Spectral decomposition of the linear elastic tensor for monoclinic symmetry

Abstract

The compliance fourth-rank tensor related to crystalline or other anisotropic media belonging to the monoclinic crystal system is spectrally decomposed for the first time, and its characteristic values and idempotent fourth-rank tensors are established. Further, it is proven that the idempotent tensors resolve the stress and strain second-rank tensors into eigentensors, thus giving rise to a decomposition of the total elastic strain-energy density into non-interacting strain-energy parts. Several examples of representative inorganic crystals of the monoclinic system illustrate the results of the theoretical analysis. It is also proven that the essential parameters required for a coordinate-invariant characterization of the elastic properties of a crystal exhibiting monoclinic symmetry are both the six characteristic values of the compliance tensor and seven dimensionless parameters. These material constants, referred to as the eigenangles, are shown to be accountable for the orientation of the stress and strain eigentensors, when represented in a stress coordinate system. Finally, the restrictions dictated by the classical thermodynamical argument on the elements of the compliance tensor, which are necessary and sufficient for the elastic strain-energy density to be positive definite, are investigated for the monoclinic symmetry.

Full text

Acta Cryst. (1999). A55, 635±647
Spectral decomposition of the linear elastic tensor for monoclinic symmetry
Pericles S. Theocaris and Dimitrios P. Sokolis
National Academy of Athens, PO Box 77230, 175 10 Athens, Greece
(Received 5 October 1998; accepted 7 December 1998)
Abstract
The compliance fourth-rank tensor related to crystalline
or other anisotropic media belonging to the monoclinic
crystal system is spectrally decomposed for the ®rst time,
and its characteristic values and idempotent fourth-rank
tensors are established. Further, it is proven that the
idempotent tensors resolve the stress and strain second-
rank tensors into eigentensors, thus giving rise to a
decomposition of the total elastic strain-energy density
into non-interacting strain-energy parts. Several exam-
ples of representative inorganic crystals of the mono-
clinic system illustrate the results of the theoretical
analysis. It is also proven that the essential parameters
required for a coordinate-invariant characterization of
the elastic properties of a crystal exhibiting monoclinic
symmetry are both the six characteristic values of the
compliance tensor and seven dimensionless parameters.
These material constants, referred to as the eigenangles,
are shown to be accountable for the orientation of
the stress and strain eigentensors, when represented in
a stress coordinate system. Finally, the restrictions
dictated by the classical thermodynamical argument
on the elements of the compliance tensor, which are
necessary and suf®cient for the elastic strain-energy
density to be positive de®nite, are investigated for the
monoclinic symmetry.
1. Introduction
Tensors of the fourth rank embodying the elastic or
other property of crystalline anisotropic substances were
initially expanded (Srinivasan & Nigam, 1969) as a
linear combination of independent elementary tensors,
corresponding to scalar coef®cients, which remain
invariant under orthogonal coordinate transformations.
Next, the algebra of fourth-rank tensors of the 32 crystal
classes was broken down to irreducible subalgebras
(Walpole, 1981, 1984), offering insight into the tensor
structure and simplifying considerably the calculations
of sums, products and inverses between the tensors.
Conversely, the spectral decomposition was proven
(Rychlewski, 1984a,b) to be the simplest possible
decomposition of the elastic compliance Sor stiffness C
fourth-rank tensors. Additionally, this decomposition
was preferable because of its ability to split these tensors
into idempotent fourth-rank tensors, which, in turn,
de®ned energy orthogonal stress and strain eigen-
tensors. Moreover, the spectral decomposition did not
correspond to the decompositions of both Walpole and
Srinivasan & Nigam, except for the trivial cases of
isotropic and cubic symmetry.
Nonetheless, the concepts of elastic eigenvalues, as
well as those of stress and strain eigentensors, were
introduced by Thomson (Lord Kelvin), who called them
the `Six principal elasticities and principal stress and
strain-types of an elastic solid' (Thomson, 1856, 1878). In
a time when access to the tensorial formulation of the
mathematical theory of elasticity was not feasible,
Thomson clearly perceived and established, using an
altogether different terminology, the unsurpassed
simplicity introduced through the notion of elastic
eigenvalues and eigentensors of compliances in the
analysis of the structure of the generally anisotropic
linearly elastic solid. It is, however, unfortunate that
Todhunter & Pearson (1886±93) criticized with a great
deal of skepticism the contribution of Thomson in this
area. In fact, Lord Kelvin's formulation was entirely
neglected for more than a century until Rychlewski
recreated the basic ideas of the analysis reported by
Lord Kelvin, exhibiting the mathematical structure of an
arbitrary linearly elastic anisotropic body.
Despite the fact that Rychlewski con®rmed the
application of the spectral decomposition principle on
the class of symmetric fourth-rank tensors, he did not
proceed to determine the eigenvalues and eigentensors
of the corresponding tensors. In fact, these were estab-
lished subsequently (Theocaris & Philippidis, 1989, 1990,
1991) and, combined with a characteristic angle, the
eigenangle !, provided an invariant speci®cation for the
elastic features of a transversely isotropic medium.
Then, the three-dimensional spectral decomposition was
extended to incorporate the two-dimensional plane
stress conditions (Theocaris & Sokolis, 1998).
In this paper, a full reduction of the compliance
fourth-rank tensor Sis developed for crystalline media
belonging to the monoclinic system, based on the
spectral decomposition principle, for the ®rst time. The
characteristic values of the compliance tensor Sare
determined and the elementary idempotent fourth-rank
tensors are established. These elementary tensors give
rise to stress and strain eigentensors, which split the
635
#1999 International Union of Crystallography Acta Crystallographica Section A
Printed in Great Britain ± all rights reserved ISSN 0108-7673 #1999

636 SPECTRAL DECOMPOSITION OF THE LINEAR ELASTIC TENSOR
elastic potential of the monoclinic medium into distinct
elements, designating the absence of a pure dilatational
strain-energy component. Further, it is proven that the
constitutive parameters, required for an invariant char-
acterization of the elastic properties of a crystal of the
monoclinic syngony, are the six distinct eigenvalues of
the compliance tensor S, in addition to a set of seven
dimensionless quantities, referred to as the eigenangles,
which are responsible for the orientation and alignment
of the stress and strain eigentensors in the six-dimen-
sional stress space. Next, the individual criteria in terms
of the elements of the compliance tensor, which are
necessary and suf®cient for the elastic strain energy to
be positive de®nite, are examined for monoclinic
symmetry. Finally, several examples of representative
inorganic crystals of the monoclinic system illustrate the
results of our theoretical analysis.
2. Linear elasticity of anisotropic media
The generalized anisotropic form of Hooke's law
(Hooke, 1678) states that each strain component is
directly proportional to each stress component or, in
symbolic indicial notation:
eSror "ij Sijklkl ;1
where i;j;k;l1;2 or 3 and the coef®cients of linearity
S
ijkl
are the coef®cients of the compliance fourth-rank
tensor Sexpressed in a Cartesian coordinate system. It is
further assumed that the deformations are measured
from the natural stress-free state and the in¯uence of
temperature and other ®elds is insigni®cant. In addition,
it is presumed that the stress tensor r, whose compo-
nents are 
ij
, and the linear strain tensor e, whose
components are "
ij
, are symmetric (Sokolnikoff, 1956).
As equations (1) stand, there are 81 components of the
compliance tensor Sbut, owing to the symmetry of the
stress rand strain etensors, two important symmetry
restrictions are imposed on the compliance tensor S,
namely:
Sijkl Sjikl;Sijkl Sijlk ;2
which reduce the number of independent components of
Sto 36. Next, another symmetry constraint is imposed
on the compliance tensor S, based on the thermo-
dynamical argument that no work is produced by an
elastic medium in a closed loading cycle. This is the
symmetry which necessitates that the components with
subscripts ijkl and klij are equal:
Sijkl Sklij:3
These reciprocal relations further reduce the number of
distinct compliance components to 21 in the most
general case. In addition, reciprocal relations (3) are of
thermodynamic origin, hence they are not dependent
upon the actual mechanism of elastic behaviour.
It should be noted that many different notations have
been proposed for the stress and strain components at
various times (Todhunter & Pearson, 1886±93; Voigt,
1910; Love, 1927; Timoshenko & Goodier, 1951;
Southwell, 1941; Cady, 1946; Wooster, 1949; Mason,
1950; Lekhnitskii, 1963; Nye, 1957; Hearmon, 1961). We
have used Nye's notation throughout this paper.
The generalized Hooke's law, expressed in (1)
utilizing a Cartesian fourth-rank tensor index notation,
may be represented in matrix notation as follows:
"11
"22
"33
2"23
2"13
2"12
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5

S1111 S1122 S1133 2S1123 2S1113 2S1112
S1122 S2222 S2233 2S2223 2S2213 2S2212
S1133 S2233 S3333 2S3323 2S3313 2S3312
2S1123 2S2223 2S3323 4S2323 4S2313 4S2312
2S1113 2S2213 2S3313 4S2313 4S1313 4S1312
2S1112 2S2212 2S3312 4S2312 4S1312 4S1212
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
11
22
33
23
13
12
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
:
4
Hooke's law is often expressed in its contracted
notation, the well known Voigt notation, which is
represented in the form
esror "pspqq5
or alternatively in the form
"1
"2
"3
"4
"5
"6
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5

s11 s12 s13 s14 s15 s16
s12 s22 s23 s24 s25 s26
s13 s23 s33 s34 s35 s36
s14 s24 s34 s44 s45 s46
s15 s25 s35 s45 s55 s56
s16 s26 s36 s46 s56 s66
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
1
2
3
4
5
6
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;6
where p;q1;2;...;6, utilizing a 6 6 matrix s.
However, one should be cautious employing the Voigt
notation, since this is not a tensorial notation, that is the
components s
pq
do not form the components of a tensor
as do the S
ijkl
, which constitute the components of a
Cartesian fourth-rank tensor in three dimensions.
Nevertheless, the Voigt notation is important because it
is almost invariably used in experimental work of elas-
ticity and has become the standard in anisotropic elas-
ticity (Voigt, 1910; Nye, 1957; Hearmon, 1961). Then, the
equivalence between the components of the compliance
fourth-rank tensor Sand the components of the 6 6
matrix sof the Voigt notation is shown to be
Sijkl spq for p;q1;2or3 7a
2Sijkl spq for p1;2 or 3 and q4;5or6 7b
4Sijkl spq for p;q4;5or6;7c
in which the following contraction rule is applied for
replacing a pair of indices by a single contracted index:
11 !1, 22 !2, 33 !3, (23, 32) !4, (13, 31) !5,
(12, 21) !6.
Furthermore, the full tensor suf®xes of the stresses r
and strains eare contracted according to the scheme:

PERICLES S. THEOCARIS AND DIMITRIOS P. SOKOLIS 637
11 1;
22 2;
33 3;
23 4;
13 5;
12 68a
"11 "1;"
22 "2;"
33 "3;
2"23 "4;2"13 "5;2"12 "6:8b
The occurrence of the factor 2 in the equations relating
to the shear strains in (8b) should be particularly noted
and the shear strains "
ij
,i;j1;2or3,i6 j, carefully
distinguished from the contracted shear strains "
p
,
p4;5 or 6, which do not form the components of a
tensor, as do the "
ij
.
3. Spectral decomposition of the monoclinic compliance
fourth-rank tensor
In this paper, our attention is restricted to the mono-
clinic crystal system, which is characterized by a plane of
elastic symmetry. In the following, the compliance
fourth-rank tensor Sof a monoclinic linear elastic solid
is decomposed spectrally for the ®rst time. We assume
the Cartesian coordinate system, where the stress and
strain tensors are referred to, with the 3 axis oriented
normal to the plane of elastic symmetry. The compo-
nents of the compliance fourth-rank tensor S, associated
with the adopted Cartesian system, with respect to the
components of the 6 6 matrix sof the Voigt notation,
are given by:
S1111 s11;S2222 s22 ;S3333 s33 ;9a
S1122 S2211 s12;S2233 S3322 s23 ;
S1133 S3311 s13;9b
S2323 S2332 S3223 S3232 1
4s44;9c
S1313 S1331 S3113 S3131 1
4s55;9d
S1212 S1221 S2112 S2121 1
4s66;9e
S1323 S1332 S3123 S3132 S2313 S3213
S2331 S3231 1
4s45;9f
S1112 S1121 S1211 S2111 1
2s16;9g
S2212 S2221 S1222 S2122 1
2s26;9h
S3312 S3321 S1233 S2133 1
2s36;9i
and all the remaining S
ijkl
components are zero.
The eigenvalues of the square matrix of rank six
associated with the tensor Swere determined by solving
its characteristic equation:
det
s11 ÿs12 s13 00s16=21=2
s12 s22 ÿs23 00s26=21=2
s13 s23 s33 ÿ00s36=21=2
000s44=2ÿs45 =20
000s45 =2s55=2ÿ0
s16=21=2s26 =21=2s36=21=200s66=2ÿ
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
010
introducing factors 1=21=2and 1=2 in order to operate
with the 6 6 matrices associated with the compliance S
and stiffness Cfourth-rank tensors using tensorial rules.
In fact, with this modi®cation, the components of the
associated 6 6 matrices form the components of a
Cartesian second-rank tensor in six dimensions. Equa-
tion (10) is then equivalent to
4A3B2CD
2ÿs44 s55
2

s44s55 ÿs2
45
4

011
with
Aÿs11 ÿs22 ÿs33 ÿs66
212a
Bs11 s22 s11 s33 s22s33 s11 s22 s33s66
2
ÿs2
12 s2
13 s2
23 s2
16
2s2
26
2s2
36
2

12b
Cs11 s2
23 s2
26
2s2
36
2

s22 s2
13 s2
16
2s2
36
2

s33 s2
12 s2
16
2s2
26
2

s66
2s2
12 s2
13 s2
23
ÿs11s12 ÿs11 s33 ÿs22s33 ÿs11s22 s33
s23s26 s36 s13 s16s36 s12 s16 s26 2s12s13 s2312c
Ds16s26 s33s12 ÿs13 s23 s26s36 s11 s23 ÿs12s13 
s16s36 s22 s13 ÿs12s23 s66
2s11s22 s33 2s12s13 s23
ÿs11s2
23 ÿs22s2
13 ÿs33s2
12s2
16
2s2
23 ÿs22s33 
s2
26
2s2
13 ÿs11s33 s2
36
2s2
12 ÿs11s22 :12d
The polynomial inside the ®rst parentheses of relation
(11) is a quartic. Therefore, by substituting yÿA=4,
the quartic polynomial is transformed to its reduced
form:
y4Py2Qy R0;13
where
PBÿ3A2
814a
QA3
8ÿAB
2C14b
Rÿ3A
4

4
A2B
16 ÿAC
4D:14c
Relation (13) may be expressed alternatively as follows:
y2z
2

2
ÿzÿPy2ÿQy z2
4ÿR

0:15

638 SPECTRAL DECOMPOSITION OF THE LINEAR ELASTIC TENSOR
The second term of (15) is a quadratic function of y,
whose discriminant is
z3ÿPz2ÿ4Rz 4PR ÿQ2:16
We choose the discriminant equal to zero (0), thus
allowing the quadratic function to become a perfect
square:
z3ÿPz2ÿ4Rz 4PR ÿQ20:17
This is a cubic equation, which has to be transformed to
its reduced form in order to be solved. Substituting
zÿP=3 gives the cubic equation in the form
3P0Q0018
with
P0ÿ4RÿP2
319a
Q0ÿ2P3
27 8PR
3ÿQ2:19b
Further, with qk=q, (18) may be recast as
q6Q0q3k30;20
in which
kP0
3:21
Moreover, with uq3, (20) becomes a quadratic
equation, which is readily solved:
u2Qu k30:22
Thus, the three solutions z
m
,m1;2;3, of the cubic
polynomial of relation (17) were determined to be
z1ÿ ÿQ0
2Q02
4ÿk3

1=2
"#
1=31i31=2
2
kÿ1i31=2
2ÿQ0=2Q02=4ÿk31=21=3P
323a
z2ÿ
Q0
2Q02
4ÿk3

1=2
"#
1=31i31=2
2
kÿ1ÿi31=2
2ÿQ0=2Q02=4ÿk31=21=3P
323b
z3ÿ
Q0
2Q02
4ÿk3

1=2
"#
1=3
k
ÿQ0=2Q02=4ÿk31=21=3P
3:23c
Substitution in relation (15) for zzm,m1;2;3,
gives the quartic equation in the form
y2zm
22
ÿyÿQ
2zmÿP2
0:24
Relation (24) leads to two quadratic equations, which
are readily solved:
y2yzm
2ÿQ
2zmÿP

0:25
The eigenvalues 
m
,m1;...;6, of the associated
square matrix of rank six to tensor Sde®ned by (9) were,
thus, evaluated to be:
1ÿzmÿP1=2
2
1
2ÿzmP 2Q
zmÿP1=2

1=2
ÿA
426a
2ÿzmÿP1=2
2
ÿ1
2ÿzmP 2Q
zmÿP1=2

1=2
ÿA
426b
3zmÿP1=2
2
1
2ÿzmPÿ 2Q
zmÿP1=2

1=2
ÿA
426c
4zmÿP1=2
2
ÿ1
2ÿzmPÿ 2Q
zmÿP1=2

1=2
ÿA
426d
5s44 s55
41
2
1
4s44 ÿs552s2
45

1=2
26e
5s44 s55
4ÿ1
2
1
4s44 ÿs552s2
45

1=2
:26f
The characteristic values 
m
,m1;...;6, de®ned by
relations (26), constitute the roots of the minimum
polynomial of the compliance tensor S, which in
factorized form may be written as
Sÿ1I...Sÿ6I0;27
where Iis the unit element of the symmetric fourth-rank
tensor Mspace, which in symbolic notation is repre-
sented as Msym(LL). Furthermore, we refer to a
symmetric fourth-rank tensor if this is described by a
symmetric 6 6 matrix of the form given in (10), that is,
if this tensor satis®es (2) and (3).
The corresponding six idempotent fourth-rank
tensors E
m
,m1;...;6, of the spectral decomposition
of Swere obtained as:
EmSÿ1I...Sÿmÿ1ISÿm1I...Sÿ6I
mÿ1...mÿmÿ1mÿm1...mÿ6:
28
Tensors E
m
were, thus, evaluated to be:

PERICLES S. THEOCARIS AND DIMITRIOS P. SOKOLIS 639
E1E1
ijkl gggijgkl 29a
E2E2
ijkl rrrijrkl 29b
E3E3
ijkl hhhijhkl 29c
E4E4
ijkl sssijskl 29d
E5E5
ijkl tttijtkl 29e
E6E6
ijkl qqqijqkl ;29f
with g,r,h,s,t,q2L, where Lrepresents the second-
rank symmetric tensor space over R
3
, which, together
with the ordinary de®nition of the scalar product,
constitutes a 6D Euclidean space. In symbolic notation,
the tensor space Lis expressed by Lsym(R
3
R
3
).
The second-rank symmetric tensors g,r,h,s,tand q,
appearing in (29) for the expressions of the idempotent
tensors E
m
,m1;...;6, are de®ned as follows:
gg3ag2bg1cg6d30a
rr3ar2br1cr6d30b
hh3ah2bh1ch6d30c
sÿsin aÿsin cos bÿsin cos cos c
cos cos cos d30d
tcos fsin e30e
qÿsin fcos e;30f
in which
g1sin sin 'cos 
ÿsin ÿ sin cos 'sin cos sin cos 31a
g2ÿsin cos 'cos ÿcos sin sin 31b
g3cos cos 31c
g6sin sin 'sin cos ÿ sin cos 'sin 
cos sin cos 31d
r1ÿsin !cos 'ÿcos sin 'cos !cos 
ÿsin ÿ sin !sin 'cos cos 'cos !sin 
cos !sin sin cos 31e
r2ÿsin !sin 'cos cos 'cos !cos 
ÿcos !sin sin sin 31f
r3cos !sin cos 31g
r6ÿsin !cos 'ÿcos sin 'cos !sin 
cos ÿ sin !sin 'cos cos 'cos !sin 
cos !sin sin cos 31h
h3cos !cos 'ÿcos sin 'sin !cos 
ÿsin cos !sin 'cos cos 'sin !sin 
sin !sin sin cos 31i
h2cos !sin 'ÿcos sin 'sin !cos 
ÿsin !sin sin sin 31j
h1sin !sin cos 31k
h6cos !cos 'ÿcos sin 'sin !sin 
cos cos !sin 'cos cos 'sin !sin 
sin !sin sin cos :31l
Furthermore, the second-rank symmetric tensors a,b,c,
d,eand femerging in relations (30), in the expressions
for the second-rank symmetric tensors g,r,h,s,tand q
are de®ned as follows:
akk;bll;cmm32a
d1
21=2lmml32b
e1
21=2kllk32c
f1
21=2kmmk32d
with k,land mbeing the unit vectors of R
3
, associated
with the 3, 2 and 1 directions of the Cartesian coordinate
system.
Further, according to (9), it is easily noted that the
components of tensor Sare both symmetrical and real,
thus, it follows that tensor Sis self-adjoint or hermitian.
Hence, the proof that all the eigenvalues 
m
and
idempotent fourth-rank tensors E
m
,m1;...;6, of the
spectral decomposition of Sare real is obtained at once,
based on the hermitian nature of the compliance fourth-
rank tensor S.
In addition, the seven angles ,,, ,,!and ',
appearing in relations (31), are called eigenangles and
are de®ned as follows:
tan Q4
Z2
4W2
411=2tan W4
Z2
411=2;
tan Z433a
cos 2 s44 ÿs55
2

s44 ÿs55
2

2
s2
45

ÿ1=2
33b
tan Z2
4W2
41
Z2
4W2
4Q2
41ÿQ2
1
Z2
1W2
1Q2
11

1=2
Q1
Z2
1W2
1Q2
111=2
"#
ÿ1
33c
tan !Z2
4W2
41
Z2
4W2
4Q2
41ÿQ2
1
Z2
1W2
1Q2
11

ÿQ2
2
Z2
2W2
2Q2
211=2
Q2
Z2
2W2
2Q2
211=2

ÿ1
33d

640 SPECTRAL DECOMPOSITION OF THE LINEAR ELASTIC TENSOR
tan 'Z2
4W2
41
Z2
4W2
4Q2
41ÿQ2
1
Z2
1W2
1Q2
11

Z2
4W2
412
Z2
4W2
4Q2
412

ÿ1
Z2
41
ÿQ2
1Q2
4W2
4
Z2
1W2
1Q2
11Z2
4W2
4121=2
W2
Z2
2W2
2Q2
211=2

Q1Q4W4
Z2
1W2
1Q2
111=2Z2
4W2
41ÿ1
;
33e
in which
QiFiÿC2
i=2Ai
2ÿ1=2EiÿBiCi=Ai

;
WiBi
Ai
FiÿC2
i=2Ai
2ÿ1=2EiÿBiCi=Ai


ÿCi
21=2Ai
34a
Ziÿ s12
s11 ÿi
Bi
Ai
FiÿC2
i=2Ai
2ÿ1=2EiÿBiCi=Ai

ÿCi
21=2Ai

s13
s11 ÿi
FiÿC2
i=2Ai
2ÿ1=2EiÿBiCi=Ai

ÿs16
21=2s11 ÿi;34b
where
Ais22 ÿiÿ s2
12
s11 ÿi

;
Bis23 ÿs12s13
s11 ÿi
 35a
Cis26 ÿs12s16
s11 ÿi

;
Dis33 ÿiÿ s2
13
s11 ÿi

35b
Eis36 ÿs13s16
s11 ÿi

;
Fis66
2ÿi

ÿs2
16
2s11 ÿi

35c
and the subscript iacquires the values 1, 2, 3 or 4.
For the eigenvalues 
m
,m1;...;6, given by rela-
tions (26), and the corresponding idempotent fourth-
rank tensors E
m
,m1;...;6, expressed by relations
(29), the compliance fourth-rank tensor Sis spectrally
decomposed. It is, hence, given the following expansion:
S1E1...6E6:36
Therefore, the six eigenvalues 
m
,m1;...;6, toge-
ther with the eigenangles ,,, ,,!and 'constitute
the 13 coordinate-invariant parameters necessary for the
characterization of the elastic properties of crystals
belonging to the monoclinic syngony.
Furthermore, the elementary idempotent tensors E
m
,
m1;...;6, decompose the unit element Iof the
fourth-rank symmetric tensor space Mand satisfy the
following set of equations:
IE1...E637a
EmEn0;m6 n37b
EmEmEm:37c
In fact, the idempotent fourth-rank tensors E
m
,
m1;...;6, provide an orthogonal expansion of the
space Mof symmetric fourth-rank tensors into orthog-
onal subspaces M
m
as follows:
MM1...M6;Mm?Mnfor m6 n;38
where E
m
is the idempotent tensor on M
m
for
m1;...;6.
4. Energy orthogonal states of stress and strain
The action of the idempotent fourth-rank tensors E
m
,
m1;...;6, on the symmetric second-rank tensor
space Lleads to a decomposition of the Lspace into
subspaces L
m
in the following manner:
LL1...L6;Lm?Lnfor m6 n:39
Therefore, the stress second-rank eigentensors rmof the
compliance fourth-rank tensor Sfor the monoclinic
symmetry are derived by the orthogonal projection of a
second-rank symmetric tensor ron subspaces L
m
,
produced by the idempotent fourth-rank tensors E
m
,as
follows:
rmEmr;m1;...;6:40
Moreover, if the second-rank stress eigentensors rm
constitute eigenstates of tensor S, they should satisfy the
eigenvalue equation
Srm1E1... 6E6rmmrm;41
in which index mvaries between 1 and 6, and the 
m
values are described in terms of relations (26).
Denoting by rthe contracted stress tensor in the form
of a 6D vector, which is expressed by
r1;
2;
3;
4;
5;
6T42
and performing the computations implied by relations
(40), it was found that, in contracted notation:

PERICLES S. THEOCARIS AND DIMITRIOS P. SOKOLIS 641
r1g11g22g33g66
g1;g2;g3;0;0;g6T43a
r2r11r22r33r66
r1;r2;r3;0;0;r6T43b
r3h11h22h33h66
h1;h2;h3;0;0;h6T43c
r4ÿsin cos cos 1ÿsin cos 2
ÿsin 3cos cos cos 6
ÿsin cos cos ; ÿsin cos ;
ÿsin ; 0;0;cos cos cos T43d
r5cos 4sin 50;0;0;cos ; sin ; 0T43e
r6ÿsin 4cos 5
0;0;0;ÿsin ; cos ; 0T;43f
where g
i
,r
i
and h
i
,i1;2;3;6, are de®ned by relations
(31).
Relations (43) assert that the stress eigentensors,
corresponding to the spectral decomposition of the
compliance tensor Sfor a medium exhibiting monoclinic
symmetry, decompose the generic stress tensor rinto six
elements, namely,
rr1...r644
with stress eigentensors r1,r2,r3and r4being a
superposition of simple shear with stressing along the 1,
2 and 3 directions of the adopted Cartesian coordinate
system, and stress eigentensors r5and r6constituting
simple shear states.
In addition, it is readily observed in relations (43) that
the contracted stress eigentensors r1,r2,r3and r4are
dependent on the values of eigenangles ,,,,!and
', expressed by relations (33), in terms of the compo-
nents of the compliance tensor Sof the monoclinic body.
On the contrary, the remaining two contracted stress
eigentensors, namely r5and r6, are dependent on the
value of eigenangle , de®ned by relation (33b).
It is of interest to note that the generalized aniso-
tropic form of Hooke's law, represented by equation (1),
may be expressed as follows:
eSr1E1... 6E6r
1r1...6r6;45
so that the strain second-rank tensor eis readily split
into six eigentensors em:
ee1...e6:46
Therefore, the expression of Hooke's law for crystals
belonging to the monoclinic system may be decomposed
into six independent laws of proportionality of stress
and strain eigentensors in a well de®ned manner:
emmrm;for m1;...;6:47
Next, considering the de®nition of the total elastic
strain-energy density, we have that:
2Trre
rSr

r1...r61E1... 6E6

r1...r6
1r1r1...6r6r6:48
Relation (48) may be recast as
2TrTr1...Tr6r1e1... r6e6;
49
that is the elastic potential is decomposed into distinct
energy components, each associated with the same stress
eigentensor. Denoting by Trmthe following quantity:
Trmmrmrm;m1;...;6;50
it is noted that any stress eigenstate rmis associated with
its own potential Trm, which does not rely on the
action of the other rm. Then, it is readily noted by
inspection of relations (43) that the elastic strain-energy-
density components Tr1;...;Tr4are dependent
upon the values of the eigenangles ,,,,!and ',
and correspond to both distortional and voluminal
alterations of the medium. On the contrary, the last two
elastic strain-energy components, namely Tr5and
Tr6, are dependent on the value of the eigenangle
and are related exclusively to shape distortion of the
medium.
5. Geometric representation of stress eigentensors
A direct geometric representation of the r5and r6
contracted stress eigentensors arises if we consider the
projections of the stress eigentensors on the shear stress
plane (
4
,
5
). Then, tensors r1to r4vanish, whereas
tensors r5and r6are represented by two orthogonal
unit vectors e
5
and e
6
, shown in Fig. 1:
Fig. 1. Geometric representation of the stress eigentensors r5and r6of
the compliance fourth-rank tensor Srelated to monoclinic media on
the shear stress plane (
4
,
5
).

642 SPECTRAL DECOMPOSITION OF THE LINEAR ELASTIC TENSOR
e5cos ; sin T;e6ÿsin ;cos T:51
It is, therefore, noted that vectors e
5
and e
6
subtend
angle with respect to axes 
4
and 
5
of the shear stress
plane (
4
,
5
). Thus, an interesting geometric inter-
pretation is offered for eigenangle as the angle
responsible for the alignment of stress eigenstates r5
and r6in the shear stress plane (
4
,
5
). Besides, by
projecting the stress eigentensors on the four-dimen-
sional space system (
1
,
2
,
3
,
6
), tensors r5and r6
disappear, whereas tensors r1to r4are represented by
the following orthonormal vectors e
m
,m1;...;4:
e1g1;g2;g3;g6T52a
e2r1;r2;r3;r6T52b
e3h1;h2;h3;h6T52c
e4ÿsin cos cos ; ÿsin cos ;
ÿsin ; cos cos cos T:52d
In fact, the unit vectors e
m
,m1;...;4, are the base
vectors of a coordinate system obtained by rotating the
stress space (
1
,
2
,
3
,
6
) successively through angles
!,,',,and , by means of the following transfor-
mation matrices A
m
,m1;...;6:
A1
cos !sin !00
ÿsin !cos !00
0010
0001
2
6
6
6
4
3
7
7
7
5
;
A2
10 00
0 cos sin 0
0ÿsin cos 0
00 01
2
6
6
6
4
3
7
7
7
5
53a
A3
cos 'sin '00
ÿsin 'cos '00
0010
0001
2
6
6
6
4
3
7
7
7
5
;
A4
10 0 0
01 0 0
0 0 cos sin 
00ÿsin cos 
2
6
6
6
4
3
7
7
7
5
53b
A5
1000
0 cos 0 sin 
0010
0ÿsin 0 cos 
2
6
6
6
4
3
7
7
7
5
;
A6
cos 0 0 sin 
0100
0010
ÿsin 0 0 cos 
2
6
6
6
4
3
7
7
7
5
:53c
Then, the complete transformation Ais obtained by
considering the product of the six transformation
matrices, namely:
AQ
6
m1
Am;54
which is an orthogonal matrix. Therefore, it is concluded
that eigenangles !,,',,and determine the
orientation of eigentensors r1;...;r4in the stress space
(
1
,
2
,
3
,
6
). In addition, the sequence of rotations
employed to de®ne the orientation of the eigentensors is
to a certain extent arbitrary. Then, the initial rotation
could be about any of the four axes, whereas, in the
subsequent ®ve rotations, the only limitation is that no
two successive rotations may be about the same axis,
namely, no two successive rotations may be taken on the
same plane. Hence, a number of different conventions is
allowable in de®ning the six eigenangles as independent
parameters specifying the orientation of eigentensors in
the stress space (
1
,
2
,
3
,
6
). However, this space is
four dimensional and, as such, eigenvectors e
1
,...,e
4
cannot be visualized. In spite of that, it is always feasible
to restrict our attention to three-dimensional pictures of
the four-dimensional stress space. Then, it is easily
observed by projecting the stress eigentensors on an
arbitrary stress space (
i
,
j
,
k
), with i;j;k1;...;4
and i6 j6 k6 i, that vectors e
1
to e
4
are nonvanishing.
Yet, vectors e
1
to e
4
are linearly dependent and, hence,
in order to acquire the three eigenvectors e
m
,
m1;2;3, corresponding to the (
i
,
j
,
k
) reference
system, one has to consider the transformation of this
system by means of three separate rotations through
angles 
1
,
2
and 
3
, expressed in matrix form A
m
,
m1;2;3, as follows:
A1
cos 1sin 10
ÿsin 1cos 10
001
2
6
43
7
555a
A2
10 0
0 cos 2sin 2
0ÿsin 2cos 2
2
6
43
7
555b
A3
cos 3sin 30
ÿsin 3cos 30
001
2
6
43
7
5:55c
Accordingly, the product matrix Ais expressed by:
AA3A2A1

cos 1cos 3sin 1cos 3sin 2sin 3
ÿsin 1cos 2sin 3cos 1cos 2sin 3
ÿcos 1sin 3ÿsin 1sin 3sin 2cos 3
ÿsin 1cos 2cos 3cos 1cos 2cos 3
sin 1sin 2ÿcos 1sin 2cos 2
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
;
56

PERICLES S. THEOCARIS AND DIMITRIOS P. SOKOLIS 643
which is an orthogonal matrix. Moreover, angles 
1
,
2
and 
3
are known as the Euler angles, which are intro-
duced into relations (55) to express in generalized
coordinates the elements of the orthogonal transfor-
mation matrix A. Therefore, the projections of stress
eigentensors ri,rjand rkare represented by a set of
three orthogonal vectors with associated unit vectors e
i
,
e
j
and e
k
having as direction cosines:
eisin 1sin 2;ÿcos 1sin 2;cos 2T57a
ejÿsin 1cos 2cos 3ÿcos 1sin 3;
cos 1cos 2cos 3ÿsin 1sin 3;sin 2cos 3T
57b
ekÿsin 1cos 2sin 3cos 1cos 3;
cos 1cos 2sin 3sin 1cos 3;sin 2sin 3T:
57c
It is possible to carry out the transformation from a
given Cartesian coordinate system to another by means
of three successive angular displacements 
1
,
2
,
3
,
performed in a speci®c sequence. Initially, frame
O
i

j

k
is rotated through an angle 
1
counterclockwise
with respect to the O
k
O00 axis. The resulting
O000 coordinate system is then rotated by an angle

2
about the OO00 axis, thus forming the subse-
quent system O000, which is ®nally rotated about
the Oaxis by an angle 
3
, hence producing the ®nal
frame Or1r2r3O000. Therefore, as seen in
Fig. 2, the unit vectors e
j
and e
k
lie on plane O000,
subtending with plane O
k
00 an angle equal to
(=2ÿ2). In addition, the 00 axis is inclined to the 
j
axis by an angle (=2ÿ1), and the e
j
and e
k
unit
vectors subtend an angle 
3
with axes 0and 00.
6. Bounds of the components of the compliance tensor
It is generally accepted within the domain of classical
elasticity that the existence of the thermodynamical
constraint of positive-de®nite elastic potential sets
restrictive bounds on the values of the components of
the compliance tensor S. These constraints entailed on
the elements of the general anisotropic compliance
matrix were established by Voigt (1910), whereas, since
then, they have been proclaimed by Born & Huang
(1954) as well as by Hearmon (1961). Considering now
the conditions imposed on the elastic constants of
isotropic media, these are all well known and found in
Love (1927). Furthermore, the restrictions applicable to
media belonging to the cubic or hexagonal crystal
systems are explicitly stated by Nye (1957).
Relations for the bounds of elastic compliances for
transversely isotropic media were determined indepen-
dently by Eubanks & Sternberg (1954), as well as by
Lempriere (1968) and Christensen (1979), employing
mathematically equivalent formulations, which guaran-
teed positive values for the elastic potential. Lempriere
(1968) also examined the restrictions on the components
of the compliance tensor Svalid for orthotropic media.
Recently, the bounds for the values of the elastic
compliances were successfully obtained by following an
alternative method based on the spectral decomposition
Fig. 2. (a), (b) The rotations de®ning the Euler angles 
1
,
2
and 
3
, and
(c) geometric representation of the stress eigentensors of the
monoclinic compliance fourth-rank tensor Sin the (
i
,
j
,
k
) stress
frame.

644 SPECTRAL DECOMPOSITION OF THE LINEAR ELASTIC TENSOR
analysis and the application of this analysis to both
transversely isotropic media (Theocaris & Philippidis,
1991) and plates (Theocaris & Sokolis, 1998).
One of the very interesting features of the spectral
analysis is its simplicity and clarity in proving the posi-
tive-de®nite character of the elastic strain energy. Given
relations (49) and (50), it is immediately noted that, in
order for the total elastic strain-energy density to be
positive de®nite, the eigenvalues of the compliance
tensor Sneed be positive de®nite:
m>0;mf1;...;6g:58
This constraint requires that
fs11;s22 ;s33;s44 ;s55 ;s66g>059a
s2
12 <s11s22 ;s2
13 <s11s33 ;s2
23 <s22s33 59b
s2
16 <s11s66 ;s2
26 <s22s66 ;s2
36 <s33s66 ;
s2
45 <s44s55 59c
s11s22 s33 ÿs2
23ÿs12 s12s33 ÿs13 s23 
s13s12 s23 ÿs13s22>059d
s11s22 s66 ÿs2
26ÿs12 s12s66 ÿs16 s26 
s16s12 s26 ÿs16s22>059e
s11s33 s66 ÿs2
36ÿs13 s13s66 ÿs16 s36 
s16s13 s36 ÿs16s33>059f
s22s33 s66 ÿs2
36ÿs23 s23s66 ÿs26 s36 
s26s23 s36 ÿs26s33>059g
s16s26 s33s12 ÿs13 s23 s26s36 s11 s23 ÿs12s13 
s16s36 s22s13 ÿs12s23 s66
2s11s22 s33 2s12 s13s23
ÿs11s2
23 ÿs22s2
13 ÿs33s2
12s2
16
2s2
23 ÿs22s33 
s2
26
2s2
13 ÿs11s33 s2
36
2s2
12 ÿs11s22 >0:59h
It is essential that inequalities (59) are all simultaneously
satis®ed in order for the elastic strain-energy density
to be positive de®nite. Hence, bounds of the elastic
constants based on partial ful®lment of these inequali-
ties are considered improper and should be excluded.
7. Numerical examples
The experimentally measured values of elastic compli-
ance-tensor components for several common repre-
sentative inorganic crystals belonging to the monoclinic
system are listed in Table 1 (Landolt-Bornstein, 1979,
1984). It must be pointed out that the multiple entries
appearing in Table 1 for dipotassium tartrate (DKT) and
for ethylenediamine tartrate (EDT) are due to
substantial disagreement between different investiga-
tors using usually reliable techniques.
Now, in order to ®x ideas, we shall try to evaluate,
using the numerical values of the compliance compo-
nents for dipotassium tartrate (DKT1), the eigenvalues,
the eigenangles and the stress and strain eigenvectors.
The experimental values, in units of 10
ÿ2
GPa
ÿ1
, are as
follows:
s11 4:75;s22 3:53;s33 2:40;s44 11:460a
s55 10:2;s66 12:3;s12 ÿ1:74;s13 ÿ0:80 60b
s23 ÿ0:62;s16 ÿ0:75;s26 0:80;
s36 ÿ1:40;s45 ÿ0:68:60c
For this dipotassium tartrate (DKT1), the eigenvalues,
in units of TPa
ÿ1
, de®ned by relations (26), are
130:575;
212:857 61a
369:183;
455:685 61b
558:534;
649:466:61c
The eigenvalues 
1
,...,
6
of the compliance tensor S,in
units of TPa
ÿ1
, for the remaining inorganic crystals
belonging to the monoclinic system are tabulated in
Table 2. Moreover, the eigenangles ,,, ,,!and ',
de®ned by relations (33), are evaluated to be
Table 1. The values of the elastic compliances (in units of 10
ÿ2
GPa
ÿ1
) for a series of crystalline media belonging to the
monoclinic system
Crystals of the monoclinic system
Elastic compliances (10
ÿ2
GPa
ÿ1
)
Symbol Material s
11
s
22
s
33
s
44
s
55
s
66
s
12
s
13
s
23
s
16
s
26
s
36
s
45
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium
tartrate (DKT1)
4.75 3.53 2.40 11.4 10.2 12.3 ÿ1.74 ÿ0.80 ÿ0.62 ÿ0.75 0.80 ÿ1.40 ÿ0.68
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium
tartrate (DKT2)
3.87 3.37 2.26 10.4 8.2 11.9 ÿ1.06 ÿ1.64 ÿ0.07 0.85 ÿ0.54 ÿ0.65 0.55
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine
tartrate (EDT1)
3.34 3.65 10.0 19.2 11.7 19.1 ÿ0.3 ÿ3.0 ÿ1.8 ÿ1.7 1.5 ÿ2.65 0.38
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine
tartrate (EDT2)
3.9 3.6 9.8 18.7 17.2 17.4 0.2 ÿ5.2 ÿ1.8 ÿ0.5 0.2 ÿ2.5 ÿ0.2
Na
2
S
2
O
3
Sodium thiosulfate 5.02 15.6 6.74 22.3 32.7 21.2 ÿ3.23 ÿ0.62 ÿ7.19 1.52 ÿ18.2 11.0 10.0

PERICLES S. THEOCARIS AND DIMITRIOS P. SOKOLIS 645
24:299;15:892;
22:641;ÿ52:72362a
129:792;!168:460;'148:32162b
and the eigenangles of the compliance tensor Sfor the
remaining representative monoclinic crystals are given
in Table 3.
Then, the eigenvectors of the compliance fourth-rank
tensor S, as de®ned in (51) and (52), are found to be
given by
e10:315;0:671;ÿ0:616;0;0ÿ0:266T63a
e20:444;0:510;0:724;0;0;0:137T63b
e30:452;ÿ0:390;0:146;0;0;ÿ0:788T63c
e40:706;ÿ0:370;ÿ0:274;0;0;0:538T63d
e50;0;0;0:912;ÿ0:411;0T63e
e60;0;0;ÿ0:411;ÿ0:912;0T:63f
Table 4 presents the eigenvectors of the remaining
monoclinic crystals.
8. Discussion
The most important property of spectral analysis is its
ability to expose in a very natural way the analogy in the
elastic characteristics of isotropic and anisotropic media.
For instance, the form of the compliance tensor Swas
established for an isotropic body during the ®rst quarter
of the last century:
S1
11 1
2I64
in terms of Lame
Âelastic moduli and . Indeed, the
analogy between this expression and the spectral
expansion (36) of the compliance tensor Sfor aniso-
tropic media belonging to the monoclinic system is
easily recognized, whereas instead of Lame
Âmoduli one
has the eigenvalues of the compliance tensor S.
Furthermore, it is well known that the stress rand
strain etensors of linearly isotropic elastic media are
decomposed into deviatoric and hydrostatic parts:
rrD1
3tr r1;eeD1
3tr e1;65
in which subscript Din r
D
and e
D
denotes the deviatoric
parts and the second terms denote the hydrostatic parts
r
P
and e
P
of the stress and strain tensors, respectively.
Then, this characteristic of isotropic elasticity encoun-
ters its analogy in the spectral decomposition of the
stress and strain tensors into six distinct non-interacting
stress states. However, it should be made clear that,
whereas the deviatoric and hydrostatic eigentensors
remain constant for all isotropic materials, the corre-
sponding eigentensors of anisotropic elasticity are
dependent on the elastic compliance components, thus
obtaining different values for different media.
Besides, the decomposition of the stress and strain
tensors in isotropic elastic bodies into hydrostatic and
deviatoric constituents results in an equivalent decom-
position of Hooke's law for isotropic materials into two
equations:
Table 3. The values of the set of eigenangles () of the compliance fourth-rank tensor Sfor a series of crystalline media
belonging to the monoclinic system
Crystals of the monoclinic system
Eigenangles ()
Symbol Material  !'
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium tartrate (DKT1) 15.89 22.64 ÿ52.72 129.79 168.46 148.32 24.29
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium tartrate (DKT2) ÿ19.47 ÿ21.58 49.73 ÿ60.57 ÿ17.70 170.90 166.72
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine tartrate (EDT1) 26.26 ÿ2.03 ÿ153.78 88.96 63.24 154.61 2.89
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine tartrate (EDT2) ÿ9.61 2.61 15.58 89.02 122.10 17.16 172.53
Na
2
S
2
O
3
Sodium thiosulfate 27.67 16.99 260.49 36.30 131.51 174.41 58.73
Table 2. The values of the six eigenvalues (in units of TPa
ÿ1
) of the compliance fourth-rank tensor Sfor a series of
crystalline media belonging to the monoclinic system
Crystals of the monoclinic system
Eigenvalues (TPa
ÿ1
)
Symbol Material 
1

2

3

4

5

6
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium tartrate (DKT1) 30.575 12.857 69.183 55.685 58.534 49.466
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium tartrate (DKT1) 31.454 10.532 65.249 47.265 52.649 40.351
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine tartrate (EDT1) 34.748 14.846 124.77 91.299 96.096 58.404
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine tartrate (EDT2) 35.004 50.395 135.15 84.809 93.631 85.869
Na
2
S
2
O
3
Sodium thiosulfate 13.716 ÿ6.678 310.69 61.864 193.86 81.144

646 SPECTRAL DECOMPOSITION OF THE LINEAR ELASTIC TENSOR
1
3tr r11
332tr e1;rD2eD66
between the hydrostatic and deviatoric stress and strain
eigentensors. Then, the generalized anisotropic Hooke's
law valid for monoclinic media, which is formulated
alternatively in the equivalent form of a system of six
non-interacting mutually orthogonal laws of direct
proportionality, expressed by relations (47), may be
thought of as a generalization of equations (66) above.
Finally, the stress and strain eigentensors were proven
to partition directly the elastic strain-energy density into
distinct strain-energy constituents. Again, an analogy is
revealed between the splitting of the total elastic strain-
energy density of the monoclinic medium and the
corresponding splitting valid for the isotropic medium,
which is given in the form
TrTrPTrD
1
18Ktr r21
2Gtr r2ÿ1
3tr r2

;67
where TrDis the deviatoric strain energy and TrPis
the hydrostatic strain energy, corresponding to the
deviatoric and hydrostatic stresses and strains respec-
tively.
However, the decomposition of the elastic potential
that is valid for the isotropic medium is not valid for the
monoclinic one. Hence, it is shown that a generalization
of the decomposition of the elastic strain-energy density
into components corresponding to sole dilatational and
distortional types of energy, valid for the isotropic
medium as well as for cubic crystals, is impossible for the
monoclinic medium, since the second-rank eigentensors
of the compliance fourth-rank tensor Sdo not include
the spherical tensor 1. Thus, an explanation is given for
the failure of the studies undertaken (Olszak &
Urbanowski, 1956; Olszak & Ostrowska-Maciejewska,
1985), which aimed to generalize the Huber±Mises±
Hencky criterion to hold for anisotropic media, there-
fore establishing the distortional component of the
elastic strain-energy density as the critical failure
quantity.
In conclusion, the spectral decomposition of the
compliance fourth-rank tensor Sallows the possibility of
generalization of well known characteristics of isotropic
linear elastic bodies to anisotropic ones, thus offering to
the theory of anisotropic media in the elastic domain a
status comparable to that of isotropic elasticity.
References
Born, M. & Huang, K. (1954). Dynamical Theory of Crystal
Lattices. Oxford: Clarendon Press.
Cady, W. G. (1946). Piezoelectricity. New York: McGraw Hill.
Christensen, R. M. (1979). Mechanics of Composite Materials.
New York: Wiley.
Eubanks, R. A. & Sternberg, E. (1954). J. Rat. Mech. Anal. 3,
89±101.
Hearmon, R. F. S. (1961). An Introduction to Applied
Anisotropic Elasticity. Oxford: Clarendon Press.
Hooke, R. (1678). Lecture de Potentia Restitutiva. London:
Martin.
Landolt-Bornstein (1979). Numerical Data and Functional
Relationships in Science and Technology. New Series. Group
III: Crystal and Solid State Physics, Vol. 11: Elastic,
Piezoelectric, Piezooptic and Electroptic Constants of
Crystals. Berlin/Heidelberg/New York: Springer.
Landolt-Bornstein (1984). Numerical Data and Functional
Relationships in Science and Technology. New Series. Group
III: Crystal and Solid State Physics, Vol. 11: Elastic,
Piezoelectric, Piezooptic and Electroptic Constants of
Crystals. Berlin/Heidelberg/New York: Springer.
Lekhnitskii, S. (1963). Theory of Elasticity of an Anisotropic
Elastic Body, translated by P. Fern. San Francisco: Holden-
Day.
Table 4. The components of the set of eigenvectors of the compliance fourth-rank tensor Sfor a series of crystalline
media belonging to the monoclinic system
Crystals of the monoclinic system
Symbol Material Eigenvectors
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium tartrate
(DKT1)
e
1
= [0.315, 0.671, ÿ0.616, 0, 0, ÿ0.266]
T
e
2
= [0.444, 0.510, 0.724, 0, 0, 0.137]
T
e
3
= [0.452, ÿ0.390, 0.146, 0, 0, ÿ0.788]
T
e
4
= [0.706, ÿ0.370, ÿ0.274, 0, 0, 0.538]
T
e
5
= [0, 0, 0, 0.912, ÿ0.411, 0]
T
e
6
= [0, 0, 0, ÿ0.411, ÿ0.912, 0]
T
K
2
(C
4
H
4
O
6
)0.5H
2
O Dipotassium tartrate
(DKT2)
e
1
= [0.215, 0.860, ÿ0.463, 0, 0, ÿ0.005]
T
e
2
=[ÿ0.556, ÿ0.282, ÿ0.782, 0, 0, ÿ0.027]
T
e
3
=[ÿ0.441, 0.243, 0.256, 0, 0, ÿ0.826]
T
e
4
=[ÿ0.670, 0.347, 0.333, 0, 0, 0.566]
T
e
5
= [0, 0, 0, ÿ0.973, ÿ0.230, 0]
T
e
6
= [0, 0, 0, 0.230, ÿ0.973, 0]
T
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine tartrate
(EDT1)
e
1
=[ÿ0.366, 0.903, 0.016, 0, 0, ÿ0.225]
T
e
2
= [0.821, 0.367, 0.404, 0, 0, 0.168]
T
e
3
=[ÿ0.188, ÿ0.221, 0.801, 0, 0, ÿ0.525]
T
e
4
= [0.396, 0.032, ÿ0.443, 0, 0, ÿ0.804]
T
e
5
= [0, 0, 0, 0.999, 0.051, 0]
T
e
6
= [0, 0, 0, 0.051, ÿ0.999, 0]
T
C
2
H
6
N
2
C
4
H
6
O
6
Ethylenediamine tartrate
(EDT2)
e
1
= [0.295, ÿ0.955, ÿ0.017, 0, 0, 0.041]
T
e
2
=[ÿ0.803, ÿ0.246, ÿ0.524, 0, 0, ÿ0.143]
T
e
3
=[ÿ0.445, ÿ0.165, 0.835, 0, 0, ÿ0.279]
T
e
4
=[ÿ0.265, ÿ0.045, 0.167, 0, 0, 0.949]
T
e
5
= [0, 0, 0, ÿ0.992, 0.130, 0]
T
e
6
= [0, 0, 0, ÿ0.130, ÿ0.992, 0]
T
Na
2
S
2
O
3
Sodium thiosulfate e
1
= [0.532, 0.454, 0.714, 0, 0, ÿ0.030]
T
e
2
=[ÿ0.092, ÿ0.492, 0.347, 0, 0, ÿ0.793]
T
e
3
=[ÿ0.101, 0.696, ÿ0.393, 0, 0, ÿ0.592]
T
e
4
= [0.835, ÿ0.259, ÿ0.464, 0, 0, ÿ0.140]
T
e
5
= [0, 0, 0, 0.519, 0.855, 0]
T
e
6
= [0, 0, 0, ÿ0.855, 0.519, 0]
T

PERICLES S. THEOCARIS AND DIMITRIOS P. SOKOLIS 647
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