ArticlePDF Available

Abstract

We show that, in general, wavefronts are more symmetric than the medium in which they propagate. This means that we cannot determine the symmetries of the medium based solely on the symmetries of the wavefronts. However, we show that we can determine the symmetries of the medium from the symmetries of wavefronts and polarizations together.
MATERIAL SYMMETRIES VERSUS WAVEFRONT SYMMETRIES
ANDREJ BÓNA, IOAN BUCATARU, AND MICHAEL A. SLAWINSKI
ABSTRACT. We show that, in general, wavefronts are more symmetric than the medium
in which they propagate. This means that we cannot determine the symmetries of the
medium based solely on the symmetries of the wavefronts. However, we show that we
can determine the symmetries of the medium from the symmetries of both wavefronts and
polarizations.
INTRODUCTION
The focus of this study is the relation between the symmetry class to which a given
medium belongs and the symmetries of wavefronts and polarizations within it. Herein, we
discuss the wavefronts and polarizations in a homogeneous linearly elastic continuum that
are generated by a point source.
Our studying the symmetries of wavefronts and polarizations is motivated by inverse
problems in seismology. The problem of determining the components of the elasticity
tensor from a series of experiments has been investigated by several researchers; notably,
by Van Buskirk et al. [2] and Norris [10].
In this paper, we show that, in general, it is not possible to determine the symmetries
of an elasticity tensor from the symmetries of wavefronts, alone; we need also information
about symmetries of polarizations.
We begin this paper with a brief review of the elasticity tensor and its symmetries.
Then, we discuss the symmetries of the Christoffel matrix and of its eigenvalues. We show
that the symmetries of the eigenvalues are equivalent to the symmetries of wavefronts.
Subsequently, we show that while the symmetry of the elasticity tensor is equivalent to
the symmetry of the Christoffel matrix, the eigenvalues can be more symmetric than the
matrix itself. Thus, symmetry of the wavefronts is not equivalent to the symmetry of the
medium. We present cases where the wavefronts are more symmetric than the medium in
which they propagate. Also, we present symmetry classes for which the wavefronts cannot
have higher symmetry. We conclude this paper by showing that information about the
wavefront symmetries combined with information about polarization symmetries, which
are associated with the eigenvectors of the Christoffel matrix, allow us to determine the
symmetry of the continuum.
1. SYMMETRIES OF ELASTICITY TENSOR
In this section, we review briefly the background necessary to discuss the relationship
between material symmetries and symmetries of wavefronts and polarizations.
A linearly elastic continuum is fully described by its elasticity tensor and mass density.
All properties associated with the symmetries of such a continuum are contained in this
fourth-rank tensor that linearly relates two second-rank tensors: stress and strain.
Key words and phrases. anisotropy, wavefront, polarization, material symmetry, symmetry group.
Department of Earth Sciences, Memorial University, St. John’s, Canada, A1B 3X5.
1
The elasticity tensor possesses several intrinsic symmetries that are independent of the
properties of a given material. Consider the elasticity tensor, c, in the three-dimensional
Euclidean space, R3. We can write this tensor as a four-linear map given by
c:R3×R3×R3×R3R
that possess the following intrinsic symmetries:
c(u, v, w, z ) = c(v, u, w, z) = c(w, z , u, v),u, v, w, z R3.(1.1)
With respect to an orthonormal basis, {e1, e2, e3}, of R3, we can write the components of
cas
cijkl =c(ei, ej, ek, el).
Using these components, we can write intrinsic symmetries (1.1) as
cijkl =cj ikl =cklij .
Thus, even though in general a fourth-rank tensor in R3has 34= 81 independent compo-
nents, cpossesses at most twenty-one independent components.
Depending on its properties, a given material can possess additional symmetries, which
are called the material symmetries.
There are several definitions of the symmetry class of the elasticity tensor. Herein, we
use the definition according to which conjugate symmetry groups belong to the same class.
In accordance with this definition, there are eight symmetry classes, as shown by Forte and
Vianello [7], Chadwick et al. [3], Ting [12], Bóna et al. [1].1
A continuum possesses a given material symmetry if cis invariant under the correspond-
ing orthogonal transformation AO(3). In other words,
c(u, v, w, z ) = c(Au, Av, Aw, Az),u, v, w, z R3.(1.2)
In terms of components, Ais a symmetry of cif
cijkl =Aim Ajn Akp Alq cmnpq .
Throughout this paper, we shall use the following notations. Repeated indices mean
summation, unless we state otherwise. Gcis the set of all orthogonal transformations
under which cis invariant. Gcis a subgroup of the orthogonal group O(3) and we refer
to Gcas the symmetry group of the elasticity tensor. Rθ,u is the rotation by angle θabout
axis u, and Ruis reflection about a plane that is orthogonal to vector u.
Two elasticity tensors, c1and c2, belong to the same symmetry class if their symmetry
groups are conjugate, which means that there exists AO(3) such that Gc2=AGc1AT.
By considering this conjugacy, we ensure that a given continuum described in different
coordinate systems belongs to the same class. Each symmetry class is completely deter-
mined by its corresponding set of conjugate symmetry groups. We choose to represent an
elasticity tensor with respect to an orthonormal basis in which it has the smallest number
of independent nonzero components. We refer to such a basis as a natural basis. For each
symmetry class, we choose, among the set of conjugate symmetry groups, one symmetry
group that corresponds to the chosen elasticity tensor and we express it using the natural
basis.
1Another definition of the symmetry class has been proposed by Yong-Zhong and Del Piero [14] and discussed
by Forte and Vianello [8]; this definition results in ten symmetry classes.
2
2. CHRISTOFFEL MATRIX
In this section, we show that the symmetries of the elasticity tensor are equivalent to the
symmetries of the Christoffel matrix. In this paper we consider wavefronts generated by a
point source located at the origin. Since such wavefronts are given by the eigenvalues of
this matrix, this equivalence will allow us to conclude that the wavefronts are at least as
symmetric as the elasticity tensor.
While solving the elastodynamic equations in anisotropic elastic continua, we obtain a
set of equations given by
[cijkl pjplρδik ]Xk= 0, i ∈ {1,2,3},
which are known as the Christoffel equations (e.g., Fedorov [6], equation (15.19) or Slaw-
inski [11], equations (7.13) and (10.1)). Here, p:= ψ(x), where the eikonal function,
ψ, describes a wavefront, ρdenotes mass density, and Xkstands for a component of the
displacement vector. Since ψ(x) = t, where xspecifies the position in R3and tstands
for time, describes the moving wavefront, we see that pis a vector that is normal to the
wavefront and whose magnitude is the slowness of the wavefront propagation. From phys-
ical considerations, the slowness, which is the reciprocal of velocity, is nonzero, so we can
rewrite the Christoffel equations as
|p|2"cijkl
pjpl
|p|21
|p|2ρδik#Xk= 0, i ∈ {1,2,3},
where |p|2:= hp, pi. Denoting the unit vector normal to the wavefront by u=p/ |p|, we
write
"cijkl ujul1
|p|2ρδik#Xk= 0, i ∈ {1,2,3}.(2.1)
Considering nontrivial solutions, we require
det "cijkl ujul1
|p|2ρδik#= 0, i, k ∈ {1,2,3}.(2.2)
We define
Γik (u) := cijkl ujul, i, k ∈ {1,2,3},(2.3)
where Γ (u)is called the Christoffel matrix. In view of its definition, Γsatisfies the fol-
lowing relation:
Γ (u) (v, w) = c(u, v , u, w).(2.4)
In view of existence of the strain-energy function, we require that c(u, v, u, v)0(e.g.,
Slawinski [11], p. 75). This implies that Γ (u)is positive-definite for all u.
For convenience and without loss of generality, in the subsequent development we set
ρ= 1.
2.1. Symmetries of Christoffel matrix and material symmetries. Examining expres-
sion (2.4), we see that Γ(u)(v, w)is not a tensor; it is quadratic in the first argument and
linear in the last two. To linearize the quadratic part, we can — in general — write
Γ(u+v)(w, z)Γ(uv)(w , z) = 2 [c(u, w, v, z) + c(v, w, u, z )] ,(2.5)
which we can rewrite with respect to an orthonormal basis {e1, e2, e3}as
Γ(ei+ej)(ek, el)Γ(eiej)(ek, el) = 2 (cik jl +cj kil),(2.6)
3
where i, j, k, l, belong to set {1,2,3}. This implies that at least two indices must be equal
to one another. In view of the intrinsic symmetries of c, we can choose these indices to be
iand jor to be iand k. Throughout Section 2.1, the equality of two indices does not imply
any summation. Using the first choice, namely, i=j, we write equation (2.6) as
cikil = Γ(ei)(ek, el).(2.7)
Using the second choice, namely i=k, we write equation (2.6) as
ciijl =1
2[Γ(ei+ej)(ei, el)Γ(eiej)(ei, el)2Γ(ei)(ej, el)] ,(2.8)
where we used also equation (2.7) and the fact that cijil =cjiil .
Using equations (2.7) and (2.8), we conclude that the Christoffel matrix determines
uniquely the elasticity tensor. This has been shown also by Norris [10]; our equations (2.7)
and (2.8) are equivalent to equations (2.4) and (2.5) in his paper. We conclude that the
symmetry groups of elasticity tensor cand Christoffel matrix Γcoincide, in other words
Gc=GΓ, where
GΓ={AO(3),Γ(Au)(Av, Aw) = Γ(u)(v , w),u, v, w R3}.(2.9)
Having shown that the symmetries of the Christoffel matrix are equivalent to the sym-
metries of the elasticity tensor, we will use the symmetries of this matrix to study the
relationship between the symmetries of the elasticity tensor and the symmetries of wave-
fronts.
2.2. Symmetries of eigenvalues and wave front symmetries. Following our discussion
at the beginning of Section 2, we can express each of the three wavefronts at time tas
ψ1
i(t), which is the level set of the eikonal function, ψi, where i∈ {1,2,3}. In media
discussed herein the velocity does not depend on x. Also, we discuss wavefronts generated
by a point source at the origin. Thus, the eikonal function, which corresponds to traveltime,
is a homogeneous function in x. Hence, the symmetry of the wavefront, which is given
by the level set of the eikonal function, is equivalent to the symmetry of eikonal function
itself. Consequently, let us consider the symmetry of these functions. The symmetry group
of the wavefronts, which we denote by Gψ, is composed of all AO(3) such that
ψi(Ax) = ψi(x),i∈ {1,2,3},xR3.(2.10)
We wish to relate the symmetry of the wavefronts to the symmetry of the eigenvalues
of the Christoffel matrix. Examining equation (2.2), we see that this equation is a cubic
polynomial in 1/|p|2. Due to the positive definiteness of Γ, this polynomial can be factored
using the three positive roots, which we denote by v2
i, where i∈ {1,2,3}. Each of these
three eigenvalues depends on direction up/ |p|. Thus, the set of all pthat satisfy
equation (2.2) can be written as
|p|2=1
v2
ip
|p|,(2.11)
where viis the squared wavefront velocity. Equation (2.11) is called the eikonal equation.
Let us consider the three wavefront velocities, vi, squares of which are the roots of
equation (2.2). In order to simplify the notation and discussion, we choose the indices
of set {vi(u), i ∈ {1,2,3}} for all uin such a way that the elements are ordered by
magnitude, namely, v1(u)v2(u)v3(u). The symmetry group that preserves all of
them and which we denote by Gv, is composed of all AO(3) such that
vi(Au) = vi(u),i∈ {1,2,3},uR3.
4
The fact that the velocities are ordered in all directions uimplies that the symmetry of
all three velocities considered together is also a symmetry of each individual velocity. The
ordering we chose for the velocities and the fact that they are eigenvalues for the Christoffel
matrix imply that GΓGv.
In Section 3.2, we will use several examples to show that, in general, the opposite is
not true. In view of GΓ=Gc, discussed in Section 2.1, we will conclude that in general
Gc6=Gv.
To complete the path of relations linking the material and wavefront symmetries, we will
show that the wavefront-velocity symmetries are equivalent to the wavefront symmetries.
Since the wavefront velocity is the reciprocal of the wavefront slowness, their symmetries
are equivalent to one another. This means that to show that the wavefront symmetries are
equivalent to the wavefront-velocity symmetries, it suffices to show that the wavefront-
slowness symmetries are equivalent to the wavefront symmetries.
Since the wavefront-slowness surfaces are given by the solutions of eikonal equations
(2.11), we will investigate their symmetries. These solutions can be represented by the
level sets H1
i(1/2) of the three Hamiltonians, Hi(p) := |p|2v2
i(p/ |p|)/2. The symme-
try group, GH, of Hamiltonians Hiis composed of all AO(3) such that
Hi(Ap) = Hi(p),i∈ {1,2,3},pR3.(2.12)
Since these Hamiltonians are homogeneous in p, the symmetry of the wavefront-slowness
surfaces is equivalent to the symmetry of the Hamiltonians themselves. Furthermore, since
the wavefront-velocity and wavefront-slowness symmetries are equivalent to one another,
it follows that GH=Gv.
To relate Gvto Gψ, we consider AGψ. Following expression (2.10) and using the
chain rule, we can write the wavefront-slowness vector for a given eikonal function as
p(x)≡ ∇ψ(x) = ψ(Ax) = ATψ|Ax ATp(Ax),xR3,
where ψ|Ax denotes ψevaluated at Ax. If we set x= 0, we see that if Ais a symmetry
of ψ, then it is a symmetry of p, which also means that Ais a symmetry of v. In other
words, GψGv.
To show the opposite relation, let us invoke Hamilton’s equations for a given Hamilton-
ian, H. In particular, let us consider
˙x=pH(p).(2.13)
Since x(t)describes rays, ˙xis the ray velocity, which describes elementary wavefronts
at a point. Hence, the symmetry of the ray velocity is equivalent to the symmetry of the
wavefront. To continue our derivation, we consider AGH. Following expressions (2.12)
and (2.13), we can write the ray-velocity vector as
˙x(p) = H(p) = H(Ap) = ATH|Ap AT˙x(Ap),pR3.
This equation means that if Ais a symmetry of H, and hence a symmetry of v, then it is a
symmetry of ˙x, and hence the symmetry of ψ. In other words, GvGψ.
Combining the two above results, we conclude that Gψ=Gv. In other words, the
wavefront symmetries are equivalent to the wavefront-velocity symmetries.
In view of Gc=GΓ, discussed in Section 2.1, and GΓGvtogether with Gv=Gψ,
we see that GcGψ. In other words, we conclude that the wavefronts are at least as
symmetric as the medium through which they propagate.
5
3. WAVEFRONT AND MATERIAL SYMMETRIES
As shown in Section 2.2, GcGv. In this section, we will discuss the case of Gc=Gv,
as well as show several examples for which this equality does not hold.
Due to the complexity of the calculations, we used computer-algebra software to study
the invariance of velocity functions under orthogonal transformations in R3. The velocity
functions are the roots of polynomial
det[Γij (u)v2(u)δij ].(3.1)
We can formally write this polynomial as
v23+f1(u)v22f2(u)v2+f3(u).(3.2)
Coefficient f1is the trace of Γ, it is a second-order polynomial in u1,u2and u3, and it is
given by
f1(u1, u2, u3)=(c1111 +c1212 +c1313)u2
1+ (c1212 +c2222 +c2323)u2
2
+ (c1313 +c2323 +c3333)u2
3+ 2 (c1112 +c1222 +c1323 )u1u2
+ 2 (c1113 +c1223 +c1333)u1u3+ 2 (c1213 +c2223 +c2333)u2u3.
Coefficient f2(u1, u2, u3)is the sum of the determinants of the second-order minors of
Γ (u1, u2, u3), namely,
Γ11Γ22 Γ2
12+Γ11 Γ33 Γ2
13+Γ22 Γ33 Γ2
23,
which is a fourth-order polynomial in u1,u2and u3. Coefficient f3(u1, u2, u3)is the
determinant of Γ (u1, u2, u3), which is a sixth-order polynomial in u1,u2and u3.
The velocity functions are invariant under a given rotation of uif and only if the three
coefficients fi(u)of polynomial (3.2) are invariant under such a rotation.2
Thus, the study of the symmetries of the wavefronts is reduced to the study of the
symmetries of the three polynomials f1(u),f2(u)and f3(u).
We use two complementary approaches in our study of invariance.
The first approach was designed to find media in which given wavefronts can propagate.
We proceeded in the following way. We set an orthogonal transformation that is a symme-
try of wavefronts. Then, we examined the effect of the transformations contained in this
wavefront-symmetry group on the elasticity tensor. The set of components that renders the
tensor invariant under these transformations determines the symmetry class of a medium in
which given wavefronts can propagate. Among the resulting media, we always found the
one whose symmetry class is the same as the symmetry class of the wavefronts. However,
we also found less symmetric media.
The second approach was designed to find wavefronts that can propagate in a given
medium. We set an orthogonal transformation that is a symmetry of the elasticity tensor.
Then, we examined the effect of the transformations contained in this symmetry group
on the wavefronts by testing the invariance of the three coefficient functions, f1,f2and
f3. Among the resulting wavefronts, we always found the ones whose symmetry class is
the same as the symmetry class of the medium. However, we also found more symmetric
wavefronts.
Below, we present the results that were obtained by combining the two approaches.
2Instead of working with function f2, we could use TraceΓ2as did Fedorov [6]. However, we prefer to
work with functions f1, f2, f3since they are the coefficients of polynomial (3.1).
6
3.1. Cases of equivalent symmetries. In this section we present cases for which the sym-
metries of the medium and the symmetries of the velocity functions are the same. This
means that knowing the symmetry class of the wavefronts is equivalent to knowing the
material symmetry of the medium in which they propagate.
Generally anisotropic wavefronts Since, as shown in Section 2.2, the wavefronts are at
least as symmetric as the medium in which they propagate, it immediately follows that if
the wavefronts are generally anisotropic, so is the medium.
Monoclinic wavefronts We assume that the velocity functions have monoclinic symme-
try, which means that they are invariant under reflection Re3. In other words, the three
coefficient functions, fi(u1, u2, u3), are invariant under transformation
(u1, u2, u3)7→ (u1, u2,u3).(3.3)
The invariance of the first coefficient function, f1(u1, u2, u3), under transformation (3.3)
implies
0 = f1(u1, u2, u3)f1(u1, u2,u3)(3.4)
= 4 (c1113 +c1223 +c1333)u1u3+ 4 (c1213 +c2223 +c2333 )u2u3,
for all (u1, u2, u3)in R3. Equation (3.4) determines two linear equations in terms of the
coefficients of the elasticity tensor
c1113 +c1223 +c1333 = 0, c1213 +c2223 +c2333 = 0.(3.5)
The invariance of the second coefficient function, f2(u1, u2, u3), under transformation
(3.3) determines six second-order equations in terms of the coefficients of the elasticity
tensor. The solution of the resulting eight equations requires that
c1113 =c1123 =c2213 =c2223 =c3313 =c3323 =c1312 =c1223 = 0.(3.6)
The invariance of the third coefficient function, f3(u1, u2, u3), under transformation (3.3)
determines another twelve third-order equations, which are satisfied already by solution
(3.6).
Solution (3.6) corresponds to a monoclinic medium. For convenience we use Voigt’s nota-
tion [13] to display our results. Hence, we write solution (3.6) as
C11 C12 C13 0 0 C16
C12 C22 C23 0 0 C26
C13 C23 C33 0 0 C36
000C44 C45 0
000C45 C55 0
C16 C26 C36 0 0 C66
.
Thus, we conclude that Gc=Gv=I3,±Re3}; in other words, if the wavefronts
are monoclinic, so is the medium. This also means that monoclinic wavefronts cannot
propagate in generally anisotropic media. Furthermore, this implies that no wavefronts
exhibiting monoclinic symmetry, can propagate in generally anisotropic media.
3.2. Cases of nonequivalent symmetries. In this section, we present examples for which
the velocity functions are more symmetric than the medium. This means that knowing the
symmetry class of the wavefronts allows us to conclude that the medium in which they
propagate is at most as symmetric as the wavefronts.
Orthotropic wavefronts We assume the wavefronts have orthotropic symmetry, which
is equivalent to the invariance under rotations Rπ,e3and Rπ,e2. The invariance of the three
7
functions fi(u1, u2, u3)under transformations
(u1, u2, u3)7→ (ζ1u1, ζ2u2, ζ3u3), ζi∈ {±1},(3.7)
results not only in orthotropic media, but also in another solution; namely,
C11 C12 C55 000
C12 C22 C44 000
C55 C44 C33 0 0 C36
000C44 0 0
0 0 0 0 C55 0
0 0 C36 0 0 C66
.
This solution corresponds to a particular case of the monoclinic medium with the symmetry
axis e3that is described by eight independent components with
C16 =C26 =C45 = 0,
C13 =C55
and
C23 =C44.
This means that orthotropic wavefronts can propagate in such monoclinic media. One
can see that these media do not have any higher symmetry by examining them under rota-
tions around e3.
Tetragonal wavefronts We assume the wavefronts have tetragonal symmetry, which is
equivalent to the invariance under rotation Rπ/2,e3. The invariance of the velocity functions
results not only in tetragonal media, but also in another solution; namely,
C11 C12 C23 2C44 0 0 C16
C12 C11 C23 0 0 C16
C23 2C44 C23 C33 0 0 C36
000C44 0 0
0 0 0 0 C44 0
C16 C16 C36 0 0 C66
.
Again, this solution corresponds to a particular case of the monoclinic medium that is
described by eight independent components. However, herein,
C11 =C22,(3.8)
C13 =C23 2C44,(3.9)
C26 =C16
and
C45 = 0.
This means that tetragonal wavefronts can propagate in such monoclinic media.
Cubic wavefronts We assume the wavefronts have cubic symmetry, which is equiva-
lent to the invariance under rotation Rπ/2,e3and Rπ/2,e1. The invariance of the velocity
functions results not only in cubic media, but also in three other solutions. First,
C11 C23 2C44 C23 2C44 000
C23 2C44 C11 C23 000
C23 2C44 C23 C11 000
000C44 0 0
0 0 0 0 C44 0
0 0 0 0 0 C44
.(3.10)
8
This solution corresponds to a particular case of the tetragonal medium, invariant under ro-
tation by π/2about e1, that is described by three independent components obeying equality
(3.9), as well as
C12 =C23 2C44.(3.11)
Furthermore,
C11 =C22 =C33 (3.12)
and
C44 =C55 =C66.(3.13)
The other two solutions correspond also to tetragonal media, but invariant under rotations
about e2, and about e3.
These results mean that cubic wavefronts can propagate in such tetragonal media.
We note that cubic wavefronts propagating in media given by matrix (3.10) are identical
to wavefronts propagating in the cubic media with the same elasticity parameters C11,C23
and C44. In this case, the number of elasticity parameters of the particular tetragonal media
given by matrix (3.10) and the cubic media is the same. This is not the case for the previous
two examples.
The above three examples show that, in general, wavefronts are more symmetric than
the medium in which they propagate. In other words, the knowledge of symmetries of
eigenvalues alone does not allow us to determine the symmetry of the Christoffel matrix.
4. WAVEFRO NT AND POLAR IZATI ON SYMMETR IES AND MATERIAL SY MME TRIES
In this section, we show that the symmetries of eigenvectors — together with the sym-
metries of eigenvalues — allow us to determine the symmetries of the Christoffel matrix,
and — hence — the symmetry of the medium.
Using matrix notation, we can write the fact that the solutions vi(u)of equation (2.2)
satisfy equations (2.1) as
Γ (u)X(i)(u) = v2
i(u)X(i)(u), i ∈ {1,2,3}, u R3,(4.1)
where Γis the Christoffel matrix, v2
iis the square of a wavefront-velocity function, and
X(i)stands for a displacement vector. (Note that in equations (2.1) Xkstood for a compo-
nent of the displacement vector.) Examining equation (4.1), we see that v2
iand X(i)are,
respectively, the eigenvalues and the eigenvectors of Γ. Since each eigenvalue specifies
the direction of the corresponding eigenvector without specifying its amplitude, we limit
our study to directions of displacement vectors; in other words, we focus our attention on
polarizations.
To investigate symmetries of polarizations, we replace uby Au in equation (4.1) to
write
Γ (Au)X(i)(Au) = v2
i(Au)X(i)(Au).(4.2)
If Ais a symmetry of Γ, following expression (2.9), we can write
Γ (u) = ATΓ (Au)A. (4.3)
Using this expression, we can rewrite equation (4.1) as
ATΓ (Au)AX(i)(u) = v2
i(u)Xi(u).
Since GΓGvand the velocities are ordered, it follows that vi(Au) = vi(u)and
ATΓ (Au)AX(i)(u) = v2
i(Au)X(i)(u).
9
Recalling the orthogonality condition, namely, ATA=I3, we multiply both sides of the
above equation by Ato get
Γ (Au)AX(i)(u) = v2
i(Au)AX(i)(u).(4.4)
Comparing this result with equation (4.2), we see that vectors AX(i)(u)and X(i)(Au)
belong to the same eigenspace. This observation motivates us to define the symmetry
group of polarizations, which we will denote by GX. Let us denote the linear eigenspace
corresponding to eigenvalue v2
i(u)by span X(i)(u). Thus, GXis composed of all
AO(3) such that
span X(i)(u)=ATspan X(i)(Au),uR3.(4.5)
In view of expression (4.5), equation (4.4) states that if Ais a symmetry of the Christoffel
matrix, it is also a symmetry of both the wavefront velocities and polarizations. In other
words, AGΓAGvGX.
To explain why the opposite is also true, namely, AGvGXAGΓ, we invoke
the eigendecomposition theorem. This theorem states that a diagonalizable matrix, such as
Γ, is uniquely determined by its eigenvalues and the corresponding eigenspaces. From this
theorem, it follows that if Ais a symmetry of the eigenvalues and the eigenspaces, then it
is also a symmetry of the matrix itself. In other words, AGvGXAGΓ.
Combining this result with the previous one, namely, AGΓAGvGX, we
conclude that AGvGXif and only if AGΓ. Since Gv=Gψand GΓ=Gc,
this conclusion means that the material-symmetry group is composed of all AO(3) that
belong to both the symmetry group of wavefronts and the symmetry group of polarizations.
5. DISCUSSION
As shown in Section 3.2, the material and wavefront symmetries are, in general, not
equivalent to one another. However, as shown in Section 4, we can determine the symmetry
class of a given medium if we consider both the wavefront and polarization symmetries.
We can explicitly state that for a linearly elastic continuum, the material symmetry group
is composed of all orthogonal transformations that belong to the symmetry groups of both
wavefronts and polarizations.
In Section 3.2, we have shown that orthotropic, tetragonal and cubic wavefronts can
propagate in less symmetric media. Also, more symmetric wavefronts can propagate in
monoclinic and tetragonal media. We have not found any example of trigonal, transversely
isotropic nor isotropic wavefronts that propagate in less symmetric media.
The results presented in this paper suggest further studies. First, one should under-
stand physical significance of cases in which the wavefronts are more symmetric than the
medium. Notably, one should understand why monoclinic wavefronts cannot propagate in
generally anisotropic media. Second, one should classify the wavefronts that propagate in
less symmetric media. In other words, one should classify these wavefronts based on their
special form that results from the particular relations among components of the tensor.
Also, one should establish a complete pattern describing media in which a given wavefront
can propagate as well as describing wavefronts that can propagate in a given medium. For
instance, as shown in Section 3.2, cubic wavefronts can propagate in cubic and tetragonal
media. However, we have not found cubic wavefronts propagating in monoclinic media —
even though the monoclinic group is a subgroup of the cubic, tetragonal and orthotropic
groups.
The results presented in this paper are yet another example of the principle stated over
a century ago by Pierre Curie [5]: “the symmetry elements of the causes must be found in
10
their effects, but the converse is not true; that is, the effects can be more symmetric than
the causes”.
ACKNOWL EDGEM ENT S
We thank Aron Murphy for his computer-algebra help during early stages of this work.
Also, we thank Michael G. Rochester for his comments on the draft of this paper.
This research was conducted within The Geomechanics Project, supported by EnCana
and Husky. The research of A.B. and M.A.S. was supported also by NSERC.
REFERENCES
[1] A. Bóna, I. Bucataru, M.A. Slawinski, Material symmetries of elasticity tensors. Q. Jl Mech. Appl. Math.
57, 4 (2004) 583 - 598.
[2] W.C. Van Buskirk, S.C. Cowin, R. Carter, Jr., A theory of acoustic measurement of the elastic constants of
a general anisotropic solid, J. Mater. Sci. 21 (1986) 2759-2762.
[3] P. Chadwick, M. Vianello, and S. C. Cowin, A new proof that the number of linear elastic symmetries is
eight. J. Mech. Phys. Solids 49 (2001) 2471-2492.
[4] S.C. Cowin and M.M. Mehrabadi, On the identification of material symmetry for anisotropic elastic mate-
rials. Quart. J. Mech. Appl. Math. 40 (1987) 451-476.
[5] P. Curie, Sur la symétrie dans les phénomènes physiques, symétrie d’un champ électrique et d’un champ
magnétique, J. de Phys. 3e série, 3, (1894) 393-415.
[6] F.I. Feodorov, Theory of Elastic Waves in Crystals. Plunum Press (1968).
[7] S. Forte and M. Vianello, Symmetry classes for elasticity tensors. J. Elasticity 43 (1996) 81-108.
[8] S. Forte and M. Vianello, Symmetry classes and harmonic decomposition for photoelasticity tensors. Inter-
nat. J. Engrg. Sci. 35 (1997) 14, 1317-1326.
[9] M.J.P. Musgrave, Crystal Acoustics. Holden-Day (1970).
[10] A.N. Norris, On the acoustic determination of the elastic moduli of anisotropic solids and acoustic conditions
for the existence of planes of symmetry. Q. Jl Mech. Appl. Math. 42, 4 (1989) 413- 426.
[11] M.A. Slawinski, Seismic Rays and Waves in Elastic Media. Pergamon (2003).
[12] T.C.T. Ting, Generalized Cowin-Mehrabadi theorems and a direct proof that the number of linear elastic
symmetries is eight. Internat. J. of Solids and Structures 40 (2003), 7129-7142.
[13] W. Voigt, Lehrbuch der Kristalphysik. Teubner (1928).
[14] H. Yong-Zhong and G. Del Piero, On the completeness of the crystallographic symmetries in the description
of the symmetries of the elastic tensor. J. Elasticity 25 (1991), 203-246.
11
... It is illustrated for wave phenomena by the fact that wavefronts propagating in a Hookean solid from a point source are at least as symmetric as the solid itself. In other words, the material symmetries of the elasticity tensor of that solid are a subgroup -possibly proper -of wavefront symmetries, as shown by Bóna et al. [1]. Herein, we prove that the elasticity tensor and the Christoffel matrix, from which wavefronts are derived, have the same symmetry groups, as suggested by Bóna et al. [1], but for reasons more subtle than presented therein. ...
... In other words, the material symmetries of the elasticity tensor of that solid are a subgroup -possibly proper -of wavefront symmetries, as shown by Bóna et al. [1]. Herein, we prove that the elasticity tensor and the Christoffel matrix, from which wavefronts are derived, have the same symmetry groups, as suggested by Bóna et al. [1], but for reasons more subtle than presented therein. Thus, it follows that the increase of symmetry may occur in passing from the Christoffel matrix to wavefronts, not in passing from the elasticity tensor to the Christoffel matrix. ...
... Each of the three eigenvalues corresponds to the velocity of one of the three types of waves that propagate in a Hookean solid, and each eigenvector to its polarization. As suggested by Bóna et al. [1], the increase of symmetries between the Hookean solid and the wavefronts propagating within it can occur only between the Christoffel matrix and the wavefronts, which is tantamount to the equality of the symmetry groups of the tensor and the matrix. This equality is proven in the next section. ...
Article
Full-text available
We prove that the symmetry group of an elasticity tensor is equal to the symmetry group of the corresponding Christoffel matrix. Comment: This note completes the argument of Bóna, A., Bucataru, I., Slawinski, M.A. (2007) Material symmetries versus wavefront symmetries. The Quarterly Journal of Mechanics and Applied Mathematics 60(2), 73-84
... Both minor symmetries, (5) and (6), are assumed to hold simultaneously. Accordingly, the tensor C ijkl can be represented by a 6 × 6 matrix with 36 independent components. ...
... Because of (5), (6), and (8), we have the following Definition: A 4th rank tensor of type 4 0 qualifies to describe anisotropic elasticity if (i) its physical components carry the dimension of force/area (in SI pascal), (ii) it obeys the left and right minor symmetries, (iii) and it obeys the major symmetry. It is then called elasticity tensor (or elasticity or stiffness) and, in general, denoted by C ijkl . ...
... We show that the longitudinal wave propagation is completely determined by the Cauchy part of the Christoffel tensor, see Proposition 15. In Proposition 16 a new result is presented on the propagation of purely polarized waves; we were led to these investigations by following up some ideas about the interrelationship of the symmetry of the elasticity tensor and the Christoffel tensor in the papers of Alshits and Lothe (2004) [1] and Bóna et al. (2004Bóna et al. ( , 2010 [5,6,7]. In Sec. 4, we investigate examples, namely isotropic media (Sec. ...
Article
Full-text available
In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor C. The decomposition of C into a partially symmetric tensor M and a partially antisymmetric tensors N is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part S of C plus the reminder A, turns out to be irreducible under the 3-dimensional general linear group. We show that the SA-decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The MN-decomposition fails to have these desirable properties and is such inferior from a physical point of view. Various applications of the SA-decomposition are discussed: the Cauchy relations (vanishing of A), the non-existence of elastic null Lagrangians, the decomposition of the elastic energy and of the acoustic wave propagation. The acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The Cauchy part governs the longitudinal wave propagation. We provide explicit examples of the effectiveness of the SA-decomposition. A complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions, similarly as in an isotropic medium.
... More recent discussions of the Cauchy relations can be found, e.g., in [1,3,4], or [7]. Different compact expressions of the Cauchy relations can be found in the literature. ...
... • material symmetries and wavefront symmetries [6,7]; ...
Article
Full-text available
We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.
... For each slowness surface, the criterion of belonging to a particular wave on either side of an intersection is not its belonging to a single root but the orientation of the corresponding eigenvectors, which are the displacement vectors of a given wave (Bóna et al., 2007b). As can be readily shown, for the innermost surface, the displacement vector is normal to it along the rotation-symmetry axis and in the plane perpendicular to it; hence, it corresponds to the qP wave. ...
Article
Full-text available
The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached qP slowness surfaces in transversely isotropic media. If the qP slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the qP slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the qP and qSV slowness surfaces are not smooth. This article is protected by copyright. All rights reserved
... For each slowness surface, the criterion of belonging to a particular wave on either side of an intersection is not its belonging to a single root but the orientation of the corresponding eigenvectors, which are the displacement vectors of a given wave (Bóna et al., 2007b). As can be readily shown, for the innermost surface, the displacement vector is normal to it along the rotation-symmetry axis and in the plane perpendicular to it; hence, it corresponds to the qP wave. ...
Preprint
Full-text available
The only restriction on the values of the elasticity parameters is the stability condition. Within this condition, we examine Christoffel equation for nondetached $qP$ slowness surfaces in transversely isotropic media. If the $qP$ slowness surface is detached, each root of the solubility condition corresponds to a distinct smooth wavefront. If the $qP$ slowness surface is nondetached, the roots are elliptical but do not correspond to distinct wavefronts; also, the $qP$ and $qSV$ slowness surfaces are not smooth.
... In other words, Hookean solids can be less symmetric than wavefronts within them. However, Bóna et al. [4] show that we can determine the symmetries of the solid if we combine the information about the symmetries of wavefronts and about symmetries of polarizations. ...
Article
Full-text available
Geophysics—similarly to astrophysics—relies on remote sensing. Inferring material properties of the Earth’s interior is akin to inferring the composition of a distant star. In both cases, scientists rely on matching theoretical predictions or explanations with observations. Notably, obtaining a sample of a material from the interior of our planet might not be less difficult than obtaining a sample from a distant celestial object. To infer the presence and orientations of subsurface fractures, seismologists might use directional properties of Hookean solids. In other words—using such a solid as a mathematical model— seismologists match its quantitative predictions with observations.
... To estimate the symmetry axis of a transversely isotropic medium, we use the fact that the wavefronts must be at least as symmetric as the material itself (Bóna et al., 2007). To find the symmetry axis, assuming this spherical shale sample has the transverse isotropy symmetry, we form an objective function ...
Article
Full-text available
Our aim is to understand the stress-dependent seismic anisotropy of the overburden shale in an oil field in the North West Shelf of Western Australia. We analyze data from measurements of ultrasonic P-wave velocities in 132 directions for confining pressures of 0.1-400 MPa on a spherical shale sample. First, we find the orientation of the symmetry axis, assuming that the sample is transversely isotropic, and then transform the ray velocities to the symmetry axis coordinates. We use two parameterizations of the phase velocity; one, in terms of the Thomsen anisotropy parameters alpha, beta, epsilon, delta as the main approach, and the other in terms of alpha, beta, eta, delta. We invert the ray velocities to estimate the anisotropy parameters alpha, epsilon, delta, and eta using a very fast simulated reannealing algorithm. Both approaches result in the same estimation for the anisotropy parameters but with different uncertainties. The main approach is robust but produces higher uncertainties, in particular for eta, whereas the alternative approach is unstable but gives lower uncertainties. These approaches are used to find the anisotropy parameters for the different confining pressures. The dependency of P-wave velocity, alpha, on pressure has exponential and linear components, which can be contributed to the compliant and stiff porosities. The exponential dependence at lower pressures up to 100 MPa corresponds to the closure of compliant pores and microcracks, whereas the linear dependence at higher pressures corresponds to contraction of the stiff pores. The anisotropy parameters epsilon and delta are quite large at lower pressures but decrease exponentially with pressure. For lower pressures up to 10 MPa, delta always is larger than epsilon; this trend is reversed for higher pressures. Despite the hydrostatic pressure, the symmetry axis orientation changes noticeably, in particular at lower pressures.
... To estimate the symmetry axis of a transversely isotropic medium, we use the fact that the wavefronts must be at least as symmetric as the material itself (Bóna et al., 2007). To find the symmetry axis, assuming this spherical shale sample has the transverse isotropy symmetry, we form an objective function ...
Article
Full-text available
The theory of first strain gradient elasticity (SGE) is widely used to model size and non-local effects observed in materials and structures. For a material whose microstructure is centrosymmetric, SGE is characterized by a sixth-order elastic tensor in addition to the classical fourth-order elastic tensor. Even though the matrix form of the sixth-order elastic tensor is well-known in the isotropic case, its complete matrix representations seem to remain unavailable in the anisotropic cases. In the present paper, the explicit matrix representations of the sixth-order elastic tensor are derived and given for all the 3D anisotropic cases in a compact and well-structured way. These matrix representations are necessary to the development and application of SGE for anisotropic materials
Article
Full-text available
The aim of this paper is to understand the seismic anisotropy of the overburden shale in an oilfield in the North West Shelf of Western Australia. To this end, we first find the orientation of the symmetry axis of a spherical shale sample from measurements of ultrasonic P-wave velocities in 132 directions at the reservoir pressure. After transforming the data to the symmetry axis coordinates, we find Thomsen's anisotropy parameters δ and ɛ using these measurements and measurements of the shear-wave velocity along the symmetry axis from a well log. To find these anisotropy parameters, we use a very fast simulated re-annealing algorithm with an objective function that contains only the measured ray velocities, their numerical derivatives and the unknown elasticity parameters. The results show strong elliptical anisotropy in the overburden shale. This approach produces smaller uncertainty of Thomsen parameter δ than more direct approaches.
Article
Two different definitions of symmetries for photoelasticity tensors are compared. Earlier for such symmetries the existence of exactly 12 classes was proved based on an equivalence relation induced on the set of subgroups of SO(3). Here, an another viewpoint is chosen, and photoelasticity tensors themselves are divided into symmetry classes, according to a different definition. By use of group-theoretical techniques, such as harmonic and Cartan decomposition, it is shown that this approach again leads to 12 classes.
Article
The twenty-one elastic moduli of a homogeneous anisotropic solid can be determined from the second-order acoustical tensors, associated with wave motion in six phase directions. The directions may be quite arbitrary as long as they cannot be contained by less than three distinct planes through the origin and do not all lie on the curves formed by the intersection of the unit sphere with an elliptical cone. Two equivalent sets of conditions necessary and sufficient for the existence of a plane of material symmetry in an elastic solid are presented. The conditions are phrased in terms of acoustic waves, the first set involving polarization vectors, the second energy-flux vectors. Some consequences of the acoustic conditions are noted.
Article
The Cowin–Mehrabadi theorem is generalized to allow less restrictive and more flexible conditions for locating a symmetry plane in an anisotropic elastic material. The generalized theorems are then employed to prove that the number of linear elastic symmetries is eight. The proof starts by imposing a symmetry plane to a triclinic material and, after new elastic symmetries are found, another symmetry plane is imposed. This process exhausts all possibility of elastic symmetries, and shows that there are only eight elastic symmetries. At each stage when a new symmetry plane is added, explicit results are obtained for the locations of the new symmetry plane that lead to a new elastic symmetry. It takes as few as three, and at most five, symmetry planes to reduce a triclinic material (which has no symmetry plane) to an isotropic material for which any plane is a symmetry plane.
Article
It is shown here that there are exactly eight different sets of symmetry planes that are admissible for an elasticity tensor. Each set can be seen as the generator of an associated group characterizing one of the traditional symmetry classes.
Article
An acoustic wave approach is presented for the measurement of the twenty-one independent elastic constants of the most general linearly elastic anisotropic solid. The method requires that one be able to measure the density of the material, the velocities of the three modes of wave propagation in each of six directions, and the particle displacements associated with each of those modes.
Article
It is shown that all symmetries possible for the elastic tensors can be reduced to the twelve symmetries already used in the description of the crystal classes. Each symmetry can be characterized by a group of rotations generated by no more than two rotations. The use of a canonical basis related to such rotations considerably simplifies the component forms of the elasticity tensor. This result applies to non-symmetric tensors; for symmetric tensors, the number of independent symmetries reduces from twelve to ten. After the present work was submitted, the following paper came to our attention: 14. S.C. Cowin and M.M. Mehrabadi, On the identification of material symmetry for anisotropic elastic materials. Q. Jl. Mech. appl. Math.40 (1987) 451–476. This paper contains an independent analysis of the partial ordering ≺ among the crystallographic elastic symmetries. However, it does not deal with the problem of the completeness of these symmetries.
Article
Harmonic and Cartan decompositions are used to prove that there are eight symmetry classes of elasticity tensors. Recent results in apparent contradiction with this conclusion are discussed in a short history of the problem.
Article
The problem considered here is that of identifying the type of elastic material symmetry of a material, given the values of the components of the fourth-rank elasticity tensor of the material relative to a known, but arbitrary, coordinate system. Four simple eigenvalue problems are posed for the determination of the normals to the planes of reflective material symmetry of the elastic material. The solution of the eigenvalue problems will determine the number and orientation of the normals to the planes of reflective material symmetry. This information is then used to determine the elastic material symmetry possessed by the material.