Irreducible decompositions of the elasticity tensor under the linear and orthogonal groups and their physical consequences
aa r X i v : . [ c ond - m a t . o t h e r] N ov Irreducible decompositions of the elasticity tensorunder the linear and orthogonal groups and theirphysical consequences
Yakov Itin and Friedrich W. Hehl Inst. Mathematics, Hebrew Univ. of Jerusalemand Jerusalem College of Technology, Jerusalem 91160, Israel Inst. Theor. Physics, Univ. of Cologne, 50923 K¨oln, Germanyand Dept. Physics & Astron., Univ. of Missouri, Columbia, MO 65211, USAE-mail: [email protected], [email protected]
Abstract.
We study properties of the fourth rank elasticity tensor C within linear elasticitytheory. First C is irreducibly decomposed under the linear group into a “Cauchy piece” S (with15 independent components) and a “non-Cauchy piece” A (with 6 independent components).Subsequently, we turn to the physically relevant orthogonal group, thereby using the metric.We find the finer decomposition of S into pieces with 9+5+1 and of A into those with5+1 independent components. Some reducible decompositions, discussed earlier by numerousauthors, are shown to be inconsistent. — Several physical consequences are discussed. TheCauchy relations are shown to correspond to A = 0. Longitudinal and transverse sound wavesare basically related by S and A , respectively. file GhentCollGroups2014 11.tex, 18 Nov 2014
1. Introduction
The constitutive law in linear elasticity theory for an anisotropic homogeneous body, thegeneralized Hooke law, postulates a linear relation between the two second-rank tensor fields,the stress σ ij and the strain ε kl (see [16, 22, 17, 12]): σ ij = C ijkl ε kl . (1)In standard linear elasticity, the stress and the strain tensors are assumed to be symmetric.Consequently the elasticity tensor has the so called minor symmetries C [ ij ] kl = C ij [ kl ] = 0 . (2)Another restriction for the elasticity tensor is based on the energy consideration. The energydensity of a deformed material is expressed as W = σ ij ε ij . When the Hooke law (2) issubstituted here, this expression takes the form W = 12 C ijkl ε ij ε kl . (3) The Bach parenthesis () and [] denote symmetrization and antisymmetrization, respectively: ( ij ) := { ij + ji } / ij ] := { ij − ji } /
2, see [20]. he right-hand side of (3) involves only those combinations of the elasticity tensor componentswhich are symmetric under permutation of the first and the last pairs of indices. Consequently,the so-called major symmetry , C ijkl − C klij = 0 (4)is assumed. In 3-dimensional space, a fourth rank tensor with the symmetries (2) and (4) has21 independent components.In the literature on elasticity, a special decomposition of C ijkl into two tensorial parts isfrequently used, see, for example, Cowin [5], Campanella & Tonton [4], Podio-Guidugli [19],Weiner [24], and Hauss¨uhl [9]. It is obtained by symmetrization and antisymmetrization of theelasticity tensor with respect to its two middle indices: M ijkl := C i ( jk ) l , N ijkl := C i [ jk ] l , with C ijkl = M ijkl + N ijkl . (5)It is straightforward to show [14] that M and N fulfill the major symmetry (2) but not theminor symmetries (4). Moreover, the tensor M can be further decomposed. Accordingly, thereducible decomposition in (5) does not correspond to a direct sum decomposition of the vectorspace defined by C . The vector spaces of M and C both turn out to be 21-dimensional, that of N is 6-dimensional. Thus, the tensors M and N are auxiliary quantities but, due to the lack ofthe minor symmetries, they they do not represent elasticity tensors, that is, they cannot be usedto characterize a certain material elastically. These quantities are placeholders without directphysical interpretation. However, in the elasticity literature some physical interpretations of thetensors M and N are offered. Still, these results are inconsistent as we showed earlier in [14].In this paper, we present various decompositions of the elasticity tensor based on group-theoretic arguments and discuss some physical applications of these decompositions.The organization of the paper is as follows: In Sec.2 we discuss the relation between thedecomposition of a tensor and the group of transformations acted on the basic vector space.Sec.3 is devoted to the GL (3) decomposition of the elasticity tensor. It is based on the Youngdiagram technique. Similarly, in Sec.4 we treat the case of the SL (3) and in Sec.5 that ofthe SO (3). In Sec.6 we delineate some physical applications of the irreducible decompositiondescribed.
2. Groups of transformations and corresponding decompositions of tensors
In a precise algebraic treatment, a tensor must be viewed as a linear map from a vector spaceto the field of real numbers, rather than as a collection of real valued components.A tensor of rank p is defined as a multi-linear map from the Cartesian product of p copies of V into the field R , T : V × · · · × V | {z } p → R . (6)The set of all tensors T of the rank p compose a vector space by itself, say T . The dimension ofthis tensor space is equal to n p , that is, a 4th rank tensor in 3 dimensions (3d) provides for T = 81 dimensions. As a basis in T , we can take the tensor products of the basis elements of V , e i ⊗ e i ⊗ · · · ⊗ e i p . (7)Accordingly, an arbitrary contravariant tensor of rank p can be expressed as T = T i i ··· i p e i ⊗ e i ⊗ · · · ⊗ e i p . (8)Thus, the properties of the tensor T in (8), in particular its proper decomposition, is relatedto the group of general linear transformations GL (3) acting on the basis of V . This groupencompasses all non-singular 3 × V has a basis e i and a dual cobasis ϑ i . Under the GL (3), the cobasistransforms as ϑ i ′ = L ii ′ ϑ i . We can attribute a volume element to this cobasis, ǫ := 13! ǫ ijk ϑ i ∧ ϑ j ∧ ϑ k ; (9)here ǫ ijk is the totally antisymmetric Levi-Civita permutation symbol, which is ǫ ijk = ± ǫ ijk = 0 otherwise. The primed volumeelement ǫ ′ for the transformed cobasis ϑ i ′ is defined analogously. If we substitute ϑ i ′ = L ii ′ ϑ i ,we find ǫ ′ = 13! ǫ i ′ j ′ k ′ L ii ′ L j j ′ L kk ′ ϑ i ∧ ϑ j ∧ ϑ k . (10)Since ǫ i ′ j ′ k ′ = ǫ ijk = 0 , ± density, ǫ ′ = (det L kk ′ ) − ǫ . (11)If we require that det L kk ′ ! = +1, then the volume element is an invariant (or scalar) and the GL (3) reduces to the special linear group SL (3).Furthermore, we know from elasticity theory that the description of a deformation is based onthe concept of the distance within a continuum. A deformation is meant to be the change of thedistances between nearby points. However, the (Euclidean) distance ds = g := g ij dx i ⊗ dx j assuch, with g ij = diag(1 , , T (3) ⋊ SO (3),the semidirect product of the translations T (3) with the rotations SO (3). Accordingly, atone point in a continuum, the special orthogonal group SO (3) is the one relevant for therepresentations of the tensors in elasticity theory.The SO (3) corresponds to all orthogonal matrices (reciprocal equals to the transpose) withdeterminant +1. If we allow also parity transformations, we arrive at the O (3), the orthogonalgroup, and, if we drop the orthogonality requirement, eventually at the GL (3). Alternatively,we can keep parity invariance in the first step and then arrive at the GL (3) via the special lineargroup SL (3), which has determinant +1: GL (3) ←− (cid:26) SL (3) , det = +1 O (3) , det = ± (cid:27) ←− SO (3) . (12)Incidentally, between the set of subgroups of GL (3) we will use only those that do not preserveany spatial direction. For physical problems that involve some special direction, elastic materialin exterior magnetic field, for instance, smaller subgroups such as GL (2) are also relevant.The process of going down from the GL (3) to the SO (3) is described in detail in Schouten[21], Chap.III. On the level of the O (3), we can define a scalar volume element according to η := p det( g kl ) ǫ . The transformations of the O (3) are volume preserving. The ǫ is a pre metricconcept, whereas the η requires the existence of a metric g .We can collect there results in a table:space volume element line element transformation group V ǫ det L = 0 GL (3) V ǫ det L = +1 SL (3) V η g det L = ± O (3) V η g det L = +1 SO (3) Table 1.
The different groups involved in the irreducible decomposition of the elasticity tensor. This is also discussed in some detail, for example, in [11], Secs.A.1.9 and C.2.3, respectively. lthough the group SO (3) is highly relevant in elasticity theory, it is convenient to providethe decomposition of the elasticity tensor relative to the bigger groups and to arrive at the SO (3) only at the last stage. In this case, we are able to identify the different origins of theirreducible parts and to derive the stratified hierarchy of the invariants. Such a method is alsouseful from the technical point of view because we need to use different algebraic methods fordifferent groups of transformations. We begin with the decomposition relative to the biggestgroup GL (3). GL (3) -decomposition Let us recall some basic facts about 4th rank tensors and their Young decomposition:1) Covariant tensor of 4th rank over the vector space V (with dim V = 3) is a linear map fromthe Cartesian product of 4 copies of the dual space V ∗ into the real numbers, T : V ∗ × V ∗ × V ∗ × V ∗ → R . (13)2) The set of the 4th rank tensors constitutes a new vector space T called a 4th rank tensorspace. Its dimension is equal to 3 = 81.3) The tensor space T can be decomposed into the direct sum of its subspaces that are invariantunder the action of the group GL (3 , R ).4) Due to the well-known Schur-Weyl duality, the irreducible decomposition of the space ofthe p th rank tensors under GL (3 , R ) corresponds to the irreducible decomposition of thepermutation group S .5) A known practical way to derive the irreducible decomposition of S is by use of Youngdiagrams.6) A generic 4th rank tensor over the 3d tensor space can be decomposed into a direct sum offour subspaces. This decomposition is depicted by the Young diagrams, see [3] or [7], ⊗ ⊗ ⊗ = ⊕ ⊕ ⊕ . (14)The left-hand side describes a generic 4th rank tensor. On the right-hand side, the diagramsrepresent the 4 subspaces.7) The diagrams in (14) come with a weight factor f λ , called the dimension of the irreduciblerepresentation λ of the permutation group S p . For a diagram of the shape λ , the f λ -factoris calculated by the use of the hook-length formula, f λ = p ! Q α hook( α ) . (15)Thus, for the diagrams depicted in (14), f λ = 1 , f λ = 3 , f λ = 2 , f λ = 3 , f λ = 1 . (16)8) The dimensions of the irreducible decomposition of GL ( n, R ) are calculated by the Stanleyhook-content formula dim V λ = Y ( α,β ) ∈ λ n + α − β hook( α, β ) . (17)Consequently, for the diagrams depicted in (14) and for the dimension n = 3 of the vectorspace, we havedim V λ = 15 , dim V λ = 15 , dim V λ = 6 , dim V λ = 3 . (18)) Every diagram represents a subspace of the tensor space T , T = (1) T ⊕ (2)
T ⊕ (3)
T ⊕ (4) T . (19)These subspaces intersect only at zero. Moreover, they all are mutually non-isomorphic.Hence we have a direct sum decomposition of the tensor space. The dimension of theinitial tensor space is separated into the dimension of the subspaces as follows:dim T = X i f λ i × dim V λ i . (20)The dimension of the tensor space is distributed between the subspaces according to81 = 1 ×
15 + 3 ×
15 + 2 × × . (21)10) The decomposition (21) is unique but reducible. In accordance with (16), the subspaces (2) T , (3) T , and (4) T can be decomposed still further into sub-subspaces: (2) T = (cid:16) (2 , T ⊕ (2 , T ⊕ (2 , T (cid:17) , (22) (3) T = (cid:16) (3 , T ⊕ (3 , T (cid:17) , (23) (4) T = (cid:16) (4 , T ⊕ (4 , T ⊕ (4 , T (cid:17) . (24)These decompositions are irreducible but not unique. C ijkl The elasticity tensor is not a general 4th rank tensor; rather, it carries its minor and majorsymmetries. Accordingly, we are looking for a a decomposition of an invariant subspace of thetensor space T into a direct sum of its sub-subspaces, C = α (1) C ⊕ β (2) C ⊕ γ (3) C ⊕ δ (4) C , (25)where α = 0 , β = 0 , , , γ = 0 , , δ = 0 , , , . (26)Using the minor and major symmetries, we find α = 1 and δ = 0. Since dim C = 21, we find asa unique solution of (25), α = γ = 1 ; β = δ = 0 . Thus, we proved
Proposition 1:
The decomposition C ijkl = S ijkl + A ijkl , (27)with S ijkl := C ( ijkl ) = 13 ( C ijkl + C iklj + C iljk ) , and A ijkl := 13 (2 C ijkl − C ilkj − C iklj ) , (28)is irreducible and unique. roposition 2: The partial tensors satisfy the minor symmetries, S [ ij ] kl = S ij [ kl ] = 0 and A [ ij ] kl = A ij [ kl ] = 0 , (29)and the major symmetry, S ijkl = S klij and A ijkl = A klij . (30) Proposition 3:
The irreducible decomposition of C signifies the reduction of the tensor space C into a direct sum of two subspaces S ⊂ C (for the tensor S ) and A ⊂ C (for the tensor A ), C = S ⊕ A . (31)In particular, we have dim C = 21 , dim S = 15 , dim A = 6 . (32) Proposition 4:
The irreducible piece A ijkl of the elasticity tensor is a fourth rank tensor.Alternatively, it can be represented as a symmetric second rank tensor density∆ mn := 14 ǫ mil ǫ njk A ijkl , (33)where ǫ ijk = 0 , ± SL (3) -decomposition Since the elasticity tensor C ijkl satisfies the symmetries (2) and (4), most of its contractionswith ǫ ijk vanish. Indeed, for the completely symmetric part the contraction in two indices isidentically zero, ǫ mij S ijkl = 0 . (34)The second part of the elasticity tensor yields a non-vanishing contraction, K mjk := 12 ǫ mil A ijkl . (35)This tensor K ijk has vanishing traces and is antisymmetric in the upper indices, K iik = 0 , K kjk = 0 , K i ( jk ) = 0 . (36)Thus, K ijk has 6 independent components, exactly like the initial tensor A ijkl .Because of the antisymmetry of K ijk , we do not loose anything if we contract the upperindices with ǫ njk : ∆ mn := 12 ǫ njk K mjk = 14 ǫ mil ǫ njk A ijkl . (37)We can check that this tensor is symmetric∆ [ mn ] = 0 . (38)Thus, it has the same 6 independent components. This proves the Proposition 4.Summing up: there are no additional SL (3) invariants of the elasticity tensor, and thistensor cannot be decomposed further under the action of the SL (3). Moreover, we derived arepresentation of A ijkl in terms of ∆ mn . Under the action of the special linear group SL (3), thelatter quantity is an ordinary tensor. . SO (3) -decomposition Since for the elasticity tensor the invariance of the volume element does not yield additionaltensor decompositions, we skip the group O (3) and pass directly to its subgroup SO (3). Considera vector space W endowed with a metric tensor g ij . The norm and the scalar product of vectorsare defined now in the conventional way by ( u, v ) = g ij u i v j and | v | = ( v, v ) / . In order topreserve the scalar product, we must restrict ourselves to the orthogonal group O (3). Invarianceof the volume element η is guaranteed if we restrict ourselves to the group SO (3). Now we canuse the metric tensor g ij and its inverse g kl .From the contraction of the metric with the totally symmetric Cauchy part S ijkl , a uniquesymmetric second rank tensor and its scalar contraction can be constructed, S ij := g kl S ijkl and S := g ij g kl S ijkl . (39)Define the traceless part of the tensor S ij as S ij := S ij − Sg ij , with g ij S ij = 0 . (40)Now we turn to the decomposition of the tensor S ijkl . We define the subtensors (2) S ijkl := α S ( ij g kl ) and (3) S ijkl := βSg ( ij g kl ) . (41)We denote the remaining part as (1) S ijkl := S ijkl − (2) S ijkl − (3) S ijkl . (42)Now we require the tensor (1) S ijkl to be traceless. This yields, α = 67 , β = 15 . (43)Hence we obtain the unique irreducible decomposition of the tensor S ijkl into three pieces, S ijkl = (1) S ijkl + (2) S ijkl + (3) S ijkl . (44)These pieces are invariant under the action of the group SO (3).In order to decompose the non-Cauchy part A ijkl , it is convenient to use its representationby the tensor density ∆ ij . The latter is irreducibly decomposed to a scalar and traceless parts.∆ ij = ∆ ij + 13 ∆ g ij , where ∆ := g ij ∆ ij . (45)Consequently, we obtain the decomposition of A ijkl into two independent parts A ijkl = (1) A ijkl + (2) A ijkl , (46)where the scalar and the traceless parts are given by (2) A ijkl := 23 ∆ (cid:16) g ij g kl − g il g jk (cid:17) and (1) A ijkl := A ijkl − (2) A ijkl , (47)respectively. This way we derived a composition of the elasticity tensor into five irreducibleparts C ijkl = (cid:16) (1) S ijkl + (2) S ijkl + (3) S ijkl (cid:17) + (cid:16) (1) A ijkl + (2) A ijkl (cid:17) . (48)etween the first parentheses we collected the terms corresponding to the Cauchy part, thesecond parentheses enclose the non-Cauchy terms. The dimension of the tensor space of theelasticity tensor is decomposed into the sum of the corresponding dimensions of the subspaces,21 = (9 + 5 + 1) + (5 + 1) . (49)Thus, C ijkl can be represented by one totally traceless and totally symmetric 4th rank tensor (1) C ijkl plus two symmetric traceless 2nd rank tensors S ij , ∆ ij plus two scalars S, ∆. Thisdecomposition is unique and irreducible under the action of the SO (3 , R ).It is quite remarkable that the same decomposition was derived Backus [1] (see also Baerheim[2]) in a rather different way. In our group-theoretical treatment, we obtained all irreduciblepieces in a covariant form. Moreover, we derived a tree of the independent invariances andtheir relations to different transformation groups. We describe this stratified structure in thefollowing diagram: Figure 1.
The decomposition tree of the elasticity tensor C ijkl . Note that ∆ ij is a tensor (and not a density) under the SL (3).
6. Applications
Some non-trivial applications were recently discussed by us in [10, 14] (for the analogous casein electrodynamics, see [13]):
The Cauchy relations are given by N ijkl = 0 or C ijkl = C ikjl , (50)for their history, see Todhunter [23]. The representation in (50) is widely used in the elasticityliterature, see, for example, [8, 5, 6, 4, 19, 24]. In [10, 14], we presented an alternative form ofthese conditions, namely A ijkl = 0 or 2 C ijkl − C ilkj − C iklj = 0 . (51)he last equation can be presented in a more economical form,∆ mn = 0 . (52)The basic difference between (50) on the one hand and (51) or (52) at the other hand is thatour conditions (51) or (52) are represented in an irreducible form. This can have decisiveconsequences.For most materials the Cauchy relations do not hold, even approximately. In fact, theelasticity of a generic anisotropic material is described by the whole set of the 21 independentcomponents, it is not restricted to the set of 15 independent components obeying the Cauchyrelations. This fact seems to nullify the importance of the Cauchy relations for modern solidstate theory and leave them only as historical artifact.However, a lattice-theoretical approach to the elastic constants shows, see Leibfried [15], thatthe Cauchy relations are valid provided (i) the interaction forces between the atoms or moleculesof a crystal are central forces, as, for instance, in rock salt, (ii) each atom or molecule is a centerof symmetry, and (iii) the interaction forces between the building blocks of a crystal can be wellapproximated by a harmonic potential, see Perrin [18]. Accordingly, a study of the violations of the Cauchy relations yields important information about the intermolecular forces of elasticbodies. Accordingly, one should look for the deviation A ijkl of the elasticity tensor C ijkl fromits Cauchy part S ijkl . If u i is the displacement field, the propagation of acoustic waves in anisotropic media isdetermined by the following equation ( ρ = mass density): ρg il ¨ u l − C ijkl ∂ j ∂ k u l = 0 . (53)With a plane wave ansatz u i = U i e i ( ζn j x j − ωt ), we obtain a system of three homogeneous algebraicequations (cid:16) v g il − Γ il (cid:17) U l = 0 , (54)where the Christoffel tensor Γ il := (1 /ρ ) C ijkl n j n k and of the phase velocity v := ω/ζ are used.Substituting the irreducible SA -decomposition into the Christoffel tensor, we obtainΓ il = S il + A il , (55)where the Cauchy and non-Cauchy Christoffel tensors are defined by S il := S ijkl n j n k = S li , and A il := A ijkl n j n k = A li . (56)Acoustic wave propagation in an elastic medium is an eigenvector problem, see (54), with thephase velocity v as the eigenvalues. In general, three distinct real positive solutions correspondto three independent waves (1) U l , (2) U l , and (3) U l , called acoustic polarizations. For isotropicmaterials, there are three pure polarizations: one longitudinal (or compression) wave with ~U × ~n = 0 and two transverse (or shear) waves with ~U · ~n = 0 . In general, for anisotropic materials, three pure modes do not exist. The identification of thepure modes and the condition for their existence is an interesting problem. Let us show howthe irreducible decomposition of the elasticity tensor, which we applied to the Christoffel tensor,can be used here. roposition 5:
Let n i denote an allowed direction for the propagation of a compression wave.Then the velocity v L of this wave in the direction of n i is determined only by the Cauchy partof the elasticity tensor: v L = q S ij n i n j . (57)Moreover, for a medium with a given elasticity tensor, all three purely polarized waves (onelongitudinal and two transverse) can propagate in the direction ~n if and only if S ij n j = S kl n k n l n i . (58)Accordingly, for a given medium, the directions of the purely polarized waves depend on theCauchy part of the elasticity tensor alone . In other words, two materials, with the same Cauchyparts S ijkl of the elasticity tensor but different non-Cauchy parts A ijkl , have the same pure wavepropagation directions and the same longitudinal velocity. Acknowledgments.
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