Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates

Vietnam Journal of Mechanics, VAST, Vol. 38, No. 3 (2016), pp. 181 – 192 DOI:10.15625/0866-7136/38/3/6055 IMPROVED ESTIMATES FOR THE EFFECTIVE ELASTIC BULKMODULUS OF RANDOM TETRAGONAL CRYSTAL AGGREGATES Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong∗ Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam ∗E-mail: vldong@imech.ac.vn Received April 10, 2015 Abstract. Particular expressions of upper and lower estimates for the macroscopic elastic bulk modulus of random cell tetragonal polycrystalline materials are derived and com- puted for a number of practical crystals. The cell-shape-unspecified bounds, based on minimum energy principles and generalized polarization trial fields, appear close to the simple bounds for specific spherical cell polycrystals. Keywords: Variational bounds, effective elastic bulk modulus, random cell polycrystal, tetragonal crystal. 1. INTRODUCTION Macroscopic (effective) elastic moduli of polycrystalline materials depend on the elastic constants of the base crystal and aggregates’ microstructure, which is often of ran- dom and irregular nature. The first simple estimations for the effective moduli are Voigt arithmetic average and Reuss harmonic average. Using minimum energy and compli- mentary energy principles and constant strain and stress trial fields, respectively, Hill [1] established that Voigt and Reuss averages are upper and lower bounds on the possible values of the effective elastic moduli of orientation-unpreferable polycrystalline aggre- gates. Assuming that the shape and crystalline orientations of the grains within a random polycrystalline aggregates are uncorrelated and using their own variational principles, Hashin and Shtrikman [2] derived the respective second order bounds for the moduli that are significantly tighter than the first order Voigt-Reuss-Hill bounds. Using Hashin- Shtrikman-type polarization trial fields, but comming directly from classical minimum energy principles, Pham [3–6] succeeded in constructing partly third order bounds on elastic moduli of random cell polycrystals, which fall strictly inside the second order Hashin-Shtrikman bounds. More general polarization trial fields have been used to de- rive even tighter bounds (still being partly third order ones, though) in [7], which are c© 2016 Vietnam Academy of Science and Technology 182 Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong specified to the cubic crystals’ case. In this work we resume the approaches and derive the explicit expressions of the estimates for the effective bulk modulus of tetragonal poly- crystals, with particular numerical results for a number of materials. 2. MINIMUM ENERGY PRINCIPLES AND THE BOUNDS Consider a representative volume element V of a polycrystalline aggregate that consists of N components occupying regions Vα ⊂ V of equal volumes vα = v0 (α = 1, . . . , N ) - each component is composed of grains of the same crystalline orientation with respective crystal elastic stiffness tensor C(x) = Cα. A random cell polycrystal (see Fig. 1) is supposed to be represented by such N-component configuration when N → ∞, vα = v0 = 1N → 0 with the crystalline orientations being distributed uniformly in all directions in the space. The effective elastic tensor Ce f f = T(Ke f f , µe f f ) of the polycrystal is defined via the minimum energy expression [3, 4] ε0 : Ce f f : ε0 = inf 〈ε〉=ε0 ∫ V ε : C : εdx , (1) or the minimum complementary energy expression σ0 : (Ce f f )−1 : σ0 = inf 〈σ〉=σ0 ∫ V σ : C−1 : σdx , (2) where the admissible (compatible) strain field ε(x) in (1) is expressed through a displace- ment field u(x) in V ε = 1 2 [∇u+ (∇u)T] , (3) while the stress field σ(x) in (2) is equilibrated in V ∇ · σ = 0 , (4) 〈·〉 means the volume average on V; ε0,σ0 are constant strain and stress fields; T(K, µ) are isotropic fourth rank tensor function Tijkl(K, µ) = Kδijδkl + µ(δikδjl + δilδjk − 23δijδkl) . (5) The strain and stress fields are related via the Hook law σ(x) = C(x) : ε(x). Fig. 1. A random cell polycrystal Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates 183 In [7], the admissible multi-parameter kinematic and static polarization trial fields more general than those in [3–6], which have contain only two free parameters, have been chosen ε ij(x) = ε0δij + N ∑ α=1 [aαikϕ α ,kj + a α jkϕ α ,ki + ba α klψ α ,ijkl ], (6) for (1), and σij(x) = σ0δij + N ∑ α=1 [aαikϕ α ,kj + a α jkϕ α ,ki − (b+ 1)δijaαklϕα,kl − aαij Iα + baαklψα,ijkl ], (7) for (2), where Iα(x) equals to 1 if x ∈ Vα, and to 0 if x /∈ Vα; and we have introduced the harmonic and biharmonic potentials ϕα(x) = − ∫ Vα 1 4pi | x− y |−1 dy , ψα(x) = − ∫ Vα 1 8pi | x− y | dy, (8) (∇2ϕα(x) = ∇4ψα(x) = δαβ , x ∈ Vβ) . The free parameters aαik in Eqs. (6) and (7) are subjected to restrictions (for 〈ε ij〉 = ε0δij, 〈σij〉 = σ0δij) N ∑ α=1 vαaαik = 0 , i, k = 1, 2, 3 . (9) Substituting the trial fields (6) and (7) into the energy functionals of (1) and (2) , and optimizing the respective energy functions over free parameters aαik restricted by Eq. (9), with the help of Lagrange multipliers, and then over b, one obtains the formal bounds on Ke f f [7] KU ≥ Ke f f ≥ KL , (10) where KU(C) = max f1,g1∈(12) min b KU f g(C, f1, g1, b) , (11) KU f g = kV + 〈CKα : A−1α 〉α : 〈A−1α 〉−1α : 〈A−1α : CKα〉α − 〈CKα : A−1α : CKα〉α , KL(C) = min f1,g1∈(12) max b KL f g(C, f1, g1, b) , KL f g = (k−1R + 〈C¯Kα : A¯−1Kα〉α : 〈A¯−1α 〉−1α : 〈A¯−1α : C¯α〉α − 〈C¯Kα : A¯−1α : C¯Kα〉α)−1, the cell (grain) shape parameters f1, g1 are restricted in the ranges 2 3 ≥ f1 ≥ 0 , 67 f1 + 8 35 ≥ g1 ≥ 67 f1 , ( f3 = 2 3 − f1 , g3 = 45 − g1) , (12) KV and KR are Voigt and Reuss averages KV = 1 9 Ciijj , KR = [(C−1)iijj]−1 , (13) 〈·〉α designates an average over all crystalline orientations α; the expressions of Aα, A¯α, CKα, C¯Kα, are given in Appendix A. The shape parameters f1, g1 defined in [4] contain 184 Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong three-point correlation information about the symmetric cell geometry of the random polycrystal. In the procedure briefly presented above, substitution of the trial field (6) [or (7)] into the energy expressions (1) [or (2)] is rather involved, which required statistical sym- metry and isotropy hypotheses for random cell polycrystals and much algebraic manip- ulation [3, 4, 7]. Still, as one can see it directly, the resulted energy expression should be a quadratic form for the free parameters aαik, which are restricted by (9). Hence, the substituted energy expression can be optimized over the free parameters analytically by the Lagrange multiplier method. The energy expression obtained still contain 3 parame- ters, including the free one b and 2 geometric parameters f1, g1 characterizing polycrystal microstructure and restricted by (12). The last optimization operation over these 3 pa- rameters in (11) have to be made numerically for particular cases, as shall be done subse- quently. The bounds will be specified for the random aggregate of tetragonal crystals in the next section. 3. BOUNDS FOR TETRAGONAL CRYSTAL AGGREGATES In the following, Cijkl (i, j, k, l = 1, 2, 3) are designated as the components of the fourth-rank elastic stiffness tensor C in the base-crystal coordinates, and Cpq (p, q = 1, . . . , 6) are the respective convenient Voigt’s two-index notations for the elastic con- stants. In the case of tetragonal crystals (classes 4, 4¯, 4/m), there are 7 independent elas- tic constants C11,C12,C13,C33,C44, C16,C66. The correspondence between the fourth-rank elasticity tensor components in the base crystal reference Cijkl and those in the two-index notation is C11 = C1111 = C2222 , C33 = C3333 , C12 = C1122 , C16 = C1112 = −C2212 , C13 = C1133 = C2233 , C44 = C1313 = C2323 , C66 = C1212 , (14) while the other independent components are zero. The elastic compliance tensor S = C−1 is often given in the respective Voigt’s two-index notation Spq (p, q = 1, . . . , 6) as S11 = S1111 , S33 = S3333 , S12 = S1122 , S16 = 2S1112 , S13 = S1133 , S44 = 4S2323 , S66 = 4S1212 . (15) As such, the 6× 6 matrix {Spq} is really the inverse to the matrix {Cpq}, with par- ticular relations between their components {Spq} = {Cpq}−1 , S11 = 1 2 [ C33 C33(C11 + C12)− 2C213 + C66 C66(C11 − C12)− 2C216 ] , S12 = 1 2 [ C33 C33(C11 + C12)− 2C213 − C66 C66(C11 − C12)− 2C216 ] , (16) S13 = −C13 (C11 + C12)C33 − 2C213 , S33 = C11 + C12 (C11 + C12)C33 − 2C213 , S44 = 1 C44 , S66 = C11 − C12 (C11 − C12)C66 − 2C216 , S16 = −C16 (C11 − C12)C66 − 2C216 . Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates 185 The Voigt and Reuss averages have the particular expressions KV = KV({Cpq}) = 19 (2C11 + 2C12 + 4C13 + C33) , (17) KR = KR({Spq}) = (2S11 + 2S12 + 4S13 + S33)−1 . Inverse (16) and functional (17) relations shall be used repeatedly afterwards. Let the tensors CKα,Aα, C¯Kα, A¯α be given as CK,A, C¯K, A¯ in the base crystal co- ordinates. Since the crystalline orientations α are distributed equally over all directions in the random polycrystalline material space, all the averages 〈·〉α in Eqs. (11) are ten- sor invariants, hence can be calculated in the base-crystal coordinates. For the upper bound KU , we derive the following particular expressions of its constituent tensors as two-index-notation matrices CK = {CKij}, (symmetric 3x3 matrix) , (18) CK11 = (C11 + C12 + C13) 1 9 (1+ 2b 5 ) + bKV 5 = CK22 , (19) CK33 = (C33 + 2C13) 1 9 (1+ 2b 5 ) + bKV 5 , CK12 = C K 13 = C K 23 = 0 , A = {CApq} , CApq = C′Apq + Dpq, (symmetric 6x6 matrices) , (20) C′A11 = C11(B1 + B2) + (C11 + C12 + C13)(B3 + B6) (21) +(C11 + C44 + C66)(B4 + B5) = C′A22 , C′A33 = C33(B1 + B2) + (C33 + 2C13)(B3 + B6) + (C33 + 2C44)(B4 + B5), C′A12 = C12B1 + C66B2 + (C11 + C12 + C13)B3 + (C11 + C44 + C66)B4, C′A13 = C13B1 + C44B2 + (C11 + C33 + C12 + 3C13) 1 2 B3 +(C11 + C33 + 3C44 + C66) 1 2 B4 = C′A23 , C′A44 = C44B1 + (C13 + C44) 1 2 B2 + (C11 + C33 + 3C44 + C66) 1 4 B5 +(C11 + C12 + 3C13 + C33) 1 4 B6 = C′A55 , C′A66 = C66B1 + (C12 + C66) 1 2 B2 + (C11 + C44 + C66) 1 2 B5 +(C11 + C12 + C13) 1 2 B6 , C′A26 = −C′A16 = −C16(B1 + B2), D11 = D22 = D33 = D1 + D2, D12 = D13 = D23 = D1, D44 = D55 = D66 = 1 2 D2, 186 Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong where B1, . . . , B6 and D1, D2 are defined in Appendix A. Further, one finds the 6× 6 sym- metric matrix A−1 = {SApq} = {CApq}−1 [according to relations (14)] . (22) Then A−1 : CK = CK : A−1 = {CACKij } (3× 3 symmetric matrix) , (23) CACK11 = (S A 11 + S A 12)C K 11 + S A 13C K 33 = C AC K22, CACK33 = S A 33C K 33 + 2S A 13C K 11 , C AC K12 = C AC K13 = C AC K23 = 0 , and CK : A−1 : CK = CCACK = 2CK11CACK11 + CK33CACK33 . (24) Now one has [isotropic tensor T is defined in Eq. (5), functionals KR - in Eq. (17)] 〈A−1α 〉α = T (1 9 K−1R ({SApq}), 1 4 M−1R ({SApq}) ) . (25) Hence 〈A−1α 〉−1α = T ( KR({SApq}),MR({SApq}) ) . (26) Also 〈A−1α : CKα〉α = 〈CKα : A−1α 〉α = I 1 3 (2CACK11 + C AC K33) , (27) I is the second rank unit tensor. Finally KU f g = KV + (2CACK11 + C AC K33) 2KR({SApq})− CCACK . (28) The upper bound KU is found from expression (28) and respective optimizing operation in (11). Similarly, for the lower bound KL in (11), we find KL f g = [K−1R + 1 9 (2¯CACK11 + C¯ AC K33) 2K−1V ({C¯Apq})− C¯CACK ]−1 , (29) where the expressions of C¯Apq, C¯ACKpq, C¯ CAC K are given in Appendix B, functional KV is de- fined in Eq. (17). The lower bound KL is found from expression (29) and respective opti- mizing operation in (11). For numerical calculations, we take the tetragonal crystals, elastic constants of which are collected in [8]. The new shape-unspecified bounds KU ,KL are compared with the old bounds KˆU , KˆL of [5], and also with bounds for specific spherical cell polycrystals KUs ,KLs and µUs , µLs , where the shape parameters f1 = g1 = 0, in Tab. 1. In Tab. 2 the respective values of b, f1, g1, where the optimal bounds are reached, are also reported. It is interesting to observe that, in the case of spherical cell polycrystals, the new bounds coincide with those obtained from our old approach [5], which have simple expressions. The new shape-unspecified bounds are closer to the bounds for spherical cell polycrystals compared to the old ones. Unfortunately, up to present, we can not find in the literature any sufficiently-high-accurate experiments on the elastic moduli of random polycrystals, Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates 187 Table 1. The new shape-unspecified bounds KL, KU on the effective bulk modulus of some tetragonal crystal aggregates compared to the old bounds KLold, K U old and the bounds for spherical cell polycrystals KLs , KUs (all in GPa) Crystal KLold K L KLs KUs KU KUold C(CH3OH)4 15.23 15.26 15.41 16.53 16.58 16.60 AgCIO3 35.31 35.32 35.32 35.33 35.33 35.33 SrMoO4 69.86 69.86 69.86 69.88 69.88 69.88 C14H8O4 4.114 4.126 4.128 4.157 4.198 4.199 CaMoO4 80.92 80.92 80.93 80.94 80.94 80.94 Table 2. bL, f L1 , g L 1 , b U , fU1 , g U 1 - the values of the free (b) and shape ( f1,g1) parameters, at which the respective extrema in the bounds on the effective bulk modulus of the random polycrystals of Tab. 1 are reached Crystal bL f L1 g L 1 b U fU1 g U 1 C(CH3OH)4 -1.6020 2/3 0.5714 -0.7875 0 0.2286 AgCIO3 -1.6070 0 0.2286 -0.7928 0 0 SrMoO4 -1.4143 2/3 0.5714 -0.7086 0 0 C14H8O4 -0.8619 0 0.2286 -0.5212 2/3 0.5714 CaMoO4 -1.3817 2/3 0.5714 -0.7025 0 0 in which all experimental data points are collected, not just the rough average one - with just 2 significant digits, for comparison with the bounds. Still, some data collected in [6] seem to support the prediction of our bounds. 4. CONCLUSION In this paper we have derived explicit expressions of the general shape-unspecified upper and lower bounds on the effective elastic bulk modulus of random tetragonal crys- tal aggregates with numerical illustrations. The estimates are expected to predict the scat- ter range of the macroscopic elastic property of practical polycrystalline materials. One can see that the much more general and flexible trial fields (compared to the old ones of [5]) lead only to small improvements indicates again that we might close to the best possible estimates. High-accuracy experiments and numerical simulations on random polycrystals are expected to access the practical value of our theoretical results. Esti- mates for the more-complex effective shear modulus of the tetragonal crystal aggregates shall be the subject of our following study. ACKNOWLEDGEMENT This work is funded by Vietnam’s National Foundation for Science and Technology Development, Project N. 107.02-2013.20. 188 Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong REFERENCES [1] R. Hill. The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society, Sec- tion A, 65, (5), (1952), pp. 349–354. [2] Z. Hashin and S. Shtrikman. A variational approach to the theory of the elastic behaviour of polycrystals. Journal of the Mechanics and Physics of Solids, 10, (4), (1962), pp. 343–352. [3] D. C. Pham. Bounds on the effective shear modulus of multiphase materials. International Journal of Engineering Science, 31, (1), (1993), pp. 11–17. [4] D. C. Pham. New estimates for macroscopic elastic moduli of random polycrystalline aggre- gates. Philosophical Magazine, 86, (2), (2006), pp. 205–226. [5] D. C. Pham. Macroscopic uncertainty of the effective properties of random media and poly- crystals. Journal of Applied Physics, 101, (2), (2007). Doi:10.1063/1.2426378. [6] D. C. Pham. On the scatter ranges for the elastic moduli of random aggregates of general anisotropic crystals. Philosophical Magazine, 91, (4), (2011), pp. 609–627. [7] D. C. Pham. Bounds on the elastic moduli of statistically isotropic multicomponent materi- als and random cell polycrystals. International Journal of Solids and Structures, 49, (18), (2012), pp. 2646–2659. [8] H. H. Landolt and R. Bo¨rnstein. Group III:: Crystal and solid state physics, Vol. 11. Springer- Verlarg, (1979). Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates 189 APPENDIX A General formula for the tensors Aα, A¯α, CKα, C¯Kα appeared in Eqs. (11) of the bounds on the effective bulk modulus of random cell polycrystals [7] CKα = {CKαij } , CKαij = Cαijkk 1 9 (1+ 2b 5 ) + bKVδij 5 ; (30) Aα = A′α +D , A′α = {A ′α ijkl} , D = {Dijkl} , (31) A′αijkl = CαijklB1 + 1 2 (Cαikjl + C α jkil)B2 + 1 2 (Cαijppδkl + C α klppδij)B3 + 1 2 (Cαipjpδkl + C α kplpδij)B4 + 1 4 (Cαipkpδjl + C α jpkpδil + C α jplpδik + C α iplpδjk)B5 + 1 4 (Cαikppδjl + C α jkppδil + C α jlppδik + C α ilppδjk)B6 , B1 = 1 9 (1+ 2b 5 )2 + f1F1 + g1G1 , B2 = f1F2 + g1G2 , B4 = f1F4 + g1G4 , B3 = 2b 45 + 4b2 225 + f1F3 + g1G3 , B5 = f1F5 + g1G5 , B6 = f1F6 + g1G6 , Dijkl = δijδklD1 + 1 2 (δikδjl + δilδjk)D2 , D1 = (KV − 23µV)( f3F1 + g3G1) + µV( f3F2 + g3G2) + 3KV( f3F3 + g3G3) +(KV + 10 3 µV)( f3F4 + g3G4) + 2 3 F7 + 4 5 G7 + b2 25 KV , D2 = 2µV( f3F1 + g3G1) + (KV + 1 3 µV)( f3F2 + g3G2) +(KV + 10 3 µV)( f3F5 + g3G5) + 3KV( f3F6 + g3G6) + 2 3 F8 + 4 5 G8 , F1 = − 115 − 8b 105 − 13b 2 315 , G1 = 43b2 1890 , F2 = 1 10 + 4b 35 + 16b2 315 , G2 = G4 = G6 = − 4b 2 189 , F3 = − 4b105 − 4b2 315 , G3 = − b 2 945 , F4 = F6 = 2b 35 + 16b2 315 , F5 = 1 10 + 4b 35 − 4b 2 315 , G5 = 10b2 189 , F7 = − 4b 2 105 KV + 4b2 63 µV , G7 = b2 630 KV − 5b 2 189 µV , F8 = b2 105 KV − 10b 2 63 µV , G8 = 2b2 63 KV + 5b2 27 µV ; C¯Kα = {C¯Kαij } , C¯Kαij = Sαijpp( 2b 15 − 1 3 )− K−1R δij( 4b 15 + 1 3 ) ; (32) 190 Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong A¯α = A¯′α + D¯ , A¯′α = {A¯ ′α ijkl} , D¯ = {D¯ijkl} , Sα = (Cα)−1 , (33) A¯′αijkl = Sαijkl B¯1 + 1 2 (Sαikjl + S α jkil)B¯2 + 1 2 (Sαipjpδkl + S α kplpδij)B¯4 + 1 2 (Sαijppδkl + S α klppδij)B¯3 + 1 4 (Sαipkpδjl + S α jpkpδil + S α jplpδik + S α iplpδjk)B¯5 + 1 4 (Sαikppδjl + S α jkppδil + S α jlppδik + S α ilppδjk)B¯6 , B¯1 = 1 9 (1− 2b 5 )2 + f1F¯1 + g1G¯1 , B¯2 = f1F¯2 + g1G¯2 , B¯4 = f1F¯4 + g1G¯4 , B¯3 = 2 9 + 4b 45 − 16b 2 225 + f1F¯3 + g1G¯3 , B¯5 = f1F¯5 + g1G¯5 , B¯6 = f1F¯6 + g1G¯6 , D¯ijkl = δijδklD¯1 + 1 2 (δikδjl + δjkδil)D¯2 , D¯1 = ( 1 9 K−1R − 1 6 µ−1R )( f3F¯1 + g3G¯1) + 1 4 µ−1R ( f3F¯2 + g3G¯2) + 1 3 K−1R ( f3F¯3 + g3G¯3) +( 1 9 K−1R + 5 6 µ−1R )( f3F¯4 + g3G¯4) + 2 3 F¯7 + 4 5 G¯7 + 1 9 (1+ 4b 5 )2K−1R , D¯2 = ( 1 2 µ−1R ( f3F¯1 + g3G¯1) + ( 1 9 K−1R + 1 12 µ−1R )( f3F¯2 + g3G¯2) +( 1 9 K−1R + 5 6 µ−1R )( f3F¯5 + g3G¯5) + 1 3 K−1R ( f3F¯6 + g3G¯6) + 2 3 F¯8 + 4 5 G¯8 , F¯1 = − 415 − 16b 105 − 13b 2 315 , G¯1 = 43b2 1890 , F¯2 = 2 5 + 8b 35 + 16b2 315 , G¯2 = G¯4 = G¯6 = − 4b 2 189 , F¯3 = 22b+ 28 105 + 2b2 315 , G¯3 = − b 2 945 , F¯4 = 4b 35 + 16b2 315 , F¯5 = 2 5 + 8b 35 − 4b 2 315 , G¯5 = 10b2 189 , F¯6 = −32b+ 2835 − 8b2 45 , F¯7 = b2 63 µ−1R − 31b2 + 90b+ 63 945 K−1R , G¯7 = b2 5670 K−1R − 5b2 756 µ−1R , F¯8 = 136b2 + 324b+ 189 945 K−1R − 5b2 126 µ−1R , G¯8 = 4b2 1134 K−1R + 5b2 108 µ−1R ; Improved estimates for the effective elastic bulk modulus of random tetragonal crystal aggregates 191 APPENDIX B For the lower bound KL, we have the following particular expressions of its con- stituent two-index-notation matrices C¯K = {C¯Kij} (symmetric 3× 3 matrix) , (34) C¯K11 = (S11 + S12 + S13)( 2b 15 − 1 3 )− 1 KR ( 4b 15 + 1 3 ) , C¯K22 = (S22 + S12 + S23)( 2b 15 − 1 3 )− 1 KR ( 4b 15 + 1 3 ) , C¯K33 = (S33 + 2S13)( 2b 15 − 1 3 )− 1 KR ( 4b 15 + 1 3 ) , C¯K12 = C¯ K 13 = C¯ K 23 = 0 ; A¯ = {S¯Apq} , S¯Apq = S¯′Apq + D¯pq (symmetric 6× 6 matrices) , (35) S¯′A11 = S11(B¯1 + B¯2) + (S11 + S12 + S13)(B¯3 + B¯6) +(S11 + 1 4 S55 + 1 4 S66)(B¯4 + B¯5) = S¯′A22 , S¯′A33 = S33(B¯1 + B¯2) + (S33 + 2S13)(B¯3 + B¯6) + (S33 + 1 2 S44)(B¯4 + B¯5) , S¯′A12 = S12B¯1 + 1 4 S66B¯2 + (S11 + S12 + S13)B¯3 + (S11 + 1 4 S44 + 1 4 S66)B¯4 , S¯′A13 = S13B¯1 + 1 4 S55B¯2 + (S11 + S12 + 3S13 + S33) 1 2 B¯3 +(S11 + S33 + 3 4 S44 + 1 4 S66) 1 2 B¯4 = S¯′A23 , S¯′A44 = S44B¯1 + (S23 + 1 4 S44)2B¯2 + (S11 + S33 + 3 4 S44 + 1 4 S66)B¯5 +(S11 + S12 + 3S13 + S33)B¯6 = S¯′A55 , S¯′A66 = S66B¯1 + (S12 + 1 4 S66)2B¯2 + (2S11 + 1 2 S44 + 1 2 S66)B¯5 +2(S11 + S12 + S13)B¯6 , D¯11 = D¯22 = D¯33 = D¯1 + D¯2, D¯12 = D¯13 = D¯23 = D¯1; D¯44 = D¯55 = D¯66 = 2D¯2 , (36) where B¯1, . . . , B¯6 and D¯1, D¯2 are defined Appendix A. A¯−1 = {C¯Apq} = {S¯Apq}−1 (6× 6 symmetric matrix) . (37) Then A¯−1 : C¯K = C¯K : A¯−1 = {C¯ACKij } (3× 3 symmetric matrix) , (38) C¯ACK11 = (C¯ A 11 + C¯ A 12)C¯ K 11 + C¯ A 13C¯ K 33 = C¯ AC K22 , C¯ACK33 = C¯ A 33C¯ K 33 + 2C¯ A 13C¯ K 11 , C¯ AC K12 = C¯ AC K13 = C¯ AC K23 = 0 , 192 Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong and C¯K : A¯−1 : C¯K = C¯CACK = 2C¯K11C¯ACK11 + C¯K33C¯ACK33 . (39) Now one has 〈A¯−1α 〉α = T ( KV({C¯Apq}),MV({C¯Apq}) ) , (40) 〈A¯−1α 〉−1α = T (1 9 K−1V ({C¯Apq}), 1 4 M−1V ({C¯Apq}) ) , (41) 〈A¯−1α : C¯Kα〉α = 〈C¯Kα : A¯−1α 〉α = I 1 3 (2C¯ACK11 + C¯ AC K33) . (42)