A Irreducible decomposition of tensors of rank p

The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation


In linear anisotropic elasticity, the elastic properties of a medium are described by the fourth rank elasticity tensor . The decomposition of into a partially symmetric tensor and a partially antisymmetric tensors is often used in the literature. An alternative, less well-known decomposition, into the completely symmetric part of plus the reminder , turns out to be irreducible under the 3-dimensional general linear group. We show that the -decomposition is unique, irreducible, and preserves the symmetries of the elasticity tensor. The -decomposition fails to have these desirable properties and is such inferior from a physical point of view. Various applications of the -decomposition are discussed: the Cauchy relations (vanishing of ), the non-existence of elastic null Lagrangians, the decomposition of the elastic energy and of the acoustic wave propagation. The acoustic or Christoffel tensor is split in a Cauchy and a non-Cauchy part. The Cauchy part governs the longitudinal wave propagation. We provide explicit examples of the effectiveness of the -decomposition. A complete class of anisotropic media is proposed that allows pure polarizations in arbitrary directions, similarly as in an isotropic medium.

Key index words: anisotropic elasticity tensor, irreducible decomposition, Cauchy relations, null Lagrangians, acoustic tensor

1 Introduction and summary of the results

Consider an arbitrary point , with coordinates , in an undeformed body. Then we deform the body and the same material point is named , with coordinates . The position of is uniquely determined by its initial position , that is, . The displacement vector is then defined as


We distinguish between lower (covariant) and upper (contravariant) indices in order to have the freedom to change to arbitrary coordinates if necessary.

1.1 Strain and stress

The deformation and the stress state of an elastic body is, within linear elasticity theory, described by means of the strain tensor and the stress tensor . The strain tensor as well as the stress tensor are both symmetric, that is, and , see Love (1927)[25], Landau & Lifshitz (1986)[22], Haussühl (2007)[16], Marsden & Hughes (1983)[26], and Podio-Guidugli (2000)[29]; thus, and both have 6 independent components.

The strain tensor can be expressed in terms of the displacement vector via


whereupon is the metric of the three-dimensional (3d) Euclidean background space and . In Cartesian coordinates, we have , with .

The stress tensor fulfills the momentum law


here is the mass density, the body force density, and a dot denotes the time derivative.

1.2 Elasticity tensor

The constitutive law in linear elasticity for a homogeneous anisotropic body, the generalized Hooke law, postulates a linear relation between the two second-rank tensor fields, the stress and the strain :


The elasticity tensor has the physical dimension of a stress, namely force/area. Hence, in the International System of Units (SI), the frame components of are measured in pascal P, with P:=N/m.

In 3d, a generic fourth-order tensor has 81 independent components. It can be viewed as a generic matrix. Since and are symmetric, certain symmetry relations hold also for the elasticity tensor. Thus,


This is the so-called right minor symmetry. Similarly and independently, the symmetry of the stress tensor yields the so-called left minor symmetry,


Both minor symmetries, (5) and (6), are assumed to hold simultaneously. Accordingly, the tensor can be represented by a matrix with 36 independent components.

The energy density of a deformed material is expressed as . When the Hooke law is substituted, this expression takes the form


The right-hand side of (7) involves only those combinations of the elasticity tensor components which are symmetric under permutation of the first and the last pairs of indices. In order to prevent the corresponding redundancy in the components of , the so-called major symmetry,


is assumed. Therefore, the matrix becomes symmetric and only 21 independent components of are left over.

As a side remark we mention that the components of a tensor are always measured with respect to a local frame ; here numbers the three linearly independent legs of this frame, the triad. Dual to this frame is the coframe , see Schouten (1989)[33] and Post (1997)[30]. For the elasticity tensor, the components with respect to such a local coframe are . They are called the physical components of . For simplicity, we will not set up a frame formalism since it doesn’t provide additional insight in the decomposition of the elasticity tensor, that is, we will use coordinate frames in the rest of our article.

Because of (5), (6), and (8), we have the following

Definition: A 4th rank tensor of type qualifies to describe anisotropic elasticity if

(i) its physical components carry the dimension of force/area (in SI pascal),

(ii) it obeys the left and right minor symmetries,

(iii) and it obeys the major symmetry.

It is then called elasticity tensor (or elasticity or stiffness) and, in general, denoted by .

In Secs. 2.1 and 2.2 we translate our notation into that of Voigt, see Voigt (1928)[39] and Love (1927)[25], and discuss the corresponding 21-dimensional vector space of all elasticity tensors, see also Del Piero (1979)[12].

Incidentally, in linear electrodynamics, see Post (1962)[30], Hehl & Obukhov (2003)[18], and Itin (2009)[20], we have a 4-dimensional constitutive tensor , with . Surprisingly, this tensor corresponds also to a matrix, like in elasticity. The major symmetry is the same, the minor symmetries are those of an antisymmetric pair of indices.

1.3 Decompositions of the 21 components elasticity tensor

In Sec. 2.3, we turn first to an algebraic decomposition of that is frequently discussed in the literature: the elasticity tensor is decomposed into the sum of the two tensors and , which are symmetric or antisymmetric in the middle pair of indices, respectively. We show that and fulfill the major symmetry but not the minor symmetries and that they can be further decomposed. Accordingly, this reducible decomposition does not correspond to a direct sum decomposition of the vector space defined by , as we will show in detail. Furthermore we show that the vector space of is 21-dimensional and that of 6-dimensional.

Often, in calculation within linear elasticity, the tensors and emerge. They are auxiliary quantities, but due to the lack of the minor symmetries, they are not elasticities. Consequently, they cannot be used to characterize a certain material elastically. These quantities are placeholders that are not suitable for a direct physical interpretation.

Subsequently, in Sec. 2.4, we study the behavior of the physical components of under the action of the general linear 3d real group . The commutes with the permutations of tensor indices. This fact yields the well-known relation between the action of and the action of the symmetry group . Without restricting the generality of our considerations, we will choose local coordinate frames for our considerations.

Figure 1. A tensor of rank 4 in 3-dimensional (3d) space has independent components. The 3 dimensions of our image represent this 81d space. The plane depicts the 21 dimensional subspace of all possible elasticity (or stiffness) tensors. This space is span by its irreducible pieces, the 15d space of the totally symmetric elasticity (a straight line) and the 6d space of the difference (also depicted, I am sad to say, as a straight line).— Oblique to is the 21d space of the reducible -tensor and the 6d space of the reducible -tensor. The “plane” is the only place where elasticities (stiffnesses) are at home. The spaces and represent only elasticities, provided the Cauchy relations are fulfilled. Then, and and cut in the 15d space of . Notice that the spaces M and C are intersecting exactly on S.

In this way, we arrive at an alternative decomposition of into two pieces and , which is irreducible under the action of the . The known device to study the action of is provided by Young’s tableaux technique. For the sake of completeness, we will be briefly describe Young’s technique in the Appendix. In an earlier paper, see Hehl & Itin (2002)[17], we discussed this problem already, but here we will present rigorous proofs of all aspects of this irreducible decomposition. It turns our that the space of the -tensor is 15-dimensional and that of the -tensor 6-dimensional.

In Sec. 2.5, we compare the reducible -decomposition of Sec. 2.3 with the irreducible -decomposition of Sec. 2.4 and will show that the latter one is definitely to be preferred from a physical point of view. The formulas for the transition between the - and the -decomposition are collected in the Propositions 9 and 11. The irreducible 4th rank tensor can alternatively be represented by a symmetric 2nd rank tensor (Proposition 10). We visualized our main results with respect to the reducible and the irreducible decompositions of the elasticity tensor in Figure 1, for details, please see Sec.2.5.

The group , which we are using here, provides the basic, somewhat coarse-grained decomposition of the elasticity tensor. For finer types of irreducible decompositions under the orthogonal subgroup of , see, for example, Rychlewski (1984)[31], Walpole (1984)[41], Surrel (1993)[36], Xiao (1998)[43], and the related work of Backus (1970)[2] and Baerheim (1993)[3]. As a result of our presentation, the great number of invariants, which emerge in the latter case, can be organized into two subsets: one related to the piece and the other one to the piece if .

1.4 Physical applications and examples

Having now the irreducible decomposition of at our command, Sec. 3 will be devoted to the physical applications. In Sec.3.1, we discuss the Cauchy relations and show that they correspond to the vanishing of one irreducible piece, namely to or, equivalently, to . As a consequence, the totally symmetric piece can be called the Cauchy part of the elasticity tensor , whereas the piece measures the deviation from this Cauchy part; it is the non-Cauchy part of . The reducible pieces and defy such an interpretation and are not useful for applications in physics. In this sense, we can speak of two kinds of elasticity, a Cauchy type and a non-Cauchy type.

In Sec. 3.2, this picture is brought to the elastic energy and the latter decomposed in a Cauchy part and its excess, the “non-Cauchy part”. In other words, this distinction between two kinds of elasticity is also reflected in the properties of the elastic energy.

A null Lagrangians is such an Lagrangian whose Euler-Lagrange expression vanishes, see Crampin & Saunders (2005)[11]; we also speak of a “pure divergence”. In Sec. 3.3, elastic null Lagrangians are addressed, and we critically evaluate the literature. We show—in contrast to a seemingly widely held view—that for an arbitrary anisotropic medium there doesn’t exist an elastic null Lagrangian. The expressions offered in the corresponding literature are worthless as null Lagrangians, since they still depend on all components of the elasticity tensor. We collect these results in Proposition 12.

In Sec. 3.4, we define for acoustic wave propagation the Cauchy and non-Cauchy parts of the Christoffel (or acoustic) tensor ( unit wave covector, mass density). We find some interesting and novel results for the Christoffel tensor(, see Propositions 13 and 14). In Sec. 3.5, we investigate the polarizations of the elastic wave. We show that the longitudinal wave propagation is completely determined by the Cauchy part of the Christoffel tensor, see Proposition 15. In Proposition 16 a new result is presented on the propagation of purely polarized waves; we were led to these investigations by following up some ideas about the interrelationship of the symmetry of the elasticity tensor and the Christoffel tensor in the papers of Alshits and Lothe (2004)[1] and Bóna et al. (2004, 2007, 2010)[5, 6, 7].

In Sec. 4, we investigate examples, namely isotropic media (Sec. 4.1) and media with cubic symmetry (Sec. 4.2). Modern technology allows modeling composite materials with their effective elastic properties, see Tadmore & Miller (2011)[37]. Irreducible decomposition of the elasticity tensor can be used as a guiding framework for prediction of certain features of these materials. As an example, we presented in Proposition 17 a complete new class of anisotropic materials that allow pure polarizations to propagate in arbitrary directions, similarly as in isotropic materials.

1.5 Notation

We use here tensor analysis in 3d Euclidean space with explicit index notation, see Sokolnikoff (1951)[34] and Schouten (1954, 1989)[32, 33]. Coordinate (holonomic) indices are denoted by Latin letters ; they run over . Since we allow arbitrary curvilinear coordinates, covariant and contravariant indices are used, that is, those in lower and in upper position, respectively, see Schouten (1989)[33]. Summation over repeated indices is understood. We abbreviate symmetrization and antisymmetrization over indices as follows:

The Levi-Civita symbol is given by , for even, odd, and no permutation of the indices , respectively; the analogous is valid for . The metric has Euclidean signature. We can raise and lower indices with the help of the metric. In linear elasticity theory, tensor analysis is used, for example, in Love (1927)[25], Sokolnikoff (1956)[35], Landau & Lifshitz (1986)[22], and Haussühl (2007)[16]. For a modern presentation of tensors as linear maps between corresponding vector spaces, see Marsden & Hughes (1983)[26], Podio-Guidugli (2000)[29], and Hetnarski & Ignaczak (2011)[19].

In Hehl & Itin (2002)[17], we denoted the elasticity moduli differently. The quantities of our old paper translate as follows into the present one: , , , and .

2 Algebra of the decompositions of the elasticity tensor

2.1 Elasticity tensor in Voigt’s notation

The standard “shorthand” notation of is due to Voigt, see Voigt (1928)[39] and Love (1927)[25]. One identifies a symmetric pair of 3d indices with a multi-index that has the range from to :


Then the elasticity tensor is expressed as a symmetric matrix . Voigt’s notation is only applicable since the minor symmetries (5) and (6) are valid. Due to the major symmetry, this matrix is symmetric, . Explicitly, we have


For general anisotropic materials, all the displayed components are nonzero and independent of one another. The stars in both matrices denote those entries that are dependent due to the symmetries of the matrices.

2.2 Vector space of the elasticity tensor

The set of all generic elasticity tensors, that is, of all fourth rank tensors with the minor and major symmetries, builds up a vector space. Indeed, a linear combination of two such tensors, taken with arbitrary real coefficients, is again a tensor with the same symmetries. We denote this vector space by .

Proposition 1.

For the space of elasticity tensors,


The basis of can be enumerated by the elements of the matrix . For instance, the element is related to the basis vector


the element corresponds to the basis vector


the element corresponds to the basis vector


To the element , we relate the basis vector


In this way, a set of vectors is constructed. Since there are no relations between the 21 components , all these vectors are linearly independent. Moreover every elasticity tensor can be expanded as a linear combination of . Thus a basis of of the vector space consists of 21 vectors. ∎

2.3 Reducible decomposition of

Definitions of and and their symmetries

In the literature on elasticity, a special decomposition of into two tensorial parts is frequently used, see, for example, Cowin (1989)[9], Campanella & Tonton (1994)[8], Podio-Guidugli (2000)[29], Weiner (2002)[42], and Haussühl (2007)[16]. It is obtained by symmetrization and antisymmetrization of the elasticity tensor with respect to its two middle indices:


Sometimes the same operations are applied for the second and the fourth indices. Due to the symmetries of the elasticity tensor, these two procedures are equivalent to one another.

We recall that the elasticity tensor fulfills the left and right minor symmetries and the major symmetry:

Proposition 2.

The major symmetry holds for both tensors and .


We formulate the left-hand side of the major symmetries and substitute the definitions given in (16):


Proposition 3.

In general, the minor symmetries do not hold for the tensors and .


We formulate the left minor symmetries for and and use again the definitions from (16):


here indices that are excluded from the (anti)symmetrization are enclosed by vertical bars. Both expressions don’t vanish in general. Moreover, we are immediately led to .

Using the major symmetry of Proposition 2, we recognize that the right minor symmetries and do not hold either, since and . ∎

Consequently, the tensors and do not belong to the vector space and cannot be written in Voigt’s notation. Thus, these partial tensors and themselves cannot serve as elasticity tensors for any material.

Vector spaces of the - and -tensors

We will denote the set of all -tensors by . It is a vector space. Indeed, is defined as a fourth rank tensor which is skew-symmetric in the middle indices and constructed from the elasticity tensor. A linear combination of such tensors will be also skew-symmetric. Moreover, it can be constructed from the tensor , which satisfies the basic symmetries of the elasticity tensor.

A simplest way to describe a finite dimensional vector space, is to write-down its basis. In the case of a tensor vector space, it is enough to enumerate all the independent components of the tensor.

Proposition 4.

For the space of -tensors, .


We can write-down explicitly six components of the “antisymmetric” tensor as


All other components vanish or differ from the components given in (2.3.2) only in sign. Since all the components with are assumed to be independent, then the components of of (2.3.2) are also independent. ∎

The set of all -tensors is also a vector space, which we denote by .

Proposition 5.

For the space of -tensors, .


The dimension of the vector space can also be calculated by considering the independent components of a generic tensor . We find,


All these 21 components are linearly independent, thus the dimension of the vector space is at least 21. However, since every element of is defined in terms of 21 independent elastic constants , the dimension of cannot be greater than 21. ∎

Algebraic properties of and tensors

Observe some principal features of the tensors and :

(i) Inconsistency. In general, a certain component of , say , can be expressed in different ways in terms of the components of a :


On the other hand, we have




we recover the result in (24), but is was achieved with the help of a non-vanishing component of .

(ii) Reducibility. Since, in general, the tensor is not completely symmetric, a finer decomposition is possible,


Accordingly, can be decomposed into three tensorial pieces:


(iii) Vector spaces. The “partial” vector spaces, and , are not subspaces of the vector space and their sum is not equal to .

Thus, the -decomposition is problematic from an algebraic point of view. Our aim is to present an alternative irreducible decomposition with better algebraic properties.

2.4 Irreducible decomposition of

Definitions of and and their symmetries

In Eq.(148) of the Appendix, we decomposed a fourth rank tensor irreducibly. Let us apply it to the elasticity tensor . Since the dimension of 3d space is less than the rank of the tensor, the last diagram in (148), representing , is identically zero. Also the minor symmetries remove some of the diagrams. Dropping the diagrams which are antisymmetric in the pairs of the indices and , we are eventually left with the decomposition


Let us now apply the major symmetry. It can be viewed as pair of simultaneous permutations and . The first and the last diagrams in (29) are invariant under those transformations. The two diagrams in the middle change their signs and are thus identically zero. Thus, we are left with irreducible parts:


In correspondence with the first table, the first subtensor of is derived by complete symmetrization of its indices


Here the parentheses denote the cycles of permutations. Consequently,


On the right hand side, we have a sum of terms with all possible orders of the indices. If we take into account the symmetries (5), (6), and (8) of , we can collect the terms:


According to (30), the second irreducible piece of can be defined as


This result can also be derived by evaluating the last diagram in (30). Substitution of (33) into the right-hand side of (34) yields


If we totally symmetrize the left- and the right-hand sides of (34), an immediate consequence is


If we symmetrize (35) with respect to the indices , we recognize that its right-hand side vanishes. Accordingly, we have the

Proposition 6.

The tensor fulfills the additional symmetry


Vector spaces of the - and -tensors

We denote the -dimensional vector space of by . The irreducible decomposition of signifies the reduction of to the direct sum of its two subspaces, for the tensor , and for the tensor ,


The vector spaces and have only zero in their intersection and the decomposition of the corresponding tensors is unique. According to Proposition 1, the sum of the dimensions of the subspaces is, equal to 21. The two irreducible parts and preserve their symmetries under arbitrary linear frame transformations. In particular, they fulfill the minor and major symmetries of likewise. Accordingly, and , or alone (but not alone, as we will see later) can be elasticity tensors for a suitable material—in contrast to and .

The dimensions of the vector spaces of and can now be easily determined.

Proposition 7.

For the vector space of the tensors ,


The number of independent components of a totally symmetric tensor of rank in n dimensions is or, for dimension 3 and rank 4, , see Schouten (1954)[32]. ∎

According to Proposition 1, we have . Because of (38) and (39), we have

Proposition 8.

For the vector space of the tensors ,


Irreducible parts in Voigt’s notation

In Voigt’s 6d notation we have


The 6 6 matrix has 15 independent components. We choose the following ones [see Voigt (1928), Eq. (36) on p.578],


The 6 6 matrix of has 6 independent components. We choose the following ones,


The decomposition (41) can be explicitly presented as


Here, we use boldface for the independent components of the tensors. Note that all three matrices are symmetric.

2.5 Comparing the - and the -decompositions with each other

We would now like to compare the two different decompositions:


The dimensions of the corresponding vector spaces are displayed explicitly. This makes it immediately clear that can be expressed in terms of and vice versa. Take the antisymmetric part of (45) with respect to and and find:

Proposition 9.

The auxiliary quantity can be expressed in terms of the irreducible elasticity as follows:


Its inverse reads,


Resolve (46) with respect to . For this purpose we recall that obeys the right minor symmetry: . This suggests to take the symmetric part of (46) with respect to and . Then,


Both, and have only 6 independent components. In 3d this means that it must be possible to represent them as a symmetric tensor of 2nd rank. With the operator , we can always map an antisymmetric index pair to a corresponding vector index . The tensor has 4 indices, that is, we have to apply the operator twice. Since obeys the left and right minor symmetries, that is, , the has always to transvect one index of the first pair and one index of the second pair. This leads, apart from trivial rearrangements, to a suitable definition.

Proposition 10.

The irreducible elasticity can be equivalently described by a symmetric 2nd rank tensor


with the inverse


The symmetry of can be readily established:


Eq. (51) can be derived by substituting (50) into its right-hand side and taking care of (46). Eq. (51) then follows by applying (47). ∎

The symmetric 2nd rank tensor (differing by a factor 2) was introduced by Haussühl (1983, 2007), its relation to the irreducible piece was found by Hehl & Itin (2002). By means of (34), can be calculated directly from the undecomposed elasticity tensor:


The tensor , like , can also be expressed in terms of irreducible pieces: We substitute (46) into (45),


and resolve it with respect to :


We collect the terms and find

Proposition 11.

The auxiliary quantity can be expressed in terms of the irreducible elasticities as follows: