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Abstract

In the physical realm, an elasticity tensor that is computed based on measured numerical quantities with resulting numerical errors does not belong to any symmetry class for two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry classes in question are properties of Hookean solids, which are mathematical objects, not measured physical materials. To consider a good symmetric model for the mechanical properties of such a material, it is useful to compute the distance between the measured tensor and the symmetry class in question. One must then of course decide on what norm to use to measure this distance. The simplest case is that of the isotropic symmetry class. Typically, in this case, it has been common to use the Frobenius norm, as there is then an analytic expression for the closest element and it is unique. However, for other symmetry classes this is no longer the case: there are no analytic formulas and the closest element is not known to be unique. Also, the Frobenius norm treats an n×n matrix as an n 2-vector and makes no use of the matrices, or tensors, as linear operators; hence, it loses potentially important geometric information. In this paper, we investigate the use of an operator norm of the tensor, which turns out to be the operator Euclidean norm of the 6×6 matrix representation of the tensor, in the expectation that it is more closely connected to the underlying geometry. We characterize the isotropic tensors that are closest to a given anisotropic tensor, and show that in certain circumstances they may not be unique. Although this may be a computational disadvantage in comparison to the use of the Frobenius norm—which has analytic expressions—we suggest that, since we work with only 6×6 matrices, there is no need to be extremely efficient and, hence, geometrical fidelity must trump computational considerations.
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Journal of Elasticity
The Physical and Mathematical Science
of Solids
ISSN 0374-3535
Volume 120
Number 1
J Elast (2015) 120:1-22
DOI 10.1007/s10659-014-9497-y
2-Norm Effective Isotropic Hookean Solids
Len Bos & Michael A.Slawinski
1 23
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J Elast (2015) 120:1–22
DOI 10.1007/s10659-014-9497-y
2-Norm Effective Isotropic Hookean Solids
Len Bos ·Michael A. Slawinski
Received: 9 January 2014 / Published online: 16 September 2014
© Springer Science+Business Media Dordrecht 2014
Abstract In the physical realm, an elasticity tensor that is computed based on measured nu-
merical quantities with resulting numerical errors does not belong to any symmetry class for
two reasons: (1) the presence of errors, and more intrinsically, (2) the fact that the symmetry
classes in question are properties of Hookean solids, which are mathematical objects, not
measured physical materials. To consider a good symmetric model for the mechanical prop-
erties of such a material, it is useful to compute the distance between the measured tensor
and the symmetry class in question. One must then of course decide on what norm to use to
measure this distance.
The simplest case is that of the isotropic symmetry class. Typically, in this case, it has
been common to use the Frobenius norm, as there is then an analytic expression for the
closest element and it is unique. However, for other symmetry classes this is no longer the
case: there are no analytic formulas and the closest element is not known to be unique.
Also, the Frobenius norm treats an n×nmatrix as an n2-vector and makes no use of
the matrices, or tensors, as linear operators; hence, it loses potentially important geometric
information. In this paper, we investigate the use of an operator norm of the tensor, which
turns out to be the operator Euclidean norm of the 6 ×6 matrix representation of the tensor,
in the expectation that it is more closely connected to the underlying geometry.
We characterize the isotropic tensors that are closest to a given anisotropic tensor, and
show that in certain circumstances they may not be unique. Although this may be a compu-
tational disadvantage in comparison to the use of the Frobenius norm—which has analytic
expressions—we suggest that, since we work with only 6 ×6 matrices, there is no need to
be extremely efficient and, hence, geometrical fidelity must trump computational consider-
ations.
L. Bos
Dipartimento di Informatica, Università di Verona, Verona, Italy
e-mail: leonardpeter.bos@univr.it
M.A. Slawinski (B)
Department of Earth Sciences, Memorial University of Newfoundland, St. John’s, Newfoundland,
Canada
e-mail: mslawins@mun.ca
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2L. Bos, M.A. Slawinski
Keywords Elasticity tensor ·Effective Hookean solid ·Anisotropy ·Isotropy ·2-Norm ·
Frobenius norm
Mathematics Subject Classification 86A15 ·86-08 ·74B04
1 Introduction
The concept of a Hookean solid emerges from Hooke’s law, which in three dimensions is
σij =
3
k,=1
cij kl εk ,i,j∈{1,2,3},(1)
where σij ,εk and cij k are the components of the stress, strain and elasticity tensors, respec-
tively. The Hookean solid is a mathematical object defined by cij k and the mass density.
These solids are used commonly as analogies for physical materials in quantitative analysis
within a large scope of subjects such as engineering and seismology. They belong to the
theory of elasticity within continuum mechanics.
Among properties of Hookean solids that provide fruitful analogies for physical materials
is their anisotropy, which herein is the directional dependence of mechanical properties. As
shown by Forte and Vianello [12], Chadwick et al. [7], and Bóna et al. [5], there are eight
material-symmetry classes of cij k . Six among these classes form a partial ordering between
general anisotropy and isotropy; the former is described by twenty-one parameters, the latter
by two. The concept of distance of a given elasticity tensor to a material-symmetry class was
introduced by Gazis et al. [13] using orthogonal projections of elasticity tensors on the space
of a particular class, and by Fedorov [11], who minimized the mean-square difference of the
slowness surface.
Let us comment on the concept of symmetry, which is ubiquitous in mathematical
physics, and, in particular, on material symmetry, which allows us to consider certain analo-
gies between symmetries of continua and symmetries of crystals. The study of the latter
has been a fruitful platform for the understanding of the former and vice versa.However,we
must be aware of differences between these two types of symmetry. Symmetry of continua is
an invariance of tensors under orthogonal transformations; notably, it can contain continuous
groups. Symmetry of crystals is a lattice symmetry, which is restricted to discrete groups.
Hence, the symmetry classes are not the same for continua and for crystals, even though
there are similarities that can be used heuristically as geometrical illustrations. Also, the
relation of physical crystals to crystal symmetries is more straightforward than the relation
between noncrystalline materials to symmetries of Hookean solids. For instance, the cubic
symmetry of a rock-salt crystal is a close physical counterpart of the geometrical concept
of cubic symmetry; the transverse isotropy of a layer of shale, on the other hand, refers to
mechanical properties and has to be mediated by a mathematical concept of a Hookean solid
and symmetries of its elasticity tensor. This meditation leads to the consideration of models
and their accuracy. While we can say that rock salt exhibits cubic symmetry, we can only
say that the pattern of mechanical properties of a layer of shale is directionally akin to the
behaviour of a transversely isotropic Hookean solid. Furthermore, we might find that, in cer-
tain cases, these properties are sufficiently well described by a simpler model of directional
independence: an isotropic Hookean solid. A quantitative approach to approximate gener-
ally anisotropic Hookean solids by isotropy was introduced—more than a century ago—by
Voi gt [ 23]; he used the Frobenius norm as a measure of proximity.
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2-Norm Effective Isotropic Hookean Solids 3
Also, experimentalists and researchers in applied disciplines have been aware of the need
for quantifying the concept of similarity among models of different simplicity. An insightful
approach to proximity of isotropy results from the work of Thomsen [22] who expressed the
elasticity parameters of a transversely isotropic Hookean solid in such a manner that three
among them reduce to zero for the isotropic case; hence, their values much smaller than
unity indicate weak anisotropy. Arts et al. [1] consider the distance between the tensor and
its symmetric counterpart as the objective function. However, they do not attempt to find
the best fit. In their own words, this average as the choice of orientation “is not based on
rigorous mathematics”. Indeed, such an approach generally does not minimize the distance
between a tensor and its symmetric counterpart.
In general, since Hookean solids are mathematical analogies for physical materials, we
might choose to represent a material by a solid exhibiting a particular symmetry. Sev-
eral researchers—among them, Gazis et al. [13], Moakher and Norris [18], Kochetov and
Slawinski [16,17], Danek et al. [9]—examined relations between a generally anisotropic
Hookean solid and its symmetric counterparts, which invoke the concept of distance within
the space of Hookean solids. Voigt [23] and Norris [19] examined, in particular, relations be-
tween a generally anisotropic solid and its most symmetric counterpart: the closest isotropic
solid. Contrary to other cases, relations to isotropy possess several distinct properties. They
are orientation-independent; they result in models with the fewest parameters; in certain
cases, they allow for analytical expressions. Since, in general, the concept of closeness en-
tails the concept of a norm, Norris [19] discusses several norms used in obtaining the closest
isotropic solid. In this paper, we examine another norm. To facilitate our discussion, we
begin with a description of notation to be used within this paper.
2 Notation
The stress tensor is naturally represented by the symmetric matrix [σij ]1i,j3R3×3or else
as a vector σ:= [σ11
22
33
23
13
12]tR6. One can translate from one form to the
other by the basis expansion
σ=σ11
100
000
000
+σ22
000
010
000
+σ33
000
000
001
+σ23
000
001
010
+σ13
001
000
100
+σ12
010
100
000
.
However, it is notationally convenient—as discussed, for instance, by Bóna et al. [3]—to
use a slightly different basis of matrices and to write
σσ11
100
000
000
σ22
000
010
000
σ33
000
000
001
σ23
1
2
000
001
010
σ13
1
2
001
000
100
σ12
1
2
010
100
000
,
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4L. Bos, M.A. Slawinski
where the definition of the ˆσij is clear from the context. It is this latter expansion that we use
in the sequel, dropping, with a forgivable abuse of notation, the hats.
Since cij k satisfies index symmetries, cijk  =cjik=ckij , and is positive-definite, we
can represent it by a 6 ×6 positive-definite symmetric matrix; as discussed, for instance, by
Chapman [8] and by Bóna et al. [3], we write
C=
c1111 c1122 c1133 2c1123 2c1113 2c1112
c1122 c2222 c2233 2c2223 2c2213 2c2212
c1133 c1133 c3333 2c3323 2c3313 2c3312
2c1123 2c2223 2c3323 2c2323 2c2313 2c2312
2c1113 2c2213 2c3313 2c2313 2c1313 2c1312
2c1112 2c2212 2c3312 2c2312 2c1312 2c1212
.(2)
A rotation of cij k ,givenbyASO(3), results in C=˜
AtC˜
A,whereCis the elasticity
tensor in the rotated coordinate system, tdenotes transpose and ˜
ASO(6)is obtained from
ASO(3)as follows (e.g., Bóna et al. [3]),
˜
A=
A2
11 A2
12 A2
13 2A12A13
A2
21 A2
22 A2
23 2A22A23
A2
31 A2
32 A2
33 2A32A33
2A21A31 2A22 A32 2A23A33 A22 A33 +A23A32
2A11A31 2A12 A32 2A13A33 A12 A33 +A13A32
2A11A21 2A12 A22 2A13A23 A12 A23 +A13A22
2A11A13 2A11 A12
2A21A23 2A21 A22
2A31A33 2A31 A32
A21A33 +A23 A31 A21A32 +A22 A31
A11A33 +A13 A31 A11A32 +A12 A31
A11A23 +A13 A21 A11A22 +A12 A21
.(3)
Cexhibits a material symmetry if it is invariant under all Ain a certain subgroup G
of SO(3).Inotherwords,
C=˜
AtC˜
A, AG. (4)
It is important to note that {˜
A:ASO(3)}is a strict subgroup of SO(6);inotherwords,
Cis not required to be invariant under all orthogonal transformations in R6.IfCis invariant
under all ASO(3), then it is isotropic; its form is
Ciso =
c1111 c1111 2c2323 c1111 2c2323 000
c1111 2c2323 c1111 c1111 2c2323 000
c1111 2c2323 c1111 2c2323 c1111 000
0002c2323 00
00002c2323 0
000002c2323
.(5)
Which norm is best to use to measure the closeness of a tensor to a symmetry class is
not very well understood. There are multiple properties to take in to consideration. Ease
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2-Norm Effective Isotropic Hookean Solids 5
of computation and uniqueness of the closest element are two, but it can also be impor-
tant to use a norm that is as consistent as possible with the geometry of the problem. For
an analogy, consider the problem of finding the closest point to the origin in R2from the
“class” {(x, y ) R2:x, y 0,x2+y2=1}(a quarter-circle). If we use the maximum norm,
(x, y ):= max |x|,|y|then the closest point is uniquely (2/2,2/2). On the other
hand, with respect to the usual Euclidean norm all points in the class are equally distant and
hence there is even a continuum of closest elements. Nevertheless, the Euclidean norm is
better adapted to this problem as it consistent with the geometry, and, indeed, the lack of
unicity actually gives important information that is lost by the use of another norm. Com-
putational efficiency can be important, especially for large-scale problems. However, in this
case, we are dealing with 6 ×6 matrices, which are matrices of a limited dimension, and
hence, we are of the opinion that computational efficiency, although a virtue, should not be
a crucial consideration.
As mentioned above, Voigt [23] and Norris [19] examined relations between Cand Ciso .
The former derives analytical expressions for the closest isotropic tensor using the Frobe-
nius norm, which is tantamount to orthogonal projection in the space of elasticity tensors.
The latter examines the closest isotropic tensors in terms of the Frobenius, logarithmic and
Riemannian norms.
These norms have been introduced also for their computational convenience. In this paper
we investigate the use of a different norm: the operator 2-norm, which is naturally associated
with the underlying geometry of the problem, even though it is less convenient computation-
ally. We characterize the closest isotropic tensor to a given measured tensor and show that
nonuniqueness is possible. We also give numerical examples comparing the closest isotropic
tensors of the operator 2-norm and of the Frobenius norm.
3Norms
The Frobenius norm
AF:=n
i,j=1
A2
ij 1/2
treats a matrix in Rn×nas a Euclidean vector in Rn2and is often the computationally most
convenient norm to use. However, its use ignores the fact that a matrix is a representation of
a linear map from Rnto Rnand that the associated operator properties of the matrix may be
important for the underlying physics. It is this possibility that we wish to study in this paper.
Given a norm ·on Rnthe associated operator norm of a matrix ARn×nis defined
to be
A:=max
x=1Ax.
It may be interpreted as the “maximum expansion factor” of the mapping A:RnRnand
is an intimate property of the matrix as an operator with its domain and range linear spaces
normed by ·.
The basic example of such norms is the Euclidean operator norm,
A2:= max
x2=1Ax2,
also known as the spectral norm since—for A=At—it turns out that
A2:=max|λ|:λan eigenvalue of A.(6)
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6L. Bos, M.A. Slawinski
Indeed, for reasons that follow, this seems to be a natural choice for the norm in the space
of elasticity tensors. In view of Eq. (1), the elasticity tensor represents a linear mapping
between the strain tensor, whose components can be expressed as a symmetric 3 ×3ma-
trix, [εk], and the stress tensor, whose components can be expressed as a symmetric 3 ×3
matrix, [σij ]. Expressing A=AtR3×3as a vector in R6with distinct elements, Aij , with
ji, and equipping these vectors with the usual Euclidean norm results in what may be
thought of as the symmetric Frobenius norm on A, namely, A2
SF :=1ij3A2
ij .
The operator norm of the elasticity tensor considered as a mapping from R3×3to R3×3,
both equipped with this symmetric Frobenius norm, is precisely the operator 2-norm of the
matrix CR6×6.
We remark that, more generally, we may consider a matrix norm that is invariant under
orthogonal transformations, which is a norm, , such that for ARn×n,
UAV=A,
for all orthogonal matrices U,V Rn×n, and, hence, is invariant under the embedding of
SO(3)in SO(6), which is the operation stated by Eq. (3). Examples of such orthogonally
invariant norms are the Ky Fan k-norms and the Schatten p-norms. The Frobenius norm
is a special case of these norms; for a detailed discussion, the reader might refer to the
monograph by Bhatia [2, Chap. 4].
In this paper, we investigate the 2-norm for measuring the distance between a generally
anisotropic tensor represented by CR6×6, and expressed in matrix (2), and the space of
isotropic tensors, expressed in matrix (5).
4 Distance between Cand Ciso
Ciso, given in matrix (5), can be written as
Ciso =(c1111 2c2323)
111000
111000
111000
000000
000000
000000
+2c2323
100000
010000
001000
000100
000010
000001
,
where the first matrix is the rank-one matrix, 3vvt,andvR6is the unit vector,
v=1
3[1,1,1,0,0,0]t.(7)
The second matrix is the identity, IR6×6. Hence, we write
Ciso =αvvt+βI, (8)
where
α:=3(c1111 2c2323)and β=2c2323 .(9)
It is useful and informative to generalize this result in ndimensions. For a unit vector,
vRn, we define the subspace of matrices:
Ciso :=ARn×n:A=αvvt+βInR.(10)
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2-Norm Effective Isotropic Hookean Solids 7
Now, for CRn×n, which is a symmetric matrix, we discuss the problem of minimizing the
2-norm distance between Cand the space Ciso; using expression (8), we write
min
α,βR
Cαvvt+βI
2.(11)
Since Cis symmetric, it is orthogonally diagonalizable; in other words, there is an or-
thogonal matrix, PRn×n, and a diagonal matrix, D=diag(d1,d
2,...,d
n), such that
C=PtDP . Moreover, we can assume that
d1d2≥···dn.(12)
Hence, we write
Cαvvt+βI
2=
PtDP αvvt+βI
2
=
PtDPαvvt+βIPtP
2
=
DPαvvt+βIPt
2
=
Dα(P v)(P v )t+βI
2
=
Dαuut+βI
2,(13)
where u:=Pv is also a unit vector. Comparing the left-hand side with the last expression
on the right-hand side, we see that—in this problem—C=D, which is a diagonal matrix.
Thus, problem (11) is reduced to
min
α,βR
Dαuut+βI
2.(14)
To solve this minimization problem, we first fix α. The eigenvalues of
Mα:=Dαuut,(15)
which is a symmetric matrix, play an important role. We denote them by λj(α),j=1,...,n,
and order them in such a manner that λ1(α) λ2) ≥···λn). Even though the eigen-
values are functions of α, their ordering means that, in general, they are not analytic func-
tions of α. However, by the Rellich [20, p. 39] analytic selection principle, the set of eigen-
values of Mαcan be given by {μ1(α), μ2(α ), . . . , μn(α)},whereμj(α) are real analytic
functions of α, defined such that μj(0)=dj. These analytic curves may cross and hence
their ordering changes; hence, it follows, then that λj(α) are piecewise analytic. Let us ex-
emplify such curves.
Example 1 Let D=diag(2,1,0)and u=[1,1,0]/2. Hence,
Mα=
22αα
20
α
21α
20
000
,
and its characteristic polynomial is
p(λ) =λλ2(3α)λ +23
2α.
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8L. Bos, M.A. Slawinski
The eigenvalue curves are
μ1(α) =3α
2+1
21+α2,
μ2(α) =3α
21
21+α2,
μ3(α) =0.
The largest eigenvalue, λ1(α) =μ1(α), is an analytic function. However, μ2(α) and μ3)
cross at α=4/3; hence,
λ2(α) =μ2(α) if α4
3,
μ3(α) if α4
3,
is piecewise analytic.
Proposition 1 For fixed α,
min
βRMαβI2=λ1) λn(α)
2,
with a unique optimal value of β,
β=λ1(α) +λn)
2.(16)
Proof MαβI is a symmetric matrix; hence, its 2-norm is given by its largest eigenvalue
in absolute value,
MαβI2=max
1jnλj(α) β.
First, note that, by the ordering of λj,
max
1jnλj(α) β=maxλ1(α) β,λn) β.
Indeed, for any 2 jn1, it follows—by definition—that λn) λj(α) λ1).
Hence, if βλj(α), which also means that βλn ),then
λj(α) β=βλj(α) βλn(α) =λn(α) β
and, if βλj(α), which also means that βλ1 ),then
λj(α) β=λj(α) βλ1(α) β=λ1(α) β.
Secondly, note that
maxλ1(α) β,λn(α) β=λ1(α) β, β β,
βλn(α), β β,
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2-Norm Effective Isotropic Hookean Solids 9
Fig. 1 Illustration of
Proposition 1:|λnβ|,|λ1β|
and their maxima
with βgiven by expression (16). This is easy to verify, but is obvious if one examines
Fig. 1. Hence,
MαβI2=λ1) β, β β,
βλn(α), β β,
with unique minimum value, β=β,forwhichMαβI2=λ1(α) β=
1(α) λn))/2, as claimed.
λ1λnis known as the spread of a matrix. Thus, we write
min
βRMαβI2=1
2S(Mα),
where
S(Mα):=λ1(α ) λn(α) (17)
is the spread of Mα. Substituting the optimal value, β=β=β(α), we reduce problem (14)
to
min
αRS(Mα). (18)
Proposition 2 S(Mα)is a piecewise analytic convex function of α.
Proof That the spread is piecewise analytic follows directly from the fact that both λ1(α)
and λn(α) are piecewise analytic. The convexity follows from well-known classical con-
siderations but, for the sake of completeness, we include the simple arguments. First note
that
λ1(α) =max
x2=1xtDαuutx
=max
x2=1xtDx αxtuutx
is the maximum of a compact family of lines and hence convex. Similarly,
λn(α) =min
x2=1xtDαuutx
=min
x2=1xtDx αxtuutx
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10 L. Bos, M.A. Slawinski
is the minimum of a compact family of lines and, hence, concave. It follows that λn(α)
is convex. Since the sum of two convex functions is convex, the spread, S(Mα)=λ1(α) +
(λn(α)), is also convex.
However, the spread need not be strictly convex. Indeed, the minimizing argument need
not be unique.
Example 2 Consider D=diag(3,3,3,1,1,1)R6×6and u=[1,0,...,0]t.ThenMα=
diag(3α, 3,3,1,1,1)and its eigenvalues are 3 α, 3,3,1,1,1. Hence,
λ1(α) =max{3α, 3}
=3α, α 0,
30,
and
λ6(α) =min{3α, 1}
=12,
3α, α 2.
Therefore, the spread is
s(α) :=S(Mα)=λ1(α) λ6(α)
=
2α, α 0,
2,0α2,
α, α 2.
Clearly, s(α) is minimized for any α∈[0,2].
It is worth studying the eigenvalues and the spread in more detail, which we now proceed
to do.
5 Investigation of λj(α) and s(α)
Since Mαis a rank-one perturbation of a diagonal matrix, there are general inequalities
between its eigenvalues, λj(α), and the eigenvalues of D, namely, dj. Specifically, it follows
from Wilkinson’s monograph [24, pp. 98 and 103] (see also Golub [14, p. 325]) that
(i) if α0,
d1λ1(α) d1α,
djλj(α) dj1,j=2,...,n.
(ii) if α0,
dj+1λj(α) dj,j=1,...,n1,
dnαλn(α) dn.(19)
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2-Norm Effective Isotropic Hookean Solids 11
We may use these to prove the following lemma.
Lemma 1 We hav e
lim
α→−∞S(Mα)=+,
lim
α→+∞S(Mα)=+.
Proof First, note that the trace,
n
j=1
λj(α) =tr(Mα)=trDαuut=n
j=1
djα.
Hence, if α0—using the fact that λj(α) dj1for j2—we write
λ1(α) =tr(Mα)
n
j=2
λj(α)
=n
j=1
djαn
j=2
λj(α)
n
j=1
djαn
j=2
dj1
=dnα.
Consequently, again for α0, the spread satisfies
S(Mα)=λ1(α) λn(α) (dnα) dn1=(dndn1)α,
which tends to +∞ as α→−.
Similarly, if α0—using the fact that then λj(α) dj+1for jn1—we write
λn(α) =n1
j=1
λj(α)tr(Mα)
=n1
j=1
λj(α)
j=1
dj+α
n1
j=1
dj+1n
j=1
dj+α
=−d1+α.
Consequently, for α0, the spread satisfies
S(Mα)=λ1(α) λn(α) d2+(d1+α) =(d2d1)+α,
which tends to +∞ as α→+.
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12 L. Bos, M.A. Slawinski
Since s(α) =S(Mα)→∞as α→±and is a convex function of α, a minimum al-
ways exists. This minimum is nonunique if and only if there is a flat spot, which means that
there is a nontrivial interval, [a,b], such that s(α) =λ1(α) λn(α) is constant on [a, b].
In particular, s(α) is not strictly convex on [a,b],sinces(α) =0 on that interval. By re-
stricting to a subinterval, if necessary, we may assume that both λ1(α) and λn(α) are ana-
lytic functions on [a,b]. Then, since λ1and λnare both convex functions, λ
1(α) 0and
λ
n(α) 0sothat,forα∈[a, b],
0=s) =λ
1(α) +λ
n(α)
means that we must have
λ
1(α) =0=λ
n(α).
Now, on [a,b]—for some j,kλ1) =μj(α) and λn) =μk(α), are both analytic func-
tions. It follows that
μ
j(α) =0=μ
k(α), αR.
In other words, there are slopes, mjand mk, such that
μj(α) dj+mjαand μk) dk+mkα.
However, for the difference, μj(α) μk(α), to be constant—which means that the spread is
horizontal on [a,b]and, hence, there is no unique minimum—it is necessary that mj=mk.
In summary, the spread can have a nonunique minimum only if there are two indices,
j,k, and a common slope, m, such that
μj(α) dj+and μk) dk+mα.
There are two cases to distinguish. First, consider m=0, which means that μj(α) djand
μk(α) dk. Whether or not this is possible depends on the multiplicity of the eigenvalues
of D,which,asshowninEq.(13), are also the eigenvalues of C. Hence, for a fixed i,
1in,let
I:={j:dj=di}.(20)
The cardinality of I,#(I ), is the multiplicity of di.
For α=0, according to definition (15), Mα=D; hence, the eigenvalues of Mαare dj.
Thus, suppose that, for a certain α=0, diis an eigenvalue of Mα. Then, there exists a vector,
xRn,x2=1, such that
Dαuutx=dix
⇐⇒ Dx αutxu=dix
⇐⇒ djxjαutxuj=dixj,j=1,2,...,n
⇐⇒ (djdi)xj=αutxuj,j=1,2,...,n. (21)
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2-Norm Effective Isotropic Hookean Solids 13
There are two cases to distinguish.
(1) uj=0, jI.
Then, for jI,Eq.(21) becomes 0 =0, since dj=di; hence, the equation is satisfied.
On the other hand, for j/I,dj=di. Note that, in this case,
utx=
k/I
ukxk,
which means that utxinvolves only indices k/I. Hence, using only k/I,Eq.(21)hasthe
following form:
yk=βkztyzk,k=1,...,m:=n#(I ).
In matrix form, this is (I Bzzt)y =0, for B:=diag1,...,β
m). Now, from the determi-
nant formula for rank-one updates,
detIBzzt=1ztBz.
There are two subcases:
(1a) 1 ztBz = 0 (the generic case).
Then, y=0Rm, which means that xk=0, for all k/I. Thus, the eigenspace for
λ=diis span(ej:jI) and has dimension #(I ).Inotherwords,λ=diis an eigenvalue
of Dαuutwith multiplicity #(I).
(1b) 1 ztBz =0.
Then, z= 0since1ztBz =1= 0. Since, for any yz,(I Bzzt)y =y, it follows
that λ=1 is an eigenvalue of IBzztof multiplicity m1; this implies that the dimension
of ker(I Bzzt)is one. Let yRmbe any nonzero vector in ker(I Bzzt)and ˜yRnbe
the corresponding vector with ˜yk=0, kI. Then, the eigenspace for λ=diis
span{ej:jI}∪{˜y},
which is of dimension #(I ) +1. The multiplicity of λ=ditherefore jumps to #(I ) +1. Here
is a simple example of cases (1a) and (1b).
Example 3 Tak e
D=
100
010
002
and u=
0
0
1
.
In this case—for λ=d1=1, I={1,2}—and, since u1=u2=0,weareincase(1).Itis
easy to check that
Dαuut=
10 0
01 0
002α
,
and its eigenvalues are 1,1,2α. Hence, for α= 1, λ=d1=1 is an eigenvalue of
multiplicity 2 =#(I ), which is case (1a). However, if α=1, the multiplicity of λ=1is
3=#(I ) +1, which is case (1b).
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14 L. Bos, M.A. Slawinski
Now, case (2): jI, such that uj= 0.
For such a j, from Eq. (21), we have
(djdi)xj=αutxuj
⇐⇒ 0=αutxuj(since jI)
⇐⇒ utx=0(since uj=0andα=0).
Hence, the eigenvalue equation, (D αuut)x =λx, becomes Dx =dix,andλ=dihas
eigenspace
span{ej:jI}xRn:utx=0.
Now, since there exists a jI, such that et
ju=uj=0, which means that u⊥ ej, it follows
that u/span{ej:jI}and so
dimspan{ej:jI}∩xRn:utx=0
=dimspan{ej:jI}+dimxRn:utx=0n
=#(I ) 1.
In other words, in case (2), λ=diis an eigenvalue of multiplicity #(I) 1.
From the considerations of this section we have the following proposition.
Proposition 3 A sufficient condition for a unique minimum is that all diare distinct and all
ui=0, i=1,...,n.
This proposition means that if the eigenvalues of Care distinct and u=Pv has no zero
components, then the 2-norm closest isotropic tensor is unique. Indeed, this is the generic
case, since C, chosen at random, has these properties with probability 1; its having multiple
eigenvalues is an extra condition. This is consistent with the fact that, as shown by Bóna et
al. [4], a generally anisotropic elasticity tensor exhibits six distinct eigenvalues.
6 Positivity
6.1 Stability conditions
Until now, we have considered only that Cis symmetric. However, as a representation of an
elasticity tensor, Cmust be also positive-definite, in accordance with the stability conditions
of a Hookean solid (e.g., Slawinski [21, Chap. 4]), which are conditions of the conservation
of energy.
Herein, we remark that—for the 2-norm—Ciso that is closest to C, which is positive-
definite, need not be positive-definite. This property is distinct from the analogous property
of the Frobenius norm, since, as shown by Gazis et al. [13], in that norm, the isotropic tensor
closest to C, which is positive-definite, is also positive-definite.
Example 4 A=αvvt+βInhas eigenvalues λ=α+β, of multiplicity 1, and λ=β,of
multiplicity n1. Hence, Ais positive-definite if and only if β>0andα+β>0.
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2-Norm Effective Isotropic Hookean Solids 15
Consider, for n=3,
D=
600
020
001
and u=
0
1
0
.
Then
Dαuut=
600
02α0
001
and
λ1(α) =max{6,2α}=2α, α ≤−4,
6≥−4,
while
λn(α) =min{1,2α}=11,
2α, α 1.
Hence the spread is
s(α) =λ1(α) λn(α) =
1α, α ≤−4,
5,4α1,
4+α, α 1.
Therefore any 4α1 minimizes the spread. Hence, following expression (16)in
Proposition 1,weseethatβ=1)+λn))/2=(6+1)/2=7/2, independently
of α. Also, we see that, for 4α≤−7/2, we have α+β0 and the correspond-
ing matrix is not positive-definite. On the other hand, if 7/2
1, the corresponding
matrix is positive-definite. Thus, there are both positive-definite and nonpositive-definite
optimal matrices.
We conjecture that if the isotropic matrix closest to C, which is positive-definite, is
unique, then this isotropic matrix must be also positive-definite.
In what follows we impose positive definiteness on the elasticity tensors and on the space
of its isotropic counterparts.
Let CRn×nbe a symmetric and positive-definite matrix. Following expression (10),
we define
Ciso :=ARn×n:A=αvvt+βInR,is positive definite.
Again, A=αvvt+βInhas eigenvalues λ=α+β, of multiplicity 1, and λ=β, of multi-
plicity n1. Hence,
Ciso :=ARn×n:A=αvvt+βInR+β>0,β >0.
The positive-definiteness of Cmeans that di>0, i=1,...,n.
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16 L. Bos, M.A. Slawinski
The positive-definite case differs from the general symmetric case only if the closest
isotropic tensor to Cis on the boundary of the closure of Ciso. This boundary corresponds
to the boundary in R2of
(α, β) :α+β00,
and consists of two line segments, {(α, β) =(α, 0):α0}and {(α, β ) =(β, β) :β0}.
6.2 Segment {(α, β) =(α, 0):α0}
Herein, the relevant matrices of (the closure of) Ciso are of the form αvvt,whereα0. We
wish to find
min
α0
Cαvvt
2.
As in Sect. 4, we may orthogonally diagonalize C=PtDP , and reduce the minimization
problem to
min
α0
Dαuut
2=:min
α0Mα2,
where u=Pv and, according to expression (15), Mα:=Dαuut.
Now, Mα2=max1jn|λj(α)|. However, since the eigenvalues are ordered, λ1(α)
···≥λn),wehave
Mα2=maxλ1(α),λn(α)=maxλ1(α), λ1(α), λn(α), λn).
If λn(α) 0, then—necessarily—λ1) λn(α) 0andMα2=λ1(α). Similarly,
if λ1(α) 0, then λn(α) λ1) 0andMα2=|λn(α)|=−λn(α).Otherwise,if
λ1(α) > 0andλn(α) < 0, Mα2=max{λ1(α ), λn(α)}. In all cases, we have
N(α):=Mα2=maxλ1(α), λn).
Since λ1(α) and λn(α) are convex functions of α, it follows, from Proposition 2,thatN(α)
is also convex. Now, note that
λ1(α) =max
x2=1xtMαx
=max
x2=1xtDαuutx
=max
x2=1xtDx αutx2.
But if αα,xtDx α(utx)2xtDx α(utx)2so that λ1)λ1(α), which means that
λ1(α) is a decreasing function of α. Similarly, λn(α) is also a decreasing function of α,so
that λn(α) is increasing. At α=0, we have λ1(0)=d1and λn(0)=dnwith d1dn0.
Hence N(0)=max{d1,dn}=d1. Moreover, just as in the proof of Lemma 1, we conclude
that
lim
α→+∞λn) =∞.
Since, according to expression (19), λ1(α) d2,forα0, we see that, for sufficiently
large α,N(α) =−λn). Hence, N(α) is the maximum of the decreasing function, λ1(α),
and the increasing function, λn(α), where at the left endpoint, α=0, the greater is λ1while
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2-Norm Effective Isotropic Hookean Solids 17
at the right endpoint, α=∞, the greater is λn. It follows that there is a crossing point,
α0, such that λ1)=−λn)and
N(α)=+λ1(α), α α,
λn(α), α α.
Therefore, the minimum value of N(α) is N(α). However, it need not be attained uniquely.
There could be flat spots, under the conditions discussed in the context of Lemma 1.
Example 5 Suppose u=ej,j=1. Then, one can easily verify that
Dαuut=diag(d1,...,d
j1,d
jα, dj+1,...,d
n),
so that
N(α)=
Dαuut
2
=max|d1|,...,|,d
j1|,|djα|,|dj+1|,...,|dn|
=maxd1,...,d
j1,|djα|,d
j+1,...,d
n
=maxd1,|djα|
=d1,0αd1+dj,
αdjd1+dj,
and the minimum value of d1is attained for α∈[0,d
1+dj].
6.3 Segment {(α, β) =(β, β) :β0}
Herein, the relevant matrices of (the closure of) Ciso are of the form βvvt+βIn,where
β0. We wish to find
min
β0
Cβvvt+βI
2=
CβI +βvvt
2.
Again, we may orthogonally diagonalize C=PtDP to reduce the minimization problem to
min
β0
DβI +βuut
2=min
β0Mβ2,
where u=Pv and Mβ:=DβI +βuut.
The analysis is similar to the one in Sect. 6.2. As before, λ1) ···≥λn ) denote the
ordered eigenvalues of Mβ, with λj(0)=dj. Then, again
Mβ2=maxλ1(β), λn ).
Since λ1(β) and λn(β ) are convex functions of α, it follows, from Proposition 2,thatN(β)
is also convex.
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18 L. Bos, M.A. Slawinski
Moreover, both λ1) and λn(β) are decreasing functions of β. To see this, note that
λ1(β) =max
x2=1xtMβx
=max
x2=1xtDx β+βutx2
=max
x2=1xtDx β1utx2.
However, since both xand uare unit vectors, (utx)21and1(utx)20. Hence, if
xis fixed, xtDx β(1(utx)2)is decreasing in β. Thus, λ1 ) is also decreasing in β.
Similarly,
λn(β) =min
x2=1xtMβx
is also decreasing, so that λn) is increasing.
Furthermore, since
tr(Mβ)=
n
i=1
di+β
n
i=1
u2
i=
n
i=1
di(n 1
tends to −∞ as β→∞,wehave
lim
β→∞λn ) =+.
It follows that N(β) := Mβ2is the maximum of both the decreasing function, λ1(β),
and the increasing function, λn), with λ1(0)=d1≥−dn=−λn(0),andatβ=∞,
λn(β) > λ1 ). Hence
min
β0N(β)=N(β),
where β∗≥0 is such that λ1)=−λn).Again,βneed not be unique.
7 Experimental results
To apply the above results, in this section, we examine isotropic elasticity tensors closest—
in the 2-norm sense—to the generally anisotropic one obtained from seismic measurements.
The most accurate representation of a physical material as a Hookean solid is the generally
anisotropic elasticity tensor. Its simplest representation, on the other hand, is the isotropic
one. In principle, measurements result in a generally anisotropic tensor, since—beyond ex-
perimental errors—no physical material is a Hookean solid nor could it possess a material
symmetry of a such a solid, which is a mathematical object contained in expression (1).
Both for its simplicity and in view of resolution of measurements, it might be more con-
venient and justifiable to study physical phenomena using the closest isotropic counterpart
rather than a generally anisotropic tensor. In other words, in view of the number of elasticity
parameters in matrix (5) as opposed to matrix (2), it might be preferable, for a variety of
reasons, to work with a two-parameter model rather than a twenty-one parameter model.
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2-Norm Effective Isotropic Hookean Solids 19
Let us consider the elasticity parameters of a generally anisotropic tensor obtained by
Dewangan and Grechka [10] from seismic measurements in New Mexico,
C=
7.8195 3.4495 2.5667 2(0.1374)2(0.0558)2(0.1239)
3.4495 8.1284 2.3589 2(0.0812)2(0.0735)2(0.1692)
2.5667 2.3589 7.0908 2(0.0092)2(0.0286)2(0.1655)
2(0.1374)2(0.0812)2(0.0092)2(1.6636)2(0.0787)2(0.1053)
2(0.0558)2(0.0735)2(0.0286)2(0.0787)2(2.0660)2(0.1517)
2(0.1239)2(0.1692)2(0.1655)2(0.1053)2(0.1517)2(2.4270)
.
(22)
These entries are the density-scaled elasticity parameters; their units are km2/s2.
As described between expressions (11)and(14), the eigenvalues, di,ofCare
D=diag(13.3805,5.2281,4.9857,4.4716,4.0194,3.2665);
according to expression (6), the 2-norm of Cis C2=d1=13.3805. The eigenvalues are
distinct, which is to be expected from any measurements that result in a Hookean solid as a
representation of a physical material. Also, according to expression (13), u:=Pv, with v,
for Ciso, given by expression (7), it turns out that uhas no zero components. Hence, there
is a unique minimum, since—as stated in Sect. 4—a sufficient condition is that all diare
distinct and all ui= 0. Thus, there are no flat spots in the spread of Mα, which is given
by expression (15). Indeed, the spread is a smooth convex function of αwith a unique
minimum.
In view of uniqueness, one finds the minimum spread using, for example, the Matlab
function fminunc, for unconstrained minimization of a univariate function. The unique
optimal value of αis α=9.3641 and the optimal β, given by expression (16) in Proposi-
tion 1,isβ=4.4969. Since both βand β+αare strictly positive, the optimal isotropic
matrix, αvvt+βI6, is positive-definite; there is no need to investigate the boundary. The
2-norm distance from Cto Ciso is the optimal spread divided by two, which, in this case, is
1.2722.
The closest Ciso, in the 2-norm sense, is—in view of matrix (5)—the isotropic Hookean
solid defined by (ciso
1111)2=7.6183 and (ciso
2323)2=2.2485. Since these parameters are scaled
by the mass density, they provide a complete description of this Hookean solid; any Hookean
solid is described completely by its elasticity parameters and mass density.
In principle, it could be the case that the unconstrained closest element of Ciso is not
positive-definite. In such a case, the closest positive element would be on the boundary.
Also, in general, the spread is only piecewise smooth. It could even have a flat spot. In
such a case, it is prudent to avoid using a gradient method, such as the one implemented
in fminunc, and use a minimizer that avoids derivatives such as the golden-section search
implemented, for example, in the Matlab function fminbnd.
For discussions in the following section, let us compute the same quantities for three
other generally anisotropic elasticity tensors obtained recently from seismic measurements
by Grechka and Yaskevich [15]; in each case, the effective isotropic tensor is unique. As an
aside, let us emphasize that acquiring measurements to obtain the twenty-one parameters of
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20 L. Bos, M.A. Slawinski
cij k is a challenging process.
C=
26.57 12.14 10.00 2(0.92)2(0.27)2(0.49)
12.14 27.80 10.66 2(0.89)2(0.86)2(0.53)
10.00 10.66 22.11 2(0.94)2(0.54)2(0.10)
2(0.92)2(0.89)2(0.94)2(7.32)2(0.14)2(0.12)
2(0.27)2(0.86)2(0.54)2(0.14)2(6.95)2(0.20)
2(0.49)2(0.53)2(0.10)2(0.12)2(0.20)2(7.85)
;(23)
D=diag(47.79,15.05,13.77,7.88,7.25,6.85),(ciso
1111)2=25.60, (ciso
2323)2=7.37, distance:
4.45.
C=
15.78 1.15 1.82 2(0.09)2(0.04)2(0.08)
1.15 15.52 1.98 2(0.12)2(0.05)2(0.08)
1.82 1.98 9.42 2(0.04)2(0.00)2(0.14)
2(0.09)2(0.12)2(0.04)2(3.88)2(0.35)2(0.21)
2(0.04)2(0.05)2(0.00)2(0.35)2(4.06)2(0.03)
2(0.08)2(0.08)2(0.14)2(0.21)2(0.03)2(6.83)
;(24)
D=diag(17.68,14.50,8.56,6.83,4.33,3.60),(ciso
1111)2=12.75, (ciso
2323)2=5.43, distance:
3.68.
C=
19.81 8.62 9.00 2(2.37)2(1.44)2(0.95)
8.62 25.79 9.09 2(0.57)2(0.99)2(0.89)
9.00 9.09 20.68 2(2.10)2(0.43)2(0.49)
2(2.37)2(0.57)2(2.10)2(7.17)2(0.15)2(0.08)
2(1.44)2(0.99)2(0.43)2(0.15)2(8.14)2(0.33)
2(0.95)2(0.89)2(0.49)2(0.08)2(0.33)2(6.49)
;(25)
D=diag(40.33,15.80,11.77,7.68,6.52,5.97),(ciso
1111)2=21.80, (ciso
2323)2=6.72, distance:
5.24.
Note the number of decimal points used for the computed quantities is consistent with
the entries of matrices (22), (23), (24)and(25).
8 Discussion
To begin the discussion, let us obtain the pertinent Frobenius-norm quantities of each C
and its Ciso. As shown by Voigt [23], the closest isotropic counterpart—in the Frobenius
sense—of a generally anisotropic elasticity tensor is
ciso
1111F=1
153(c1111 +c2222 +c3333)+2(c1122 +c1133 +c2233)
+4(c1212 +c1313 +c2323)(26)
and
ciso
2323F=1
15c1111 +c2222 +c3333 (c1122 +c1133 +c2233)
+3(c1212 +c1313 +c2323).(27)
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2-Norm Effective Isotropic Hookean Solids 21
As discussed by Bucataru and Slawinski [6], the Frobenius squared distance between Cand
Ciso
Fis
λ2
1λiso
12+
6
i=2λ2
iλiso
i2,
where λ1and λiso
1are their respective largest eigenvalues; notably, λiso
1=3(ciso
1111)F
4(ciso
2323)F. Thus, for tensors (22), (23), (24)and(25), respectively, we obtain:
CF=16.6748, (ciso
1111)F=7.3662, (ciso
2323)F=2.2089, distance: 2.1354,
CF=53.49, (ciso
1111)F=22.62, (ciso
2323)F=5.12, distance: 9.67,
CF=25.97, (ciso
1111)F=10.77, (ciso
2323)F=3.86, distance: 9.56,
CF=46.39, (ciso
1111)F=19.72, (ciso
2323)F=4.82, distance: 9.79,
which can be compared with the 2-norm results stated in Sect. 7.
To gain an insight into differences resulting from the 2-norm and the Frobenius norm,
let us compare the P-wave and S-wave velocities obtained using these norms. These
velocities are commonly considered in quantitative seismology. Using standard expres-
sions (e.g., Slawinski [21, Chap. 6]), we consider (vP)2=(ciso
1111)2,(vP)F=(ciso
1111)F,
(vS)2=(ciso
2323)2and (vS)F=(ciso
2323)F, for tensors (22), (23), (24)and(25), respectively.
The units are km/s.
(vP)2=2.76 and (vP)F=2.71; (vS)2=1.50 and (vS)F=1.49;
(vP)2=5.06 and (vP)F=4.76; (vS)2=2.71 and (vS)F=2.26;
(vP)2=3.57 and (vP)F=3.28; (vS)2=2.33 and (vS)F=1.96;
(vP)2=4.67 and (vP)F=4.44; (vS)2=2.59 and (vS)F=2.20.
In most cases, the velocities are sufficiently different as to affect conclusions drawn from
seismic interpretation. Furthermore, since tensors (22), (23), (24)and(25) are only weakly
anisotropic, as discussed by Kochetov and Slawinski [17], we expect that—in general—the
differences between the two norms are even greater.
In our subsequent investigations, we will include symmetry classes beyond isotropy,
whose search—unlike the search for isotropy—is not rotation-independent. Moreover, in
these cases there is no analytic form for the closest element, even using the Frobenius norm.
Notably, in that study, computer-intensive operations require particular attention to the
issue of positivity investigated in this paper. As stated above—unlike in the case of the
Frobenius norm—the closest 2-norm symmetric counterpart of a positive-definite tensor
might not be positive-definite.
Let us conclude by saying that even though the computational convenience of the Frobe-
nius norm—in particular, the issue of positivity and the existence of expressions (26)and
(27)—results in its being the standard method to obtain effective isotropic tensors, the choice
of norm carries consequences that should not be ignored.
Also, as discussed in Sect. 3and in view of expression (1), the 2-norm is derived as an
operator norm from the norms of the stress and strain tensors; no such derivation appears to
be possible for the Frobenius norm.
Acknowledgements We wish to acknowledge discussions with Misha Kochetov, Michael Rochester, and
discussions with, and computational results of, Tomasz Danek, as well as the graphic and editorial support of
Elena Patarini and David Dalton, respectively.
This research was performed in the context of The Geomechanics Project supported by Husky Energy.
M.A. Slawinski’s research was supported partially by the Natural Sciences and Engineering Research Council
of Canada.
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22 L. Bos, M.A. Slawinski
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Author's personal copy
... One can distinguish a group of techniques, which are based on the separation of an additive part of the elasticity tensor (e.g., by way projecting), possessing one or another type of symmetry [34][35][36][37]. In this case, the identification problem is reduced to determination of the symmetric part, which is the closest in some metric to a given tensor, as e.g., in [38][39][40][41][42][43][44][45]. ...
... Nevertheless, optimization problems analogous to Problems 1-3, can be formulated in terms of quite arbitrary metric on 4 3  . Various ways to define such metric are considered, e.g., in [34,36,37,40]. In [36], the symmetric approximations of elasticity tensors are constructed with the use of different approaches, which are based on the Frobenius, Log-Euclidean and Riemannian distance functions. ...
... The last two, compared to the first one, possess some additional invariance properties: in particular, with regard to inversion of the tensor-arguments. In [34], the specific features of the operator norm, ⋅ , are investigated theoretically, when searching for the ⋅optimal elasticity tensor approximations. It was shown that the ⋅ -optimal approximation of a positive definite tensor, unlike the ⋅ 4 3  -optimal one, may be not positive definite or unique in some degenerate cases. ...
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The products made by the forming of polycrystalline metals and alloys, which are in high demand in modern industries, have pronounced inhomogeneous distribution of grain orientations. The presence of specific orientation modes in such materials, i.e., crystallographic texture, is responsible for anisotropy of their physical and mechanical properties, e.g., elasticity. A type of anisotropy is usually unknown a priori, and possible ways of its determination is of considerable interest both from theoretical and practical viewpoints. In this work, emphasis is placed on the identification of elasticity classes of polycrystalline materials. By the newly introduced concept of "elasticity class" the union of congruent tensor subspaces of a special form is understood. In particular, it makes it possible to consider the so-called symmetry classification, which is widely spread in solid mechanics. The problem of identification of linear elasticity class for anisotropic material with elastic moduli given in an arbitrary orthonormal basis is formulated. To solve this problem, a general procedure based on constructing the hierarchy of approximations of elasticity tensor in different classes is formulated. This approach is then applied to analyze changes in the elastic symmetry of a representative volume element of polycrystalline copper during numerical experiments on severe plastic deformation. The microstructure evolution is described using a two-level crystal elasto-visco-plasticity model. The well-defined structures, which are indicative of the existence of essentially inhomogeneous distribution of crystallite orientations, were obtained in each experiment. However, the texture obtained in the quasi-axial upsetting experiment demonstrates the absence of significant macroscopic elastic anisotropy. Using the identification framework, it has been shown that the elasticity tensor corresponding to the resultant microstructure proves to be almost isotropic.
... To examine the closeness between elasticity tensors, as discussed by Bos and Slawinski [3] and by Danek et al. [5,6], we consider several norms. ...
... As discussed by Bos and Slawinski [3], by treating a matrix as a vector, the Frobenius norms ignore the fact that a matrix is a representation of a linear map from n to n . In view of equation (1), the elasticity tensor represents a linear map between the strain tensor, whose components can be expressed as a symmetric 3 × 3 matrix, [ k ] , and the stress tensor, whose components can be expressed as a symmetric 3 × 3 matrix, [σ i j ] . ...
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It is common to assume that a Hookean solid is isotropic. For a generally anisotropic elasticity tensor, it is possible to obtain its isotropic counterparts. Such a counterpart is obtained in accordance with a given norm. Herein, we examine the effect of three norms: the Frobenius 36-component norm, the Frobenius 21-component norm and the operator norm on a general Hookean solid. We find that both Frobenius norms result in similar isotropic counterparts, and the operator norm results in a counterpart with a slightly larger discrepancy. The reason for this discrepancy is rooted in the very definition of that norm, which is distinct from the Frobenius norms and which consists of the largest eigenvalue of the elasticity tensor. If we constrain the elasticity tensor to values expected for modelling physical materials, the three norms result in similar isotropic counterparts of a generally anisotropic tensor. To examine this important case and without loss of generality, we illustrate the isotropic counterparts by commencing from a transversely isotropic tensor obtained from a generally anisotropic one. Also, together with the three norms, we consider the L 2 slowness-curve fit. Upon this study, we infer that-for modelling physical materials-the isotropic counterparts are quite similar to each other, at least, sufficiently so that-for values obtained from empirical studies, such as seismic measurements-the differences among norms are within the range of expected measurement errors.
... a Department of Geoinformatics and Applied Computer Science, AGH -University of Science and Technology, Kraków, Poland b Department of Earth Sciences, Memorial University of Newfoundland, St. John's, NL, Canada As discussed by Bos and Slawinski [2] ...
... This is a seven-variable problem, whose variables are the two angles that describe the orientation of the axis and the five elasticity parameters of the corresponding effective tensor, ˜ c . Furthermore, one must verify that each candidate for the effective tensor is positive-definite, as required for a Hookean solid, since—unlike for the Frobenius-effective tensor—this requirement is not intrinsically satisfied for the operator effective tensor, as discussed by Bos and Slawinski [2] ...
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A generally anisotropic elasticity tensor, which might be obtained from physical measurements, can be approximated by a tensor belonging to a particular material-symmetry class; we refer to such a tensor as the effective tensor. The effective tensor is the closest to the generally anisotropic tensor among the tensors of that symmetry class. The concept of closeness is formalized in the notion of norm. Herein, we compare the effective tensors belonging to the transversely isotropic class and obtained using two different norms: the Frobenius norm and the L2 operator norm. We compare distributions of the effective elasticity parameters and symmetry-axis orientations for both the error-free case and the case of the generally anisotropic tensor subject to errors. © 2009 - 2014. Universita degli Studi di Padova - Padova University Press - All Rights Reserved.
... Underlying this question is the accuracy of the generally anisotropic tensor, and, hence, its effect on the reliability of information provided by the chosen model. Several researchers-among them, Voigt [25], Gazis et al. [15], Moakher and Norris [20], Norris [21], Bucataru and Slawinski [6], Kochetov and Slawinski [18,19], Diner et al. [11,12], Bos and Slawinski [5]-examined relations between a given generally anisotropic Hookean solid and its symmetric counterpart by invoking the concept of a norm in the space of elasticity tensors. In this approach, the term elasticity tensor is understood in a broader sense: the requirement of positive definiteness is omitted so the possible values of c form a vector space. ...
... There are several natural norms on the space of elasticity tensors as discussed, for instance, by Norris [21] and by Bos and Slawinski [5]; the simplest among them is the Frobenius norm. Each of them has certain theoretical advantages and disadvantages; for instance, the Frobenius norm guarantees positive definiteness of the effective tensor. ...
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... In order to find the closest isotropic elasticity tensor to the orthotropic tensor given in Table 4, several different measures can be applied (e.g. Gazis et al., 1963;Bos and Slawinski, 2015). We perform a least squares inversion which leads to the following analytical solutions for the equivalent 1-D isotropic l and l c values Fig. 7. Obtained optic mode periods (1=xr ) in bridgmanite in lower mantle conditions as a function of characteristic length Lc. ...
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