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# On Choosing Effective Symmetry Classes For Elasticity Tensors

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## Abstract and Figures

We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective tensor is based on examining the features of the plot of the monoclinic-distance function of a given tensor, choosing an appropriate symmetry class, and then finding the closest tensor in that class. The concept of the effective tensor is not tantamount to the closest tensor since one always obtains a closer approximation using a monoclinic tensor than a tensor of any other nontrivial symmetry. Hence, we use qualitative features of the plot of the monoclinic-distance function to choose an effective symmetry class within which the closest tensor can be computed.
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ON CHOOSING EFFECTIVE SYMMETRY CLASSES
FOR ELASTICITY TENSORS
by C¸ A ˘
GRI D˙
INER
(Department of Earth Sciences, Memorial University of Newfoundland,
St. John’s, Newfoundland A1B 3X5, Canada)
MIKHAIL KOCHETOV
(Department of Mathematics and Statistics, Memorial University of Newfoundand,
St. John’s, Newfoundland A1C 5S7, Canada)
and
MICHAEL A. SLAWINSKI
(Department of Earth Sciences, Memorial University of Newfoundland, St. John’s,
[Received 12 March 2010. Revised 13 August 2010]
Summary
We formulate a method of representing a generally anisotropic elasticity tensor by an elasticity
tensor exhibiting a material symmetry: an effective tensor. The method for choosing the effective
tensor is based on examining the features of the plot of the monoclinic-distance function of a
given tensor, choosing an appropriate symmetry class, and then ﬁnding the closest tensor in that
class. The concept of the effective tensor is not tantamount to the closest tensor since one always
obtains a closer approximation using a monoclinic tensor than a tensor of any other nontrivial
symmetry. Hence, we use qualitative features of the plot of the monoclinic-distance function to
choose an effective symmetry class within which the closest tensor can be computed.
1. Introduction
The motivation for the present work is a relation between materials studied in seismology and their
idealization stated by Hooke’s law. In particular, we formulate a method of representing an elasticity
tensor, which might be obtained from the laboratory or seismic measurements (1), by an elasticity
tensor exhibiting a particular material symmetry, to which we refer as an effective tensor. Following
the work of researchers studying the distance in the space of elasticity tensors (2to 6), we focus our
attention on choosing the effective symmetry class of a given generally anisotropic elasticity tensor.
Gazis et al. (2) and Moakher and Norris (4) ﬁnd the closest tensor for a given symmetry class by
taking the projection of the tensor onto a linear subspace of symmetric elasticity tensors in one
coordinate system. Kochetov and Slawinski (5) take the projections of a given tensor to the linear
subspace of transversely isotropic tensors in all coordinate systems and then ﬁnd the minimum
distance in order to determine the closest transversely isotropic tensor. Kochetov and Slawinski (6)
propose a method to ﬁnd the closest orthotropic elasticity tensor for a given generally anisotropic
tensor for all orientations of Cartesian coordinate systems. In this paper, we generalize the problem
for any linear symmetric subspace expressed in any coordinate system. No a priori assumption
about the symmetry class or orientations of symmetry axes and planes is made.
The concept of the effective tensor is not tantamount to the closest tensor since one always obtains
a closer approximation using a monoclinic tensor than a tensor of any other nontrivial symmetry as
Q. Jl Mech. Appl. Math, Vol. 64. No. 1 c
Advance Access publication 29 November 2010. doi:10.1093/qjmam/hbq023
58 C¸ . D˙
INER et al.
Fig. 1 Order relation of symmetry groups of elasticity tensors
illustrated in Fig. 1. Thus, one cannot choose the effective symmetry class by considering only the
distance to different symmetry classes. In this paper, we propose a method for choosing the effective
tensor that is based on examining the features of the plot of the monoclinic-distance function of a
given tensor, choosing an appropriate symmetry class, and then ﬁnding the closest tensor in that
class. A similar approach with less detail is discussed by Franc¸ois et al. (3).
We begin this paper by stating the notation we use. In section 3, we present the symmetry
classes and symmetry groups of elasticity tensors. In section 4, we describe the distance functions
associated with the symmetry classes, except isotropy and general anisotropy, since the distance to
isotropy does not require any rotation considerations, and the distance to general anisotropy is zero:
every elasticity tensor belongs to general anisotropy. Finally, we consider numerical examples to
illustrate a choice of the effective symmetry class and the closest tensor within it.
2. Notation
In this section, we describe the notation used in this paper for studying symmetries of an elasticity
tensor, which appears in Hooke’s law,
σij =cijkl εk l , i, j ∈ {1,2,3},(2.1)
where σij ,εkl and cijkl are the components of the stress, strain and elasticity tensors, respectively.
Note that repeated indices imply summation. The elasticity tensor, which is a fourth-rank tensor in
R3, can be represented by a symmetric 6 ×6 matrix (7):
C=
c1111 c1122 c1133 2c1123 2c1113 2c1112
c1122 c2222 c2233 2c2223 2c2213 2c2212
c1133 c1133 c3333 2c3323 2c3313 2c3312
2c1123 2c2223 2c3323 2c2323 2c2313 2c2312
2c1113 2c2213 2c3313 2c2313 2c1313 2c1312
2c1112 2c2212 2c3312 2c2312 2c1312 2c1212
.(2.2)
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 59
For notational convenience, we include factors 2 and 2 in the corresponding elasticity parameters;
for instance, we write C16 =2c1112. Hence,
C=[Cij ]16i,j66(2.3)
represents matrix (2.2). An orthogonal transformation in R3, given by a matrix A=[Aij]O (3),
results in the following transformation of matrix (2.2):
C=˜
ATC˜
A, (2.4)
where ˜
Ais the following orthogonal 6 ×6 matrix (7):
A2
11 A2
12 A2
13 2A12A13 2A11 A13 2A11A12
A2
21 A2
22 A2
23 2A22A23 2A21 A23 2A21A22
A2
31 A2
32 A2
33 2A32A33 2A31 A33 2A31A32
2A21A31 2A22 A32 2A23A33 A22 A33 +A23A32 A21 A33+A23 A31 A21 A32+A22 A31
2A11A31 2A12 A32 2A13A33 A12 A33 +A13A32 A11 A33+A13 A31 A11 A32+A12 A31
2A11A21 2A12 A22 2A13A23 A12 A23 +A13A22 A11 A23+A13 A21 A11 A22+A12 A21
.
3. Symmetry classes of elasticity tensors
In view of (2.4), an elasticity tensor is invariant under an orthogonal transformation AO(3)if
and only if
C=˜
ATC˜
A. (3.1)
As shown by several researchers (8to 10), there are eight symmetry classes of elasticity ten-
sor: isotropy, cubic symmetry, transverse isotropy (TI), tetragonal symmetry, trigonal symmetry,
orthotropic symmetry, monoclinic symmetry and general anisotropy. In other words, the symmetry
group of any elasticity tensor is conjugate to one of the following eight groups:
Giso =O(3), (3.2)
Gcubic = {AO(3)|A(ei)= ±ejfor any i, j ∈ {1,2,3}},(3.3)
GTI = {±I, ±Rθ ,e3,±Mv|θ[0,2π ) and vin the e1e2-plane},(3.4)
Gtetra =±I, ±Rπ
2,e3,±Rπ
2,e3,±Me1,±Me2,±Me3,±M(1,1,0),±M(1,1,0),(3.5)
Gtrigo =±I, ±R2π
3,e3,±R2π
3,e3,±Me1,±Mcos π
3,sin π
3,0,±Mcos 2π
3,sin 2π
3,0,(3.6)
Gortho = {±I, ±Me1,±Me2,±Me3},(3.7)
Gmono = {±I, ±Me3},(3.8)
Ganiso = {±I},(3.9)
where Rθ,vdenotes the rotation around the vector vby angle θand Mvdenotes the reﬂection about
the plane whose normal is v.
The order relation among the symmetry groups is shown in Fig. 1. The dots are connected by
a line if, up to conjugation, the symmetry group corresponding to the dot below is contained in
60 C¸ . D˙
INER et al.
the symmetry group corresponding to the dot above. We have to say ‘up to conjugation’ because
the form of Gtrigo presented above, and commonly found in the literature, has e3as a three-fold
rotation axis, which is not the normal of any mirror-symmetry plane. Hence, Gmono is not contained
in Gtrigo, but it is contained in the conjugate group, Gtrigo, that has e1as a three-fold rotation axis.
With this in mind, the monoclinic symmetry group is a subgroup of all other nontrivial symmetry
groups: all except the general anisotropy.
We say that a basis, {e1,e2,e3}, is a natural coordinate system for Cif the symmetry group of
Cis Gsym (not just conjugate to Gsy m ), where Gsym is one of the groups listed above. For each
symmetry class, one can ﬁnd the form of matrix (2.2) representing an elasticity tensor in a natural
coordinate system (11,12).
4. Distance function
4.1 Deﬁnition
To consider the concept of distance in the space of elasticity tensors, we invoke the Frobenius norm:
cijk l 2:=cijk l cijkl .(4.1)
For each symmetry class, sym, we deﬁne a linear subspace of elasticity tensors, Lsy m , as the set of
all Csuch that its symmetry group includes Gsym . Hence, the distance between Cand Lsy m is the
norm of the difference of Cand Csym , the orthogonal projection of Conto Lsym ; in other words,
min
CLsym CC∥ = ∥CCsy m,
where, as shown by Gazis et al. (2),
Csym =average
AGsym ˜
ATC˜
A. (4.2)
For all classes, except for isotropy and TI, this average involves a ﬁnite sum.
Hence, the squared distance from an elasticity tensor Cto Lsym is
d(C , Lsym)= ∥CCsy m2= ∥C2− ∥Csym 2,(4.3)
where the second equality holds since CCsym and Csy m are orthogonal to one another.
4.2 Explicit expressions
In this section, for convenience of the reader, we present explicit expressions for the distance func-
tions for all nontrivial symmetry classes, except isotropy, which is well known. These expressions
can be obtained by using (4.2) to calculate the projection of a generic elasticity tensor Cto Lsym ,
where sym is any of the six symmetry classes and then using (4.3) to obtain the distance function
for sym. The calculation of projection can be simpliﬁed by observing that, since the transformation
IO(3)has no effect on C, it sufﬁces to take only elements of Gsym SO (3)in (4.2). Moakher
and Norris (4) use a different method to obtain these projections.
4.2.1 Monoclinic. Since, as stated in (3.8), Gmono = {±I, ±Me3}, the projection of Conto
Lmono is
Cmono =1
2(C +˜
MT
e3C˜
Me3). (4.4)
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 61
Evaluating the right-hand side of (4.4), we obtain
Cmono =
C11 C12 C13 0 0 C16
C12 C22 C23 0 0 C26
C13 C23 C33 0 0 C36
000C44 C45 0
000C45 C55 0
C16 C26 C36 0 0 C66
.(4.5)
The distance of Cto Lmono is found by substituting matrix (4.5) into expression (4.3),
d(C , Lmono)=2(C2
14 +C2
15 +C2
24 +C2
25 +C2
34 +C2
35 +C2
46 +C2
56).
4.2.2 Orthotropic. In a similar manner, one can ﬁnd the projection of Conto Lortho,
Cortho =1
4C+MT
e1CMe1+MT
e2CMe2+MT
e3CMe3,(4.6)
since Gortho = {±I, ±Me1,±Me2,±Me3}as stated in expression (3.7). This results in
Cortho =
C11 C12 C13 000
C12 C22 C23 000
C13 C23 C33 000
000C44 0 0
0000C55 0
00000C66
.(4.7)
According to expression (4.3), the distance of Cto Lortho is
d(C , Lortho)=2(C2
14 +C2
15 +C2
16 +C2
24 +C2
25 +C2
26 +C2
34 +C2
35 +C2
36 +C2
45 +C2
46 +C2
56).
4.2.3 Tetragonal. The projection of Conto Ltetra is
Ctetra =
1
2(C11+C22 )C12 1
2(C13+C23 )0 0 0
C12 1
2(C11+C22 )1
2(C13+C23 )0 0 0
1
2(C13+C23 )1
2(C13+C23 )C33 0 0 0
0 0 0 1
2(C44+C55 )0 0
0 0 0 0 1
2(C44+C55 )0
0 0 0 0 0 C66
.
Thus, the distance of Cto Ltetra is
d(C, Ltetra)=1
2(C11 C22)2+1
2(C44 C55)2+(C13 C23 )2
+2(C2
14 +C2
24 +C2
34 +C2
15 +C2
25 +C2
35 +C2
16 +C2
26 +C2
36 +C2
45 +C2
46 +C2
56).
62 C¸ . D˙
INER et al.
4.2.4 Trigonal. Similarly, Ctrigo is
Ctrigo =
Ctrigo
11 Ctrigo
12 Ctrigo
13 Ctrigo
14 0 0
Ctrigo
12 Ctrigo
11 Ctrigo
13 Ctrigo
24 0 0
Ctrigo
13 Ctrigo
23 Ctrigo
33 000
Ctrigo
14 Ctrigo
24 0Ctrigo
44 0 0
0 0 0 0 Ctrigo
44 Ctrigo
56
0 0 0 0 Ctrigo
56 Ctrigo
66
,(4.8)
where
Ctrigo
11 =Ctrigo
22 =1
8(3C11 +3C22 +2C12 +2C66),
Ctrigo
12 =1
8(C11 +C22 +6C12 2C66), Ctrigo
13 =Ctrigo
23 =1
2(C13 +C23),
Ctrigo
14 = −Ctrigo
24 =1
4(C14 C24 +2C56), Ctrigo
33 =C33,
Ctrigo
44 =Ctrigo
55 =1
2(C44 +C55), Ctrigo
56 =2
4(C14 C24 +2C56),
Ctrigo
66 =1
4(C11 +C22 2C12 +2C66).
Following expression (4.3), we get
d(C , Ltrigo)=2(C2
16 +C2
26 +C2
15 +C2
25 +C2
34 +C2
35 +C2
36 +C2
45 +C2
46)
+1
2(C11 C22)2+(C13 C23 )2+(C14 +C24)2+1
2(C44 C55)2
+1
21
2C11 +1
2C22 C12 C662
+1
2C14 1
2C24 C562
.
REM AR K 4.1 It is sometimes more convenient to use an alternative form of projection, Ctrigo,
which is associated with the form of trigonal symmetry group, Gtrigo, that has e1(rather than e3)
as a three-fold rotation axis. Ctrigois obtained by interchanging 1 and 3 in the subscripts of the
components of Ctrigo and adjusting matrix (4.8) accordingly. The resulting matrix is consistent with
(4.5) in the sense of having zeros in the same places.
4.2.5 Cubic. The projection of Conto Lcubic is
Ccubic =
Ccubic
11 Ccubic
12 Ccubic
12 000
Ccubic
12 Ccubic
11 Ccubic
12 000
Ccubic
12 Ccubic
12 Ccubic
11 000
0 0 0 Ccubic
44 0 0
0 0 0 0 Ccubic
44 0
0 0 0 0 0 Ccubic
44
,(4.9)
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 63
where
Ccubic
11 =1
3(C11 +C22 +C33), Ccubic
12 =1
3(C12 +C13 +C23),
Ccubic
44 =1
3(C44 +C55 +C66).
Hence, by expression (4.3), we get
d(C, Lcubic)=2(C 2
14 +C2
15 +C2
16 +C2
25 +C2
24 +C2
26 +C2
34 +C2
35 +C2
36 +C2
45 +C2
46 +C2
56)
+1
3((C11 C22)2+(C11 C33)2+(C22 C33)2+(C44 C55)2+(C44 C66)2
+(C55 C66)2)+2
3((C12 C13)2+(C12 C23 )2+(C13 C23)2).
4.2.6 Transversely isotropic. The projection of Conto LTI is
CTI =
CTI
11 CTI
12 CTI
13 0 0 0
CTI
12 CTI
11 CTI
13 0 0 0
CTI
13 CTI
23 CTI
33 0 0 0
0 0 0 CTI
44 0 0
0 0 0 0 CTI
44 0
0 0 0 0 0 CTI
11 CTI
12
,(4.10)
where
CTI
11 =1
8(3C11 +3C22 +2C12 +4C66), CTI
12 =1
8(C11 +C22 +6C12 4C66),
CTI
13 =1
2(C13 +C23), CTI
33 =C33, CTI
44 =1
4(C44 +C55).
Substituting CTI into expression (4.3), we get
d(C, LTI)=2(C 2
14 +C2
24 +C2
15 +C2
25 +C2
16 +C2
26 +C2
34 +C2
35 +C2
36 +C2
45 +C2
46 +C2
56)
+1
2(C11 C22)2+(C13 C23)2+1
2(C44 C55)2+1
2C66 C11 +C22
2+C12
2
.
4.3 Orientation
For the cases considered above, Lsym is a space that includes only those tensors whose natural
coordinate system is {e1,e2,e3}. If Cbelongs to sym but is expressed in a coordinate system dif-
ferent from its natural one, then the distance from Cto Lsym is not zero. To address this issue, we
consider the projections of Con a rotated space, Lsym , for all orientations of the coordinate system.
Hence, we deﬁne the distance function as the function of a rotation matrix, XSO (3),
d(C, ˜
XLsym ˜
XT)=d( ˜
XTC˜
X, Lsy m)= ∥ ˜
XTC˜
X2− ∥(˜
XTC˜
X)sy m2
= ∥C2− ∥(˜
XTC˜
X)sy m2.
64 C¸ . D˙
INER et al.
Thus, the squared distance from Cto the symmetry class sym is the minimum value of the dis-
tance function over all XSO(3). Any rotation matrix XSO (3)that rotates the coordinate
system {e1,e2,e3}to the coordinate system {e
1,e
2,e
3}, namely X(ei)=e
ifor i∈ {1,2,3}, can be
obtained by multiplying three elementary rotations, namely Z1, Z2and Z3:
X=Z1Z2Z3
=
cos ψcos θsin ψcos ϕsin θcos ψsin θsin ψcos ϕcos θsin ψsin ϕ
sin ψcos θ+cos ψcos ϕsin θsin ψsin θ+cos ψcos ϕcos θcos ψsin ϕ
sin ϕsin θsin ϕcos θcos ϕ
,
where Z1is the rotation around e3by angle ϕ,Z2around e1by ψand Z3around e3by θ;ϕ,ψand
θare the Euler angles. The z-axis of the coordinate system in which the tensor is expressed after
rotation is
X(e3)=cos ψπ
2sin ϕ, sin ψπ
2sin ϕ, cos ϕ.(4.11)
For monoclinic and transversely isotropic symmetry classes, the distance function depends only on
ϕand ψ, that is, Z3in Xdoes not change the value of the function. In other words, the values of
d( ˜
XTC˜
X, Lmono)and d( ˜
XTC˜
X, LTI)are determined by X(e3)(14). Hence, we can plot these two
functions on the surface of the unit sphere in R3. A code for plotting the distance functions can be
found in (14).
5. Determining effective elasticity tensor
5.1 Formulation
In this section, we propose a method of choosing the symmetry class to approximate a given gen-
erally anisotropic elasticity tensor. Then, we evaluate the closest tensor that belongs to that class
among all orientations of the coordinate systems. Hence, the orientation of the closest tensor is
found.
The choice of the effective symmetry class of a given Cis guided by the plot of the monoclinic-
distance function of C. Diner (13) and Diner et al. (14) discuss features of monoclinic-distance
plots and TI-distance plots of elasticity tensors that belong to nontrivial symmetry classes. Apart
from the fact that the monoclinic-distance function can be plotted on a sphere, it has two important
properties. First, it vanishes in the directions of the normals of the mirror planes of a given elasticity
tensor. Second, the distance function is symmetric with respect to the symmetry group of the tensor.
Hence, if a tensor is symmetric with respect to an orthogonal transformation so is the plot of the
distance function as illustrated by Diner et al. (14).
Herein, we consider Cto be a small perturbation of such a tensor. Thus, the monoclinic-distance
function of Cis also a slight perturbation of the monoclinic-distance function of the symmetric
tensor. Moreover, we prove that, for a generally anisotropic elasticity tensor that is a small per-
turbation of an orthotropic, tetragonal, trigonal or cubic symmetry, the near-zero minima of the
monoclinic-distance function are in the neighbourhood of the mirror plane normals of the symmet-
ric tensor. Hence, the pattern of near-zero minima of the monoclinic-distance function can be used
to recognize the appropriate symmetry class and to determine the orientation of a natural coordinate
system.
THE OR EM 5.1. Let C0be a tensor that belongs to one of the following symmetry classes:
orthotropic, tetragonal, trigonal or cubic. Let nbe a normal to a mirror plane of symmetry for
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 65
C0. Then, for a sufﬁciently small perturbation of C0, which we denote by C, there is a unique
near-zero minimum of d( ˜
XTC˜
X, Lmono)in a neighbourhood of n.
Proof. Referring to Fig. 1, we see that, to prove the theorem, it sufﬁces to consider two cases: 1. C0
has at least orthotropic but not transversely isotropic symmetry and 2. C0has at least trigonal but
not transversely isotropic symmetry.
Let f (u, v, w;C ) =d ( ˜
XTC˜
X, Lmono), where X(e3)=(u, v , w). Then f (u, v, w;C0)attains
its absolute minimum of zero on the unit sphere, u2+v2+w2=1, at the points corresponding
to the normals to mirror planes of C0. Taking {e1,e2,e3}to be a natural coordinate system of C0
(regarded as orthotropic or trigonal tensor, disregarding higher symmetry, if any, and using the
alternative form of the trigonal symmetry class as in Remark 4.1), we may assume n=(0,0,1)
and use (u, v) as local coordinates on the sphere, with w=1u2v2; note that in the case
of a cubic C0some of the normals are covered by Case 1 and others by Case 2. The critical points
of f (u, v, 1u2v2;C) are determined by ∂f/∂ u =0 and ∂f/∂ v =0. If Cis in a small
neighbourhood of C0, then there exists a neighbourhood of (0,0,1)where this system of equations
has a unique solution, provided we can show that the Jacobian of the system, which herein is the
Hessian of f,
H=2f
u2
2f
v22f
u∂v 2
,
does not vanish at (0,0,1)and C=C0.
Case 1 was shown by Kochetov and Slawinski (6, p. 158). In fact, the Hessian was seen to be
positive, which implies that the critical point in question is a minimum.
We consider Case 2. Using Maple to evaluate the partial derivatives at (0,0,1)and C=C0, we
obtain
2f
u2=144(C0
14)2,2f
u∂v =0,2f
v2=4(22(C0
14)2+PTM P ),
where
M=
3 0 123
0 2 0 22
1 0 3 2 1
222 6 4
32 1 4 5
,(5.1)
and P=[C0
11 C0
33 C0
12 C0
13 C0
44 ]T. Therefore, the Hessian is
H=576(C0
14)2(22(C0
14)2+PTM P ).
Since matrix (5.1) is positive semideﬁnite, we have H>0 and the vanishing of the Hessian requires
C0
14 =0, which implies that C0is transversely isotropic with rotation axis e2, a contradiction.
Hence, H > 0, and the proof is complete.
REM AR K 5.2 Theorem 5.1 holds if C0is a transversely isotropic tensor and ncoincides with the
rotation axis of C0.
66 C¸ . D˙
INER et al.
Fig. 2 Plot of the monoclinic-distance function of Cgiven in matrix (5.2)
5.2 Trigonal tensor
In this section, we discuss the problem of choosing the trigonal symmetry class as an effective
symmetry class of a generally anisotropic elasticity tensor. Consider
C=
46.52147 11.54680 14.54473 6.37327 13.99002 12.86110
11.54680 43.17348 12.03186 24.84764 4.73753 10.74845
14.54473 12.03186 14.05824 3.04593 2.44397 1.84001
6.37327 24.84764 3.04593 28.54111 1.72684 2.06005
13.99002 4.73752 2.44397 1.72684 35.73905 14.10898
12.86110 10.74845 1.84001 2.06005 14.10898 33.96664
.(5.2)
Figures 2and 3are plots of the monoclinic-distance function of C.
One can observe that the plot in Fig. 2is close to a three-fold symmetry. Moreover, Fig. 3contains
three near-zero minima that are close to the plane that is perpendicular to what would be the three-
fold symmetry axis. Moreover, unit vectors pointing to these minima are roughly 60apart. In view
of Theorem 5.1, these features suggest that the effective symmetry class of Ccan be chosen as the
trigonal symmetry. The orientations of the minima shown in Fig. 3and the values of the monoclinic-
distance function along those directions can be found using the code presented in (14). One seeks
the minimum of the distance function in the neighbourhood of dark regions shown in Fig. 3to get
v1=sin 117.80238π
180 cos 102.09231π
180 ,sin 117.80238π
180 sin 102.09231π
180 ,cos 117.80238π
180 ,
v2=sin 88.86122π
180 cos 157.12134π
180 ,sin 88.86122π
180 sin 157.12134π
180 ,cos 88.86122π
180 ,
v3=sin 119.08658π
180 cos 33.08494π
180 ,sin 119.08658π
180 sin 33.08494π
180 ,cos 119.08658π
180 ,
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 67
Fig. 3 A projection of the sphere shown in Fig. 2, from a perpendicular direction. We see three minima
coloured by dark shades of grey, which means that the values are close to zero
where v1is in the middle of Fig. 3,v2is to the right of v1and v3is to the left of v1. The angle
between v1and v2is 58.285and between v1and v3is 69.015. The values of the monoclinic-
distance function in those directions are
d( ˜
XT
1C˜
X1,Lmono)=24.53272,where X1(e3)=v1,
d( ˜
XT
2C˜
X2,Lmono)=47.10467,where X2(e3)=v2,
d( ˜
XT
3C˜
X3,Lmono)=74.03831,where X3(e3)=v3.
One can search for the absolute minimum of the trigonal distance function, d( ˜
XTC˜
X, Ltrigo), with
Xsuch that X(e1)is close to v1and X (e3)is close to being normal to the plane spanned by v2and
v3. (Unlike the rotation axes of tetragonal, cubic and TI tensors, the rotation axis of the trigonal
rotation tensor is not aligned with a normal of a mirror plane. Thus, there is no extremum along the
would be three-fold rotation axis of Cin Fig. 2.) The search for the absolute minimum is done by the
code shown in the Appendix, and results in d( ˜
XT
0C˜
X0,Ltrigo)=97.18723, where the Euler angles
of X0are ϕ=33.34485,ψ=65.38897+90and θ=122.05007. Hence, the effective trigonal
elasticity tensor expressed in its natural coordinates can be found by, ﬁrst, rotating the tensor Cto
a coordinate system where its minimum is achieved, namely the Euler angles of X0, and evaluating
the closest trigonal tensor, by (4.8), in that coordinate system. We get
(XT
0CX0)trigo =
58.00539 13.83418 3.00375 12.62102 0 0
13.83418 58.00539 3.00375 12.62102 0 0
3.00375 3.00375 24.30584 0 0 0
12.62102 12.62102 0 8.75607 0 0
0 0 0 0 8.75607 17.84882
0 0 0 0 17.84882 44.17122
.
68 C¸ . D˙
INER et al.
5.3 Transversely isotropic and orthotropic tensors
In this section, we discuss the problem of choosing either the transversely isotropic symmetry class
or the orthotropic symmetry class as the effective class of a given generally anisotropic elasticity
tensor. The distinction between the two effective classes might depend on the required accuracy.
Since the orthotropic group is a subgroup of the transversely isotropic group, we can view the
effective TI tensor as a less accurate representation of the effective orthotropic tensor.
The symmetry group of a transversely isotropic tensor has one mirror plane whose normal is
aligned with the rotation axis and inﬁnitely many mirror planes whose normals are perpendicular
to the rotation axis. Since the value of the monoclinic-distance function is zero along the normal of
a mirror plane, the monoclinic-distance function plot for a TI tensor exhibits an equatorial plane,
so that the distance to Lmono is zero for any direction in that plane and along its normal. Thus,
for a generally anisotropic elasticity tensor that is a small perturbation of a TI tensor, the plot of
a monoclinic-distance function has an equatorial plane with values close to zero and a near-zero
minimum close to the normal of that plane (Remark 5.2). Consider
C=
185.8183 59.8984 67.5846 0.4290 0.0272 0.0268
59.8984 191.5953 64.2541 0.2710 0.8688 1.5136
67.5846 64.2541 161.5560 0.2707 1.5353 0.8571
0.4290 0.2710 0.2707 106.9964 1.6032 1.6301
0.0272 0.8688 1.5353 1.6032 98.1860 0.1026
0.0268 1.5136 0.8571 1.6301 0.1026 126.1984
.(5.3)
Figure 4contains the plot of the monoclinic-distance function of C, with an equatorial plane. Since
the equatorial plane is between two contours, the distance function along that plane can take only
values in a small, near-zero range.
Fig. 4 Plot of the monoclinic-distance function of Cgiven in matrix (5.3). Note the equatorial plane and the
direction perpendicular to that plane. Dark shades of grey mean that the values are close to zero
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 69
Fig. 5 Plot of the TI-distance function of Cgiven in matrix (5.3). Note the near-zero minimum shown in black
If we choose the TI symmetry class to approximate C, then the search for the absolute minimum
of the TI-distance function among all orientations in R3gives the closest TI tensor to C. One
can plot the TI-distance function and make a search for the minimum guided by its plot. A code for
plotting this distance function and searching for the minimum is given in (14). Figure 5shows the
plot of the distance function. The search results in the rotation axis
v=sin 2.6915π
180 cos 10.000π
180 ,sin 2.6915π
180 sin 10.000π
180 ,cos 2.6915π
180 .
Then, the orthogonal transformation matrix, X0, that maps e3to vhas the Euler angles, ϕ=
2.6915, ψ =10.0000+90;θcan be set to zero since the TI-distance function does not
depend on it. Hence, the effective TI tensor evaluated in that coordinate system can be found using
(4.10) to get
(˜
XT
0C˜
X0)TI =
219.5173 28.9930 66.0139 0.0000 0.0000 0.0000
28.9930 219.5173 66.0139 0.0000 0.0000 0.0000
66.0139 66.0139 161.3675 0.0000 0.0000 0.0000
0.0000 0.0000 0.0000 51.3374 0.0000 0.0000
0.0000 0.0000 0.0000 0.0000 51.3374 0.0000
0.0000 0.0000 0.0000 0.0000 0.0000 190.5243
.
The squared distance of the elasticity tensor ˜
XT
0C˜
X0to LTI is d( ˜
XT
0C˜
X0,LTI)=87.5430, which
is the absolute minimum of the distance of ˜
XTC˜
Xto LTI among all XSO (3).
The existence of an equatorial plane and a near-zero minimum close to the normal of that plane
suggested the effective symmetry class to be TI. However, depending on the accuracy with which
Cis known, one may choose a less symmetric class to approximate C.
Consider Fig. 6, which is the monoclinic plot of Cdrawn with more contours than shown in
Fig. 4. One of the features of the plot is the existence of three near-zero minima that are almost
70 C¸ . D˙
INER et al.
Fig. 6 Plot of the monoclinic-distance function shown in Fig. 4, but exhibiting more contours
perpendicular to each other. In view of Theorem 5.1, this suggests orthotropic symmetry. To facili-
tate the search for the orientation of the coordinate system where the orthotropic distance function
achieves its absolute minimum, we may use the following theorem due to Diner (13).
THE OR EM 5.3. Let Cbe an elasticity tensor. Let Rπ
2,e2and Rπ
2,e1denote the matrices of rotations
by π
2around e1and e2. Then, the following equation holds:
d(C , Lortho)=1
2d(C , Lmono)+d˜
RT
π
2,e1C˜
Rπ
2,e1,Lmono+d˜
RT
π
2,e2C˜
Rπ
2,e2,Lmono.
The theorem states that the distance to orthotropic symmetry class is equal to the sum of the
monoclinic-distances along the three perpendicular directions. Hence, in order to ﬁnd the absolute
minimum of the orthotropic distance function, d( ˜
XTC˜
X, Lortho), one may restrict the search to
Xthat maps e1,e2and e3to the neighbourhoods of the three minima of the monoclinic-distance
function, namely, v1,v2and v3. A search guided by Fig. 6results in
v1=sin 92.1252π
180 cos 9.9907π
180 ,sin 92.1252π
180 sin 9.9907π
180 ,cos 92.1252π
180 ,
v2=sin 90.0003π
180 cos 99.9915π
180 ,sin 90.0003π
180 sin 99.9915π
180 ,cos 90.0003π
180 ,
v3=sin 2.1252π
180 cos 10.0048π
180 ,sin 2.1252π
180 sin 10.0048π
180 ,cos 2.1252π
180 .
The values of the monoclinic-distance function in these directions are
d( ˜
XT
1C˜
X1,Lmono)=8.98037,where X1(e3)=v1,
d( ˜
XT
2C˜
X2,Lmono)=0.00000,where X2(e3)=v2,
d( ˜
XT
3C˜
X3,Lmono)=8.98034,where X3(e3)=v3.
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 71
The restricted search for the minimum of the orthotropic distance function results in d( ˜
XT
0C˜
X0,
Lortho)=8.9804, where the Euler angles of X0are ϕ=2.1252,ψ=100.0028and θ=
0.0117. Hence, the effective orthotropic tensor expressed in its natural coordinates is
(˜
XT
0C˜
X0)ortho =
191.9780 59.6635 64.1847 0 0 0
59.6635 185.7667 67.7993 0 0 0
64.1847 67.7993 161.4039 0 0 0
0 0 0 98.1199 0 0
0 0 0 0 107.1766 0
0 0 0 0 0 125.9062
.
As expected, the absolute minimum of d( ˜
XTC˜
X, Lortho)is less than the absolute minimum of
d( ˜
XTC˜
X, LTI). The vanishing of d( ˜
XT
2C˜
X2,Lmono)means that Cis, in fact, monoclinic.
6. Conclusions
Since Hookean solids exist only in mathematical formulations, their relation to seismic measure-
ments can be achieved only as an approximation. Hence, no real material belongs to a given sym-
metry class of a Hookean solid, it can only be approximated by such a solid. The concept of distance
in the space of elasticity tensors allows us to quantify this approximation.
The importance of considering a material symmetry is at least two-fold. First, these symmetries
provide us with an insight into the materials, such as their laminations, layering or fractures. Sec-
ondly, we might be able to describe the material using simpler expressions without loss of accuracy.
In many cases, the assumption of isotropy, whose description requires only 2 parameters, is sufﬁ-
cient, without invoking the 21 parameters of a generally anisotropic Hookean solid. The choice of
a more or less symmetric tensor to represent the given material depends on accuracy available and
required.
Prior to a quantitative analysis of ﬁnding the closest tensor, we qualitatively examine the plot of
the monoclinic-distance function to choose the effective symmetry class. Notably, the example in
section 5.3 was obtained by combining two transversely isotropic tensors whose rotation axes are
oriented at 80to one another. Since the combination of two such tensors whose rotation axes are
90apart results in an orthotropic tensor, we expect our example to be close to orthotropic symmetry
as indeed revealed by Fig. 6. In our example, one of the transversely isotropic summands was much
greater in norm than the other one. For this reason, our tensor admits also a good TI approximation.
Such a tensor could represent a structure of the subsurface; for instance, one TI summand could be
due to layering and the other to fractures.
Once the effective symmetry class of a given tensor is chosen, the plot of the monoclinic-distance
function allows us to guide the numerical search for the elasticity parameters of the effective tensor
and the orientation of its natural coordinates, by virtue of Theorem 5.1 and, for the orthotropic
symmetry, also Theorem 5.3, which does not require that the given tensor be a small perturbation
of the orthotropic one.
Acknowledgements
C¸ . Diner’s research was supported by a graduate fellowship of Memorial University of Newfound-
land and M. A. Slawinski’s research grant. M. Kochetov’s and M. A. Slawinski’s research was
72 C¸ . D˙
INER et al.
supported by the Natural Sciences and Engineering Research Council of Canada. The authors wish
to acknowledge fruitful discussions with David Dalton.
References
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for the stiffness tensor, Geophysics 68 (2003) 1022–1031.
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an anisotropic elastic tensor, Acta Crystallogr. 16 (1963) 917–922.
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tally determined stiffness tensor: application to acoustic measurements, Int. J. Solids Struct.
35 (1998) 4091–4106.
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tensor of lower symmetry, J. Elast. 85 (2006) 215–263.
5. M. Kochetov and M. A. Slawinski, On obtaining effective transversely isotropic elasticity ten-
sors, ibid. 94 (2009) 1–13.
6. M. Kochetov and M. A. Slawinski, On obtaining effective orthotropic elasticity tensors, Q. Jl
Mech. Appl. Math. 62 (2009) 149–166.
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Cambridge 2004).
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9. P. Chadwick, M. Vianello and S. C. Cowin, A new proof that the number of linear elastic
symmetries is eight, J. Mech. Phys. Solids 49 (2001) 2471–2492.
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ona, I. Bucataru and M. A. Slawinski, Material symmetries of elasticity tensors, Q. Jl
Mech. Appl. Math. 57 (2004) 583–598.
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ona, I. Bucataru and M. A. Slawinski, Coordinate-free characterization of the symmetry
classes of elasticity tensors, J. Elast. 87 (2007) 109–132.
12. M. A. Slawinski, Seismic Waves and Rays in Elastic Media (Elsevier, Oxford 2003).
13. C¸ . Diner, Identifying the symmetry class and determining the closest symmetry class of an
elasticity tensor, Ph.D. Thesis, (2009, Memorial University, Newfoundland, Canada).
14. C¸ . Diner, M. Kochetov and M. A. Slawinski, Identifying symmetry classes of elasticity tensors
by using monoclinic-distance function, J. Elast. 102 (2011).
APPENDIX A
Maple Codes: Finding the Minimum of Distance Functions
We introduce the Maple packages to be used in the code.
restart;
with(LinearAlgebra);
with(Optimization);
The rotation matrix that transforms e3to the new z-axis:
X(e_3) := Matrix([[cos(psi), -sin(psi), 0], [sin(psi),
cos(psi), 0], [0, 0, 1]]).
Matrix([[1, 0, 0], [0, cos(phi), -sin(phi)], [0, sin(phi), cos(phi)]]));
CHOOSING EFFECTIVE SYMMETRY CLASSES FOR ELASTICITY TENSORS 73
The rotation matrix that transforms e1to the new x-axis:
X(e_1) := Matrix([[cos(theta), -sin(theta), 0], [sin(theta),
cos(theta), 0], [0, 0, 1]]);.
The rotation matrix that transforms the coordinate system:
X := X(e_3)*X(e_1)
Finding ˜
XSO (6)from XSO(3)that is going to act on elasticity tensor. Note that ˜
Xin the article is
represented as Xtilde in the code.
Xtilde:= simplify(Matrix([[X[1, 1]ˆ2, X[1, 2]ˆ2, X[1, 3]ˆ2,
sqrt(2)*X[1, 2]*X[1, 3],
sqrt(2)*X[1, 1]*X[1, 3], sqrt(2)*X[1, 1]*X[1, 2]], [X[2, 1]ˆ2,
X[2, 2]ˆ2, X[2, 3]ˆ2,
sqrt(2)*X[2, 2]*X[2, 3], sqrt(2)*X[2, 1]*X[2, 3],
sqrt(2)*X[2, 1]*X[2, 2]],
[X[3, 1]ˆ2, X[3, 2]ˆ2, X[3, 3]ˆ2, sqrt(2)*X[3, 2]*X[3, 3],
sqrt(2)*X[3, 1]*X[3, 3],
sqrt(2)*X[3, 1]*X[3, 2]], [sqrt(2)*X[2, 1]*X[3, 1],
sqrt(2)*X[2, 2]*X[3, 2],
sqrt(2)*X[2, 3]*X[3, 3], X[2, 3]*X[3, 2]+X[2, 2]*X[3, 3],
X[2, 3]*X[3, 1]+X[2, 1]*X[3, 3], X[2, 2]*X[3, 1]+X[2, 1]*X[3, 2]],
[sqrt(2)*X[1, 1]*X[3, 1],
sqrt(2)*X[1, 2]*X[3, 2], sqrt(2)*X[1, 3]*X[3, 3], X[1, 3]*X[3, 2]+
X[1, 2]*X[3, 3],
X[1, 3]*X[3, 1]+X[1, 1]*X[3, 3], X[1, 2]*X[3, 1]+X[1, 1]*X[3, 2]],
[sqrt(2)*X[1, 1]*X[2, 1], sqrt(2)*X[1, 2]*X[2, 2],
sqrt(2)*X[1, 3]*X[2, 3],
X[1, 3]*X[2, 2]+X[1, 2]*X[2, 3], X[1, 3]*X[2, 1]+X[1, 1]*X[2, 3],
X[1, 2]*X[2, 1]+
X[1, 1]*X[2, 2]]])):
Introducing the elasticity matrix Cin its general form.
C := Matrix(6, symbol = c, shape = symmetric);
This line is for changing the Voigt notation to the Kelvin notation. Herein, the entries of Cis expressed already
in the Kelvin notation thus, we do not change anything. Otherwise, the ﬁrst line should be multiplied by 2
and the second line by 2.
C[1 .. 3, 4 .. 6] := C[1 .. 3, 4 .. 6];
C[4 .. 6, 4 .. 6] := C[4 .. 6, 4 .. 6];
Introducing the parameters of elasticity matrix.
c := Matrix(6, 6, {(1, 1) = 42.03357978, (1, 2) = 32.49305220,
(1, 3) = 26.42291512, (1, 4) = 5.799531000, (1, 5) = 9.938160420,
(1, 6) = 21.33549018, (2, 1) = 32.49305219, (2, 2) = 36.72218489,
(2, 3) = 34.84774953, (2, 4) = -5.221667422, (2, 5) = -35.36967653,
(2, 6) = -23.60019509, (3, 1) = 26.42291511, (3, 2) = 34.84774954,
(3, 3) = 31.71680164, (3, 4) = 12.16688779, (3, 5) = 3.767088965,
74 C¸ . D˙
INER et al.
(3, 6) = 10.24209672, (4, 1) = 5.799531000, (4, 2) = -5.221667442,
(4, 3) = 12.16688780, (4, 4) = 32.84181099, (4, 5) = 20.98631307,
(4, 6) = 2.36291613, (5, 1) = 9.938160440, (5, 2) = -35.36967653,
(5, 3) = 3.767088965, (5, 4) = 20.98631308, (5, 5) = -22.64105584,
(5, 6) = -9.709913499, (6, 1) = 21.33549018, (6, 2) = -23.60019510,
(6, 3) = 10.24209672, (6, 4) = 2.36291613, (6, 5) = -9.709913499,
(6, 6) = 11.32667859});
Expressing the elasticity matrix Cin any coordinate system by rotating it with a generic orthogonal transfor-
mation that is introduced in the above lines.
R := evalf((Transpose(Xtilde).C.Xtilde);
Evaluating the trigonal-distance function, namely d( ˜
XTC˜
X, Ltrigo). Note that in the code, we denote the
trigonal distance function d( ˜
XTC˜
X.Lmono)by Dist-trigo.
Dist-trigo := simplify(2*(R_{16}ˆ2+R_{26}ˆ2+R_{15}ˆ2+R_{25}ˆ2+R_{34}ˆ2
+R_{35}ˆ2+R_{36}ˆ2+R_{45}ˆ2+R_{46}ˆ2)+1/2(R_{11}-R_{22})ˆ2+(R_{13}
-R_{23})ˆ2+(R_{14}+R_{24})ˆ2+1/2(R_{44}-R_{55})ˆ2+1/2(1/2R_{11}
+1/2R_{22}-R_{12}-R_{66})ˆ2+(1/\sqrt{2})R_{14}-(1/\sqrt{2})R_{24}
-R_{56})ˆ2);.
To ﬁnd a local minimum of the trigonal-distance function, observe dark regions on the plot. Read its orientation
from the locater bar of the plot that gives the location of the minimum approximately. To ﬁnd its exact location,
search for the minimum in the neighbourhood of the approximate location:
Min_data := Minimize(Dist-trigo,
phi = (40-20)*(1/180)*Pi .. (40+20)*(1/180)*Pi,
psi = (1/180)*(90-34-20)*Pi .. (1/180)*(90-34+20)*Pi,
theta = (1/180)*(128-20)*Pi .. (1/180)*(128+20)*Pi);
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Based on the theory derived in part 1, in which we obtained the azimuthally dependent fourth-order normal-moveout (NMO) velocity functions for layered orthorhombic media in the slowness- azimuth/slowness and the slowness-azimuth/offset domains, in part 2, we extend the theory to the offset-azimuth/slowness and offset-azimuth/offset domains. We reemphasize that this paper does not suggest a new nonhyperbolic traveltime approximation; rather, it provides exact expressions of the NMO series coefficients, computed for normal-incidence rays, which can then be further used within known azimuthally dependent traveltime approximations for short to moderate offsets. The same type of models as in part 1 are considered, in which the layers share a common horizontal plane of symmetry, but the azimuths of their vertical symmetry planes are different. The same eight local (single- layer) and global (overburden multilayer) effective parameters are used. In addition, we have developed an alternative set of global effective parameters in which the "anisotropic" effective parameters are normalized, classified into two groups: two "azimuthally isotropic" parameters and six "azimuthally anisotropic" parameters. These parameters have a clearer physical interpretation and they are suitable for inversion purposes because they can be controlled and constrained. Next, we propose a special case, referred to as "weak azimuthal anisotropy," in which only the azimuthally anisotropic effective parameters are assumed to be weak. The resulting NMO velocity functions are considerably simplified, reduced to the form of the slowness-azimuth/slowness formula. We verify the correctness of our method by applying it to a multilayer orthorhombic medium with strong anisotropy. We introduce our derived, fourth-order slowness-azimuth/offset domain NMO velocity function into the well-known nonhyperbolic asymptotic traveltime approximation, and we compare the approximate traveltimes with exact traveltimes obtained by twopoint ray tracing. The comparison shows an accurate match up to moderate offsets. Although the accuracy with the weak azimuthal anisotropic formula is inferior, it can still be considered reasonable for practical use.
... The ideas of Kelvin have been rediscovered in the context of anisotropic elasticity, see (Mehrabadi and Cowin 1990;Kowalczyk-Gajewska and Ostrowska-Maciejewska 2014), and put into the context of modern tensor algebra by several authors (Mehrabadi and Cowin 1990;Kowalczyk-Gajewska and Ostrowska-Maciejewska 2014;Moakher 2008). Major topics of interest in which the concept has been used are: the use of six eigenstiffnesses and orthogonal eigenstates for a better understanding of material behaviour (Rychlewski 1984;Annin and Ostrosablin 2008); different aspects of a spectral decomposition of the stiffness tensor (Theocaris and Philippidis 1991;Theocaris 2000;Bolcu et al. 2010); the investigation of material symmetries and preferred deformation modes of anisotropic media, e.g., composite materials (Mehrabadi and Cowin 1990;Bóna et al. 2007) including the relationship to fabric tensors (Moesen et al. 2012) and deformation-induced anisotropy (Cowin 2011); the transformation of the properties of one anisotropic medium to the closest effective medium from a differing symmetry group (Norris 2006;Diner et al. 2011;Kochetov and Slawinski 2009;Moakher and Norris 2006); wave attenuation and elastic constant inversion from wave traveltime data (Carcione et al. 1998;Dellinger et al. 1998). The inversion of Hooke's law in the case of incompressible or slightly compressible materials was studied by Itskov and Aksel (2002), while the use of the spectral decomposition of the stiffness tensor in a constitutive formulation for finite hyperelasticity in a finite element context was described in Dłuzewski and Rodzik (1998). ...
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