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On symmetries of elasticity tensors and Christoffel matrices

Authors:

Abstract

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
arXiv:1011.4975v2 [physics.geo-ph] 28 Nov 2010
ON SYMMETRIES OF ELASTICITY TENSORS AND
CHRISTOFFEL MATRICES
ANDREJ B
´
ONA, C¸ A
˘
GRI D
˙
INER, MIKHAIL KOCHETOV, AND MICHAEL A. SLAWINSKI
Abstract. We prove that the symmetry group of an elasticity tensor is equal
to the symmetry group of the corresponding Christoffel matrix.
1. Introduction
The Curie principle states that the results are a t least as symmetric a s the causes.
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 symmetr ie s of the e lasticity tensor of that solid are
a subgroup p ossibly proper of wavefront symmetries, as shown by ona et
al. [1]. Herein, we prove that the elasticity tensor and the Chris toffel matrix, from
which wavefronts are derived, have the same symmetry groups, as suggested by
ona et al. [1], but for reasons more subtle than pr e sented therein. Thus, it follows
that the increase of symmetry may occ ur in pas sing from the Christoffel matrix to
wavefronts, not in passing from the elasticity tensor to the Christoffel matrix. The
importance of this result is our being able to obtain information about ma terial
symmetries of a Hookean solid fro m measurements that are common in seismology,
which allow us to estimate the Christoffel matrix, not the elasticity tensor directly.
We beg in this paper with a brief des cription of relations between the elasticity
tensor and the Christoffel matrix and betwee n the matrix and the wavefronts. A
more detailed description can be found in many sources, nota bly,
ˇ
Cerven´y [3]; the
description below follows Bos and Slawinski [2].
2. Background
A Hookean solid is defined by the following constitutive equation:
(1) σ
ij
=
3
X
k,ℓ=1
c
ijkℓ
ε
kℓ
,
where c is the elasticity tensor relating the stress tensor, σ, and the strain tensor,
ε. As shown by Forte and Vianello [5], c belongs to one of eight symmetry groups;
the symmetry group of c is the set of orthogonal transformations under which c is
invariant. Also, c possesses index symmetries: c
ijkℓ
= c
jikℓ
= c
kℓij
.
Inserting constitutive equation (1) into the equations of motion,
(2)
3
X
j=1
σ
ij
x
j
= ρ
2
u
i
t
2
,
2000 Mathematics Subject Classification. P rimary 74E10, secondary 74Q15, 86A15, 86A22.
Key words and phrases. anisotropy; Chris toffel matrix; elasticity tensor; H ookean solid.
1
2 A. B
´
ONA, C¸ . D
˙
INER, M. KOCHETOV, AND M. A. SLAWINSKI
where i {1, 2, 3}, ρ is the mass density, x and t are the spatial and temporal
variable s, respectively, and u is the displacement vector whose components are
related to the strain tensor by ε
ij
:= (u
i
/∂x
j
+u
j
/∂x
i
)/2, we obtain the equations
of motion in a Hookean solid,
(3) ρ (x)
2
u
i
(x, t)
t
2
=
3
X
j,k,ℓ=1
c
ijkℓ
(x)
x
j
u
k
(x, t)
x
+ c
ijkℓ
(x)
2
u
k
(x, t)
x
j
x
,
where i {1, 2 , 3}, which describe propagation of displacement in an anisotropic
inhomogeneous elastic medium. Wavefronts and polarizations of waves pr opagating
in such a medium stem fro m the characteristic equation of these equa tions, which
is
(4) det
3
X
j,ℓ=1
c
ijkℓ
n
j
n
ρ
v
2
δ
ik
= 0, i, k {1, 2, 3},
where n is the unit vector normal to the wavefront and v is the wavefront velocity.
It is common to let Γ
ik
(n) := c
ijkℓ
n
j
n
, which is the Christoffel matrix. The
symmetry group of the Chris toffel matrix is the set of orthogonal transformations ,
A, such that Γ(Au)(Av, Aw ) = Γ(u)(v, w), for all u , v, w in R
3
. Note that Γ is
quadratic in the first variable, u; also note that u and v herein and in the next
section ar e gene ric variables , not the physical entities of equations (2), (3) and
(4). Also, the Christoffel matrix possesses index symmetries, Γ
ij
= Γ
ji
, which ar e
inherited from the index s ymmetries of c. Equation (4) is a third-degree polynomial
in 1/v
2
. Its root are the eigenvalues of Γ sc aled by ρ. E ach of the three eigenvalues
corres ponds to the velocity of one of the three types of waves that propagate in a
Hookean solid, and each e igenvec tor to its polarization.
As suggested by ona et al. [1], the increase of symmetries b etween the Hookean
solid and the wavefronts propagating within it can o ccur only between the Christof-
fel matrix and the wave fronts, which is tantamount to the equality of the symmetry
groups of the tensor and the matrix. This equality is proven in the nex t section.
The proof is based on the fact that the Christoffel matrix determines uniquely the
elasticity tensor, as shown by ona et al. [1], and also on the fact that this one-to-
one corres po ndenc e respects the action of O(3 ), which guarantees the equality of
the symmetry groups.
3. Equality of symmetry groups
Explicitly, the correspondence between c and Γ is
(5) c
ikiℓ
= Γ(e
i
)(e
k
, e
)
and
(6) c
iijℓ
=
1
2
[Γ(e
i
+ e
j
)(e
i
, e
) Γ(e
i
e
j
)(e
i
, e
) 2Γ(e
i
)(e
j
, e
)] ,
which are equations (3.7) and (3.8) in ona et al. [1].
For a given Γ, these equatio ns allow us to determine the components of c due to
repetitions among i, j, k, , but they have different forms for differe nt components.
Thus, it is not obvious that the relatio n between c and Γ respects the action of
O(3). To establish this property, we restate this rela tion in a coordinate-fre e form.
We can r e gard c as a quadrilinea r form, c : R
3
× R
3
× R
3
× R
3
R
3
, and Γ as a
ON SYMMETRIES OF ELASTICITY TENSORS AND CHRISTOFFEL MATRICES 3
function, Γ : R
3
×R
3
×R
3
R
3
, which is quadratic in the firs t variable and linear
in the other two. Let us linearize Γ by defining
(7)
˜
Γ(x, y, v, w) :=
1
2
[Γ(x + y)(v, w) Γ(x)(v, w) Γ(y)(v, w)] ,
for all x, y, v, w R
3
. Then Γ can be recovered from
˜
Γ as follows:
(8) Γ(u)(v, w) =
˜
Γ(u, u, v, w).
Theorem. For all x, y, v, w R
3
, we have
c(x, y, v, w) =
˜
Γ(x, v, y, w) +
˜
Γ(y, v, x, w)
˜
Γ(x, y, v, w);(9)
˜
Γ(x, y, v, w) =
1
2
[c(x, v, y, w ) + c(y, v, x, w)].(10)
Proof. Equatio n (7) c an be restated as Γ(x + y)(v, w) = Γ(x)(v, w) + Γ(y)(v, w) +
2
˜
Γ(x, y, v, w). Hence we can rewrite formula (6) as
(11) c(e
i
, e
i
, e
j
, e
) = 2
˜
Γ(e
i
, e
j
, e
i
, e
) Γ(e
i
)(e
j
, e
).
In view of the linearity of c and
˜
Γ in each variable and the linearity of Γ in the
second and third variables, we obtain:
(12) c(e
i
, e
i
, v, w) = 2
˜
Γ(e
i
, v, e
i
, w) Γ(e
i
)(v, w),
for all v, w R
3
. Now suppose k 6= i. Then we get:
(13) c(e
k
, e
k
, v, w) = 2
˜
Γ(e
k
, v, e
k
, w) Γ(e
k
)(v, w),
and, replacing {e
i
, e
k
} by {(e
i
+ e
k
)/
2, (e
i
e
k
)/
2} in the orthonormal basis,
we also get
(14) c(e
i
+ e
k
, e
i
+ e
k
, v, w) = 2
˜
Γ(e
i
+ e
k
, v, e
i
+ e
k
, w) Γ(e
i
+ e
k
)(v, w).
Subtracting equations (12) and (13) fr om equation (14) and using the index sym-
metry of c, we obtain
(15) c(e
i
, e
k
, v, w) =
˜
Γ(e
i
, v, e
k
, w) +
˜
Γ(e
k
, v, e
i
, w)
˜
Γ(e
i
, e
k
, v, w).
Equations (12) and (15) imply equation (9) in view of the linearity of both sides of
equation (9).
Note that by substituting x = v = e
i
, y = e
k
and w = e
in equation (9), we
obtain expression (5), and by substituting x = y = e
i
, v = e
j
and w = e
, we obtain
expression (11), which is an equivalent form of expression (6).
Finally, equation (10) follows immediately from definition (7) and equation (5).
The mappings
˜
Γ 7→ c and c 7→
˜
Γ given by expressions (9) and (10), respectively,
are linear isomorphisms that are inverses of one another. Clearly, these mapping s
respect the action of O(3).
Corollary. The symmetries of the elasticity tensor, c, are the same as the symme-
tries of the Christoffel matr ix, Γ .
Proof. Any symmetry of Γ is a symmetry of
˜
Γ a nd vice versa by equations (7) and
(8), respectively. It remains to invo ke the isomorphisms (9) and (10) between
˜
Γ
and c.
4 A. B
´
ONA, C¸ . D
˙
INER, M. KOCHETOV, AND M. A. SLAWINSKI
4. Conclusions
As stated by the corollary, the material symmetries of the elasticity tensor, c,
are indeed the same as the symmetries of the Christoffel matrix, Γ, as sugessted
by ona et al. [1]. This means that the increase of symmetry occurs between the
Christoffel matrix and the wavefronts, not between the elasticity tensor and the
Christoffel matrix. As shown by ona et al. [1], this result entails tha t the sym-
metry group of c is the intersection of the symmetry groups of wavefronts and
polarizations. Otherwise if the increase of symmetry could occur between c and
Γ the knowledge of the wavefront and p olarization symmetries would be insuf-
ficient to infer the symmetry of the Hookean solid. In mathematical language, the
equality of symmetry groups of c and Γ ensures that the knowledge of e igenvalues
and eigenvectors, which allows us to reconstruct Γ, entails the knowledge of the
symmetries o f c. Furthermore, measure ments might result in computing Γ without
considering wavefronts and polarizations, as shown by Dewangan and Grechka [4].
According to the corollary, the material symmetries of the Hookean solid can be
obtained directly from Γ.
Finally, the formulæ r elating c and Γ that are stated in the theorem are of
interest, since unlike equations (3.7) and (3.8) in ona et al. [1] they have
the same form for all components of c and Γ.
Acknowledgements
M. Kochetov’s and M.A. Slawinski’s research was supported by the Na tural
Sciences and Engineering Resear ch Council of Canada.
References
[1] ona, A., Bucataru, I., Slawinski, M.A. (2007) Material symmetries versus wavefr ont symme-
tries. The Quarterly Journal of Mechanics and Applied Mathematics 60(2), 73–84
[2] Bos, L., Slawinski, M.A. (2010) Elastodynamic equations: Characteristics, wavefronts and
rays. The Quarterly Journal of Mechanics and Applied Mathematics 63(1), 23–37
[3]
ˇ
Cervey, V. (2001). Se ismic ray theory. Cambridge Universi ty Press.
[4] D ewangan, P. and Grechka, V. (2003) Inversion of multicomponent, multiazimuth walkaway
VSP data for the stiffness tensor, Geophysics, 68, 1022-1031
[5] Forte, S., Vianello, M . (1996) Symmetry Class es for Elasticity Tensors: Journal of Elasticity
43(2), 81–108
Department of Exploration Geophysics, Curtin Technical University, Perth, Aus-
tralia
E-mail address: a.bona@curtin.edu.au
Geophysics Department, Kandilli Observatory and Earthquake Research Institute,
Bo
˘
gazic¸i University, Istanbul, Turkey
E-mail address: cadiner@yahoo.com
Department of Mathematics and Statistics, Memorial University of Newfoundland,
St. John’s, Newfoundland, Canada
E-mail address: mikhail@mun.ca
Department of Earth Sciences, Me morial University of Newfoundand, St. John’s,
Newfoundland, Canada
E-mail address: mslawins@mun.ca
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