Abstract In this paper we derive necessary and sufficient conditions for strongellipticity in several classes of anisotropic linearly elastic materials. Our results coverall classes in the rhombic system (nine elasticities), four classes of the tetragonalsystem (six elasticities) and all classes in the cubic system (three elasticities). As aspecial case we recover necessary and sufficient conditions for strong ellipticity intransversely isotropic materials. The central result shows that for the rhombic systemstrong ellipticity restricts some appropriate combinations of elasticities to take valuesinside a domain whose boundary is the third order algebraic surface defined byx2 + y2 + z2 − 2xyz − 1 = 0 situated in the cube |x| < 1, |y| < 1, |z| < 1. For moresymmetric situations, the general analysis restricts combinations of elasticities torange inside either a plane domain (for four classes in the tetragonal system) orin an one-dimensional interval (for the hexagonal systems, transverse isotropy andcubic system). The proof involves only the basic statement of the strong ellipticitycondition.
Key words strong ellipticity · anisotropic materials · rhombic · tetragonal · cubic ·hexagonal systems · transverse isotropy
Mathematics Subject Classifications (2000) 74 B 05 · 74 E 10
S. Chirita (B)Faculty of Mathematics, “Al. I. Cuza” University, Blvd. Carol I, no. 11, 700506-Iasi, Romaniae-mail: [email protected]
A. DanescuDepartment Mécanique des Solides, Génie Mécanique, Génie Civil,Ecole Centrale de Lyon, Av. Guy de Collongue, BP 163-69131 Ecully Cedex, France
M. CiarlettaDepartment of Engineering Information and Applied Mathematics, University of Salerno,84084 Fisciano (SA), Italy
2 S. Chirita, et al.
1 Introduction
For a linearly anisotropic elastic solid the components Cijkl of the tensor of elasticmoduli satisfy the symmetries
Cijkl = Cklij = Cijlk, (1.1)
while the indices i, j, k, l take values 1, 2, 3.The strong ellipticity condition states that
Cijklninkm jml > 0 (1.2)
for all non-zero vectors m = (m1, m2, m3) and n = (n1, n2, n3), where summation isimplied by index repetition. This condition is of importance in discussing uniqueness,wave propagation (see e.g. Gurtin [1, p. 86]), loss of ellipticity in the context of thenonlinear elasticity of fibre–reinforced materials (see Merodio and Ogden [2]) andalso in the study of spatial behaviour of the constrained anisotropic cylinders (seeChirita and Ciarletta [3]).
In the particular case of an isotropic linearly elastic material with Lamé moduli λ
and μ the strong ellipticity is equivalent to the fact that (see e.g. Gurtin [1, p. 86])
μ > 0, λ + 2μ > 0. (1.3)
Simpson and Spector [4] treat the strong ellipticity condition for isotropic non-linearly elastic materials. They establish necessary and sufficient conditions for thestrong ellipticity of the equations governing an isotropic (compressible) nonlinearlyelastic material at equilibrium. The main tool in their analysis consists in a represen-tation theorem for copositive matrices. A review on strong ellipticity for isotropicnonlinearly elastic materials is given by Dacorogna [5].
Concerning the strong ellipticity for the class of anisotropic linearly elastic mate-rials very few is known. A notable exception is the investigation given by Payton [6]and the recent results by Padovani [7], Merodio and Ogden [8] and Chirita [9] whoexamine the strong ellipticity for transversely isotropic linearly elastic solids. In termsof the elastic constants c11, c33, c55, c12 and c13 of the transversely isotropic linearlyelastic solid, the strong ellipticity condition is equivalent to the following inequalities
c11 > 0, c33 > 0, c55 > 0, c11 > c12,
|c13 + c55| < c55 + √c11c33. (1.4)
We recall that equivalence with respect to the restrictions that material symmetryplaces on the elasticity tensor for 32 crystal classes provides the following distinctsituations (see Gurtin [1, pp. 87–89]):
(i) Triclinic system (21 elasticities);(ii) Monoclinic system (all classes), (13 elasticities);
(iii) Rhombic system (all classes), (nine elasticities);(iv) Tetragonal system (tetragonal–disphenoidal, tetragonal–pyramidal, tetra-
gonal–dipyramidal), (seven elasticities);
Strong ellipticity of anisotropic materials 3
(v) Tetragonal system (tetragonal–scalenohedral, ditetragonal–pyramidal, tetra-gonal–trapezohedral, ditetragonal–dipyramidal), (six elasticities);
(vi) Cubic system (tetartoidal, diploidal, hexatetrahedral, gyroidal, hexoctahe-dral), (three elasticities);
(vii) Hexagonal system (trigonal–pyramidal, rhombohedral), (seven elasticities);(viii) Hexagonal system (ditrigonal–pyramidal, trigonal–trapezohedral, hexag-
onal–scalenohedral), (six elasticities);(ix) Hexagonal system (trigonal–dipyramidal, hexagonal–pyramidal, hexagonal–
dipyramidal, ditrigonal–dipyramidal, dihexagonal–pyramidal, hexagonal–trapezohedral, dihexagonal–dipyramidal, i.e., the hexagonal division), (fiveelasticities).
The form of the elasticity tensor imposed by the material symmetry requirementsin the last case above is identical with that imposed by the transverse isotropy andthis is the only anisotropic situation where necessary and sufficient conditions forstrong ellipticity are known (see [6–9]).
In what follows we provide necessary and sufficient conditions equivalent tostrong ellipticity that cover anisotropic materials in all classes of the rhombic system,i.e. the case (iii) above, a part of the tetragonal system covered by classes in (v) andall classes of the cubic system.
Our main results show that for rhombic materials and, in particular, more symmet-ric situations, strong ellipticity restricts some combinations among the classical elasticcoefficients to take values inside a domain whose boundary has a simple algebraicform.
The paper is organized as follows: the second section provides statements of allresults but the proof of Theorem 1, which solves the problem for the rhombic systemis presented in detail in Section 3. In Section 4 we specialize the general results ofSection 3 in order to cover the above mentioned symmetry classes. We end the paperwith some general comments and remarks concerning various possible extensions ofour main results and their intrinsic difficulties.
2 Problem Setting and Main Results
Throughout this paper we shall discuss the strong ellipticity condition (1.2) for therhombic system and for all other more symmetric systems. In the present sectionwe shall write the form of the strong ellipticity condition for each system underdiscussion and then we shall state the restrictions that it places on the correspondingelastic coefficients.
2.1 Strong Ellipticity for the Rhombic System
Let us consider the class of rhombic elastic materials with the group C3 generated byRπ
e3, Rπ
e2(here Rθ
e is the orthogonal tensor corresponding to a right-handed rotationthrough the angle θ, 0<θ <2π , about an axis in the direction of the unit vector e).
4 S. Chirita, et al.
According to Gurtin [1, p. 88] such a class of materials is characterized by theconditions
For convenience, we shall introduce a standard notation in linear elasticity, inorder to denote the only non–zero components of the elasticity tensor as follows
The part of the surface S (x, y, z) situated in the cube |x| ≤ 1, |y| ≤ 1, |z| ≤ 1 isgiven in Fig. 1, while the Fig. 2 gives an image of the same surface in the extendedcube |x| ≤ 2, |y| ≤ 2, |z| ≤ 2.
2.2 Strong Ellipticity for the Hexagonal Division and Transverse Isotropy
The restrictions placed on the elasticity tensor by material symmetry in seven crystalclasses of the hexagonal systems are identical with those imposed by transverseisotropy (see Gurtin [1, p. 89]). The explicit restrictions on elasticities for symmetry
These last conditions have been previously established by Payton [6], Padovani[7], Merodio and Ogden [8] and Chirita [9].
2.3 Strong Ellipticity for the Tetragonal System
Restrictions placed by symmetries in the seven crystal classes of the tetragonal systemare covered by two distinct situations. Three classes, i.e., tetragonal–disphenoidal,tetragonal–pyramidal, tetragonal–dipyramidal provide an elasticity tensor with sevenelastic coefficients while the others, i.e. tetragonal–scalenohedral, ditetragonal–pyramidal, tetragonal–trapezohedral, ditetragonal–dipyramidal provide an elasticitytensor with six elastic coefficients. This subsection cover the last situation.
The tetragonal system C5 generated by Rπ/2e3 , Rπ
e1is characterized by the relations
(2.1) and (2.2) and
c22 = c11, c23 = c13, c44 = c55. (2.10)
Moreover, the strong ellipticity condition (1.2) becomes
Fig. 3 Strong ellipticity regionfor tetragonal system
8 S. Chirita, et al.
Theorem 4 The strong ellipticity condition (2.18) for a cubic system is equivalent withthe following conditions
c11 > 0, c66 > 0, |c12 + c66| < c11 + c66. (2.19)
3 Proof of the Theorem 1
This section contains the proof for our main result described by the Theorem 1. Westart considering the strong ellipticity condition (2.3) as a quadratic in one of thevariables m1, m2, m3, n1, n2 , n3. Thus, we rewrite the relation (2.3) in the followingform
Regarding (3.1) as a quadratic in n3 ∈ R, we deduce
c55m21 + c44m2
2 + c33m23 > 0 for all m = (m1, m2, m3) �= 0, (3.2)
c11n21m2
1 + c22n22m2
2 + c66n21m2
2 + c66n22m2
1 + 2 (c12 + c66) n1m1n2m2 ++ (
c55n21 + c44n2
2
)m2
3 > 0, (3.3)
and the discriminant has to be strictly negative for all non–zero vectors m =(m1, m2, m3) and n = (n1, n2, n3).
As a direct consequence, from the relation (3.2) we obtain that
c33 > 0, c44 > 0, c55 > 0, (3.4)
while the relation (3.3) implies
c11n21m2
1 + c22n22m2
2 + c66n21m2
2 + c66n22m2
1 + 2 (c12 + c66) n1m1n2m2 > 0, (3.5)
that is
(c11m2
1 + c66m22
)n2
1 + 2 (c12 + c66) m1m2n2n1 + (c66m2
1 + c22m22
)n2
2 > 0, (3.6)
for all non-zero vectors N = (n1, n2), M = (m1, m2). Considering (3.6) as a quadraticin n1 ∈ R, we deduce
c11m21 + c66m2
2 > 0, c66m21 + c22m2
2 > 0 for all (m1, m2) �= 0, (3.7)
(c12 + c66)2 m2
1m22 <
(c11m2
1 + c66m22
) (c66m2
1 + c22m22
)for all (m1, m2) �= 0. (3.8)
Consequently, the relation (3.7) implies
c11 > 0, c22 > 0, c66 > 0, (3.9)
Strong ellipticity of anisotropic materials 9
and the relation (3.8) becomes
c11c66m41 + [
c11c22 + c266 − (c12 + c66)
2]m21m2
2 + c22c66m42 > 0, (3.10)
that is,
c66(√
c11m21 − √
c22m22
)2 +[(
c66 + √c11c22
)2 − (c12 + c66)2]
m21m2
2 > 0. (3.11)
Therefore, setting√
c11m21 = √
c22m22 in (3.11), we obtain
(c66 + √
c11c22)2 − (c12 + c66)
2 > 0 (3.12)
and hence we have
|c12 + c66| < c66 + √c11c22. (3.13)
We now return to the equation (3.1) and note that the condition expressing thenegativeness of the discriminant gives
[(c13 + c55) n1m1 + (c23 + c44) n2m2]2 m23 <
(c55m2
1 + c44m22 + c33m2
3
) ·· [(c55n2
1 + c44n22
)m2
3 + c11n21m2
1 + c22n22m2
2 + c66n21m2
2++ c66n2
2m21 + 2 (c12 + c66) n1m1n2m2
](3.14)
and hence
c33(c55n2
1 +c44n22
)m4
3 + {(c55m2
1 + c44m22
) (c55n2
1 + c44n22
)+ c33[c11n2
1m21+
+ c22n22m2
2 + c66n21m2
2 + c66n22m2
1 + 2 (c12 + c66) n1m1n2m2]−
− [(c13 + c55) n1m1 + (c23 + c44) n2m2]2}
m23 + (
c55m21 + c44m2
2
) ·· [c11n2
1m21 + c22n2
2m22 + c66n2
1m22 + c66n2
2m21 + 2 (c12 + c66) n1m1n2m2
]> 0.
(3.15)
Therefore, if we set
m23 = 1√
c33
√c55m2
1 + c44m22
c55n21 + c44n2
2
.
·√
c11n21m2
1 + c22n22m2
2 + c66n21m2
2 + c66n22m2
1 + 2 (c12 + c66) n1m1n2m2,
(3.16)
into relation (3.15), then we deduce
|(c13 + c55) n1m1 + (c23 + c44) n2m2| <
√(c55m2
1 + c44m22
) (c55n2
1 + c44n22
) +
+ √c33
√c11n2
1m21 + c22n2
2m22 + c66n2
1m22 + c66n2
2m21 + 2 (c12 + c66) n1m1n2m2, (3.17)
for all non-zero vectors N = (n1, n2), M = (m1, m2).Next step is the study of restrictions placed on the elastic coefficients by the
inequality (3.17). We start noting that the explicit expression of the term on the left
10 S. Chirita, et al.
hand side of the relation (3.17) depends on the sign of the terms c13 + c55, c23 + c44,n1m1 and n2m2. For this reason we rewrite it in an equivalent form more tractable inthe subsequent analysis.
Let us assume that the terms c13 + c55 and c23 + c44 are prescribed. Then theexplicit expression of the term on the left hand side of the relation (3.17) dependson the sign of the terms n1m1 and n2m2 only. In fact, when n1m1 and n2m2 have thesame sign the relation (3.17) becomes
|(c13 + c55) |n1m1| + (c23 + c44) |n2m2|| <
√(c55m2
1 + c44m22
) (c55n2
1 + c44n22
) +
+ √c33
√c11n2
1m21 + c22n2
2m22 + c66n2
1m22 + c66n2
2m21 + 2 (c12 + c66) |n1m1| |n2m2|,
(3.18)
while when (n1m1) · (n2m2) < 0 we have
|− (c13 + c55) |n1m1| + (c23 + c44) |n2m2|| <
√(c55m2
1 + c44m22
) (c55n2
1 + c44n22
) +
+ √c33
√c11n2
1m21 + c22n2
2m22 + c66n2
1m22 + c66n2
2m21 − 2 (c12 + c66) |n1m1| |n2m2|,
(3.19)
for all non-zero vectors N = (n1, n2), M = (m1, m2).Further, we can prove that the relation (3.18) is equivalent with the following
relation
∣∣(c13 + c55) m21 + (c23 + c44) m2
2
∣∣ < c55m21 + c44m2
2 +
+ √c33
√c11m4
1 + c22m42 + 2 (c12 + 2c66) m2
1m22, for all (m1, m2) �= 0, (3.20)
and the relation (3.19) is equivalent with the following one
∣∣− (c13 + c55) m2
1 + (c23 + c44) m22
∣∣ < c55m2
1 + c44m22 +
+ √c33
√c11m4
1 + c22m42 − 2c12m2
1m22, for all (m1, m2) �= 0. (3.21)
Let us first assume that the inequality (3.18) holds for all non–zero vectors (n1, n2),(m1, m2). If we set n1 = αm1, n2 = αm2, with α ∈ R , into relation (3.18), then weobtain the relation (3.20). In a similar manner we can prove that (3.19) implies (3.21).
Conversely, assume that the relation (3.20) holds. If we set m1 → √|n1m1| andm2 → √|n2m2| into (3.20), we obtain
for all non–zero vectors (n1, n2), (m1, m2). Further, we note that by means of therelation (3.4) and the Schwarz inequality, we have
c55 |n1m1| + c44 |n2m2| ≤√
c55m21 + c44m2
2
√c55n2
1 + c44n22, (3.23)
while the relation (3.9) and the arithmetic–geometric mean inequality furnish
2c66 |n1m2| |n2m1| ≤ c66n21m2
2 + c66n22m2
1. (3.24)
Thus, when used in (3.22), the relations (3.23) and (3.24) give the relation (3.18).Assuming now m1 → √|n1m1| and m2 → √|n2m2| into relation (3.21), we obtain
for all non-zero vectors (n1, n2), (m1, m2). When we use the relations (3.23) and (3.24)into relation (3.25) we are led to the relation (3.19).
On this basis we will proceed in what follows to find necessary and sufficientconditions in order to satisfy the relations (3.20) and (3.21).
For later convenience we note that if we set m2 = 0 into relation (3.20) then weobtain
|c13 + c55| < c55 + √c11c33, (3.26)
while, for m1 = 0, the relation (3.21) furnishes
|c23 + c44| < c44 + √c22c33. (3.27)
At this point we note that the restrictions (3.13), (3.26) and (3.27) are equivalentto the following estimates
−1 <c12 + 2c66√
c11c22,
c12√c11c22
< 1,
−1 <c23 + 2c44√
c22c33,
c23√c22c33
< 1,
−1 <c13 + 2c55√
c11c33,
c13√c11c33
< 1. (3.28)
We have now all the preliminary material to establish the restrictions impliedby the inequality (3.17). The discussion is guided by the sign of the two termsentering the left hand side of the relations (3.20) and (3.21), that is the sign ofthe coefficients c13 + c55 and c23 + c44. As we will see later, when c12, c13 and c23
vary according with the restrictions (3.28), the relations (3.20) and (3.21) furnishimproved lower bounds for c12+2c66√
c11c22, c23+2c44√
c22c33and c13+2c55√
c11c33and upper bounds for c12√
c11c22,
c23√c22c33
and c13√c11c33
. In fact, we will prove that all points P (x, y, z), with coordinates
x ∈ { c23+2c44√c22c33
, c23√c22c33
}, y ∈ { c13+2c55√c11c33
, c13√c11c33
}, z ∈ { c12+2c66√c11c22
, c12√c11c22
}, are situated inside of
the region limited by the surface S (x, y, z) ≡ x2 + y2 + z2 − 2xyz − 1 = 0, (x, y, z) ∈(−1, 1) × (−1, 1) × (−1, 1) (see Fig. 1).
12 S. Chirita, et al.
We have to develop our analysis of the relations (3.20) and (3.21) for the followingcases: 1) (c13 + c55) · (c23 + c44) > 0 and 2) (c13 + c55) · (c23 + c44) < 0. When c13 +c55 = 0 or c23 + c44 = 0 the discussion follows the same procedure.
3.1 The Case (c13 + c55) · (c23 + c44) > 0
Throughout this subsection we will assume that (c13 + c55) · (c23 + c44) > 0. Forconvenience we first study the case when
c13 + c55 > 0, c23 + c44 > 0, (3.29)
while the case c13 + c55 < 0, c23 + c44 < 0 will be discussed in short at the end of thissubsection. We recall that the relations (3.26) and (3.27) are equivalent with
−2c55 − √c11c33 < c13 <
√c11c33, − 2c44 − √
c22c33 < c23 <√
c22c33, (3.30)
and therefore, the case in discussion is characterized by
−c55 < c13 <√
c11c33, − c44 < c23 <√
c22c33. (3.31)
Under such hypothesis we will study the restrictions imposed upon the elasticcoefficients by the relations (3.20) and (3.21).
3.1.1 Analysis of Relation (3.20)
In view of the assumption (3.31), relation (3.20) becomes
c13m21 + c23m2
2 <√
c33
√c11m4
1 + c22m42 + 2 (c12 + 2c66) m2
1m22 for all (m1, m2) �= 0.
(3.32)In order to study the above inequality we have to consider the following four cases:
In case (1) inequality (3.32) is trivially satisfied. So we shall proceed to study the case(2). Then relation (3.32) implies
c213m4
1 + c223m4
2 + 2c13c23m21m2
2 < c11c33m41 + c22c33m4
2 + 2c33 (c12 + 2c66) m21m2
2
(3.33)
and hence we have(c11c33 − c2
13
)m4
1 + (c22c33 − c2
23
)m4
2 + 2 [c33 (c12 + 2c66) − c13c23] m21m2
2 > 0. (3.34)
The relation (3.34) is trivially satisfied when c33 (c12 + 2c66) − c13c23 ≥ 0 so we shallassume that
c33 (c12 + 2c66) − c13c23 < 0. (3.35)
Strong ellipticity of anisotropic materials 13
Then we can write the relation (3.34) as(√
c11c33 − c213m2
1 −√
c22c33 − c223m2
2
)2
+ 2 [c33 (c12 + 2c66) − c13c23 +
+√(
c11c33 − c213
) (c22c33 − c2
23
)]
m21m2
2 > 0 for all (m1, m2) �= 0. (3.36)
Therefore, if we set
m22 =
√c11c33 − c2
13
c22c33 − c223
m21 (3.37)
in (3.36), we deduce that
c12 + 2c66√c11c22
>c13√c11c33
c23√c22c33
−√(
1 − c213
c11c33
)(1 − c2
23
c22c33
)> −1, (3.38)
provided the assumption (3.35) holds true. Thus, the relation (3.20) imposes a moresevere restriction upon the term c12+2c66√
c11c22with respect to that described by the relation
(3.28), in which we have only c12+2c66√c11c22
> −1.Let us consider the case (3). If we set m2 = βm1, with β ∈ R, into relation (3.32)
then we can write
c13 + c23β2 <
√c33
√c11 + c22β4 + 2 (c12 + 2c66) β2 for all β ∈ R. (3.39)
From (3.13) we have
−√c11c22 < c12 + 2c66 < 2c66 + √
c11c22 (3.40)
so that if c12 + 2c66 ∈ [0, 2c66 + √
c11c22)
then
c13 + c23β2 ≤ c23β
2 =√
c223β
4 <√
c22c33β4 <
<√
c11c33 + c22c33β4 + 2c33 (c12 + 2c66) β2 (3.41)
and hence inequality (3.39) is trivially satisfied.In what follows we shall discuss the case (3) under the assumption
c12 + 2c66 ∈ (−√c11c22, 0
). (3.42)
We first observe that the relation (3.39) is trivially satisfied when c13 + c23β2 ≤ 0.
Further, for c13 + c23β2 > 0 we set
t =√
β2 + c13
c23(3.43)
into relation (3.39) to obtain
c23t2<√
c33
√
c11 + c22
(t2 − c13
c23
)2
+2 (c12+2c66)
(t2 − c13
c23
)for all t > 0. (3.44)
14 S. Chirita, et al.
It follows that
(c22c33 − c2
23
)t4 + 2
[c33 (c12 + 2c66) − c22c33
c13
c23
]t2 +
+ c33
c223
[c11c2
23 + c22c213 − 2 (c12 + 2c66) c13c23
]> 0 for all t > 0. (3.45)
The relation (3.45) is trivially satisfied when c23 (c12 + 2c66) − c22c13 ≥ 0 so that in thesubsequent analysis we will assume that
c23 (c12 + 2c66) − c22c13 < 0. (3.46)
Since c23 ∈ (0,
√c22c33
)we can write the relation (3.45) in the following form
(√c22c33 − c2
23t2 −√
c33
c223
[c11c2
23 + c22c213 − 2 (c12 + 2c66) c13c23
])2
+
+ 2
{
c33 (c12 + 2c66) − c22c33c13
c23+
+√
c22c33 − c223
√c33
c223
[c11c2
23 + c22c213 − 2 (c12 + 2c66) c13c23
]}
t2 > 0, (3.47)
thus, if we set
t2 = 1√
c22c33 − c223
√c33
c223
[c11c2
23 + c22c213 − 2 (c12 + 2c66) c13c23
], (3.48)
we obtain the following restriction
c33 (c12 + 2c66) − c22c33c13
c23+
+√
c22c33 − c223
√c33
c223
[c11c2
23 + c22c213 − 2 (c12 + 2c66) c13c23
]> 0, (3.49)
that is,
√
1 − c223
c22c33
√c2
13
c11c33+ c2
23
c22c33− 2
c13√c11c33
c23√c22c33
c12 + 2c66√c11c22
>
>c13√c11c33
− c23√c22c33
c12 + 2c66√c11c22
, (3.50)
under the assumption (3.46). Further, from the relation (3.50) we obtain
(c12 + 2c66)2
c11c22+ c2
13
c11c33+ c2
23
c22c33− 2
c13√c11c33
c23√c22c33
c12 + 2c66√c11c22
− 1 < 0, (3.51)
Strong ellipticity of anisotropic materials 15
which can be solved with respect to c12+2c66√c11c22
in order to obtain
− 1 <c13√c11c33
c23√c22c33
−√(
1 − c213
c11c33
)(1 − c2
23
c22c33
)<
c12 + 2c66√c11c22
<
<c13√c11c33
c23√c22c33
+√(
1 − c213
c11c33
)(1 − c2
23
c22c33
), (3.52)
provided that c55 <√
c11c33. Thus, we have again an improvement of the estimategiven by the relation (3.28) for c12+2c66√
c11c22.
The case (4) can be treated in the same manner and hence we are led to
√
1 − c213
c11c33
√c2
13
c11c33+ c2
23
c22c33− 2
c13√c11c33
c23√c22c33
c12 + 2c66√c11c22
>
>c23√c22c33
− c13√c11c33
c12 + 2c66√c11c22
, (3.53)
under the assumption
c13 (c12 + 2c66) − c11c23 < 0. (3.54)
It follows that if we solve inequality (3.53) in terms of c12+2c66√c11c22
, then we are led to thesame expression as in (3.52), provided that c44 <
√c22c33.
Concluding this subsection we note that the relation (3.20) furnishes, in thecase under discussion, the sharper estimates (3.38) and (3.52) for c12+2c66√
c11c22. They
are more restrictive than those described by the relation (3.28) since
c12 + 2c66√c11c22
> κi3 > −1 (3.55)
where
κi3 = xy −
√(1 − x2
) (1 − y2
), (x, y) ∈
(− c44√
c22c33, 1
)×(
− c55√c11c33
, 1
),
(3.56)
and x, y denotes respectively, c23√c22c33
and c13√c11c33
, provided c44 <√
c22c33, c55 <√c11c33.
3.1.2 Analysis of (3.21)
We now study the relation (3.21) under the assumption (3.31). If we set m2 = γ m1,then the relation (3.21) yields
∣∣−(c13 + c55) + (c23 + c44) γ 2∣∣< c55 + c44γ
2+√
c33(c11 + c22γ 4 − 2c12γ 2
)for all γ ∈R.
(3.57)
16 S. Chirita, et al.
Let us assume first that
|γ | ≤√
c13 + c55
c23 + c44. (3.58)
Then, relation (3.57) becomes
c13 + c55 − (c23 + c44) γ 2 < c55 + c44γ2 +
√c33
(c11 + c22γ 4 − 2c12γ 2
)(3.59)
and hence we have
c13 − (c23 + 2c44) γ 2 <
√c33
(c11 + c22γ 4 − 2c12γ 2
)for |γ | ≤
√c13 + c55
c23 + c44. (3.60)
It can be easily seen that inequality (3.60) is satisfied trivially when c13 ∈ (−c55, 0], sowe shall assume that
0 < c13 <√
c11c33. (3.61)
Further, we set
δ = 1
|γ | (3.62)
in (3.60) to rewrite it as
c13δ2 − (c23 + 2c44) <
√c33
(c11δ4 + c22 − 2c12δ2
)for all δ ≥
√c23 + c44
c13 + c55. (3.63)
Inequality (3.63) is trivially satisfied for√
c23+c44c13+c55
≤ δ ≤√
c23+2c44c13
, or when c12 ∈(−2c66 − √
c11c22, 0], so that, in what follows, we shall consider only the case when
δ >
√c23 + 2c44
c13and c12 ∈ (
0,√
c11c22)
(3.64)
and we set
r =√
δ2 − c23 + 2c44
c13. (3.65)
Inequality (3.63) yields
(c11c33 − c2
13
)r4 + 2
c33
c13[c11 (c23 + 2c44) − c12c13] r2 +
+ c33
c213
[c11 (c23+2c44)
2+c22c213−2c12c13 (c23+2c44)
]>0 for all r > 0. (3.66)
Note that the relation (3.66) is trivially satisfied for c11 (c23 + 2c44) − c12c13 ≥ 0, sothat we shall discuss only the situation
c11 (c23 + 2c44) − c12c13 < 0. (3.67)
Strong ellipticity of anisotropic materials 17
We use the relation (3.61) and rewrite (3.66) in the form
(√c11c33 − c2
13r2 −√
c33
c13
√c11 (c23 + 2c44)
2 + c22c213 − 2c12c13 (c23 + 2c44)
)2
+
+ 2
c13c11c33
√c22c33
⎡
⎣c23 + 2c44√c22c33
− c12√c11c22
c13√c11c33
+
+√
1 − c213
c11c33
√(c23 + 2c44)
2
c22c33+ c2
13
c11c33− 2
c12√c11c22
c13√c11c33
c23 + 2c44√c22c33
⎤
⎦ r2 > 0,
(3.68)
for all r ≥ 0. If we set
r2 =√
c33
c13
√c11c33 − c2
13
√c11 (c23 + 2c44)
2 + c22c213 − 2c12c13 (c23 + 2c44), (3.69)
then (3.68) gives
c23 + 2c44√c22c33
− c12√c11c22
c13√c11c33
+
+√
1 − c213
c11c33
√(c23 + 2c44)
2
c22c33+ c2
13
c11c33− 2
c12√c11c22
c13√c11c33
c23 + 2c44√c22c33
> 0
(3.70)
and hence, by solving with respect to c23+2c44√c22c33
, we obtain
− 1 <c13√c11c33
c12√c11c22
−√(
1 − c213
c11c33
)(1 − c2
12
c11c22
)<
c23 + 2c44√c22c33
<
<c13√c11c33
c12√c11c22
+√(
1 − c213
c11c33
)(1 − c2
12
c11c22
)< 1. (3.71)
In a similar manner we can treat the case when
|γ | >
√c13 + c55
c23 + c44, (3.72)
in order to show that inequality (3.57) implies
−1 <c23√c22c33
c12√c11c22
−√(
1 − c223
c22c33
)(1 − c2
12
c11c22
)<
c13 + 2c55√c11c33
<
<c23√c22c33
c12√c11c22
+√(
1 − c223
c22c33
)(1 − c2
12
c11c22
)< 1. (3.73)
18 S. Chirita, et al.
3.1.3 Short Study of the Case c13 + c55 < 0, c23 + c44 < 0
It can be easily seen that (3.77) and (3.78) are similar to (3.31) and (3.32),respectively, and hence we can use the previous analysis in order to obtain thefollowing improved estimates for c12+2c66√
Thus, in the present case the relation (3.20) furnishes the estimates described by(3.79) and (3.80).
Concluding the analysis of this case we can assert that the relation (3.20) furnishesthe following lower bound
c12 + 2c66√c11c22
> κi3 > −1, (3.83)
Strong ellipticity of anisotropic materials 19
where now
κi3 = xy −
√(1 − x2
) (1 − y2
), (x, y) ∈
(−1,− c44√
c22c33
)×(
−1,− c55√c11c33
)
(3.84)
and x is c23√c22c33
and y is c13√c11c33
, provided c44 <√
c22c33, c55 <√
c11c33.Concerning (3.21), we have to note that the analysis follows the same procedure
as in the case c13 + c55 > 0, c23 + c44 > 0 leading to conclusions (3.71) and (3.73).
3.2 The Case (c13 + c55) · (c23 + c44) < 0
For convenience we will first suppose that
c13 + c55 < 0, c23 + c44 > 0. (3.85)
In view of the relation (3.30), we have to study the following situation
−2c55 − √c11c33 < c13 < −c55, − c44 < c23 <
√c22c33. (3.86)
We note that the analysis of (3.20) and (3.21) in this case has some symmetry withrespect to the case (c13 + c55) · (c23 + c44) > 0. Thus, we start our study with therelation (3.21).
3.2.1 Analysis of (3.21)
The relation (3.21) gives
− (c13+2c55) m21 + c23m2
2 <√
c33
√c11m4
1 + c22m42 − 2c12m2
1m22 for all (m1, m2) �= 0.
(3.87)
Using a notation previously introduced in (3.76), that is c13 = − (c13 + 2c55),relation (3.86) becomes
−c55 < c13 <√
c11c33, − c44 < c23 <√
c22c33, (3.88)
while the relation (3.87) reads
c13m21 + c23m2
2 <√
c33
√c11m4
1 + c22m42 − 2c12m2
1m22 for all (m1, m2) �= 0. (3.89)
Thus, we have to discuss the consequences of the relation (3.89) in the followingfour cases:
We note that this inequality is obvious satisfied if c23 ∈ (−c44, 0] or c12 ∈ [0, 2c66 +√c11c22), so that in what follows we shall assume
c23 ∈ (0,
√c22c33
), c12 ∈ (−√
c11c22, 0). (3.112)
The method used to study (3.73) provides here√
1 − c223
c22c33
√c2
23
c22c33+ c2
13
c11c33− 2
c12 + 2c66√c11c22
c13√c11c33
c23√c22c33
>
>c13√c11c33
− c23√c22c33
c12 + 2c66√c11c22
, (3.113)
which when solved with respect to c12+2c66√c11c22
, gives
c13√c11c33
c23√c22c33
−√(
1 − c213
c11c33
)(1 − c2
23
c22c33
)<
c12 + 2c66√c11c22
<
<c13√c11c33
c23√c22c33
+√(
1 − c213
c11c33
)(1 − c2
23
c22c33
). (3.114)
If we consider the case
0 < τ ≤√
−c13 + c55
c23 + c44, (3.115)
Strong ellipticity of anisotropic materials 23
inequality (3.109) gives
− (c13+2c55)−(c23+2c44) τ 2<√
c11c33+c22c33τ 4+2c33 (c12 + 2c66) τ 2, for all τ ∈R.
(3.116)
Using for (3.116) the same procedure as the one used to study (3.59) leads to
√
1−(c23 + 2c44)2
c22c33
√(c23 + 2c44)
2
c22c33+ (c13 + 2c55)
2
c11c33−2
c12 + 2c66√c11c22
c13 + 2c55√c11c33
c23 + 2c44√c22c33
>
>c13 + 2c55√
c11c33− c23 + 2c44√
c22c33
c12 + 2c66√c11c22
, (3.117)
which further gives
c13 + 2c55√c11c33
c23 + 2c44√c22c33
−√√√√[
1 − (c13 + 2c55)2
c11c33
][
1 − (c23 + 2c44)2
c22c33
]
<c12 + 2c66√
c11c22<
<c13 + 2c55√
c11c33
c23 + 2c44√c22c33
−√√√√[
1 − (c13 + 2c55)2
c11c33
][
1 − (c23 + 2c44)2
c22c33
]
. (3.118)
Combining the results obtained in the last two cases we see that (3.20) gives anupper bound for c12√
c11c22as
c12√c11c22
< κs3 < 1, (3.119)
where
κs3 = xy +
√(1 − x2
) (1 − y2
), (x, y) ∈ (−1, 1) × (−1, 1) . (3.120)
3.3 Overview
Collecting together the previously obtained results we conclude that the strongellipticity condition (2.3) is equivalent to the set of restrictions described by therelations (3.4), (3.9) and
−2c66 + κi3
√c11c22 < c12 < κ
s3
√c11c22, (3.121)
where κi3 < κ
s3 are the solutions of the equation in the unknown z
Taking into account the symmetry of the problem, we can deduce that
−2c44 + κi1
√c22c33 < c23 < κ
s1
√c22c33, (3.123)
24 S. Chirita, et al.
where κi1 < κ
s1 are the solutions in x of (3.122). The same argument gives further
−2c55 + κi2
√c11c33 < c13 < κ
s2
√c11c33, (3.124)
where κi2 < κ
s2 are the solutions in y of the equation (3.122), and this completes the
proof.
4 Consequences of Theorem 1
4.1 Strong Ellipticity for Transversely Isotropic Materials
A direct consequence of the above analysis gives the proof of the Theorem 2.Consider the crystal class of the hexagonal division. Then elasticities are restricted byeither the group C10 generated by Rπ/3
e3 , or C11 generated by Rπ/3e3 and Rπ
e2. As noted
in the Introduction, the restrictions imposed on the tensor of elasticity are identicalwith those imposed by transverse isotropy C12 generated by Rϕ
e3 , 0 < ϕ < 2π . In thiscase the relation (3.17) becomes
|c13 + c55| |n1m1 + n2m2| < c55
√(m2
1 + m22
) (n2
1 + n22
)+
+ √c33
√c11 (n1m1 + n2m2)
2 + c66 (n1m2 − n2m1)2. (4.1)
It can be easily shown that (4.1) is a direct consequence of (3.9) and (3.26). Thus,(3.4), (3.9), (3.13), (3.26) and (3.27) show that the strong ellipticity for a transverselyisotropic elastic material is equivalent with
which is exactly (2.9). This completes the proof of Theorem 2 and recover thepreviously results of Payton [6], Padovani [7], Merodio and Ogden [8] and Chirita[9].
4.2 Strong Ellipticity for 4 Classes in the Tetragonal System
Among the seven crystal classes in the tetragonal system, for those listed at item(v) in the Introduction (four classes) restrictions on elasticities are imposed by thesymmetry group C5 generated by Rπ/2
e3 , Rπe1
. The number of elasticities is reduced tosix and an explicit form of the elasticity tensor is obtained collecting together (2.1),(2.2) and (2.10).
In view of the relation (3.26) and by using the inequality c11 > c12, we have
∣∣(c13+c55)(m2
1−m22
)∣∣<(√
c11c33+c55)∣∣m2
1−m22
∣∣≤√c11c33
∣∣m21 − m2
2
∣∣+c55(m2
1 + m22
)≤
≤√
c11c33(m2
1 − m22
)2 + 2c33 (c11 − c12) m21m2
2 + c55(m2
1 + m22
), (4.6)
so that (4.5) is trivially satisfied. It follows that the strong ellipticity condition (2.11)imposes restrictions upon the elastic coefficients only by means of the relation (4.4).
As a consequence, the strong ellipticity condition (2.11) is equivalent with thefollowing conditions
c11 > 0, c33 > 0, c55 > 0, c66 > 0, (4.7)
− 2c66 + κi3c11 < c12 < κ
s3c11, (4.8)
− 2c55 + κi2
√c11c33 < c13 < κ
s2
√c11c33, (4.9)
where
κi3 = 2x2 − 1, κ
s3 = 1, for x ∈ (−1, 1) , (4.10)
κi2 = −
√1 + z
2, κ
s2 =
√1 + z
2, for z ∈ (−1, 1) , (4.11)
and x = c13√c11c33
or x = c13+2c55√c11c33
while z = c12√c11c22
or z = c12+2c66√c11c22
. Thus the proof ofTheorem 3 is complete.
Inequality (4.4) shows that the term c12 − c11 + 2c66 is responsible for the restric-tions described by the relations (4.8) and (4.9). In fact, for c12 − c11 + 2c66 = 0, i.e.,for a transversely isotropic elastic material, the relation (4.4) is a consequence of(3.28) and therefore it imposes no additional restriction other than that alreadyestablished up to that point. In this case we have c12+2c66√
c11c22= 1 and hence κ
i3 = 1, that is
z = 1 and (4.11) gives κi2 = −1 and κ
s2 = 1. Thus, the relation (4.9) reduces to (4.2).
Finally we note that the cubic system can by analyzed by the same method as fortransverse isotropy and so we have the Theorem 4.
5 Concluding Remarks
Collecting together, the results of this paper give a complete characterization forstrong ellipticity in terms of elastic coefficients for exactly 16 crystal classes (among32). The key point is the proof of these conditions in the rhombic case and moresymmetric situations are dealt with as particular cases.
26 S. Chirita, et al.
We have to note that, at a first glance, imposing positivity of the acoustic tensorseems to be a more natural way to address this problem as long as this conditionalready eliminates one of the two vectors m or n. On this line, the Sylvester criteriumprovides a straightforward way toward conditions of positivity and one can obtaineasily both (2.4) and three inequalities similar to (2.5) but where κ
sj = κ
ij = 1 which
is weaker than (2.5). The remaining difficulty is the sixth degree homogeneouspolynomial expression of the determinant of the acoustic tensor, which as provedabove, gives a sharper estimate than the second-order minors of the acoustic tensor.Obviously, the determinant of the acoustic tensor is bi-cubic so that one may expectsome tractable restrictions from the study of conditions imposed on the coefficientsof the third order polynomial in order to insure positivity on R
+. When explicitlywritten, these conditions provide a 12-degree polynomial in two variables and despiteour efforts we could not recover our (known) result on this way.
A careful inspection of the proof shows that one of the key points is the equiva-lence between (3.18)–(3.19) and (3.20)–(3.21) which in itself reduce the number ofindependent variables. Also we note that the intrinsic symmetry of the problem inthe rhombic case leads to a restriction that inherits some symmetry of the problem,and the restrictions imposed by strong ellipticity lead to the simple algebraic surfacex2 + y2 + z2 − 2xyz − 1 = 0.
A natural question is the way to extend to the other crystal classes this result, or,at least, to adapt it in order to cover other anisotropic situations, a problem that willbe addressed in a future work.
Finally, we have to point out that such characterization proved to be useful indiscussing the uniqueness and wave propagation problems [1], loss of ellipticityin the context of nonlinear elasticity of fibre-reinforced materials [2] and in thestudy of spatial behaviour of the constrained anisotropic cylinder [3]. Particularly,we emphasize that our results can be useful in the study of the properties of theauxetic materials, that is materials with extreme and unusual physical properties.Among them, one has polymeric foams with low density, Evans [10], having a highcapacity of absorbtion energy and high resistance to flexural deformations, Gibsonand Ashby [11] thus leading to a significant loss of acoustic wave propagation, Scarpaand Tomlinson [12].
Acknowledgements The authors are very grateful to the reviewers for their suggestions andobservations which led to an essential improvement of the paper. The work of S.C. was supported bygrant CERES–2–Cex06–11–56.
References
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2. Merodio, J., Ogden, R.W.: Instabilities and loss of ellipticity in fiber-reinforced compressiblenonlinearly elastic solids under plane deformation. Int. J. Solids Struct. 40, 4707–4727 (2003)
3. Chirita, S., Ciarletta, M.: Spatial estimates for the constrained anisotropic elastic cylinder. J.Elast. 85, 189–213 (2006)
4. Simpson, H.C., Spector, S.J.: On copositive matrices and strong ellipticity for isotropic elasticmaterials. Arch. Ration. Mech. Anal. 84, 55–68 (1983)
5. Dacorogna, B.: Necessary and sufficient conditions for strong ellipticity of isotropic functions inany dimension. Dyn. Syst. Ser. B 1, 257–263 (2001)
Strong ellipticity of anisotropic materials 27
6. Payton, R.G.: Elastic Wave Propagation in Transversely Isotropic Media. Martinus Nijhoff, TheHague (1983)
12. Scarpa, F., Tomlinson, G.: Theoretical characteristics of the vibration of sandwich plates within-plane negative Poisson’s ratio values. J. Sound Vib. 230, 45–67 (2000)
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