International Journal of Engineering Science 49 (2011) 727–731

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International Journal of Engineering Science

journal homepage: www.elsevier .com/locate / i jengsci

On the spectral decomposition of the elasticity tensor for the cubiccrystal system

Lewis WheelerDepartment of Mechanical Engineering, University of Houston, United States

a r t i c l e i n f o

Article history:Received 11 June 2010Received in revised form 12 July 2010Accepted 10 August 2010

Dedicated to colleague and scholar K.R.Rajagopal on the occasion of his 60thbirthday.

Keywords:Elasticity tensorCubic symmetrySpectral decomposition

0020-7225/$ - see front matter � 2011 Elsevier Ltddoi:10.1016/j.ijengsci.2011.03.005

E-mail address: mece16@Central.UH.EDU

a b s t r a c t

The spectral decompostion of the elasticity tensor is presented in direct notation and usedto discuss choices of the principal (eigen-) vectors that yield a dyadic form. The principalvectors, also recognized as 2-tensors, are used to facilitate the physical interpretation ofterms that appear in the spectral representation.

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1. Introduction

The idea of exploiting the major symmetry of the elasticity tensor, the symmetry associated with the existence of a storedenergy function, began perhaps with the observation by Thomson (1856) (Lord Kelvin) that the elasticity tensor possessesprincipal values, which he termed principal elasticities. The notions of principal values and spectral decomposition of theelasticity tensor have been visited often in recent years (Mehrabadi & Cowin, 1990; Mehrabadi, Cowin, & Horgan, 1993;Theocaris & Sokolis, 1998; Theocaris & Sokolis, 1999; Theocaris & Sokolis, 2000; Sutcliffe, 1992), in some cases being con-cerned primarily with the relationship between material symmetry and principal values and in others with representations.Alternatives to the spectral representation appear in a comprehensive study by Walpole (1984), where in particular a form ofthe spectral decomposition appears for crystals possessing cubic symmetry. This representation is based upon an equivalentearlier one due to Thomas (1965). The decomposition found in Walpole (1984) expresses the elasticity tensor in terms ofprojections in a manner that reveals the principal values and the associated invariant subspaces, but does not place imme-diately in evidence an orthonormal set of principal tensors (eigenvectors) of the elasticity tensor. The spectral form as itapplies to the elasticity tensor for anisotropic materials is carefully and rigorously revealed by Rychlewski (1984) in apublication that appeared almost simultaneously with Walpole’s work (1984). Although the scope of Rychlewski (1984)is general, the case of transverse isotropy is discussed in detail.

In the present article we express the spectral decomposition in a dyadic form that explicitly reveals an orthonormal set ofprincipal tensors that serves as a basis for the linear space of symmetric 2-tensors. The work of Sutcliffe (1992) emphasizesdyadic representation, and discusses the spectral theorem in matrix terms. In the present note, we aim within reason to

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728 L. Wheeler / International Journal of Engineering Science 49 (2011) 727–731

remain coordinate free in order to make the conventional crystal basis, in the words of Thomas (1965) the frame of orien-tation vectors, stand out.

2. Preliminaries

We write C for the elasticity operator on the linear space Sym of all symmetric 2-tensors that describes the elastic re-sponse of stress to strain in the linear theory. The operator C is required to possess the major symmetry C ¼ C�, wherethe adjoint C� is defined by

1 In Tvectors

hC½U�;Vi ¼ hU;C�½V �i ð1Þ

for all symmetric 2-tensors U and V. Here the inner product is defined on Sym by

hU;Vi ¼ trUVT : ð2Þ

We use the notation (e1,e2,e3) for a generic right-handed orthonormal set and refer to it as a cartesian frame. For a crystal,we write (a1,a2,a3) for a crystallographic cartesian frame.1 It is with respect to this basis that the conventional components of C

appear in reduced form for a particular symmetry class. For the cubic class presently under consideration the crystallographicbasis is aligned with a basis of primitive lattice vectors. Define Eij 2 Sym through

Eij ¼ symðei � ejÞ ð3Þ

and similarly define Aij in terms of (a1,a2,a3). Then, {Eij} and {Aij} are bases in Sym. Such bases are orthogonal in the innerproduct (2). They are however not normalized, but rather obey

hEij;Ekli ¼12ðdikdjl þ dildjkÞ; ð4Þ

which yields

jEijj2 ¼1; i ¼ j;12 ; i – j:

(ð5Þ

An orthonormal set bE ij is thus formed from Eij by setting

bE ij ¼Eij for i ¼ j;ffiffiffi

2p

Eij for i – j:

(ð6Þ

The set fbE ijg furnishes an orthonormal basis for Sym, as does fbAijg:With the aid of the foregoing bases, we may express the tensor components of C in the inner product form Gurtin (1972)

Cijkl ¼ hEij;C½Ekl�i; ð7Þ

while those for U 2 Sym appear as

Uij ¼ hU; bE iji: ð8Þ

Corresponding to a pair of 2-tensors U and V in Sym, we introduce the dyad of U and V, written as U � V and defined as anoperator on Sym by

ðU � VÞ½W� ¼ hW;ViU for all W 2 Sym: ð9Þ

The product of two dyads is given by

ðU � VÞðM � NÞ ¼ hM;ViU � N: ð10Þ

We write I for the identity 2-tensor and to distinguish it from the identity operator on Sym we denote the identity on Symby I. The components Iijkl of I are given by

Iijkl ¼ hEij; I½Ekl�i ¼ hEij;Ekli ¼12ðdikdjl þ dildjkÞ: ð11Þ

Further, I admits the dyadic decomposition

I ¼ bE11 � bE11 þ bE22 � bE22 þ bE33 � bE33 þ bE12 � bE12 þ bE13 � bE13 þ bE23 � bE23; ð12Þ

ordinarily referred to as the resolution of the identity. Alternatively, in terms of the orthogonal basis {Eij},

I ¼ E11 � E11 þ E22 � E22 þ E33 � E33 þ 2 E12 � E12 þ E13 � E13 þ E23 � E23ð Þ: ð13Þ

homas (1965), a1, a2, a3 are referred to as orientation vectors, since they describe the orientation of the crystal lattice. For the cubic case, the orientationcoincided in direction with primitive lattice vectors.

L. Wheeler / International Journal of Engineering Science 49 (2011) 727–731 729

These result holds for an arbitrary cartesian frame (e1,e2,e3) and therefore hold in particular for a crystallographic frame.

3. Decomposition into orthogonal projections

In the presence of cubic symmetry, the nonvanishing components of C in a cartesian crystallographic frame (a1,a2,a3) are

C1111 ¼ C2222 ¼ C3333 ¼ c11;

C1122 ¼ C1133 ¼ C2233 ¼ c12;

C1212 ¼ C1313 ¼ C2323 ¼ c44;

ð14Þ

where cIJ (I, J = 1, . . . ,6) refer to matrix elements in the Voigt system (Nye, 1957).In the presence of cubic symmetry, the operator C admits the following spectral decomposition due to Thomas (1965)

(see also Walpole, 1984)

C ¼ a1P1 þ a2P2 þ a3P3; ð15Þ

where

a1 ¼ c11 þ 2c12; a2 ¼ c11 � c12; a3 ¼ 2c44 ð16Þ

and

P1 ¼13

I � I; P2 ¼ K� 13

I � I; P3 ¼ I�K; ð17Þ

K ¼ A11 � A11 þ A22 � A22 þ A33 � A33: ð18Þ

The operators P1, P2, P3 are orthogonal projections having the respective ranks 1, 2, 3. Being orthogonal projections, Pk havethe properties

P2k ¼ Pk;

PkPl ¼ 0; k – l:ð19Þ

The principal elasticities of C are a1, a2, a3, which are of multiplicity 1, 2, 3, respectively. We may thus write

C ¼ ðc11 þ 2c12Þ13

I � I þ ðc11 � c12ÞðK�13

I � IÞ þ 2c44ðI�KÞ: ð20Þ

An expression for C as a linear combination of I; K and P1 appears in indicial form in Thomas (1965), but the properties (19)are not emphasized. The spectral form (20) is established in Walpole (1984). Positive definiteness of the elasticity tensor cor-responds to positive principal elasticities. The corresponding restrictions on the Voigt components can be read directly from(20),

c11 þ 2c12 > 0; c11 � c12 > 0; c44 > 0:

The compliance tensor S, the inverse of the operator C, is found from (20) by replacing the coefficients with their reciprocals,

S ¼ 1c11 þ 2c12

13

I � I þ 1c11 � c12

K� 13

I � I� �

þ 12c44

ðI�KÞ:

The operator K is easily shown to be a projection, in other words it is self adjoint (symmetric) and obeys

K2 ¼ K: ð21Þ

This operator conveys the dependence C on the imbedded frame (a1,a2,a3) of orientation vectors. If c44 = (c11 � c12)/2 then C

is isotropic, the operator K drops out.

4. Dyadic decomposition of P2 and P3:

As it stands, the decomposition (15) does not readily reveal the principal tensors of C, though it would if P2 and P3 weresuitably expressed in dyadic form similar to the rank one projection P1: The dyadic representation of P1 enables us to see byinspection that I is a principal tensor. Our goal is thus to express P2 and P3 as combinations of dyads that are mutuallyorthogonal projections. In this regard, we recall that

rankP1 ¼ 1; rankP2 ¼ 2; rankP3 ¼ 3: ð22Þ

Concerning P3, we note that

P3 ¼ bA12 � bA12 þ bA13 � bA13 þ bA23 � bA23: ð23Þ

The dyads expressed here represent orthogonal projections whose products vanish. Each furnishes a principal tensor. Theform (23) is easily confirmed with the aid of (17) and by resolving the identity in terms of the orthonormal basis fbAijg:As for P2, we assert

730 L. Wheeler / International Journal of Engineering Science 49 (2011) 727–731

P2 ¼16ð2A11 � A22 � A33Þ � ð2A11 � A22 � A33Þ þ

12ðA22 � A33Þ � ðA22 � A33Þ: ð24Þ

This form is based upon results in Theocaris and Sokolis (2000). It easily verified by expanding the right hand member andthat of (17) to see that they yield identical linear combinations of Aij � Akl. An alternative form,

P2 ¼13½ A11 � A22ð Þ � ðA11 � A22Þ þ ðA11 � A33Þ � ðA11 � A33Þ þ ðA22 � A33Þ � ðA22 � A33Þ� ð25Þ

is ascribed in Theocaris (2000) to Walpole (1984), Walpole (1984). One way to arrive at the form (24) is to apply the Gram-Schmidt process to a pair of the principal tensors made evident by (25). In this manner, the pair

A22 � A33;A11 � A22 ð26Þ

generates the orthonormal pair

1ffiffiffi2p ðA22 � A33Þ;

1ffiffiffi6p ð2A11 � A22 � A33Þ: ð27Þ

The result (24) expresses P2 in terms of two rank one orthogonal projections whose product vanishes. We may write

C ¼ a11ffiffiffi3p I � 1ffiffiffi

3p Ia2

1ffiffiffi6p ð2A11 � A22 � A33Þ �

1ffiffiffi6p ð2A11 � A22 � A33Þ þ

1ffiffiffi2p ðA22 � A33Þ �

1ffiffiffi2p ðA22 � A33Þ

� �þ a3

ffiffiffi2p

A12 �ffiffiffi2p

A12 þffiffiffi2p

A13 �ffiffiffi2p

A13 þffiffiffi2p

A23 �ffiffiffi2p

A23

� �; ð28Þ

from which an orthonormal set of principal tensors (eigenvectors) VK (K = 1, . . . ,6) of C is immediately evident:

V1 ¼1ffiffiffi3p I;

V2 ¼1ffiffiffi6p ð2A11 � A22 � A33Þ; V3 ¼

1ffiffiffi2p ðA22 � A33Þ;

V4 ¼ffiffiffi2p

A23; V5 ¼ffiffiffi2p

A13; V6 ¼ffiffiffi2p

A12:

ð29Þ

5. Alternative derivation of the principal tensors V2 and V3

If U is a 2-tensor such that P2½U�– 0, then P2½U� is a principal tensor associated with the principal elasticity a2. Expand U,

U ¼ UijbAij ð30Þ

and apply P2 as expressed in (17) and (18) to arrive at

V ¼ P2½U� ¼ U11 �13

Ukk

� �A11 þ U22 �

13

Ukk

� �A22 þ U33 �

13

Ukk

� �A33: ð31Þ

Clearly trV = 0, so V has the form

V ¼ V11A11 þ V22A22 � ðV11 þ V22ÞA33: ð32Þ

The choices of V11 and V22 that yield V2 and V3 as in (29) are clear.

6. Interpretation of the projections P1;P2 and P3

Consider elements in ranges of Pk: By (17) and (18), if U 2 Sym, then

P1½U� ¼13ðtrUÞI; ð33Þ

P2½U� ¼ U11 �13

Ukk

� �A11 þ U22 �

13

Ukk

� �A22 þ U33 �

13

Ukk

� �A33; ð34Þ

P3½U� ¼ffiffiffi2p

U12A12 þ U13A13 þ U23A23ð Þ: ð35Þ

Thus, the range of P1 is a subspace comprised of 2-tensors that are hydrostatic (sometimes referred to as ‘‘spherical’’ or ‘‘iso-tropic’’). The range of P2 consists of traceless tensors, while that of P3 is made up of pure shear elements for which (a1,a2,a3)are pure shear axes that are aligned with the lattice vectors.

Strictly speaking, an element V of Sym is a pure shear if there is an orthonormal set {k, l,m} such that

hV ;k� ki ¼ hV ; l� li ¼ hV ;m�mi ¼ 0: ð36Þ

The triad {k, l,m} is then a pure shear basis. It is obvious that a pure shear is traceless. The converse, that a traceless tensor isalso a pure shear, is established in Norris (2006) by exhibiting a pure shear basis. The equivalence between pure shear

L. Wheeler / International Journal of Engineering Science 49 (2011) 727–731 731

tensors and traceless tensors has received considerable attention recently. Norris’ work (2006) should be consulted for athorough discussion and further references.

An interesting result is proved in Norris (2006) involving states of stress that extremize the strain energy. If a3 > a2

(a2 > a3) then the strain energy is maximum (minimum) when the pure shear axes of the deviatoric stress are aligned withthe crystallographic directions (a1,a2,a3). The strain energy is minimum (maximum) when the principal axes of stress arealigned with (a1,a2,a3).

References

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15–41.Mehrabadi, M. M., Cowin, S. C., & Horgan, C. O. (1993). Strain energy density bounds for linear anisotropic elastic materials. Journal of Elasticity, 30, 191–196.Norris, Andrew N. (2006). Pure shear axes and elastic strain energy. Quaterly Journal of Mechanics and Applied Mathematics, 59(4). doi:10.1093/qjmam/

hbl015.Nye, J. F. (1957). Physical Properties of Crystals. Oxford University Press (reprinted 2000).Rychlewski, J. (1984). On Hooke’s Law. Journal of Applied Mathematics and Mechanics, 48, 303–314.Sutcliffe, S. (1992). Spectral decomposition of the elasticity tensor. Journal of Applied Mechanics, 59, 762–773.Theocaris, P. S., & Sokolis, D. P. (1998). Spectral decomposition of the compliance tensor for anisotropic plates. Journal of Elasticity, 51, 89–103.Theocaris, P. S., & Sokolis, D. P. (1999). Spectral decomposition of the linear elastic tensor for monoclinic symmetry. Acta Crystallographica, A55, 635–647.Theocaris, P. S., & Sokolis, D. P. (2000). Invariant elastic constants and eigentensors of orthorhombic, tetragonal, hexagonal and cubic crystalline media. Acta

Crystallographica, A56, 319–331.Thomas, T. Y. (1965). On the stress-strain relations for cubic crystals. Proceedings of the National Academy of Sciences, U.S.A., 55, 235–239.Thomson, W. K. (1856). Elements of a mathematical theory of elasticity. Philosophical Transactions of the Royal Society of London, 146, 481–498.Walpole, L. J. (1984). Fourth-rank tensors of the thirty two crystal classes: multiplication tables. Proceedings of the Royal Society of London, A, 391, 149–179.