improved estimates for the effective elastic bulk modulus of random ...

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 BULK MODULUS 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: [email protected]

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 computed 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 random and irregular nature. The first simple estimations for the effective moduli are Voigt arithmetic average and Reuss harmonic average. Using minimum energy and complimentary 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 aggregates. 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 HashinShtrikman-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 derive even tighter bounds (still being partly third order ones, though) in [7], which are c 2016 Vietnam Academy of Science and Technology

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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 polycrystals, 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 = N1 → 0 with the crystalline orientations being distributed uniformly in all directions in the space. The effective elastic tensor Ce f f = T(K e f f , µe f f ) of the polycrystal is defined via the minimum energy expression [3, 4] ε0 : Ce f f : ε0 = inf hεi=ε0

Z

ε : C : εdx ,

(1)

V

or the minimum complementary energy expression 0

σ : (C

e f f −1

)

0

:σ =

inf

hσ i=σ 0

Z

σ : C−1 : σdx ,

(2)

V

where the admissible (compatible) strain field ε(x) in (1) is expressed through a displacement field u(x) in V 1 (3) ε = [∇u + (∇u)T ] , 2 while the stress field σ (x) in (2) is equilibrated in V

∇·σ = 0 ,

(4)

h·i means the volume average on V; ε0 , σ 0 are constant strain and stress fields; T(K, µ) are isotropic fourth rank tensor function 2 Tijkl (K, µ) = Kδij δkl + µ(δik δjl + δil δjk − δij δkl ) . (5) 3 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 N

α α α + aαjk ϕ,ki + baαkl ψ,ijkl ], ∑ [aikα ϕ,kj

(6)

α α α α + aαjk ϕ,ki − (b + 1)δij aαkl ϕ,kl − aijα I α + baαkl ψ,ijkl ], ∑ [aikα ϕ,kj

(7)

ε ij (x) = ε0 δij +

α =1

for (1), and N

σij (x) = σ0 δij +

α =1

for (2), where I α (x) equals to 1 if x ∈ Vα , and to 0 if x ∈ / Vα ; and we have introduced the harmonic and biharmonic potentials R 1 R 1 ϕα (x) = − 4π | x − y |−1 dy , ψα (x) = − 8π | x − y | dy, (8) Vα

Vα

(∇2 ϕα (x)

=

∇4 ψ α ( x )

= δαβ ,

x ∈ Vβ ) .

α in Eqs. (6) and (7) are subjected to restrictions (for h ε i = The free parameters aik ij ε0 δij , hσij i = σ0 δij ) N

∑ vα aikα = 0 ,

i, k = 1, 2, 3 .

(9)

α =1

Substituting the trial fields (6) and (7) into the energy functionals of (1) and (2) , and α restricted by Eq. (9), optimizing the respective energy functions over free parameters aik with the help of Lagrange multipliers, and then over b, one obtains the formal bounds on K e f f [7] KU ≥ K e f f ≥ K L ,

(10)

where KU (C) = max min KU f g (C, f 1 , g1 , b) , f 1 ,g1 ∈(12)

(11)

b

1 −1 −1 −1 −1 KU f g = kV + hCKα : A− α iα : hAα iα : hAα : C Kα iα − h C Kα : Aα : C Kα iα ,

K L (C) = K

Lfg

=

min max K L f g (C, f 1 , g1 , b) ,

f 1 ,g1 ∈(12)

1 (k− R

b

−1

−1

−1

−1

1 −1 ¯ Kα iα : hA ¯ α i− ¯ ¯ ¯ Kα : A ¯ ¯ ¯ + hC α : h Aα : Cα iα − h C Kα : Aα : C Kα iα ) ,

the cell (grain) shape parameters f 1 , g1 are restricted in the ranges 6 8 6 2 ≥ f1 ≥ 0 , f1 + ≥ g1 ≥ f 1 , 3 7 35 7 KV and K R are Voigt and Reuss averages

( f3 =

2 4 − f 1 , g3 = − g1 ) , 3 5

(12)

1 Ciijj , K R = [(C−1 )iijj ]−1 , (13) 9 ¯ α, h·iα designates an average over all crystalline orientations α; the expressions of Aα , A ¯ CKα , CKα , are given in Appendix A. The shape parameters f 1 , g1 defined in [4] contain KV =

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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 symmetry and isotropy hypotheses for random cell polycrystals and much algebraic manipulation [3, 4, 7]. Still, as one can see it directly, the resulted energy expression should α , which are restricted by (9). Hence, the be a quadratic form for the free parameters aik substituted energy expression can be optimized over the free parameters analytically by the Lagrange multiplier method. The energy expression obtained still contain 3 parameters, including the free one b and 2 geometric parameters f 1 , g1 characterizing polycrystal microstructure and restricted by (12). The last optimization operation over these 3 parameters in (11) have to be made numerically for particular cases, as shall be done subsequently. 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 C pq ( p, q = 1, . . . , 6) are the respective convenient Voigt’s two-index notations for the elastic con¯ 4/m), there are 7 independent elasstants. In the case of tetragonal crystals (classes 4, 4, 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 , (14) C13 = C1133 = C2233 , C44 = C1313 = C2323 , C66 = C1212 , 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 S pq ( 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 {S pq } is really the inverse to the matrix {C pq }, with particular relations between their components

{S pq } = {C pq }−1 , i C33 1h C66 + , S11 = 2 2 2 C33 (C11 + C12 ) − 2C13 C66 (C11 − C12 ) − 2C16 i 1h C33 C66 S12 = − , (16) 2 2 2 C33 (C11 + C12 ) − 2C13 C66 (C11 − C12 ) − 2C16 −C13 C11 + C12 S13 = , S33 = , 2 2 (C11 + C12 )C33 − 2C13 (C11 + C12 )C33 − 2C13 1 C11 − C12 −C16 S44 = , S66 = , S16 = . 2 2 C44 (C11 − C12 )C66 − 2C16 (C11 − C12 )C66 − 2C16

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

185

The Voigt and Reuss averages have the particular expressions 1 (2C11 + 2C12 + 4C13 + C33 ) , 9 K R = K R ({S pq }) = (2S11 + 2S12 + 4S13 + S33 )−1 . KV = KV ({C pq }) =

(17)

Inverse (16) and functional (17) relations shall be used repeatedly afterwards. ¯ α be given as CK , A, C ¯ in the base crystal co¯ Kα , A ¯ K, A Let the tensors CKα , Aα , C ordinates. Since the crystalline orientations α are distributed equally over all directions in the random polycrystalline material space, all the averages h·iα in Eqs. (11) are tensor 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 = {CijK },

(symmetric 3x3 matrix) ,

(18)

1 2b bK K K C11 = (C11 + C12 + C13 ) (1 + ) + V = C22 , 9 5 5 1 2b bK K K K K C33 = (C33 + 2C13 ) (1 + ) + V , C12 = C13 = C23 =0, 9 5 5 A A = {C pq },

A C pq = C 0pqA + D pq ,

(symmetric 6x6 matrices) ,

0A = C11 ( B1 + B2 ) + (C11 + C12 + C13 )( B3 + B6 ) C11

+(C11 + C44 + C66 )( B4 + B5 ) = 0A C33 0A C12

0A C22

,

= C33 ( B1 + B2 ) + (C33 + 2C13 )( B3 + B6 ) + (C33 + 2C44 )( B4 + B5 ),

= C12 B1 + C66 B2 + (C11 + C12 + C13 ) B3 + (C11 + C44 + C66 ) B4 , 1 0A C13 = C13 B1 + C44 B2 + (C11 + C33 + C12 + 3C13 ) B3 2 1 0A +(C11 + C33 + 3C44 + C66 ) B4 = C23 , 2 1 1 0A C44 = C44 B1 + (C13 + C44 ) B2 + (C11 + C33 + 3C44 + C66 ) B5 2 4 1 0A +(C11 + C12 + 3C13 + C33 ) B6 = C55 , 4 1 1 0A C66 = C66 B1 + (C12 + C66 ) B2 + (C11 + C44 + C66 ) B5 2 2 1 +(C11 + C12 + C13 ) B6 , 2 0A 0A C26 = −C16 = −C16 ( B1 + B2 ), D11 = D22 = D33 = D1 + D2 , D12 = D13 = D23 = D1 , 1 D44 = D55 = D66 = D2 , 2

(19)

(20) (21)

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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 symmetric matrix A A −1 A−1 = {S pq } = {C pq }

[according to relations (14)] .

(22)

Then AC A−1 : CK = CK : A−1 = {CKij } (3 × 3 symmetric matrix) , AC CK11 AC CK33

= =

A A K A K AC (S11 + S12 )C11 + S13 C33 = CK22 , A K A K AC AC S33 C33 + 2S13 C11 , CK12 = CK13

(23)

AC = CK23 =0,

and K AC K AC CK : A−1 : CK = CKCAC = 2C11 CK11 + C33 CK33 .

Now one has [isotropic tensor T is defined in Eq. (5), functionals K R - in Eq. (17)]  1 1 −1 1 −1 A A K ({ S }) , M ({ S }) . hA− i = T pq pq α α 9 R 4 R Hence   1 −1 A A hA− i = T K ({ S }) , M ({ S }) . R R α α pq pq

(24)

(25)

(26)

Also 1 1 −1 AC AC hA− α : C Kα iα = h C Kα : Aα iα = I (2CK11 + CK33 ) , 3 I is the second rank unit tensor. Finally AC AC 2 A KU f g = KV + (2CK11 + CK33 ) K R ({S pq }) − CKCAC .

(27)

(28)

The upper bound KU is found from expression (28) and respective optimizing operation in (11). Similarly, for the lower bound K L in (11), we find 1 ¯ AC ¯ AC 2 −1 ¯ A ¯ CAC ]−1 , (29) K L f g = [K R−1 + (2C K11 + CK33 ) KV ({C pq }) − CK 9 A,C ¯ AC , C¯ CAC are given in Appendix B, functional KV is dewhere the expressions of C¯ pq Kpq K fined in Eq. (17). The lower bound K L is found from expression (29) and respective optimizing 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 , K L are compared with the old bounds Kˆ U , Kˆ L of [5], and also with bounds for specific spherical cell polycrystals L KsU , KsL and µU s , µs , where the shape parameters f 1 = g1 = 0, in Tab. 1. In Tab. 2 the respective values of b, f 1 , 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 K L , KU on the effective bulk modulus L , KU of some tetragonal crystal aggregates compared to the old bounds Kold old L U and the bounds for spherical cell polycrystals Ks , Ks (all in GPa)

Crystal C(CH3 OH)4 AgCIO3 SrMoO4 C14 H8 O4 CaMoO4

L Kold 15.23 35.31 69.86 4.114 80.92

KL 15.26 35.32 69.86 4.126 80.92

KsL 15.41 35.32 69.86 4.128 80.93

KsU 16.53 35.33 69.88 4.157 80.94

KU 16.58 35.33 69.88 4.198 80.94

U Kold 16.60 35.33 69.88 4.199 80.94

Table 2. b L , f 1L , g1L , bU , f 1U , g1U - the values of the free (b) and shape ( f 1 ,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 C(CH3 OH)4 AgCIO3 SrMoO4 C14 H8 O4 CaMoO4

bL -1.6020 -1.6070 -1.4143 -0.8619 -1.3817

f 1L 2/3 0 2/3 0 2/3

g1L 0.5714 0.2286 0.5714 0.2286 0.5714

bU -0.7875 -0.7928 -0.7086 -0.5212 -0.7025

f 1U 0 0 0 2/3 0

g1U 0.2286 0 0 0.5714 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 crystal aggregates with numerical illustrations. The estimates are expected to predict the scatter 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. Estimates 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.

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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, Section 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 aggregates. Philosophical Magazine, 86, (2), (2006), pp. 205–226. [5] D. C. Pham. Macroscopic uncertainty of the effective properties of random media and polycrystals. 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 materials and random cell polycrystals. International Journal of Solids and Structures, 49, (18), (2012), pp. 2646–2659. ¨ [8] H. H. Landolt and R. Bornstein. Group III:: Crystal and solid state physics, Vol. 11. SpringerVerlarg, (1979).

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APPENDIX A ¯ α , CKα , C ¯ Kα appeared in Eqs. (11) of the General formula for the tensors Aα , A bounds on the effective bulk modulus of random cell polycrystals [7] CKα = {CijKα } ,

Aα = A0α + D ,

bKV δij 2b α 1 CijKα = Cijkk (1 + ) + ; 9 5 5 0

α A0α = {Aijkl },

D = { Dijkl } ,

(30)

(31)

0α 1 α 1 α α α α + Cjkil ) B2 + (Cijpp δkl + Ckl Aijkl = Cijkl B1 + (Cikjl pp δij ) B3 2 2 1 α 1 α α α α α + (Cipjp δkl + Ckpl p δij ) B4 + (Cipkp δjl + C jpkp δil + C jpl p δik + Cipl p δjk ) B5 2 4 1 α α + (Cikpp δjl + Cjkpp δil + Cjlα pp δik + Cilα pp δjk ) B6 , 4 1 2b B1 = (1 + )2 + f 1 F1 + g1 G1 , B2 = f 1 F2 + g1 G2 , B4 = f 1 F4 + g1 G4 , 9 5 2b 4b2 B3 = + + f 1 F3 + g1 G3 , B5 = f 1 F5 + g1 G5 , B6 = f 1 F6 + g1 G6 , 45 225 1 Dijkl = δij δkl D1 + (δik δjl + δil δjk ) D2 , 2 2 D1 = (KV − µV )( f 3 F1 + g3 G1 ) + µV ( f 3 F2 + g3 G2 ) + 3KV ( f 3 F3 + g3 G3 ) 3 10 2 4 b2 +(KV + µV )( f 3 F4 + g3 G4 ) + F7 + G7 + KV , 3 3 5 25 1 D2 = 2µV ( f 3 F1 + g3 G1 ) + (KV + µV )( f 3 F2 + g3 G2 ) 3 2 4 10 +(KV + µV )( f 3 F5 + g3 G5 ) + 3KV ( f 3 F6 + g3 G6 ) + F8 + G8 , 3 3 5 2 2 2 1 8b 13b 43b 1 4b 16b F1 = − − − , G1 = , F2 = + + , 15 105 315 1890 10 35 315 4b2 4b 4b2 b2 G2 = G4 = G6 = − , F3 = − − , G3 = − , 189 105 315 945 2b 16b2 1 4b 4b2 10b2 F4 = F6 = + , F5 = + − , G5 = , 35 315 10 35 315 189 4b2 4b2 b2 5b2 F7 = − KV + µV , G7 = KV − µV , 105 63 630 189 b2 10b2 2b2 5b2 F8 = KV − µV , G8 = KV + µV ; 105 63 63 27

¯ Kα = {C¯ Kα } , C ij

2b 1 4b 1 α C¯ ijKα = Sijpp ( − ) − KR−1 δij ( + ) ; 15 3 15 3

(32)

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Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong

¯α=A ¯ 0α + D ¯ , A

¯ 0α = {A¯ 0 α } , A ijkl

¯ ijkl } , ¯ = {D D

S α = ( C α ) −1 ,

(33)

0α 1 α 1 α α α ¯ ¯ + Sαjkil ) B¯ 2 + (Sipjp δkl + Skpl A¯ ijkl = Sijkl B1 + (Sikjl p δij ) B4 2 2 1 α 1 α α α α α ¯ ¯ δkl + Skl + (Sijpp pp δij ) B3 + ( Sipkp δjl + S jpkp δil + S jpl p δik + Sipl p δjk ) B5 2 4 1 α + (Sikpp δjl + Sαjkpp δil + Sαjl pp δik + Silα pp δjk ) B¯ 6 , 4 1 2b B¯ 1 = (1 − )2 + f 1 F¯1 + g1 G¯ 1 , B¯ 2 = f 1 F¯2 + g1 G¯ 2 , B¯ 4 = f 1 F¯4 + g1 G¯ 4 , 9 5 2 4b 16b2 B¯ 3 = + − + f 1 F¯3 + g1 G¯ 3 , B¯ 5 = f 1 F¯5 + g1 G¯ 5 , B¯ 6 = f 1 F¯6 + g1 G¯ 6 , 9 45 225 ¯ ijkl = δij δkl D ¯ 1 + 1 (δik δjl + δjk δil ) D ¯2 , D 2 ¯ 1 = ( 1 K −1 − 1 µ−1 )( f 3 F¯1 + g3 G¯ 1 ) + 1 µ−1 ( f 3 F¯2 + g3 G¯ 2 ) + 1 K −1 ( f 3 F¯3 + g3 G¯ 3 ) D 9 R 6 R 4 R 3 R 4 ¯ 1 4b 2 −1 5 1 2¯ 1 ¯ ¯ ) KR , +( KR−1 + µ− R )( f 3 F4 + g3 G4 ) + F7 + G7 + (1 + 9 6 3 5 9 5 ¯ 2 = ( 1 µ−1 ( f 3 F¯1 + g3 G¯ 1 ) + ( 1 K −1 + 1 µ−1 )( f 3 F¯2 + g3 G¯ 2 ) D 2 R 9 R 12 R 5 1 1 −1 2¯ 4 ¯ 1 ¯ ¯ ¯ ¯ +( KR−1 + µ− R )( f 3 F5 + g3 G5 ) + K R ( f 3 F6 + g3 G6 ) + F8 + G8 , 9 6 3 3 5 2 2 2 4 16b 13b 43b 2 8b 16b F¯1 = − − − , G¯ 1 = , F¯2 = + + , 15 105 315 1890 5 35 315 22b + 28 2b2 b2 4b2 , F¯3 = + , G¯ 3 = − , G¯ 2 = G¯ 4 = G¯ 6 = − 189 105 315 945 2 8b 4b2 10b2 32b + 28 8b2 4b 16b2 + , F¯5 = + − , G¯ 5 = , F¯6 = − − , F¯4 = 35 315 5 35 315 189 35 45 b2 −1 31b2 + 90b + 63 −1 ¯ b2 −1 5b2 −1 F¯7 = µR − K R , G7 = K − µ , 63 945 5670 R 756 R 136b2 + 324b + 189 −1 5b2 −1 4b2 −1 5b2 −1 F¯8 = KR − µ R , G¯ 8 = K + µ ; 945 126 1134 R 108 R

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

191

APPENDIX B For the lower bound K L , we have the following particular expressions of its constituent two-index-notation matrices ¯ K = {C¯ K } C ij

(symmetric 3 × 3 matrix) ,

(34)

2b 1 1 4b 1 K C¯ 11 = (S11 + S12 + S13 )( − ) − ( + ), 15 3 K R 15 3 1 4b 1 2b 1 K ( + ), C¯ 22 = (S22 + S12 + S23 )( − ) − 15 3 K R 15 3 2b 1 1 4b 1 K K K K C¯ 33 = (S33 + 2S13 )( − ) − ( + ) , C¯ 12 = C¯ 13 = C¯ 23 =0; 15 3 K R 15 3 A A ¯ = {S¯ pq A } , S¯ pq = S¯ 0pqA + D¯ pq (symmetric 6 × 6 matrices) , 0A S¯11 = S11 ( B¯ 1 + B¯ 2 ) + (S11 + S12 + S13 )( B¯ 3 + B¯ 6 )

(35)

1 1 0A , +(S11 + S55 + S66 )( B¯ 4 + B¯ 5 ) = S¯22 4 4 1 0A S¯33 = S33 ( B¯ 1 + B¯ 2 ) + (S33 + 2S13 )( B¯ 3 + B¯ 6 ) + (S33 + S44 )( B¯ 4 + B¯ 5 ) , 2 1 1 1 0A S¯12 = S12 B¯ 1 + S66 B¯ 2 + (S11 + S12 + S13 ) B¯ 3 + (S11 + S44 + S66 ) B¯ 4 , 4 4 4 1 1 0 A S¯13 = S13 B¯ 1 + S55 B¯ 2 + (S11 + S12 + 3S13 + S33 ) B¯ 3 4 2 3 1 1¯ 0A +(S11 + S33 + S44 + S66 ) B4 = S¯23 , 4 4 2 1 3 1 0A S¯44 = S44 B¯ 1 + (S23 + S44 )2 B¯ 2 + (S11 + S33 + S44 + S66 ) B¯ 5 4 4 4 0A +(S11 + S12 + 3S13 + S33 ) B¯ 6 = S¯55 , 1 1 1 0A S¯66 = S66 B¯ 1 + (S12 + S66 )2 B¯ 2 + (2S11 + S44 + S66 ) B¯ 5 4 2 2 +2(S11 + S12 + S13 ) B¯ 6 , ¯ 11 = D ¯ 22 = D ¯ 33 = D ¯1 + D ¯ 2, D ¯ 12 = D ¯ 13 = D ¯ 23 = D ¯ 1; D ¯ 44 = D ¯ 55 = D ¯ 66 = 2 D ¯2 , D (36) ¯ 1, D ¯ 2 are defined Appendix A. where B¯ 1 , . . . , B¯ 6 and D A A −1 ¯ −1 = {C¯ pq A } = {S¯ pq }

(6 × 6 symmetric matrix) .

(37)

Then ¯ −1 : C ¯ −1 = {C¯ AC } (3 × 3 symmetric matrix) , ¯K = C ¯K : A A Kij AC C¯ K11 AC C¯ K33

= =

A A ¯K A ¯K AC (C¯ 11 + C¯ 12 )C11 + C¯ 13 C33 = C¯ K22 , A ¯K A ¯K AC AC ¯ ¯ ¯ ¯ C33 C33 + 2C13 C11 , CK12 = CK13

AC = C¯ K23 =0,

(38)

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Vuong Thi My Hanh, Pham Duc Chinh, Vu Lam Dong

and K ¯ AC K ¯ AC ¯ −1 : C ¯K : A ¯ K = C¯ KCAC = 2C¯ 11 C CK11 + C¯ 33 CK33 .

(39)

  1 A A ¯− ¯ ¯ hA i = T K ({ C }) , M ({ C }) , V V α α pq pq

(40)

Now one has

−1

1 ¯ α i− hA α = T

 1 A A KV−1 ({C¯ pq }), MV−1 ({C¯ pq }) , 9 4

1

1 ¯ AC 1 ¯ ¯− ¯ −1 ¯ AC ¯ hA α : C Kα iα = h C Kα : Aα iα = I (2CK11 + CK33 ) . 3

(41) (42)