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Classes of Anisotropic Media: A Tutorial

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Abstract

The purpose of the present article is to give a precise definition and analysis from first principals of anisotropy, as the term applies to elastic media, taking care to avoid unnecessary assumptions. Two fundamental concepts, material invariance and symmetry group of a material, are defined purely in terms of the stress-strain relation. The implications of material symmetry, or in other words, of anisotropy, for the structure of the stiffness tensor are then investigated. Using the reduced notation of Voigt, these results are presented as the well-known simplifications in the form taken by the six-by-six stiffness matrix that represents the material's stiffness tensor. A new, simple proof is given for the remarkable fact that an elastic medium cannot have rotational symmetry by an angle of less than 90 without being transversely isotropic. In addition, the mutual relation that the notions of elastic symmetry and crystal symmetry have with respect to the so-called orthogonal group is sketched. Despite the historical association between anisotropic elastic materials and the study of crystals, the given presentation shows that conceptually the notion of anisotropy in elastic media is entirely independent of that of crystal symmetry.
CLASSES OF ANISOTROPIC MEDIA: A TUTORIAL
L. Bos1, P. Gibson2, M. Kotchetov3, M. Slawinski4
1 Dept. of Mathematics and Statistics, University of Calgary, Canada
2 Mathematisches Institut II, Universit¨at Karlsruhe, Germany
3 Dept. of Mathematics and Statistics, University of Saskatchewan, Canada
4 Dept. of Earth Sciences, Memorial University of Newfoundland, Canada
Received: 23 May 2003; Revised: 5 November 2003; Accepted: 25 November 2003
ABSTRACT
The purpose of the present article is to give a precise definition and analysis
from first principals of anisotropy, as the term applies to elastic media, taking care
to avoid unnecessary assumptions. Two fundamental concepts, material invariance
and symmetry group of a material, are defined purely in terms of the stress-strain
relation. The implications of material symmetry, or in other words, of anisotropy,
for the structure of the stiffness tensor are then investigated. Using the reduced
notation of Voigt, these results are presented as the well-known simplifications in
the form taken by the six-by-six stiffness matrix that represents the material’s stiff-
ness tensor. A new, simple proof is given for the remarkable fact that an elastic
medium cannot have rotational symmetry by an angle of less than 90without being
transversely isotropic. In addition, the mutual relation that the notions of elastic
symmetry and crystal symmetry have with respect to the so-called orthogonal group
is sketched. Despite the historical association between anisotropic elastic materials
and the study of crystals, the given presentation shows that conceptually the notion
of anisotropy in elastic media is entirely independent of that of crystal symmetry.
K e y w o r d s : Anisotropy, elasticity, material invariance, symmetry group
Stud. Geophys. Geod., 48 (2004), 265–287
c
°2004 StudiaGeo s.r.o., Prague 265
L. Bos et al.
1. INTRODUCTION
1.1. Overview
Anisotropy has emerged as a topic of intensive research within the geophysics
community. In modeling the mechanical properties of the earth, as elsewhere, there
is a tradeoff between more complicated models, which represent more accurately the
natural phenomena under consideration, and simpler models, which are easier to
analyze. While it may not be feasible to analyze a fully anisotropic Earth, it may
nevertheless be desirable for the sake of accuracy to allow some level of anisotropy
in one’s mathematical model. As a prerequisite to finding the right balance, it is
necessary to understand precisely the menagerie of anisotropies that are possible
within the context of linearized elasticity. The purpose of this tutorial is to explain
in fundamental terms the meaning, and classification, of anisotropy in this context.
The tutorial is organized as follows. We define material invariance, and then
derive what we call the Fundamental Invariance Condition, in Section 2. In Sec-
tion 3 we introduce the key notion of symmetry group, and define precisely several
symmetry classes, and in particular anisotropy classes, of elastic materials. Then
in Section 4 we derive the form of the stiffness matrix corresponding to each class.
In addition we prove a remarkable fact concerning transverse isotropy, namely, that
an elastic medium cannot have rotational symmetry by an angle of less than 90
without being transversely isotropic. Lastly, in Section 5, we comment on the place
of elastic symmetry within the broader context of subgroups of the orthogonal group
and in relation to crystallographic symmetries.
The results brought together in this tutorial are well known and are intended to
be presented at an accessible level. We hope that they serve as a useful resource to
readers wishing to re-acquaint themselves with the technical details of anisotropy.
1.2. Differences from the standard literature
As Helbig (1994) writes, “Historically the study of anisotropic elastic materials
has been synonymous with the study of crystals.” In the present article we wish to
emphasize that this historical association does not stem from any sort of conceptual
necessity. On the contrary, it is entirely possible to define and to analyze anisotropy
in elastic media from first principles without any mention of crystals or of crystal
symmetry – that is what we do here. Pedagogically this makes it clear that the
notion of anisotropy rests solely on the equations of elasticity, and does not involve
any additional assumptions to the effect that an elastic medium can be modeled as a
piecewise collection of crystals, which it cannot. Our approach thus differs from the
standard literature, which typically invokes crystal symmetries as a starting point
for the analysis of anisotropy, or is directly concerned with crystalline materials.
The association between crystal symmetries and anisotropy appears in the liter-
ature at least as early as the monumental work of Voigt (1928). More recently, the
paper of Bond (1943) and the books by Fedorov (1968) and Musgrave (1970) deal
266 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
explicitly with crystalline materials. While Helbig (1994) makes no such explicit as-
sumption, he nevertheless invokes crystal symmetries at the outset of his discussion
of material symmetry.
Our presentation also differs from the standard literature in the way that we
use the mathematical result known as Herman’s Theorem. We provide a new and
comparatively simple proof of this result (more precisely, of the special case of this
result that we need) and use it to determine the form of the stiffness matrices for
certain classes of anisotropic material. In fact, it is possible to derive a complete
classification of elastic symmetries using Herman’s Theorem as a principal tool,
although the technical details are beyond the scope of the present tutorial. That
being said, we have given Herman’s Theorem a prominent place because of its power
as tool for analyzing symmetry. Note that Helbig (1994) cites Herman’s Theorem,
but his approach to analyzing symmetry, which is ultimately based on eigentensors,
is very different from the one given here.
As mentioned above, our presentation makes no assumption that the elastic ma-
terial in question is a crystal. Before proceeding with our analysis we discuss briefly
the notion of crystal versus that of elastic continuum in order to emphasize the
contrast between the two corresponding mathematical models. Consider for the mo-
ment a particular physical substance, such as a large block of crystalline salt. As is
very well known, one may represent the salt geometrically as a system of regularly
spaced points in space, called a lattice, corresponding to the actual microscopic ar-
rangement of sodium and chlorine ions. This is of course an idealization – typically
the mathematical lattice extends infinitely far in space, for instance – but it is still
a very useful and accurate model. The lattice serves as one possible conceptual
starting point for crystal symmetries, which may be defined in terms of rotations,
translations etc., with respect to which the lattice is invariant. See Weyl (1952) for
a detailed discussion. A completely different mathematical model, consisting of an
elastic continuum together with particular equations of motion, provides the basis
for the analysis of elastic symmetry and anisotropy – an elastic continuum mod-
els salt geometrically as if it were a uniform solid at all scales, with no underlying
atomic structure. The two models, a lattice and an elastic continuum, are obvi-
ously inconsistent with one another; they serve to represent different aspects of the
original physical substance. For the present tutorial it is important to note that
there are separate notions of symmetry for the lattice, namely crystal symmetry,
and for the elastic continuum, namely elastic symmetry or anisotropy. Neumann’s
Principle asserts that the so-called point symmetry of the lattice model (which ex-
cludes translational symmetry) must be manifest in any physical property of the
salt. In particular any physical property captured by the continuum model should
also manifest this same point symmetry. However in this tutorial we do not consider
the lattice model at all; we instead focus purely on the analysis of symmetry as it
applies to the elastic continuum.
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L. Bos et al.
1.3. Preliminaries
To begin, we need some basic facts from the theory of linearized elasticity. The
most fundamental fact is that the material properties at a particular point pof a
three-dimensional elastic medium are characterized by the linear relation between
stress and strain, and by the mass density (which does not enter into our present
considerations). The stress-strain relation is usually written
σij =cijkl εkl ,(1)
where σij , εkl (1 i, j, k, l 3) denote variables of stress and strain, respectively,
and cijkl are the components of the stiffness tensor c. The summation convention,
whereby a summation is taken corresponding to each instance of a repeated index
(on the same side of the equation), applies to (1).
We emphasize that the equation (1) applies at a particular material point p, and
that, along with mass density, the values of the components cijkl characterize the
material properties at pcompletely. In general, the tensor cvaries from point to
point in the medium, so that cis in fact a function of p. However in what follows we
are mainly interested in a single p. The equation (1) refers implicitly to Cartesian
coordinates
x=
x1
x2
x3
,
that is, to a fixed orthonormal coordinate system. (Indeed, for a given configuration
of the medium, both σij and εkl may be regarded as the elements of 3 ×3 matrices σ
and εrespectively, and hence they represent linear maps in the x-coordinate system.)
Another basic fact which we need is that stress and strain variables are symmetric.
This means that
σij =σji and εij =εji (1 i, j 3) .(2)
In matrix notation this same fact is expressed by writing
σ=σTand ε=εT.
Also, the stiffness tensor csatisfies the symmetry relations*
cijkl =cj ikl =cijlk =cklij (1 i, j, k, l 3) .(3)
*This and other details concerning the basic framework of linearized elasticity are ex-
plained in Sokolnikoff (1956)(and also in numerous other texts).
268 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
We wish to consider the effect of a transformation of coordinates on the funda-
mental relation (1). To this end, let Abe a matrix representing a linear transfor-
mation in the x-coordinate system. Let σ0and ε0denote the resulting transformed
stress and strain variables:
σ0=AσA1and ε0=AεA1.
The assumption that the medium is linearly elastic requires that
σ0
ij =c0
ijkl ε0
kl ,
for some stiffness tensor c0.
Definition 1 The material (or medium) is said to be invariant at pwith respect to
the transformation Aif c=c0, in other words, if the transformed stiffness tensor
is identical to the original stiffness tensor.
It turns out that material invariance places constraints on the the stiffness tensor
c, and this provides a natural way to classify elastic media. Our objective for the
present article is to study these constraints in detail. The first step is to simplify
the notation, so that instead of working with a fourth-order stiffness tensor c, we
can work with a stiffness matrix C.
1.4. Contracted notation
It is convenient to work with the so-called Voigt notation* which takes advantage
of (2) and (3). Replace pairs of indices ij (1 ij3) with a single index m
(1 m6) according to the rule
ij m
11 7→ 1
22 7→ 2
33 7→ 3
23 7→ 4
13 7→ 5
12 7→ 6
.(4)
Define variables σm=σij and constants Cmn =cijk l, where ij 7→ mand kl 7→ n
according to (4). A slightly more complicated notation is required for the variables
εij in order to make the summation convention carry over to this contracted notation.
Namely, define
εm=εij if 1 m3
εm= 2εij if 4 m6,
*The notation is usually attributed to Voigt (1928).
Stud. Geophys. Geod., 48 (2004) 269
L. Bos et al.
where, as above, ij 7→ maccording to (4).
The stress-strain relation (1) can be rewritten in terms of this new notation,
using the summation convention, as
σm=Cmnεn.
In matrix notation this is simply
σ=Cε.
Thus σand εare vectors (in six-dimensional space) representing stress and strain
variables, and Cis a six-by-six matrix which, by (3), is symmetric (i.e. C=CT).
There is no inherent physical reason to represent the stress-strain relation in this
compressed form, rather we use the Voigt notation merely as a convenient shorthand.
It saves space to replace the full stiffness tensor with a 6 ×6 matrix which, when
written out in full, takes up much less room on the page, and this makes for a clearer
presentation. It is important to note that the notational transformation (4) and the
transformation of stress and strain variables that follow are fully reversible. Thus
anything we express using compressed notation may be easily translated back in
terms of the original tensors.
2. A CHARACTERIZATION OF MATERIAL INVARIANCE
Our objective is to analyze the constraints which material invariance with respect
to various transformations places on the stiffness matrix. We are interested specifi-
cally in distance-preserving transformations of our x-coordinate system based at p.
Such transformations are represented by orthogonal matrices, that is, by three-by-
three matrices Asatisfying the condition AAT=I, or equivalently, the condition
A1=AT. Orthogonality of Acan be expressed in terms of its entries Aij , using
the summation convention, as
Aij Ajk =½1 if i=k
0 if i6=k.
Now, if Ais an orthogonal matrix, and x0=Ax (so that x=ATx0) is a new
coordinate system based at p(obtained from xby the transformation A), then the
vector σx, expressed in x0-coordinates, is
A(σx) = Aσ(ATx0) = (AσAT)x0.
Thus in terms of the x0-coordinates, the stress variables become
σ0=AσAT.(5)
Similarly, the strain variables become
270 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
ε0=AεAT.(6)
It is a straightforward matter to rewrite the relations (5) and (6) in terms of the
contracted notation. The relation (5) is
σ0=Aσ,(7)
where, in terms of the entries Aij of A,
A=
A11A11 A12 A12 A13A13 2A12 A13 2A11 A13 2A11A12
A21A21 A22 A22 A23A23 2A22 A23 2A21 A23 2A21A22
A31A31 A32 A32 A33A33 2A32 A33 2A31 A33 2A31A32
A21A31 A22 A32 A23A33 A22 A33 +A23A32 A21 A33+A23A31 A21 A32+A22A31
A11A31 A12 A32 A13A33 A12 A33 +A13A32 A11 A33+A13A31 A11 A32+A12A31
A11A21 A12 A22 A13A23 A12 A23 +A13A22 A11 A23+A13A21 A11 A22+A12A21
.
(8)
As an aside, let us calculate A1. By (5),
σ=ATσ0A.(9)
Thus we may rewrite (9) in contracted notation, exactly as above, as
σ=ATσ0,
where ATis constructed as in (8), but with the entries AT
ij =Aji of ATused in
place of the entries Aij of A. On the other hand, it follows immediately from (7)
that
AT=A1.
(Note: ATis not the same as (A)T.) Writing this out in full, we obtain
A1=
A11A11 A21 A21 A31A31 2A21 A31 2A11 A31 2A11A21
A12A12 A22 A22 A32A32 2A22 A32 2A12 A32 2A12A22
A13A13 A23 A23 A33A33 2A23 A33 2A13 A33 2A13A23
A12A13 A22 A23 A32A33 A22 A33 +A32A23 A12 A33+A23A31 A21 A32+A22A31
A11A13 A21 A23 A31A33 A21 A33 +A31A23 A11 A33+A31A13 A11 A23+A21A13
A11A12 A21 A22 A31A32 A21 A32 +A31A22 A11 A32+A31A12 A11 A22+A21A12
.
(10)
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L. Bos et al.
We now return to the task of writing (6) in contracted notation. Recall that ε
was defined with additional factors of 2 to accommodate the summation convention.
Because of this, we get a result slightly different from (7), namely,
ε0=FA F1ε,(11)
where Ais the matrix (8) and
F=
100000
010000
001000
000200
000020
000002
.
The coefficient matrix FAF1appearing in (11) will be needed below. To streamline
notation, we call it MA; that is,
MA=FA F1=
A11A11 A12 A12 A13A13 A12 A13 A11 A13 A11A12
A21A21 A22 A22 A23A23 A22 A23 A21 A23 A21A22
A31A31 A32 A32 A33A33 A32 A33 A31 A33 A31A32
2A21A31 2A22 A32 2A23A33 A22 A33 +A23A32 A21 A33 +A23A31 A21A32 +A22A31
2A11A31 2A12 A32 2A13A33 A12 A33 +A13A32 A11 A33 +A13A31 A11A32 +A12A31
2A11A21 2A12 A22 2A13A23 A12 A23 +A13A22 A11 A23 +A13A21 A11A22 +A12A21
.
(12)
Note that as soon as we are given an orthogonal transformation matrix A(with
entries Aij), we can write down the corresponding matrix MAusing (12). Two
simple facts concerning MAwill be useful for future reference. Firstly,
A1= (MA)T.(13)
To see this, just compare (10) with (12). Secondly, if Aand Bare any two orthogonal
transformations, then
MAB =MAMB.(14)
We are now ready formulate precisely the implications of material invariance for
the stiffness matrix C.
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Classes of Anisotropic Media: a Tutorial
Fundamental Invariance Condition:The elastic properties of a material at a
point pare invariant with respect to an orthogonal transformation Aif and only if
C=MT
ACMA,(15)
where Cis the stiffness matrix at p, and
MA=
A11A11 A12 A12 A13A13 A12 A13 A11 A13 A11A12
A21A21 A22 A22 A23A23 A22 A23 A21 A23 A21A22
A31A31 A32 A32 A33A33 A32 A33 A31 A33 A31A32
2A21A31 2A22 A32 2A23A33 A22 A33 +A23A32 A21 A33 +A23A31 A21A32 +A22A31
2A11A31 2A12 A32 2A13A33 A12 A33 +A13A32 A11 A33 +A13A31 A11A32 +A12A31
2A11A21 2A12 A22 2A13A23 A12 A23 +A13A22 A11 A23 +A13A21 A11A22 +A12A21
.
Explanation: We know that the elastic properties of the material at pare expressed
by the relation
σ=Cε.(16)
Invariance with respect to Ameans, by definition, that if
σ0=Aσand ε0=MAε(17)
are the transformed stress and strain variables (as in (7) and (11)), then σ0and ε0
are related by the original stiffness matrix Cin (16). That is,
σ0=Cε0.(18)
In other words, material invariance with respect to Ais equivalent to the simulta-
neous validity of (16) and (18). Now, substituting (17) into (18) yields
σ0=Cε0
Aσ=C MAε
σ=A1C MAε
σ=MT
AC MAε.
(19)
Note that the last equivalence follows by equation (13). Now, the two equations (16)
and (19) hold simultaneously for every possible εif and only if condition (15) holds.
The condition (15) is very easy to apply. And it implies, depending on the
transformation Ain question, various simplifications in the structure of the stiffness
matrix C. We explore this in later sections. For now let us illustrate the Funda-
mental Invariance Condition by a simple example. Let I3and I6denote the identity
matrices,
Stud. Geophys. Geod., 48 (2004) 273
L. Bos et al.
I3=
1 0 0
0 1 0
0 0 1
and I6=
100000
010000
001000
000100
000010
000001
.
What does it mean for a material to be invariant with respect to the transforma-
tion matrix A=I3, which represents a reflection through the origin in three-
dimensional space? Constructing MAaccording to (12) yields
MA=I6.
So, by the Fundamental Invariance Condition, a material is invariant with respect
to I3if and only if
C= (I6)TCI6,
that is, if and only if C=C. This is automatically true for any C! We have
thus shown that every elastic medium is invariant (at every point p) with respect to
reflection through the origin.
3. SYMMETRY GROUPS
3.1. Definition of symmetry group
Going a conceptual step beyond the notion of invariance with respect to a trans-
formation, we can consider the set of all transformations with respect to which a
medium is invariant at a given point. This set is of central importance to our present
considerations.
Definition 2: The set Gof all orthogonal transformations Awith respect to
which the elastic properties at a point pare invariant, is called the symmetry group
of the material (or medium) at p. In the special case of uniform symmetry, where
an elastic medium has the same symmetry group at every material point, we call G
the symmetry group of the medium.
It is easy to see that if A,Bare any two transformations in a particular symmetry
group G, then their product and their inverses are also in G. (This is what makes Ga
group in the the mathematical sense of the word. Recall that the matrices MA,MB
respect this multiplicative structure in the sense that MAB =MAMB.) It follows
that if A∈ G, then so are all the transformations in the list
. . . , A3,A2,A1,I3,A,A2,A3, . . . (20)
274 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
These need not all be different: if A=I3, or A=I3, then the list (20) consists of
just one, or two, matrices, respectively. The set of transformations in (20) is called
the group generated by A.
Let us consider a fixed x-coordinate system at a point pand the associated
symmetry group Gof matrices. If we change coordinates by an orthogonal matrix
Bto x0-coordinates
x0=Bx ,
we get a new symmetry group G0associated to the new coordinates. The exact
relation is
G0=BGBT,(21)
where
BGBT=©BABT|A G ª.
Note that Gand G0represent exactly the same set of transformations, albeit
with respect to different coordinate systems. With this in mind we agree to regard
any two sets G,G0of transformation matrices as equivalent if, for some orthogonal
matrix B, (21) holds. And in particular we do not distinguish between symmetry
groups related by (21).
3.2. Classification of anisotropic
media by symmetry groups
Symmetry groups provide the basis for the standard classification of the various
types of anisotropy in elastic media. In general the symmetry group of a medium may
vary from point to point. But in ordinary parlance, and particularly in the context
of geophysical applications, one often refers to the symmetry class of a medium, the
underlying assumption being that the medium has uniform symmetry. We abide by
this convention in what follows: when we refer to the symmetry group of a medium,
it is to be understood that the medium has uniform symmetry. Of course all our
considerations apply equally well to the symmetry groups of elastic media restricted
to a single point.
Anisotropic media are classified according to which orthogonal transformations
belong to their symmetry group – in a suitably chosen orthonormal coordinate sys-
tem. (Recall that we consider as equivalent, symmetry groups that are related by
an orthogonal transformation.) Before giving the classification scheme, we note a
basic fact from linear algebra: For any given orthogonal transformation of three-
dimensional space, there exists an orthonormal coordinate system in terms of which
the transformation is represented by a matrix either of the form
Stud. Geophys. Geod., 48 (2004) 275
L. Bos et al.
Aθ=
cos θsin θ0
sin θcos θ0
0 0 1
,(22)
or of the form
A0
θ=
cos θsin θ0
sin θcos θ0
0 0 1
,(23)
for some angle θ. As long as we consider just a single transformation A∈ G, we
are free to assume it has the form (22) or (23). On the other hand, given two or
more matrices A,B, . . ., we can only assume a priori that one of them has the stated
structure.
Using the notation in (22) and (23), and writing Bfor the matrix
B=
1 0 0
01 0
0 0 1
,
we define six classes of elastic media in terms of their symmetry groups as follows.
Each of the classes corresponds to a type of symmetry, and, with the exception of
the isotropic class, each class defines a type of anisotropy.
Type of symmetry Transformations belonging G
Full anisotropy A0,A0
πonly
Monoclinic A0
0
Orthotropic A0
0,B
Tetragonal Aπ/2,B
Transverse isotropy Aθfor all θ
Isotropy all orthogonal transformations (24)
In the next section we will examine in detail how the symmetry (or anisotropy)
type of a medium is manifest in the structure of its stiffness matrix. In general
the larger the symmetry group of a medium, the fewer the number of independent
parameters required to specify its stiffness matrix. Before proceeding, however, a
few comments about the above table are in order. Note that, except for the case
of full anisotropy, the definition of symmetry type is not exclusive. For example
the symmetry group Gof a monoclinic medium is allowed to contain – in fact, does
contain – transformations other than A0
0; the requirement is just that A0
0be among
the members of G. With this definition, a medium having orthotropic symmetry, for
example, automatically has monoclinic symmetry too. A second comment is that
there are many different, but equivalent, ways to define the various classes. We
276 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
explore this point in more detail in the next section, where we show, for instance,
that an equivalent definition to the one given above for transverse isotropy is simply
that the transformation A2π/5belong to G.
4. FORM OF THE STIFFNESS MATRIX
FOR VARIOUS SYMMETRY CLASSES
Given the stiffness matrix Cof a uniform medium, one can determine the corre-
sponding symmetry group Gusing the Fundamental Invariance Condition: Gis the
set of transformations Afor which the equation
C=MT
ACMA(25)
holds. Conversely, supposing that Cis unknown, we cannot recover Cfrom knowl-
edge of G. However, for each A∈ G, the equation (25) imposes a set of linear
equations on the entries of C. Thus, given G, one has a system of linear equations in
the entries of the stiffness matrix C, and this system determines Cup to a certain
number, depending on G, of arbitrary parameters. In the case where Gis as large as
possible, corresponding to isotropy, two parameters arise; at the other extreme, 21
parameters are needed. We now look in more detail at these systems of equations.
4.1. Full anisotropy
What is the smallest symmetry group that an elastic medium can have? We know
from the example at the end of Section 2 that every symmetry group Gcontains I3
and I3. Note that in terms of the notation established in (22) and (23), I3=A0
and I3=A0
π, so according to table (24) a fully anisotropic medium is defined to
have symmetry group
G={I3,I3},
the smallest possible. Invariance with respect to the above two transformations
places no constraints whatsoever on the stiffness matrix – 21 parameters are required
to describe the stiffness matrix of a fully anisotropic medium.
4.2. Monoclinic symmetry
A uniform medium whose symmetry group Gcontains (some) reflection in a
plane through the origin is said to be monoclinic. Without loss of generality, we
may assume that this reflection is represented by a matrix A0
θof the form (23) with
θ= 0,
Stud. Geophys. Geod., 48 (2004) 277
L. Bos et al.
A0
0=
1 0 0
0 1 0
0 0 1
.
This corresponds to choosing coordinates such that reflection takes place through
the x1x2-plane (or equivalently, along the x3-axis). The corresponding matrix MA0
0
is
MA0
0=
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0001 0 0
0 0 0 0 1 0
0 0 0 0 0 1
.
the Fundamental Invariance Condition requires that the stiffness matrix Cshould
satisfy C= (MA0
0)TCMA0
0; written out in full,
C11 C12 C13 C14 C15 C16
C12 C22 C23 C24 C25 C26
C13 C23 C33 C34 C35 C36
C14 C24 C34 C44 C45 C46
C15 C25 C35 C45 C55 C56
C16 C26 C36 C46 C56 C66
=
C11 C12 C13 C14 C15 C16
C12 C22 C23 C24 C25 C26
C13 C23 C33 C34 C35 C36
C14 C24 C34 C44 C45 C46
C15 C25 C35 C45 C55 C56
C16 C26 C36 C46 C56 C66
.
This implies that
C14 =C15 =C24 =C25 =C34 =C35 =C46 =C56 = 0 .(26)
Thus, the stiffness matrix of a medium that is invariant with respect to reflection
along the x3-axis has the form
CMONOx3=
C11 C12 C13 0 0 C16
C12 C22 C23 0 0 C26
C13 C23 C33 0 0 C36
0 0 0 C44 C45 0
0 0 0 C45 C55 0
C16 C26 C36 0 0 C66
.(27)
The simplified form of the stiffness matrix means that eight fewer constants (for
a total of 13) are needed to characterize a monoclinic medium than in the fully
anisotropic case.
278 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
4.3. Orthotropic symmetry
A medium whose symmetry group Gcontains two reflections, with respect to
planes that are orthogonal to one-another, is said to be orthotropic. Without loss
of generality we may represent these reflections by matrices of the form
A0
0=
1 0 0
0 1 0
0 0 1
B=
1 0 0
01 0
0 0 1
.(28)
This corresponds to choosing coordinates such that the planes of reflection are the
x1x2-plane and the x1x3-plane. Recall that Gnecessarily contains the transformation
I3, in addition to A0
0and B. The fact that Gis a group then implies that it also
contains the product transformation
I3A0
0B=
1 0 0
0 1 0
0 0 1
,
that is, reflection in the x2x3-plane. This shows that there cannot exist a symmetry
group G0of an elastic medium such that G0contains reflections with respect to two,
but not three, orthogonal planes. For this reason we could have just as well defined
the symmetry group of an orthotropic medium to contain three mutually orthogonal
reflections.
As in the monoclinic case, applying the Fundamental Invariance Condition to
the stiffness matrix Cimplies a simplified structure. To begin, since A0
0belongs to
G, the same matrix as in the monoclinic case, we know that
C=MA0
0CMA0
0,
which implies that Chas the form (27). But not every matrix of the form (27)
corresponds to an orthotropic medium. Since Bbelongs to G, it is also true that
C=MBCMB.(29)
Using (12),
MB=
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0001 0 0
0 0 0 0 1 0
0 0 0 0 0 1
,
and so, in addition to the identities (26), (29) implies that C16 =C26 =C36 =C45 =
= 0. Thus in the coordinates we have chosen Chas the form
Stud. Geophys. Geod., 48 (2004) 279
L. Bos et al.
CORTHO =
C11 C12 C13 0 0 0
C12 C22 C23 0 0 0
C13 C23 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C55 0
00000C66
.(30)
The remaining third reflection I3A0
0Bdoes not engender any further simplification
in the stiffness matrix. (This is not a coincidence but rather stems from the fact that
CORTHO is already invariant with respect to the individual factors of I3A0
0B.)
4.4. Tetragonal symmetry
Instead of starting with two reflections, as in the case of orthotropic symmetry, let
us consider a medium whose symmetry group Gcontains a rotation and a reflection.
More precisely, let Gcontain the rotation represented by the matrix (22) with θ=
π/2,
Aπ/2=
01 0
1 0 0
0 0 1
,
as well as the reflection
B=
1 0 0
01 0
0 0 1
occurring in the previous section. Such a medium is said to be tetragonal, or to
have tetragonal symmetry. Applying the Fundamental Invariance Condition, the
transformations Aπ/2and Bforce the stiffness matrix Cto have a structure even
simpler than that of an orthotropic medium. The most direct way to obtain this
simplified structure is, as before, to simply solve the system of equations
C=MT
Aπ/2CMAπ/2C=MT
BCMB
in the entries of C. An alternative approach is to make use of the earlier results, as
follows. Referring to (28), note that the reflection A0
0is generated by the transfor-
mations I3,Aπ/2,B, that is,
A0
0=I3Aπ/2BA3
π/2B.
Therefore the tetragonal group Gautomatically contains two orthogonal reflections
A0
0,B, and consequently has the form (30), possibly with further simplifications.
Indeed the equation
280 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
C=MT
Aπ/2CMAπ/2
implies the additional identities C22 =C11, C23 =C13, C55 =C44 , resulting in the
final form
CTET =
C11 C12 C13 000
C12 C11 C13 000
C13 C13 C33 000
000C44 0 0
0000C44 0
00000C66
.(31)
Thus only 6 constants are needed to completely characterize a medium having tetrag-
onal symmetry.
4.5. Transverse isotropy
We come now to a particularly interesting case. According to table (24), an
elastic medium is transversely isotropic if, in a suitably chosen orthogonal frame of
reference, the rotation Aθbelongs to Gfor every angle θ.
Now suppose that a medium is invariant with respect to a single rotation of the
form (22), by an angle slightly smaller than the value π/2 considered in the previous
section. Let us consider, for example, θ= 2π/5, and assume that the symmetry
group Gcontains
A2π/5=
cos(2π/5) sin(2π/5) 0
sin(2π/5) cos(2π/5) 0
0 0 1
.
We will prove the following remarkable fact: if A2π/5∈ G then the medium is
transversely isotropic. To begin with, we know by the Fundamental Invariance
Condition that
C=MT
A2π/5CMA2π/5.(32)
Recall that
Aθ=
cos θsin θ0
sin θcos θ0
0 0 1
,
and consider the stiffness matrix Cθdefined by
Cθ=MT
AθCMAθ,
Stud. Geophys. Geod., 48 (2004) 281
L. Bos et al.
corresponding to a rotation by an arbitrary angle θabout the x3-axis. (With this
notation, C=C0=C2π=C4πetc.) Two key properties of Cθwill come into play.
Firstly, each element Cθ
ij of Cθis a trigonometric polynomial in θof degree at most
four. This follows immediately from the definition of Cθ, given that
MAθ=
cos2θsin2θ0 0 0 cos θsin θ
sin2θcos2θ0 0 0 cos θsin θ
0 0 1 0 0 0
0 0 0 cos θsin θ0
0 0 0 sin θcos θ0
2 cos θsin θ2 cos θsin θ0 0 0 cos2θsin2θ
.
For example, the (1,1)-element of Cθis
Cθ
11 =C11 cos4θ+ (2C12 + 4C66) cos2θsin2θ
+ 4C16 cos3θsin θ+C22 sin4θ+ 4C26 cos θsin3θ .
Secondly, Cθis 2π/5periodic; that is,
Cθ+2π/5=Cθ.(34)
This latter fact follows simply by expanding Cθ+2π/5. In detail,
Cθ+2π/5=MT
Aθ+2π/5CMAθ+2π/5
= (MA2π/5MAθ)TC(MA2π/5MAθ) (by equation(14))
=MT
Aθ(MT
A2π/5CMA2π/5)MAθ
=MT
AθCMAθ(by equation(32))
=Cθ,
as desired.
Since it is a fourth degree trigonometric polynomial, each element Cθ
jk has a
representation of the form
Cθ
jk =
4
X
N=4
αjk
NeiNθ ,(35)
where the 9 coefficients (αjk
4, αjk
3, . . . , αjk
4) are uniquely determined complex num-
bers. Applying periodicity, we see that Cθ
jk can also be written as
282 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
Cθ
jk =Cθ+2π/5
jk =
4
X
N=4
αjk
NeiN(θ+2π /5)
=
4
X
N=4
(αjk
Ne2πiN/5)eiN θ .
(36)
Since the coefficients in the representation (35) are unique, comparing the represen-
tations (35) and (36) implies that for each N,4N4,
αjk
N=αjk
Ne2πiN/5.
Since |N/5|<1, provided N6= 0 we have that e2πiN/56= 1, and this is only possible
if, for each N6= 0, αjk
N= 0. Thus Cθ
jk =αjk
0is a constant function, independent of
θ.
We have shown that each element of Cθis constant; this means that for all θ,
Cθ=C0=C. The equality C=Cθis, by the Fundamental Invariance Condition,
equivalent to the definition of transverse isotropy. So the medium in question is
transversely isotropic, which is what we set out to prove.
The above argument works equally well for any positive integer n5 in place
of 5. (All that is needed is that |N/n|<1 for each N,4N4.) Thus we have
the following more general fact: if A2π/n ∈ G for any n > 4, then Aθ∈ G for every
θ; i.e., the medium is transversely isotropic. We shall refer to this fact as Herman’s
Theorem. (In fact Herman’s Theorem is more general still. Quoting from Herman
(1943), “If the medium has a rotating axis of symmetry CNof order N, it is axially
isotropic relative to this axis for all the physical properties defined by the tensors of
the rank r= 0,1,2, . . . N 1.”)
Let us return to the question of the special form of the stiffness matrix for a
transversely isotropic medium. Identifying the coefficients of the trigonometric poly-
nomials* on the right-hand side of the equation
C=MT
AθCMAθ(37)
with the coefficients of the (constant!) polynomials on the left-hand side gives us
a system of linear equations in the entries of C. Solving these equations yields
relations amongst the entries of Cthat are equivalent to Chaving the special form:
CTRANS =
C11 C12 C13 0 0 0
C12 C11 C13 0 0 0
C13 C13 C33 0 0 0
0 0 0 C44 0 0
0 0 0 0 C44 0
00000C11C12
2
.(38)
*Note that these polynomials need to be represented in terms of a suitable basis such as
{eiθ/N }4
N=4; the monomials cosmθsinnθare not linearly independent.
Stud. Geophys. Geod., 48 (2004) 283
L. Bos et al.
Another way to derive the above form of CTRANS is to choose particular values for
θand then to determine the consequent relations amongst the entries of Cimplied
by the equation (37). In this regard the angle θ=π/3 turns out to be a particularly
good choice: the linear algebra works out quite simply, and all the necessary relations
can be derived from this single value of θ. (Indeed the sufficiency of the single value
π/3 is a consequence of Herman’s Theorem.)
4.6. Isotropy
A medium whose symmetry group Gis the orthogonal group, that is, the group
consisting of all orthogonal transformations, is said to be isotropic. An isotropic
medium has the maximum possible invariance, and consequently its stiffness matrix
Chas the simplest possible form, which we now derive. To begin, since Gcontains
all rotations about the x3-axis (for any chosen system of coordinates), the stiffness
matrix Chas the form (38), possibly with further simplifications. Indeed invariance
with respect to the transformation that interchanges the x1and x3-coordinates,
Ax1x3=
0 0 1
0 1 0
1 0 0
,
which by the Fundamental Invariance Condition implies the equation
C=MT
Ax1x3CMAx1x3,
imposes the additional identities C11 =C33, C12 =C13 , C44 =C66. Incorporating
the latter yields the form
CISO =
C11 C12 C12 000
C12 C11 C12 000
C12 C12 C11 000
0 0 0 C11 C12
20 0
0 0 0 0 C11C12
20
0 0 0 0 0 C11C12
2
.(39)
The structure of (39) is such that for any transformation A,CISO satisfies the
equation
C=MT
ACMA.
This demonstrates the well-known fact that precisely two constants are required to
characterize an isotropic medium.
284 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
5. THE ORTHOGONAL GROUP AND CRYSTAL GROUPS
In this section we return to the definition of symmetry groups with two objectives
in mind. We wish firstly to reiterate the basic definition of anisotropy class in terms
of groups. Secondly, we wish to mention the connection to crystallography, since in
the literature nomenclature borrowed from crystallography is occasionally applied
to symmetry classes of elastic media. Our purpose here is to give just a sketch of
the bigger picture, without getting into details or giving proofs.
5.1. The complete list of anisotropy classes
Let us re-examine our definition of symmetry groups after first establishing some
additional terminology. Let O(3) denote the orthogonal group, i.e., the set of all
orthogonal transformations of 3-dimensional euclidean space. Given any set Dof
stiffness matrices, let G(D) denote the group consisting of all orthogonal transfor-
mations AO(3) such that, for every C∈ D,
C=MT
ACMA.(40)
Call a set G ⊆ O(3) an elastic group if Ghas the form G(D) for some set Dof
stiffness matrices. Classes of anisotropy correspond exactly to elastic groups (where
we abide by our earlier convention whereby two elastic groups related by a change of
coordinates are identified). What is the complete list of elastic groups? In principle
this can be derived using the techniques that we have already presented, and in
particular using Herman’s Theorem, together with known results concerning finite
subgroups of the orthogonal group. Rather than giving the technical details however,
which are in any case beyond the scope of such a tutorial, we present just the
end result. The following extension of table (24) represents every possible elastic
symmetry group, or equivalently, every possible anisotropy class.
Type of symmetry Transformations belonging G
Full anisotropy A0,A0
πonly
Monoclinic A0
0
Orthotropic A0
0,B
Week trigonal A2π/3
Trigonal A2π/3,B
Week tetragonal Aπ/2
Tetragonal Aπ/2,B
Transverse isotropy Aθfor all θ
Cubic all transformations that leave invariant
a cube centred at the origin
Isotropy all orthogonal transformations (41)
Stud. Geophys. Geod., 48 (2004) 285
L. Bos et al.
Note that the weak trigonal class is sometimes grouped together with the trigonal
class, as is the weak tetragonal class with the tetragonal class. There exist no other
possible anisotropy classes for an elastic medium.
5.2. Crystal groups
Given a crystal lattice L, and fixing an orthonormal system of coordinates cen-
tered on a molecule of the lattice, one can consider the group of all orthogonal
transformations that carry Linto itself. This group does not completely capture the
symmetry of the crystal, since a crystal is by definition also invariant with respect
to translations, but it does give rise to a group G O(3), called a crystal group.
Is there any relation between crystal groups and elastic groups? The two do not
exactly coincide, but in fact they are related. Briefly, every crystal group is finite,
so the two infinite elastic groups, corresponding to isotropy and transverse isotropy
respectively, are not crystal groups. And there are 32 crystal groups, so not every
crystal group is an elastic group – the table (41) doesn’t have 32 entries! However,
every finite elastic group is a crystal group. To some extent this justifies an over-
lap in the nomenclature, despite the fact that a crystal and an elastic medium are
utterly different types of mathematical objects.
6. CONCLUSION
An elastic continuum is a well-established model which with to represent and to
analyze wave phenomena in actual materials. In this context anisotropy is funda-
mental notion, and is defined purely in terms of the elasticity tensor. As has been
already emphasized, it is a local concept in that the anisotropy class is a charac-
teristic of a single material point. As such the idea of inhomogeneity, or in other
words, the variation of material properties from one point to another, is irrelevant
to anisotropy. The proper interpretation of a phrase such as “transverse isotropy of
the Pierre Shale in Colorado,” which one encounters in the applied geophysics lit-
erature, is of course that, in an appropriate model of the Pierre Shale, the stiffness
tensor should conform to the definition of transverse isotropy at each point. Note
that even this interpretation does not imply homogeneity, for two reasons. Firstly,
at each point the stiffness matrix must, with respect to appropriate coordinates,
have the form of the matrix (38) in Section 4.5, but the actual values of the element
C11, for instance, can still vary from one point to the next. Secondly, the material
density, which is not involved in the definition of anisotropy, may also vary from one
point to the next.
In analyzing material symmetry, the key mathematical concept is that of a sym-
metry group; a key technical role is also played by Herman’s thereom. It is a
remarkable fact of nature that the list of all possible anisotropy classes is not only
finite, but relatively short. Moreover, one can derive this list in large measure using
precisely the type of arguments that we have presented here. In closing, the reader is
286 Stud. Geophys. Geod., 48 (2004)
Classes of Anisotropic Media: a Tutorial
invited to consult the references below for alternate perspectives and further details
concerning anisotropy in elastic media.
References
Bond, W.L., 1943. The mathematics of the physical properties of crystals. Bell System
Technical Journal,22, 1-72.
Fedorov, F.I., 1968. Theory of Elastic Waves in Crystals. Plenum Press, New York.
Helbig, K., 1994. Foundations of Anisotropy for Exploration Seismics. Pergamon, Trow-
bridge.
Herman, B., 1945. Some theorems of the theory of anisotropic media. Comptes Rendus
(Doklady) de l’Acad´emie des Sciences de l’URSS,48 No. 2, 89-92.
Musgrave, M.J.P., 1970. Crystal Acoustics. Holden-Day, San Francisco.
Sokolnikoff, I.S., 1956. Mathematical Theory of Elasticity, 2nd ed. McGraw-Hill, New York.
Voigt, W., 1928. Lehrbuch der Kristallphysik. Teubner, Leipzig.
Weyl, H., 1952. Symmetry. Princeton University Press, Princeton.
Stud. Geophys. Geod., 48 (2004) 287
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... We approximate the corresponding isotropic velocities: v p = (A 11 + A 22 + A 33 )/3, v s = (A 44 + A 55 + A 66 )/3, where A i j represent the density normalised elastic tensor in the Voigt notation (e.g. Bos et al. 2004). Using (10) and (11) in (9) the influence of elastic properties at the source and along the ray path can be separated. ...
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Anisotropic material properties are commonly neglected during moment tensor inversion. On the other hand, anisotropy is a widely observed rock property. We show that anisotropy may greatly influence characteristics of moment tensors. For the inversion we apply a method based on amplitude spectra of waveforms in isotropic media. We investigate effects of anisotropy on seismic moment, moment-tensor components, and apparent slip inclination of dislocation point sources. The direct calculation of moment tensors for shear sources in anisotropic regions shows spurious non-double-couple components that may be mistaken as an indication of (apparent) opening or closing of the fracture plane. On the other hand real volumetric components may be increased but also hidden in the presence of anisotropy. These effects as well as the seismic moment depend on the orientation of the elastic tensor relative to the fault plane and the slip direction. If anisotropy is present near the source but isotropy is assumed during inversion, the properties of the moment tensor can still be obtained in a good approximation. In the case where anisotropy extends to the medium along the ray path, only the fault orientation can be successfully retrieved by inverting qP waves to derive the deviatoric moment tensor. The inversions show that retrieved moment tensors can deviate systematically from moment tensors of shear and tensile sources expected in isotropic media. Further complications may arise when qS waves are included in the inversion process. We account for near-source anisotropy to re-interpret moment tensors derived for two events at the KTB super deep drill hole, SE-Germany. The obtained source mechanisms are close to shear faulting although the moment tensors comprise non-double-couple components. We interpret the volumetric moment-tensor components partly as a result of anisotropy. This indicates that for detailed studies of volumetric source components anisotropy should be considered during inversion. In addition, we show that for shear sources in anisotropic media the elastic properties near the source can also be derived from inverted moment tensors in media where anisotropy is restricted to the source region.
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One hundred and fifty five years ago, Kelvin published the first part of a fundamental analysis of the elastic tensor, in which he proposed a coordinate‐free representation through its eigensystem. His thoughts were apparently far ahead of his time, since it took 125 years before the paper elicited a positive reaction (it is now accessible through several modern reviews). Science not only lost track for 125 years of the original paper but also lost the ideas Kelvin might have proposed in the second part, a publication that was never put to paper, presumably in view of the lack of appreciation of the first part.In an attempt to establish what might have been on Kelvin's mind for a second part, one has to ‘forget’ the progress of mathematical physics in the intervening time and base all arguments strictly on the content of the first part and on the state of science in the second half of the 19thcentury.The theory of elasticity would certainly have developed faster, had Kelvin's paper peen appreciated by his ‘peers’. But a theory based on Kelvin's ideas would be fruitful even today.
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In this work we aim to develop expressions for the calculation of biaxial and triaxial stresses in polycrystalline anisotropic materials, and to determine their elastic constants using the theory of elasticity for continuum isochoric deformations; thus, we also derive a model to determine residual stress. The constitutive relation between strain and stress in these models must be assumed to be orthotropic, obeying the generalized Hooke's law. One technique that can be applied with our models is that of X-ray diffraction, because the experimental conditions are similar to the assumptions in the models, that is, it measures small deformations compared with the sample sizes and the magnitude of the tensions involved, and is insufficient to change the volume (isochoric deformation). Therefore, from the equations obtained, it is possible to use the sin 2ψ technique for materials with texture or anisotropy by first characterizing the texture through the pole figures to determine possible angles ψ that can be used in the equation, and then determining the deformation for each diffraction peak with the angles ψ obtained from the pole figures.
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Papers dealing with the generalized Hooke’s law for linearly elastic anisotropic media are reviewed. The papers considered are based on Kelvin’s approach disclosing the structure of the generalized Hooke’s law, which is determined by six eigenmoduli of elasticity and six orthogonal eigenstates.