Isotropic orientation distributions of cubic crystals

Transcription

1 Journal of the Mechanics and Physics of Solids 49 ( Isotropic orientation distributions of cubic crystals T. Bohlke, A. Bertram Otto-von-Guericke-University of Madebur, Institute of Mechanics, Postfach 4120, Madebur, Germany Abstract The determination of the eective elastic properties of areates of crystalline rains with or without texture is a lon standin problem. Presently, such averaes are investiated and their anisotropy is quantied by an anisotropy tensor. The harmonic decomposition of fourth-order tensors is applied to both the elasticity tensors of sinle crystals with cubic symmetry as well as to the eective elasticity tensors of areates of cubic sinle crystals. It is shown that the anisotropic parts of the dierent estimates of the eective stinesses are irreducible. A set of four discrete crystal orientations is presented, which ensures the isotropy of the eective elastic properties, i.e., a vanishin of the harmonic parts of dierent averaes.? 2001 Elsevier Science Ltd. All rihts reserved. Keywords: B. Anisotropic material; B. Elastic material; B. Inhomoeneous material; A. Microstructures; B. Polycrystalline material 1. Introduction Many materials are polycrystals consistin of a lare number of crystalline rains. The eective properties of such areates are dominated by the statistical characteristics of the microstructure, e.., the orientation distributions of the crystal lattice, the rain boundary surfaces, and rain edes. In most cases, the distribution of crystal orientations is of prime importance. Deviations from a uniform crystal orientation distribution cause the eective mechanical or physical properties to be anisotropic (Kocks et al., Usually, for a iven crystal orientation distribution, one needs to determine the correspondin macroscopic properties. In the present paper, our concern is to nd orientations that uarantee isotropic eective properties, where only the elastic Correspondin author. Tel.=fax: address: boehlke@mb.uni-madebur.de (T. Bohlke /01/$ - see front matter? 2001 Elsevier Science Ltd. All rihts reserved. PII: S (

2 2460 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( properties are considered. The main purpose is to formulate an isotropy condition for areates of cubic crystals and to construct exact solutions of that condition with the minimal number of crystal orientations. The outline of the paper is as follows. In Section 2, the harmonic decomposition of the elasticity tensors is specied for the case of cubic sinle crystals. In Section 3, dierent averaes based on the orientation distribution function are used in order to ive simple estimates of the eective elastic properties of textured polycrystals. These estimates correspond to the arithmetic, eometric, and harmonic averaes of the stiness tensors. This approach, althouh less accurate than more sophisticated averain schemes like the self-consistent estimates or nite element calculations, allows the derivation of simple expressions describin the macroscopic anisotropy in terms of the microstructure and ives a rst insiht into the dominatin mechanisms. In the case of areates of cubic crystals, the harmonic decomposition of all these averaes consists of isotropic parts correspondin to a dilatational and a distortional deformation mode, and a complete symmetric and traceless fourth-order tensor with nine independent components. The reduction in the number of independent elastic constants on the macroscale from 21 (eneral triclinic to 11 is caused by the cubic crystal symmetry on the microscale. In Section 4, we discuss the case when the orientation distribution function is such that all of the aforementioned eective stiness tensors are isotropic. An exact solution based only on four sinle crystals with equal volumes is presented. Throuhout the text the summation convention is used. Linear mappins between second-order tensors are written as A = C[B]. The scalar product, the dyadic product, and the Frobenius norm are denoted by A B, A B, and A =(A A 1=2, respectively. Orth represents the set of proper orthoonal second-order tensors (special orthoonal roup. Irreducible, i.e., complete symmetric and traceless tensors are provided with a prime, e.. C. The basic isotropic second-, fourth-, and sixth-order tensors (Zhen and Betten, 1995 are denoted by I, I, and J, respectively I = e i e i ; I = 1 2 e i e j (e i e j + e j e i ; (1 J = 1 8 (e i e j + e j e i (e j e k + e k e j (e k e i + e i e k ; (2 {e i } bein an orthonormal basis. The Rayleih product A? C of a second-order tensor A = A ij e i e j and a fourth-order tensor C = C ijkl e i e j e k e l is dened by A? C = C ijkl Ae i Ae j Ae k Ae l : (3 2. Elastic properties of sinle crystals In classical elasticity, the stress tensor T is iven as a linear map of the strain tensor E (Hooke s law, and vice versa T = C[E]; E = S[T]: (4 For hyperelastic materials, the elasticity tensors possess the major symmetry. The symmetry in the rst and last pair of indices is assumed for convenience as usual. The

3 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( fourth-order stiness tensor C and the correspondin compliance tensor S are specied by the symmetry roup S of the material bein a subroup of the special orthoonal roup, e.., in terms of stinesses C = H? C; H S Orth: (5 As mentioned by Forte and Vianello (1996, to each symmetry transformation H Orth corresponds to H. Therefore, it is sucient to consider a priori only symmetry transformations with a positive determinant. All kinds of physically possible two- and three-dimensional symmetry roups have been classied by Zhen and Boehler (1994. Forte and Vianello (1996 classied all the dierent forms of the linear operators C due to three-dimensional symmetry roups and showed that in the context of fourth-order linear operators only eiht dierent symmetry classes can be distinuished. Due to the major symmetry of the elasticity tensors, the followin projector representations exist (Halmos, 1958; Rychlewski, 1995 C = P ; S = =1 =1 1 P : (6 Here, ( is the number of distinct eienvalues of C. The projectors P map each symmetric second-order tensor into the th eienspace of C. They are idempotent P P = P, biorthoonal P P = O (, and complete =1 P = I. In the case of isotropy, i.e. S = Orth, only two dierent projectors exist P I 1 = 1 3 I I; PI 2 = I P I 1: (7 It is possible to decompose all elasticity tensors of arbitrary symmetry into a direct sum of orthoonal subspaces C = h 1 P I 1 + h 2 P I 2 + H 1 I + I H 1 +4J[H 2]+H ; (8 where H 1, H 2, and H are irreducible, i.e., completely symmetric and traceless, tensors. This representation is called the harmonic decomposition of fourth-order tensors (Schouten, 1924; Spencer, 1970; Cowin, 1989; Boehler et al., 1994; for an overview see Forte and Vianello, h 1;2 are called the rst and second isotropic parts, H 1;2 the rst and second deviatoric parts, and H the harmonic part. Irreducible secondorder tensors have ve and irreducible fourth-order tensors have nine independent components. In the followin, we consider the case of a cubic crystal symmetry. In this case, S consists of 24 elements. They are iven here with respect to the orthonormal crystal lattice vector { i }: (1 the identity; (2 rotations of order 2 about the axes ( i ± j = 2; (i j; (3 rotations of order 3 about the axes ( 1 ± 2 ± 3 = 3; and (4 rotations of order 4 about the axes i (i =1; 2; 3 and their powers of order 2 and 3. The anle of the rotation is determined by # =2=k, k bein the order of the rotation. The cubic elasticity tensors have three distinct eienvalues ( = 3 which can be iven by the components of C with respect to the lattice vectors { i } (i =1; 2; 3:

4 2462 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( = C C 1122, 2 = C 1111 C 1122, and 3 =2C The cubic projectors (Rychlewski and Zhan, 1989; Bertram and Olschewski, 1991 are P C 1 = P I 1; P C 2 = D P C 1 ; P C 3 = I P C 2 P C 1 : (9 The anisotropic part D is specied by a dyadic product of lattice vectors 3 D = i i i i : (10 i=1 The symmetry roup of C is the intersection of the symmetry roups of its harmonic and deviatoric parts (Forte and Vianello, As a result, cubic symmetry forces H 1;2 = 1;2I, and from tr(h 1;2 = 0 one concludes that 1;2 = 0. Therefore, the tensors H 1;2 vanish and the harmonic decomposition of the sinle crystal stiness reduces to C = h 1 P I 1 + h 2 P I 2 + H : (11 The rst and second isotropic parts h 1;2 are determined by a projection of C onto the space of isotropic fourth-order tensors, i.e. P I h = C P I ; =1; 2: (12 2 From Eqs. (7, (9, (11, and (12 the harmonic part H can be determined. As a result, the followin relations hold h 1 = 1 ; h 2 = ; H 1;2 = 0; H = 1 5 ( 3 2 (I I +2I 5D: (13 3. Eective elastic properties Based on the orientation distribution function f dened on Orth, dierent orientation averaes of elasticity tensors can be introduced. The most common mean values are the arithmetic and the harmonic means of the local stiness tensors, which were rst suested by Voit and Reuss, respectively C V = f(c(d; S R = f(s(d C V 1 ; (14 d bein the volume element in Orth (Voit, 1928; Reuss, 1929; Bune, 1968; Morris, The arithmetic and the harmonic means correspond to the assumption of homoeneous strain and stress elds, respectively. These approaches ive upper and lower bounds for the strain enery density (Hill, 1952; Nemat-Nasser and Hori, An approach that focuses on homoenization resultin in unique eective properties, such that the inverse of the eective compliance is equal to the eective stiness, was iven by Aleksandrov and Aisenber (1966 and further developed by Matthies and Humbert (1995. Here, the eometric mean of the local elastic moduli is used ( ( C A = exp f(ln(c( d ; S A = exp f(ln(s( d C A 1 : (15

5 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( The natural loarithm is applied to the eienvalues of C and S, as usual. Predictions based on this approach are similar to self-consistent estimates and close to experimental data (Matthies and Humbert, For a uniform distribution of crystal orientations, f( = 1 holds for all Orth, and the interals ( C VI = C(d; C AI = exp ln(c( d ; S RI = S(d (16 can be expressed solely in terms of the sinle crystal constants. Based on the aforementioned formulae, it is possible to decompose the enerally anisotropic averaes into isotropic and anisotropic parts C V = C VI + V A V ; ln(c A =ln(c AI + A A A ; S R = S RI + R A R : (17 By denition, the norms A V;A;R are zero for a uniform orientation distribution. The scalars V;A;R may be dened such that the norms A V;A;R are equal to one for a sinle crystal orientation. For areates of cubic crystals, the arithmetic mean (14 1 reads (see Eqs. (11 and (13 C V = f(c(d = h 1 P I 1 + h 2 P I 2 + f(h (d ( 2 = 1 P I P I ( 5 ( 3 2 I I +2I 5 f(d(d : (18 In a similar way, the eometric and the harmonic means are obtained ln(c A =ln( 1 P I 1 +ln( 2=5 2 3=5 3 PI (ln( 3 ln( 2 ( I I +2I 5 f(d(d ; S R = 1 ( 2 P I P I ( ( I I +2I 5 f(d(d : (19 It can be easily seen that for a sinle crystal orientation I I +2I 5 f(d(d = 30 (20

6 2464 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( holds. Therefore, for areates of cubic crystals, the scalar factors in Eqs. (17 are iven by ( V 6 = 5 ( 3 2 ; A 6 = 5 (ln( 3 ln( 2 ; R 6 1 = 5 1 : ( Moreover, from Eqs. (18, (19, and (21 it follows that A V = A A = A R =:A ; (22 where ( 30 A = I I +2I 5 f(d(d (23 30 (Bertram and Bohlke, Inspection shows that the tensor A is irreducible, i.e., completely symmetric and traceless A ijkl = A jikl = A klij = A kjil = ; A iike =0: (24 A harmonic decomposition of the dierent averaes (14 and (15 is obtained in a similar way as in the case of sinle crystals. Comparison of Eqs. (8, (11, and (14 ives the arithmetic mean C V = h V 1 P I 1 + h V 2 P I 2 + H 1 V I + I H 1 V +4J[H 2 V ]+H V ; (25 where h V 1 = h 1 ; h V 2 = h 2 ; H 1;2 V = 0; H = V A : (26 In order to determine the elastic anisotropy experimentally, in most cases Youn s modulus E is measured in tensile tests. E is dened as the ratio of stress and strain in the tensile direction d which in the case of the harmonic mean is equivalent to 1 E R (d = d d SR [d d] = 1 3 hr hr 2 + R d d A [d d]; (27 h R 1;2 bein the isotropic parts of the harmonic decomposition of SR. Experiments con- rm that the tensor A actually represents the elastic anisotropy. Texture data of rolled copper have been used to determine the Voit and Reuss averae as well as the Hill approximation of Youn s modulus (Alers and Liu, 1966; Kallend and Davies, A comparison of these predictions with the measured modulus shows that the absolute value of Youn s modulus stronly depends on the type of the averae, but the variation of the modulus with respect to the sample orientation, i.e. E(d, is not very sensitive to the type of averae and corresponds to the experimental ndins. It is well known that for each of the 32 crystal symmetry roups and the ve transversely isotropic roups a sinle tensor can be constructed that characterizes this roup (Zhen and Spencer, Here, the sinle tensor A covers all possible cases of symmetries that can be induced by a texture in an areate of cubic crystals, e.., a triclinic behavior, orthotropy, or isotropy. This description of anisotropy diers from the method based on second-order structure tensors (Boehler, 1987, where the number of structure tensors depends on the specic symmetry to be modeled. In Bertram

7 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( and Bohlke (1999 the texture induced elastic anisotropy is studied numerically for polycrystalline copper. In Bohlke and Bertram (2001 an evolution equation for A is formulated based on the theory of isotropic tensor functions in order to describe the texture induced elastic anisotropy that accompanies metal formin operations. The decomposition (17 with the special properties (22 and (24 may be considered from the point of a Fourier series expansion of the distribution function of crystal orientations f(. If f( is continuous and square interable over Orth then it can be represented by a Fourier series (Adams et al., 1992; Bune, Moreover, the eneralized spherical harmonics constitute a complete orthoonal basis for the continuous and square interable functions over Orth. The tensor spherical harmonics of the expansion are irreducible, i.e. completely symmetric and traceless. The Fourier series expansion of f( in the case of areates of cubic crystals has the property that only even-rank spherical harmonics occur and that the rst tensor spherical harmonic is of rank four (Adams et al., 1992; Bune, Since the strain enery density of the polycrystal is expected to be overned by the tensor coecients of that expansion, the decomposition of the elasticity tensors in an isotropic and a fourth-order irreducible anisotropic part corresponds to the Fourier series expansion. 4. Discrete isotropic orientation distributions In the previous section, it was shown that in the case of areates of cubic crystals the eective elastic behavior was completely determined by 11 constants. Two parameters describe the isotropic behavior and the other nine reect the inuence of the crystalloraphic texture. The reduction in the number of independent elastic constants on the macroscale from 21 (eneral triclinic to 11 (reduced triclinic is caused by the cubic crystal symmetry on the microscale. In this section, we present a set of discrete orientations that satisfy the isotropy condition (Bohlke and Bertram, 1999 A = O; which ives isotropic arithmetic, eometric, and harmonic means. The averaes (14 and (15 can be rearded as special cases of power means (e.., Marcus and Minc, 1992 which are applied to positive denite fourth-order tensors. Inspection shows that the isotropy condition (28 holds irrespective of the order of the power mean. Hence, the sets of crystal orientations satisfyin Eq. (28 uarantee isotropic eective elastic properties for a broad class of micromechanical estimates. Due to the above mentioned irreducibility of A, Eq. (28 contains nine independent equations. The irreducibility of A is equivalent to the followin constraints upon its components: A 45 = 2A 36; A 11 + A 12 + A 13 =0; A 46 = 2A 25; A 12 + A 22 + A 23 =0; A 56 = 2A 14; A 13 + A 23 + A 33 =0; A 44 =2A 23; A 14 + A 24 + A 34 =0; (28

8 2466 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( A 55 =2A 13; A 15 + A 25 + A 35 =0; A 66 =2A 12; A 16 + A 26 + A 36 =0: (29 The components A = B A [B ] refer to the orthonormal basis B of a six-dimensional space (Federov, 1968; Cowin, 1990 B 1 = e 1 e 1 ; 2 B 4 = 2 (e 2 e 3 + e 3 e 2 ; B 2 = e 2 e 2 ; 2 B 5 = 2 (e 1 e 3 + e 3 e 1 ; B 3 = e 3 e 3 ; 2 B 6 = 2 (e 1 e 2 + e 2 e 1 : (30 A formulation of the tensor D with respect to the basis {e i } is obtained by introducin the orthoonal tensor Q = i e i = Q ij e i e j that maps the basis {e i } onto the crystalloraphic basis { i } 3 D ijmn = Q ik Q jk Q mk Q nk : (31 k=1 If only discrete orientations are considered, the averae over the orientation space can be formulated as a weihted sum N f(d(d = D( ; (32 =1 bein the volume fraction correspondin to the orientation. N denotes the total number of orientations. Then the isotropy condition (28 reads N 3 ij mn + im jn + in jm 5 QikQ jkq mkq nk =0: (33 =1 k=1 Each of the orientations Q has three independent components as a result of the orthoonality condition. In Bertram et al. (2000 Eq. (28 is discussed in a roup theoretical context. In that work, it is shown that exact solutions of Eq. (28 can be enerally found for N rains with equal size ( =1=N for all even N larer than 3. In this paper, such a minimal solution for N = 4 is derived by a dierent method. Eq. (28 can be solved numerically with the rst orientation bein equal to the identity tensor, so that nine derees of freedom remain correspondin to the nine independent components of A. The numerical solutions are of the followin kind: 100 +p 1 p 5 p 4 Qij I = 010 ; QII ij = p 2 +p 3 p 5 ; 001 +p 6 +p 2 p 1

9 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( p 5 +p 2 +p 3 p 2 +p 1 +p 6 Qij III = p 1 p 6 +p 2 ; QIV ij = p 5 p 4 p 1 ; (34 +p 4 +p 1 +p 5 +p 3 p 5 +p 2 with the property p 1 + p 2 = p 5 of the components occurrin twice. These constraints, indicated by the numerical solutions, allow the complete determination of a solution with N =4. The quaternion representation of orthoonal tensors is convenient for the subsequent arumentation and implies for the orientation Q II y1 2 + y2 2 y3 2 y 2 4 2y 1 y 4 +2y 2 y 3 2y 1 y 3 +2y 2 y 4 Qij II = 2y 1 y 4 +2y 2 y 3 y1 2 y2 2 + y3 2 y4 2 2y 1 y 2 +2y 3 y 4 (35 2y 1 y 3 +2y 2 y 4 2y 1 y 2 +2y 3 y 4 y1 2 y2 2 y3 2 + y4 2 with y1 2 + y2 2 + y3 2 + y4 2 =1: (36 The constraints iven by Eq. (34 2, i.e. Q21 II = QII 32, QII 11 = QII 33, and QII 12 = QII 23, are fullled identically by settin y 1 = y 3. With the substitutions y 3 = r sin(= 2 and y 4 = r cos(, Eq. (36 ives y 2 = 1 r. One of the two remainin unknowns r and can be eliminated with the aforementioned constraint p 1 + p 2 = p 5, i.e. Q12 II + Q21 II QII 11 = 0. One obtains 1 r = 1 + cos 2 ( 2 (37 2 sin(cos( or, with z = tan(, equivalently z 2 +1 r = z 2 2z 2+2 : (38 Now y i read 2 z y 1 = y 3 = 2 z 2 ; y 2 = 1 2 2( 2 4z ; y 4 = 1 : ( z 2 z By substitutin Eq. (39 into Eqs. (35 and (34 the Frobenius norm of A reduces to A = z3 +7z 2 2+6z 2 2 (z : ( The discriminant of the cubic polynomial in the numerator has the value D = 2600=27. Since the discriminant is neative, three real solutions exist. They have the followin values z 1 = 8 ( 1 5 cos ( 47 3 arccos ; z 2 = 8 ( ( cos 3 3 arccos ; 160 3

10 2468 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( Fi. 1. {100} pole ure. Fi. 2. {111} pole ure. z 3 = 8 3 ( 1 5 sin 6 1 ( 47 3 arccos : ( The symmetry roup S induces equivalent crystal orientations by i H i ; H S Orth Q HQ; H S Orth: (42 The three solutions z 1;2;3 are equivalent in the sense that the correspondin sets of orthoonal matrices Q1;2;3 ( =I; II; III; IV can be transformed by elements of S such that Q1 = Q 2 = Q 3 ( =I; II; III; IV hold. In Fis. 1 and 2, the {100} and the {111} pole ures of this exact solution are shown. In Fi. 3, a three-dimensional plot of cubes representin the four crystal orientations is iven.

11 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( Fi. 3. Three-dimensional plot of cubes representin the four crystal orientations. 5. Conclusions It is shown that certain parts of the harmonic decomposition of the eective stinesses vanish as a result of the cubic symmetry of the constituents formin the areate. An isotropy condition is formulated for areates of cubic crystals. An exact solution with the minimal number of crystal orientations is presented. Solutions of that kind can be used as input data for micro-macro simulations based on Taylor-type models or nite element approaches. Acknowledements The authors thank Prof. Q.-S. Zhen for his interest and helpful comments. References Adams, B.L., Boehler, J.P., Guidi, M., Onat, E.T., Group theory and representation of microstructure and mechanical behavior of polycrystals. J. Mech. Phys. Solids 40 (4, Aleksandrov, K., Aisenber, L., A method of calculatin the physical constants of polycrystalline materials. Sov. Phys. Dokl. 11, Alers, G., Liu, Y.C., Calculation of elastic anisotropy in rolled sheet. Trans. Met. Soc. AIME 236, Bertram, A., Bohlke, T., Simulation of texture induced elastic anisotropy. Compos. Mater. Sci. 16, 2 9. Bertram, A., Olschewski, J., Formulation of anisotropic linear viscoelastic constitutive laws by a projection method. In: Freed, A., Walker, K. (Eds., Hih Temperature Constitutive Modelin: Theory and Application, MD-Vol. 26, AMD-Vol ASME, New York, pp Bertram, A., Bohlke, T., Gake, N., Heiliers, B., Oner, R., On the eneration of discrete isotropic orientation distributions for linear elastic cubic crystals. J. Elast. 58,

12 2470 T. Bohlke, A. Bertram / J. Mech. Phys. Solids 49( Boehler, J., Applications of Tensor Functions in Solid Mechanics. CISM Advanced School, Spriner, Berlin. Boehler, J., Kirillov, A., Onat, E., On the polynomial invariants of the elasticity tensor. J. Elast. 34, Bohlke, T., Bertram, A., An isotropy condition for discrete sets of sinle crystals. Z. Anew. Math. Mech. 79 (S2, S Bohlke, T., Bertram, A., The evolution of Hooke s law due to texture development in polycrystals. Int. J. Sol. Struct., to appear. Bune, H.-J., Uber die elastischen Konstanten kubischer Materialien mit beliebier Textur. Krist. Tech. 3 (3, Bune, H.-J., Texture Analysis in Material Science. Cuviller, Gottinen. Cowin, S., Properties of the anisotropic elasticity tensor. Q. J. Mech. Appl. Math. 42, Cowin, S., Eientensors of linear anisotropic elastic materials. Q. J. Mech. Appl. Math. 43, Federov, F., Theory of Elastic Waves in Crystals. Plenum Press, New York. Forte, S., Vianello, M., Symmetry classes for elasticity tensors. J. Elast. 43, Halmos, P., Finite-Dimensional Vector Spaces. Van Nostrand, New York. Hill, R., The elastic behaviour of a crystalline areate. Proc. Phys. Soc. London A 65, Kallend, J., Davies, G., The elastic and plastic anisotropy of cold-rolled sheets of copper, ildin metal, and -brass. J. Inst. Met. 5, Kocks, U., Tome, C., Wenk, H., Texture and Anisotropy: Preferred Orientations in Polycrystals and Their Eect on Materials Properties. Cambride University Press, Cambride. Marcus, M., Minc, H., A Survey of Matrix Theory and Matrix Inequalities. Dover, New York. Matthies, S., Humbert, M., On the principle of a eometric mean of even-rank symmetric tensors for textured polycrystals. J. Appl. Crystallor. 28, Morris, P., Averain fourth-order tensors with weiht functions. J. Appl. Phys. 40 (2, Nemat-Nasser, S., Hori, M., Micromechanics: Overall Properties of Heteroeneous Materials. Elsevier Science, Amsterdam. Reuss, A., Berechnun der Flierenze von Mischkristallen auf Grund der Plastizitatsbedinun fur Einkristalle. Z. Anew. Math. Mech. 9, Rychlewski, J., Unconventional approach to linear elasticity. Arch. Mech. 47 (2, Rychlewski, J., Zhan, J., Anisotropy deree of elastic materials. Arch. Mech. 47 (5, Schouten, J., Der Ricci-Kalkul. Spriner, Berlin. Spencer, A.J.M., A note on the decomposition of tensors into traceless symmetric tensors. Int. J. En. Sci. 8, Voit, W., Lehrbuch der Kristallphysik. Teubner, Leipzi. Zhen, Q.-S., Betten, J., On the tensor function representations of 2nd-order and 4th-order tensors. Z. Anew. Math. Mech. 75 (4, Zhen, Q.-S., Boehler, J.P., The description, classication, and reality of material and physical symmetries. Acta Mech. 102, Zhen, Q.-S., Spencer, A.J.M., Tensors which characterize anisotropies. Int. J. En. Sci. 31,