
+ c
bb
+ c
cc
achieves it smallest
value. This implies that the best cubic approximation in the Euclidean sense occurs in the
coordina te system with s mallest value of (c
′
11
+ c
′
22
+ c
′
33
).
The closest cubic material for log-Euclidean distance function is that which minimizes
d
L
( C, C
cub
). Using the Euclidean property, see (136), that
d
2
L
( C, C
cub
) = d
2
L
( C, C
iso
) − d
2
L
( C
cub
, C
iso
) , (164)
and the fact that d
L
( C, C
iso
) is unchanged under rotation, it follows that the optimal
C
cub
maximizes d
L
( C
cub
, C
iso
). Let µ
Li
denote the isotropic modulus, from eq. (139)
2
,

Moakher and Norris February 4, 2008 35
then since the bulk modulus κ
L
is the same for isotropy and cubic symmetry, it follows
that
d
L
( C
cub
, C
iso
) =
q
15
2
|lo g
µ
L
µ
Li
| (165)
Therefore, the optimal orientation is that for which
1
2
(
µ
L
µ
Li
+
µ
Li
µ
L
) achieves its largest value.
Alternatively, if we define the Euclidean cubic approximation fo r C as C
cub
F
= 3κ
F
J +
2µ
F
L + 2η
F
L, then κ
F
= κ and the optimal µ
F
and η
F
maximize |µ
F
− η
F
| subject to
the additive constraint
3
5
µ
F
+
2
5
η
F
= µ, where µ is t he isotropic (Fedorov) shear modulus.
By comparison, the optimal lo g-Euclidean moduli maximize |log µ
L
− log η
L
| subject to
the constraint
3
5
log µ
L
+
2
5
log η
L
= log µ.
The closest cubic tensor in the Riemannian norm, C
cub
R
= 3κ
R
J + 2µ
R
L + 2η
R
M,
is obtained using a numerical scheme similar to t hat for t he isotropic case. Define the
functions
F
1
(3κ
R
, 2µ
R
, 2η
R
) = tr[Log( C
−1
(3κ
R
J + 2µ
R
L + 2 η
R
M)) J],
F
2
(3κ
R
, 2µ
R
, 2η
R
) = tr[Log( C
−1
(3κ
R
J + 2µ
R
L + 2 η
R
M)) L].
Using the condition (127) of equality of determinants we can eliminate one of the three
unknowns, and therefore only need to solve two equations for κ
R
and µ
R
F
1
(3κ
R
, 2µ
R
, (
det C
3κ
R
(2µ
R
)
3
)
1/2
) = 0, F
2
(3κ
R
, 2µ
R
, (
det C
3κ
R
(2µ
R
)
3
)
1/2
) = 0.
These equations are solved by the two-dimensional Newton’s method.
The numerical search for the optimal cubic approximation indicates that the orienta-
tion which yields the maximum ρ
∗
cub
coincides for the three distance functions fo r this set
of moduli. The optimal cubic moduli are as follows
κ µ η
Eucl, C 170.1 139.7 32.6
Eucl, S
169.3 135.1 29.7
log −Eucl 169.7 137.5 31.2
Riemannian
169.8 138.1 30.9
(166)
The Euclidean moduli of the first row are in agreement with Fran¸cois et al. [7], who used a
method based on transformation groups to effect the projection. Note that the Euclidean
projection for compliance again gives a different answer than f or stiffness. Also, the bulk
modulus for the log-Euclidean distance is the same as for the isotropic approximation, see
(158).
This example illustrates the practical application of all three distance functions. The
procedure is similar for materials in which the closest or best symmetry for approximation
might b e, for instance, transversely isotropic. The Euclidean projection follow from eq.
(68c) a nd the log- Euclidean moduli fro m eqs. (152) and (153). The closest transversely
isotropic tensor in the Riemannian norm can be found in a similar manner as above - in
this case a system of four nonlinear equations for four unknowns is obtained, which can
be solved by Newton’s method.

Moakher and Norris February 4, 2008 36
9 Conclus i ons
Three distance functions for elastic tensors have been introduced and their application to
elasticity examined. The Euclidean distance function is the most commonly used, but it
lacks the property of invariance under inversion. Two distance functions with this impo r-
tant property have been described in detail: the log-Euclidean and the Riemannian norms.
For each of the three distance functions we have developed a coordinate-independent pro-
cedure to find the closest elasticity tensor of a given symmetry to a set of moduli of a
lower symmetry.
For the Euclidean distance f unction we have described a projection scheme using basis
tensors, similar to vector space proj ections. Explicit forms for the projection operator have
been given for isotropic, cubic, transversely isotropic, tetragonal and trigonal symmetries.
The proj ection method was compared with the group transformation scheme and with
the 21-dimensional a pproa ch. We also described the form of the projection operator using
6×6 matrices, which is easily implement ed on a computer, and derived explicit expressions
for the tensor complements/residues, and for the lengths of the projections. This allows
one to decompose, in a Pythagorean sense, the elastic stiffness o r compliance, although
with different results for each.
Detailed and practical results have been presented for applying the logarithmic based
distance functions to the same problem of approximating using a higher elastic symmetry.
Both the log-Euclidean and the Riemannian distance functions are invariant under inver-
sion. As such they offer an unambiguous method for determining the closest moduli of a
given symmetry, providing unique results regardless of whether stiffness or compliance is
considered.
Calculation of the logarithmic distance functions requires evaluation of the logarithm,
square root and exponential of tensors. Semi-analytical procedures for obtaining these
for the most important, higher, elastic symmetries have been derived. For the first time,
practical expressions are available for computing the log arithmic distances for symme-
tries higher than orthotropic, that is: isotropic, cubic, hexagonal (transversely isotropic),
tetragonal, and trigonal. The numerical example considered illustrates how a data set
with no symmetry can be reduced to find the optimally fitting cubic moduli. The Eu-
clidean/Fro benius norm yields different results based on whether stiffness or compliances
is use, but the log-Euclidean and the Riemannian distance functions provide unique an-
swers.

Moakher and Norris February 4, 2008 37
Appendix
A The close st tensors using the Euclid ean norm
Results for the Euclidean projection are summarized. We first give the explicit form of
the projection for the various symmetries.
A.1 S ummary of the Euclidean projections
The projections onto the symmetry classes are expressed in terms of the element s of
the six-dimensional second-order tensor
b
C of eq. (7) . For the purpose of subsequent
calculations, we ta ke {a, b, c} = {e
1
, e
2
, e
3
}. The symmetry classes are characterized
by a distinct direction, the direction of the monoclinic symmetry plane, t he direction
perpendicular to the plane spanned by the normals to the planes of reflection symmetry
for trigona l a nd tetrago nal symmetry, the axis of transversely isotropic symmetry, a cube
axis. In each case we take this direction as e
3
.
A.1.1 Monoclinic and orthorhombic symmetry
b
C
mon
=








ˆc
11
ˆc
12
ˆc
13
0 0 ˆc
16
ˆc
22
ˆc
23
0 0 ˆc
26
ˆc
33
0 0 ˆc
36
ˆc
44
ˆc
45
0
S Y M ˆc
55
0
ˆc
66








, (A.1)
b
C
ort
=








ˆc
11
ˆc
12
ˆc
13
0 0 0
ˆc
22
ˆc
23
0 0 0
ˆc
33
0 0 0
ˆc
44
0 0
S Y M ˆc
55
0
ˆc
66








. (A.2)
A.1.2 Tetragonal symmetry
C
tet
as defined in (37) becomes in 6 × 6 notatio n
b
C
tet
=








1
2
(b + q)
1
2
(b − q)
1
√
2
c 0 0
r
√
2
1
2
(b + q)
1
√
2
c 0 0 −
r
√
2
a 0 0 0
g 0 0
S Y M g 0
p








, (A.3)
and the nor m is
kC
tet
k
2
= a
2
+ b
2
+ 2c
2
+ p
2
+ q
2
+ 2r
2
+ 2g
2
. (A.4)

Moakher and Norris February 4, 2008 38
Performing the inner products using (64), with N = 7 and {V
1
, V
2
, V
3
, V
4
, V
5
, V
6
, V
6
} =
{E
1
, E
2
, ( E
3
+ E
4
), F
1
, F
2
, ( F
3
+ F
4
), G} gives
a = ˆc
33
, b =
1
2
(ˆc
11
+ ˆc
22
+ 2ˆc
12
), c =
1
√
2
(ˆc
13
+ ˆc
23
), g =
1
2
(ˆc
44
+ ˆc
55
),
p = ˆc
66
, q =
1
2
(ˆc
11
+ ˆc
22
− 2ˆc
12
), r =
1
√
2
(ˆc
16
− ˆc
26
).
(A.5)
In summary, the projection is
b
C
tet
=








1
2
(ˆc
11
+ ˆc
22
) ˆc
12
1
2
(ˆc
13
+ ˆc
23
) 0 0
1
2
(ˆc
16
− ˆc
26
)
1
2
(ˆc
11
+ ˆc
22
)
1
2
(ˆc
13
+ ˆc
23
) 0 0
1
2
(ˆc
26
− ˆc
16
)
ˆc
33
0 0 0
1
2
(ˆc
44
+ ˆc
55
) 0 0
SYM
1
2
(ˆc
44
+ ˆc
55
) 0
ˆc
66








.
(A.6)
A.1.3 Trigonal symmetry
C
trig
as defined in (45) becomes
b
C
trig
=








1
2
(b + p)
1
2
(b − p)
1
√
2
c
1
√
2
r
1
√
2
s 0
1
2
(b + p)
1
√
2
c −
1
√
2
r −
1
√
2
s 0
a 0 0 0
q 0 −s
SYM q r
p








, (A.7)
and the nor m is
kC
tet
k
2
= a
2
+ b
2
+ 2c
2
+ 2p
2
+ 2q
2
+ 4r
2
+ 4s
2
, (A.8)
where
a = ˆc
33
, b =
1
2
(ˆc
11
+ ˆc
22
+ 2ˆc
12
), c =
1
√
2
(ˆc
13
+ ˆc
23
), p = ˆc
66
,
q =
1
2
(ˆc
44
+ ˆc
55
), r =
1
2
√
2
(ˆc
14
− ˆc
24
) +
1
2
ˆc
56
, s =
1
2
√
2
(ˆc
15
− ˆc
25
) −
1
2
ˆc
46
.
(A.9)
In this case,
b
C
trig
=








ˆc
∗
11
ˆc
∗
11
− ˆc
∗
66
1
2
(ˆc
13
+ ˆc
23
)
1
√
2
ˆc
∗
56
−
1
√
2
ˆc
∗
46
0
ˆc
∗
11
1
2
(ˆc
13
+ ˆc
23
) −
1
√
2
ˆc
∗
56
1
√
2
ˆc
∗
46
0
ˆc
33
0 0 0
1
2
(ˆc
44
+ ˆc
55
) 0 ˆc
∗
46
SYM
1
2
(ˆc
44
+ ˆc
55
) ˆc
∗
56
ˆc
∗
66








, (A.10)

Moakher and Norris February 4, 2008 39
where
ˆc
∗
11
=
1
8
(3ˆc
11
+ 3ˆc
22
+ 2ˆc
12
+ 2ˆc
66
) , (A.11a)
ˆc
∗
66
=
1
4
(ˆc
11
+ ˆc
22
− 2ˆc
12
+ 2ˆc
66
) , (A.11b)
ˆc
∗
46
=
1
2

ˆc
46
−
1
√
2
ˆc
15
+
1
√
2
ˆc
25

, (A.11c)
ˆc
∗
56
=
1
2

ˆc
56
+
1
√
2
ˆc
14
−
1
√
2
ˆc
24

. (A.11d)
A.1.4 Transverse isotropy
With C
hex
defined as in (33), we have
b
C
hex
=








1
2
(b + f )
1
2
(b − f)
1
√
2
c 0 0 0
1
2
(b + f)
1
√
2
c 0 0 0
a 0 0 0
g 0 0
SYM g 0
f








, (A.12)
and
kC
hex
k
2
= a
2
+ b
2
+ 2c
2
+ 2f
2
+ 2g
2
. (A.13)
The projection onto the basis tensors {V
1
, V
2
, V
3
, V
4
, V
5
} = {E
1
, E
2
, ( E
3
+ E
4
), F, G},
yields a, b, c and g as given in (A.5) and
f =
1
2
(p + q) =
1
4
(ˆc
11
+ ˆc
22
+ 2ˆc
66
− 2ˆc
12
) . (A.14)
Thus,
b
C
hex
=








ˆc
∗
11
ˆc
∗
11
− ˆc
∗
66
1
2
(ˆc
13
+ ˆc
23
) 0 0 0
ˆc
∗
11
1
2
(ˆc
13
+ ˆc
23
) 0 0 0
ˆc
33
0 0 0
1
2
(ˆc
44
+ ˆc
55
) 0 0
SYM
1
2
(ˆc
44
+ ˆc
55
) 0
ˆc
∗
66








. (A.15)
A.1.5 Cubic symmetry
The 6-D matrix associated with C
cub
defined in (28) is
b
C
cub
=








1
3
(a + 2c)
1
3
(a − c)
1
3
(a − c) 0 0 0
1
3
(a + 2c)
1
3
(a − c) 0 0 0
1
3
(a + 2c) 0 0 0
b 0 0
SYM b 0
b








, (A.16)
with length
kC
cub
k
2
= a
2
+ 3b
2
+ 2c
2
. (A.17)

Moakher and Norris February 4, 2008 40
The Euclidean projection onto cubic symmetry follows f r om (64) with N = 3 and {V
1
, V
2
, V
3
}
= {J, L, M}, and gives
a =
1
3
(ˆc
11
+ ˆc
22
+ ˆc
33
+ 2ˆc
12
+ 2ˆc
13
+ 2ˆc
23
) (A.18a)
b =
1
3
(ˆc
44
+ ˆc
55
+ ˆc
66
) , (A.18b)
c =
1
3
(ˆc
11
+ ˆc
22
+ ˆc
33
− ˆc
12
− ˆc
23
− ˆc
31
). (A.18c)
Note that a = 3κ where κ defined in eq. (23)
1
. In summary, the proj ection is
b
C
cub
=








ˆc
C
11
ˆc
C
12
ˆc
C
12
0 0 0
ˆc
C
11
ˆc
C
12
0 0 0
ˆc
C
11
0 0 0
ˆc
C
66
0 0
SYM ˆc
C
66
0
ˆc
C
66








, (A.19)
where
ˆc
C
11
=
1
3
(ˆc
11
+ ˆc
22
+ ˆc
33
), ˆc
C
12
=
1
3
(ˆc
12
+ ˆc
13
+ ˆc
23
), ˆc
C
66
=
1
3
(ˆc
44
+ ˆc
55
+ ˆc
66
). (A.20)
A.1.6 Isotr opy
With C
iso
as given by ( 22), we have
b
C
iso
=








κ +
4
3
µ κ −
2
3
µ κ −
2
3
µ 0 0 0
κ +
4
3
µ κ −
2
3
µ 0 0 0
κ +
4
3
µ 0 0 0
2µ 0 0
SYM 2µ 0
2µ








, (A.21)
where κ and µ are defined in (23). These follow from the general projection formula (64)
with N = 2 and {V
1
, V
2
} = {J, K}.
A.2 The space between isotropy and transverse isotropy
We use a Gram-Schmid approach to define an orthogona l subspace between isotropy a nd
transverse isotropy (hexagonal symmetry). Noting that
J =
1
3
[ E
1
+2 E
2
+
√
2( E
3
+ E
4
)], K =
1
3
[2 E
1
+ E
2
−
√
2( E
3
+ E
4
)+3 F+3 G] , (A.22)
we introduce
L
1
= G − F , (A.23a)
L
2
= 2 E
1
− 2 E
2
+
1
√
2
( E
3
+ E
4
) , (A.23b)
L
3
= 8 E
1
+ 4 E
2
− 4
√
2( E
3
+ E
4
) − 3 F − 3 G. (A.23c)

Moakher and Norris February 4, 2008 41
Thus, hL
i
, L
j
i = 0, i 6= j, and hL
i
, Ji = 0, hL
i
, Ki = 0, i = 1, 2, 3. Hence {V
i
, i =
1, 2, . . . 5} = {J, K, L
1
, L
2
, L
3
} form an basis for Hex, with D = diag(1, 5, 4, 9, 180).
Equation (67) gives
P
hex
= J ⊗ J +
1
5
K ⊗ K +
1
4
L
1
⊗ L
1
+
1
9
L
2
⊗ L
2
+
1
180
L
3
⊗ L
3
= P
iso
+ P
hex/iso
. (A.24)
Thus, we have the explicit decomposition into orthogonal complements C
hex
= C
iso
+
C
hex/iso
, where
C
hex/iso
= d
1
L
1
+ d
2
L
2
+ d
3
L
3
,


C
hex/iso


2
= 4d
2
1
+ 9d
2
2
+ 180d
2
3
, (A.25)
with
d
1
=
1
2
(g −f), d
2
=
1
9
(2a −2b +
√
2c), d
3
=
1
90
(4a + 2b −4
√
2c − 3f −3g), (A.26)
and a, . . . , g are given by ( A.5) and (A.14). Eliminating the latter gives
d
1
=
1
8
[2(ˆc
44
+ ˆc
55
+ ˆc
12
) − (ˆc
11
+ ˆc
22
+ 2ˆc
66
)] , (A.27a)
d
2
=
1
9
[ˆc
13
+ ˆc
23
+ 2ˆc
33
− (ˆc
11
+ ˆc
22
+ 2ˆc
12
)] , (A.27b)
d
3
=
1
360
[ˆc
11
+ ˆc
22
+ 14ˆc
12
+ 16(ˆc
33
− ˆc
13
− ˆc
23
) −6(ˆc
44
+ ˆc
55
+ ˆc
66
)] . (A.27c)
A.3 Complements
Some orthogonal complements or residues [3] between the symmetry classes are presented
here. These lead to the explicit expressions for elastic lengths in the next subsection, eqs.
(A.30) and (A.31 ) . Thus,
b
C
⊥mon
=








0 0 0 ˆc
14
ˆc
15
0
0 0 ˆc
24
ˆc
25
0
0 ˆc
34
ˆc
35
0
0 0 ˆc
46
S Y M 0 ˆc
56
0








, (A.28a)
b
C
mon/ort
=








0 0 0 0 0 ˆc
16
0 0 0 0 ˆc
26
0 0 0 ˆc
36
0 ˆc
45
0
S Y M 0 0
0








, (A.28b)
b
C
mon/tet
=








1
2
(ˆc
11
− ˆc
22
) 0
1
2
(ˆc
13
− ˆc
23
) 0 0
1
2
(ˆc
16
+ ˆc
26
)
1
2
(ˆc
22
− ˆc
11
)
1
2
(ˆc
23
− ˆc
13
) 0 0
1
2
(ˆc
16
+ ˆc
26
)
0 0 0 ˆc
36
1
2
(ˆc
44
− ˆc
55
) ˆc
45
0
SYM
1
2
(ˆc
55
− ˆc
44
) 0
0








,
(A.28c)

Moakher and Norris February 4, 2008 42
b
C
ort/hex
=








r
0
+
1
2
(ˆc
11
− ˆc
22
) −r
0
1
2
(ˆc
13
− ˆc
23
) 0 0 0
r
0
+
1
2
(ˆc
22
− ˆc
11
)
1
2
(ˆc
23
− ˆc
13
) 0 0 0
0 0 0 0
1
2
(ˆc
44
− ˆc
55
) 0 0
SYM
1
2
(ˆc
55
− ˆc
44
) 0
−2r
0








,
(A.28d)
b
C
tet/hex
=








r
0
−r
0
0 0 0
1
2
(ˆc
16
− ˆc
26
)
r
0
0 0 0
1
2
(ˆc
26
− ˆc
16
)
0 0 0 0
0 0 0
S Y M 0 0
−2r
0








, (A.28e)
b
C
tet/cub
=








r
1
−2r
2
r
2
0 0
1
2
(ˆc
16
− ˆc
26
)
r
1
r
2
0 0
1
2
(ˆc
26
− ˆc
16
)
−2r
1
0 0 0
r
3
0 0
S Y M r
3
0
−2r
3








, (A.28f)
b
C
cub/iso
=








2r −r −r 0 0 0
2r −r 0 0 0
2r 0 0 0
−2r 0 0
S Y M −2r 0
−2r








, (A.28g)
where
r
0
=
1
8
(ˆc
11
+ ˆc
22
− 2ˆc
12
− 2ˆc
66
), r
1
=
1
6
(ˆc
11
+ ˆc
22
− 2ˆc
33
), (A.29a)
r
2
=
1
6
(ˆc
13
+ ˆc
23
− 2ˆc
12
), r
3
=
1
6
(ˆc
44
+ ˆc
55
− 2ˆc
66
), (A.29b)
r =
1
15
(ˆc
11
+ ˆc
22
+ ˆc
33
− ˆc
12
− ˆc
13
− ˆc
23
− ˆc
44
− ˆc
55
− ˆc
66
) . (A.29c)
Also,
b
C
hex/iso
follows from (A.27) and (A.25) but is not given explicitly because of its
length, and neither are the complements involving
b
C
trig
.
A.4 The Euclidean distances
Three projections of the elastic length are as follows
kCk
2
= kC
iso
k
2
+


C
cub/iso


2
+


C
tet/cub


2
+


C
mon/tet


2
+ kC
⊥mon
k
2
= kC
iso
k
2
+


C
hex/iso


2
+


C
tet/hex


2
+


C
mon/tet


2
+ kC
⊥mon
k
2
= kC
iso
k
2
+


C
hex/iso


2
+


C
ort/hex


2
+


C
mon/ort


2
+ kC
⊥mon
k
2
.
(A.30)

Moakher and Norris February 4, 2008 43
These can be explicitly calculated using



b
C
⊥mon



2
=2

ˆc
2
14
+ ˆc
2
15
+ ˆc
2
24
+ ˆc
2
25
+ ˆc
2
34
+ ˆc
2
35
+ ˆc
2
46
+ ˆc
2
56

, (A.31a)



b
C
mon/ort



2
=2

ˆc
2
16
+ ˆc
2
26
+ ˆc
2
36
+ ˆc
2
45

, (A.31b)



b
C
ort/hex



2
=
1
8
(ˆc
11
+ ˆc
22
− 2ˆc
12
− 2ˆc
66
)
2
+
1
2
(ˆc
11
− ˆc
22
)
2
+ (ˆc
13
− ˆc
23
)
2
+
1
2
(ˆc
44
− ˆc
55
)
2
,
(A.31c)



b
C
mon/tet



2
=
1
2
(ˆc
11
− ˆc
22
)
2
+ (ˆc
13
− ˆc
23
)
2
+
1
2
(ˆc
44
− ˆc
55
)
2
+ (ˆc
16
+ ˆc
26
)
2
+ 2ˆc
2
45
+ 2ˆc
2
36
,
(A.31d)



b
C
tet/cub



2
=
1
6
(ˆc
11
+ ˆc
22
− 2ˆc
33
)
2
+
1
3
(ˆc
13
+ ˆc
23
− 2ˆc
12
)
2
+
1
6
(ˆc
44
+ ˆc
55
− 2ˆc
66
)
2
+ (ˆc
16
− ˆc
26
)
2
, (A.31e)



b
C
cub/iso



2
=
2
15
(ˆc
11
+ ˆc
22
+ ˆc
33
− ˆc
12
− ˆc
13
− ˆc
23
− ˆc
44
− ˆc
55
− ˆc
66
)
2
, (A.31f)



b
C
tet/hex



2
=
1
8
(ˆc
11
+ ˆc
22
− 2ˆc
12
− 2ˆc
66
)
2
+ (ˆc
16
− ˆc
26
)
2
(A.31g)



b
C
hex/iso



2
=
1
16
[2(ˆc
44
+ ˆc
55
+ ˆc
12
) − (ˆc
11
+ ˆc
22
+ 2ˆc
66
)]
2
+
1
9
[ˆc
13
+ ˆc
23
+ 2ˆc
33
− (ˆc
11
+ ˆc
22
+ 2ˆc
12
)]
2
+
1
720
[ˆc
11
+ ˆc
22
+ 14ˆc
12
+ 16(ˆc
33
− ˆc
13
− ˆc
23
) − 6(ˆc
44
+ ˆc
55
+ ˆc
66
)]
2
,
(A.31h)
and eqs. (23) and (58) give
kC
iso
k
2
=
1
9
(ˆc
11
+ ˆc
22
+ ˆc
33
+ 2ˆc
12
+ 2ˆc
13
+ 2ˆc
23
)
2
+
1
45
[2(ˆc
11
+ ˆc
22
+ ˆc
33
− ˆc
12
− ˆc
23
− ˆc
31
) + 3(ˆc
44
+ ˆc
55
+ ˆc
66
)]
2
.
(A.32)
All versions of the elastic length in (A.30) are the sum of 21 positive numbers, the same
as the raw form of the squared length in any rectangular coordinate system using the
Voigt elements. However, t he three sets of 21 positive numbers in (A.30) contain far more
information abo ut the underlying elastic symmetry of the material.
B Exponential, logarithm and square root of Hermi-
tian matrices
Let A be the 2 × 2 Hermitian matrix
A =

a c + id
c − id b

, (B.1)
where a, b, c and d are real numbers. The matrix can be decompo sed into the sum o f an
isotropic part and a deviatoric part,
A = A
iso
+ A
dev
=

α 0
0 α

+

β c + id
c −id −β

, (B.2)

Moakher and Norris February 4, 2008 44
where α = (a + b)/2, β = (a −b)/2 and γ =
p
β
2
+ c
2
+ d
2
. Note that A
dev
vanishes if
and only if γ = 0, i.e., a = b and c = d = 0, and that A
2
dev
= γ
2
I. The eigenvalues of A
are real and given by
λ
±
= α ±γ. (B.3)
For convenience we introduce the matrix
A
′
=
(
I if γ = 0,
1
γ
A
dev
if γ > 0,
(B.4)
so that A
′2
= I. The exponential of A is then
exp A = e
α
(cosh γI + sinh A
′
) =



e
α
I if γ = 0,
e
α
(cosh γI +
sinh γ
γ
A
dev
) if γ > 0.
(B.5)
When ab − c
2
6= 0 the matrix A is invertible and its inverse is
A
−1
=
1
ab − c
2
− d
2

b −(c + id)
−(c − id) a

=
1
ab − c
2
− d
2
(αI − γA
′
)
=
1
ab − c
2
− d
2
(αI − A
dev
). (B.6)
If a > 0, b > 0 and ab − c
2
− d
2
> 0, then A is po sitive definite. Let δ =
√
ab − c
2
− d
2
,
then the positive-definite square root of A is
A
1/2
=
1
p
2(α + δ)

a + δ c + id
c − id b + δ

=
1
p
2(α + δ)
[(α + δ) I + A
dev
] , (B.7)
and the inverse of its square root is given by
A
−1/2
=
1
δ
p
2(α + δ)

b + δ −(c + id)
−(c − id) a + δ

=
1
δ
p
2(α + δ)
[(α + δ) I − A
dev
] . (B.8)
The logarithm is
Log A = ln δI + ln
r
α + γ
α −γ
A
′
=



ln δI if γ = 0,
ln δI +
1
γ
ln
r
α + γ
α −γ
A
dev
if γ > 0.
(B.9)
More generally, for any analytic function f we have
f(A) =



f(α)I if γ = 0,
f(α + γ) + f(α −γ)
2
I +
f(α + γ) −f (α −γ)
2γ
A
dev
if γ > 0.
(B.10)
The Cayley-Hamilton theorem states that
det(A)I − tr(A)A + A
2
= 0. (B.11)

Moakher and Norris February 4, 2008 45
Therefore, any analytic function of A can be written as a linear combination of I, A and
A
2
, or, if A is invertible, as a linear combination of I, A and A
−1
. Using the fact that
2A
dev
= A − δ
2
A
−1
, we have the alternative expressions for the expo nential, logarithm
and square roots
exp A = e
α

cosh γI +
sinh γ
2γ
(A − δ
2
A
−1
)

, (B.12a)
Log A = ln δI +
1
4γ
ln
α + γ
α −γ

A − δ
2
A
−1

, (B.12b)
A
1/2
=
1
p
2(α + δ)
[δI + A] , (B.12c)
A
−1/2
=
1
p
2(α + δ)

I + δA
−1

. (B.1 2d)
We recall that for any n ×n matrix M we have
det(exp M) = e
tr M
. (B.13)
Therefore, when M = Log P for some matrix P we get
det P = e
tr(Log P )
. (B.14)

Moakher and Norris February 4, 2008 46
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