Construction of bounds on the effective shear modulus of isotropic multicomponent materials

Volume 35 Number 4 4 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp. 275 – 283 CONSTRUCTION OF BOUNDS ON THE EFFECTIVE SHEAR MODULUS OF ISOTROPIC MULTICOMPONENT MATERIALS Vu Lam Dong∗, Pham Duc Chinh Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam ∗E-mail: vldong@imech.ac.vn Abstract. In our previous paper, we constructed bounds on the effective bulk modulus of isotropic multicomponent composites using minimum energy principles and modi- fied Hashin-Shtrikman polarization trial fields. In this paper, following the variational approach, we construct more sophisticated bounds on the effective shear modulus. Ap- plications to particular models are presented. Keywords: Isotropic multicomponent material, effective shear elastic modulus, three- point correlation parameters. 1. INTRODUCTION Macroscopic (effective) elastic moduli keff and µeff of isotropic multicomponent materials are important mechanical properties of the materials. It is difficult to find ex- actly these moduli because of complicated micro-geometries of composites. The most well- known estimates are the volume-weighted arithmetic or harmonic average formulae of Voigt and Reuss (Hill first order) bounds and Hashin-Shtrikman (second order) bounds [1–5]. Pham [3] extended Hashin-Shtrikmans inequalities to incorporate a number of coefficients depending on the fluctuation fields to improve the bounds. In [1] we had constructed new bounds for effective bulk elastic modulus of isotropic multicomponent materials which involve three-point correlation parameters. Continuing the research in this direction we will use more general multi-free parameter trial fields to construct new tight bounds on effective shear elastic properties of isotropic multicompo- nent materials. Applications of the bounds are performed for some representative material models. 2. CONSTRUCTION OF NEW BOUNDS The α-component of the multicomponent composite has elastic moduli kα, µα, α = 1, ..., N . The local elastic tensor C(x) is expressible as C(x) = N∑ α=1 T(kα, µα)Iα(x) , x ∈ V, (1) 276 Vu Lam Dong, Pham Duc Chinh where Iα is the indicator function Iα(x) = { 1, x ∈ Vα 0, x /∈ Vα (2) T is the isotropic fourth rank tensor with components Tijkl(k, µ) = kδijδkl + µ(δikδjl + δilδjk − 2 3 δijδkl), (3) δij is Kro¨necker symbol. The effective elastic moduli C eff = T(keff , µeff ) of the composite can be defined via the minimum energy expression [1] ε 0 : Ceff : ε0 = inf 〈ε〉=ε0 ∫ V ε : C : εdx, (4) while the strain field is expressible via the displacement field u(x) ε(x) = 1 2 [∇u+ (∇u)T ]. (5) To find the best possible upper bound on µeff from the minimum energy principle (4), we choose the following admissible compatible strain trial field εij = ε˜ 0 ij + N∑ α=1 [aα 1 2 (ϕα,ikε˜ 0 kj + ϕ α ,jk ε˜ 0 ki) + bαψ α ,ijklε˜ 0 kl], i, j = 1, ..., 3; (6) where ε0ij = ε˜ 0 ij (ε˜ 0 ii = 0) is a constant deviatoric strain; ϕ α and ψα are the harmonic and biharmonic potentials, Latin indices after comma designate differentiation with respective Cartesian coordinates; ϕα(x) = ∫ Vα Γϕ(x− y)dy ; ∇ 2ϕα(x) = δαβ , x ∈ Vβ; Γϕ(r) = − 1 4pir , ∇2Γϕ = δ(r); (7) r =| x− y | ; δ(r) is the Delta Dirac function; ψα(x) = ∫ Vα Γψ(x− y)dy ; ∇ 4ψα(x) = δαβ , x ∈ Vβ ; Γψ(r) = − 1 8pi r , ∇4Γψ = δ(r). (8) In [3] we have introduced the three-point correlation parameters Aβγα = ∫ Vα ϕβαij ϕ γα ij dx , ϕ βα ij = ϕ β ,ij − 1 vα ∫ Vα ϕβ,ijdx, Bβγα = ∫ Vα ψβαijklψ γα ijkldx , ψ βα ijkl = ψ β ,ijkl − 1 vα ∫ Vα ψβ,ijkldx. (9) Construction of bounds on the effective shear modulus of isotropic multicomponent materials 277 2N free scalars aα, bα in (6) are subjected to restrictions N∑ α=1 vαaα = 0, (10) N∑ α=1 vαbα = 0, (11) for the trial field (6) to satisfy the restriction 〈ε〉 = ε0 of Eq. (4). Substituting the trial field (6) into the energy functional of Eq. (4) and taking into account (9) and the respective expressions in [3–5], one gets Wε = ∫ V ε : C : εdx = N∑ α=1 ∫ vα [ kαεiiεjj + µα ( 2εijεij − 2 3 εiiεkk )] dx = { µV + N∑ α=1 vαµα [ 2 3 aα + 4 15 bα + 1 9 ( aα + 2bα 5 )2] + N∑ α,β,γ=1 [ Aβγα ( 1 10 (kα − 2 3 µα)(aβ + bβ)(aγ + bγ) + 11 60 µαaβaγ + 4 15 µαaβbγ − 1 15 µαbβbγ ) + 1 5 µαbβbγB βγ α ]} 2ε˜0ij ε˜ 0 ij, (12) where µV = N∑ α=1 vαµα is Voigt arithmetic average. We minimize the expression (12) over variable aα, bα restricted by Eqs. (10), (11) with the help of Lagrange multipliers λ and κ and get the equations 1 3 vαµα + vα 9 ( aα + 2bα 5 ) µα + N∑ β,γ=1 Aαβγ [ kγ − 2 3µγ 10 (aβ + bβ) + 11 60 µγaβ + 2 15 µγbβ ] − λvα = 0, α = 1, ..., N ; (13) 2vαµα 15 + 2vαµα 45 ( aα + 2bα 5 ) + N∑ β,γ=1 { Aαβγ [kγ − 23µγ 10 (aβ + bβ) + 2µγaβ 15 − µγbβ 15 ] +Bαβγ µγbβ 5 } − κvα = 0, α = 1, ..., N. (14) Summing Eqs. (13) multiplied by µ−1α on α from 1 to N and taking into account Eq. (10), one gets 1 3 + N∑ α=1 2bαvα 45 + N∑ α,β,γ=1 Aαβγ µ −1 α [ aβ ( kγ 10 + 7µγ 60 ) + bβ ( kγ 10 + µγ 15 )] − λµ−1R = 0, (15) 278 Vu Lam Dong, Pham Duc Chinh where µR is Reuss harmonic average µR = ( N∑ α=1 vαµ −1 α )−1 . (16) Also summing Eqs. (14) multiplied by µ−1α on α from 1 to N and taking into account Eq. (11), one obtains 2 15 + N∑ α=1 2aαvα 45 + N∑ α,β,γ=1 { Aαβγ µ −1 α [ aβ ( kγ 10 + µγ2 15 ) +bβ ( kγ 10 − 2µγ 15 )] + Bαβγ µ −1 α µγbβ 5 } − κµ−1R = 0. (17) Now substituting λ and κ from Eqs. (15) and (17) into Eqs. (13) and (14), finally leads to equations containing only the unknown aα and bα vµ +Aµ · a = 0. (18) In (18) we have introduced vectors vµ, a and matrix Aµ in 2N -space a = {a1, . . . , aN , b1, . . . , bN} T , (19) vµ = { v1 3 (µ1 − µR), . . . , vN 3 (µN − µR), 2v1(µ1 − µR) 15 , . . . , 2vN(µN − µR) 15 }T , (20) Aµ = { Aµαβ } , α, β = 1, . . . , 2N ; (21) where (in the following α, β = 1, ..., N ; α̂= N + α; β̂ = N + β) Aµαβ = vα 9 µαδαβ + N∑ γ=1 ( Aαβγ − vαµR N∑ δ=1 µ−1δ A δβ γ )[ kγ 10 + 7µγ 60 ] , Aµ bαbβ = 4vα 225 µαδαβ + N∑ γ=1 [( Aαβγ − vαµR N∑ δ=1 µ−1δ A δβ γ )( kγ 10 − 2µγ 15 ) + ( Bαβγ − vαµR N∑ δ=1 µ−1δ B δβ γ ) µγ 5 ] , A αbβ =Abαβ = 2vα 45 (µαδαβ − µRvβ) + N∑ γ=1 ( Aαβγ − vαµR N∑ δ=1 µ−1δ A δβ γ )[ kγ 10 + µγ 15 ] . (22) From Eq. (18), we find the necessary solutions for aα, bα a = −A−1µ · vµ . (23) Construction of bounds on the effective shear modulus of isotropic multicomponent materials 279 From Eq. (12), with Eqs. (13), (14) and (23), one finds Wε = ∫ V ε : C : εdx = [ µV + 1 3 N∑ α=1 vαµα ( aα + 2bα 5 )] 2ε˜0ij ε˜ 0 ij = ( µV + v ′ µ · a ) 2ε˜0ij ε˜ 0 ij = ( µV − v ′ µ ·A −1 µ · vµ ) 2ε˜0ij ε˜ 0 ij , (24) where v′µ = { v1µ1 3 , . . . , vNµN 3 , 2v1µ1 15 , . . . , 2vNµN 15 }T . (25) From Eqs. (2), (24), finally we obtain the upper bound on the effective shear modulus µeff ≤MUAB({kα, µα, vα}, {A βγ α , B βγ α }) = µV − v ′ µ ·A −1 µ · vµ . (26) To construct the lower bound on the effective shear modulus we use the minimum complementary energy principle σ 0 : (Ceff)−1 : σ0 = inf 〈σ〉=σ0 ∫ V σ : C−1 : σdx , (27) where σ0 is a constant stress field, and the stress field σ should satisfy equilibrium equation ∇ · σ(x) = 0 , x ∈ V (28) To find a lower bound on the effective shear modulus µeff from the minimum com- plementary energy principle (27), we take the admissible equilibrated stress trial field σij = σ˜ 0 ij + N∑ α=1 [aα(ϕ α ,ikσ˜ 0 kj + ϕ α ,jkσ˜ 0 ki − Iασ˜ 0 ij) − (aα + bα)δijϕ α ,klσ˜ 0 kl + bαψ α ,ijklσ˜ 0 kl], i, j = 1, ..., 3; (29) where σ0ij = σ˜ 0 ij(σ˜ 0 ii = 0) is a constant deviatoric stress, the free scalars aα, bα are subjected to the same restrictions (10) and (11). Substituting the trial field (29) into (27) and following procedure similar to that form (12) to (26), one obtains the best possible lower bound on µeff µeff ≥MLAB ( {kα, µα, vα}, {A βγ α , B βγ α } ) = (µ−1R − v¯ ′ µ · A¯ −1 µ · v¯µ) −1 , (30) where v¯µ = { − v1 3 (µ−1 1 − µ−1V ), . . . ,− vN 3 (µ−1N − µ −1 V ), 2v1(µ −1 1 − µ−1V ) 15 , . . . , 2vN (µ −1 N − µ −1 V ) 15 }T , (31) v¯ ′ µ = { − v1µ −1 1 3 , . . . ,− vNµ −1 N 3 , 2v1µ −1 1 15 , . . . , 2vNµ −1 N 15 } , (32) A¯µ = { A¯µαβ } , α, β = 1, . . . , 2N ; (33) 280 Vu Lam Dong, Pham Duc Chinh [in (34) α, β = 1, . . . , N ; α̂= N + α; β̂ = N + β] A¯µαβ = vα 9 µ−1α δαβ + N∑ γ=1 ( Aαβγ − vα µV N∑ δ=1 µδA δβ γ )[ 2k−1γ 45 + 7 15 µ−1γ ] , A¯µ bαbβ = 4vα 225 µ−1α δαβ + N∑ γ=1 [( Aαβγ − vα µV N∑ δ=1 µδA δβ γ )( 8k−1γ 45 − 2µ−1γ 15 ) + ( Bαβγ − vα µV N∑ δ=1 µδB δβ γ ) µ−1γ 5 ] , A¯µ αbβ = A¯µ bαβ = − 2vα 45 ( µ−1α δαβ − µ −1 V vβ ) + N∑ γ=1 ( Aαβγ − vα µV N∑ δ=1 µδA δβ γ )[ 4k−1γ 45 + 2µ−1γ 15 ] . (34) 3. APPLICATIONS In the case of symmetric cell material without distinct inclusion and matrix phases [4] (Fig. 1a), the three-point correlation parameters A βγ α , B βγ α have particular forms [4, 5] (α 6= β 6= γ 6= α) Aβγα = vαvβvγ(f1 − f3) , A αα α = vα(1− vα)[(1− vα)f1 + vαf3] , Aαβα = vαvβ [(vα − 1)f1 − vαf3] , A ββ α = vαvβ [(1− vβ)f3 + vβf1] , Bβγα = vαvβvγ(g1 − g3) , B αα α = vα(1− vα)[(1− vα)g1 + vαg3] , Bαβα = vαvβ [(vα − 1)g1− vαg3] , B ββ α = vαvβ [(1− vβ)g3 + vβg1] , (35) which depend on just 4 shape parameters f1, f3, g1, g3. One also has 6 7 f1 + 8 35 ≥ g1 ≥ 6 7 f1 f1 + f3 = 2 3 , 0 ≤ f1, f3 ≤ 2 3 , g1 + g3 = 4 5 , 0 ≤ g1, g3 ≤ 4 5 . (36) The three-point correlation bounds (26), (30) are specialized to MUfg ≥ µ eff ≥MLfg, (37) where MUfg({kα, µα, vα}, f1, g1) =M U AB({kα, µα, vα}, {A βγ α , B βγ α } ∈ (35)), MLfg({kα, µα, vα}, f1, g1) =M L AB({kα, µα, vα}, {A βγ α , B βγ α } ∈ (35)); (38) and then the shape-unspecified bounds for all symmetric cell materials read MUsym ≥ µ eff ≥MLsym , (39) Construction of bounds on the effective shear modulus of isotropic multicomponent materials 281 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 7 8 9 10 v 1 µ HS SYM SPHE (b) Fig. 1. The bounds on the effective shear modulus of three-component symmetric cell materials (SYM), compared to bounds for the specific symmetric spherical cell materials (SPHE) and Hashin-Shtrikman (HS) bounds. (a) A symmetric cell mixture; (b) The bounds where MUsym({kα, µα, vα}) = max f1,g1∈(36) MUfg({kα, µα, vα}, f1, g1), MLsym({kα, µα, vα}) = min f1,g1∈(36) MLfg({kα, µα, vα}, f1, g1). (40) Numerical result for the shape-unspecified bounds on the effective shear modulus of three-phase symmetric cell materials with same data of [1] at the range v1 = 0.1 → 0.9, v2 = v3 = 1 2(1 − v1) with k1 = 1, µ1 = 0.3, k2 = 12, µ2 = 8, k3 = 30, µ3 = 15 , are presented in Fig. 1b, which fall inside Hashin-Shtrikman bounds for the larger class of isotropic composites. The bounds µUs , µ L s (with f1 = g1 = 0) for the specific spherical cell materials are also presented, which lie inside both presented bounds. The next examples involve two-phase random suspensions of equisized hard spheres (Fig. 2a ) and overlaping spheres (Fig. 3a). The parametersAβγα , B βγ α are expressed through just two parameters ζ1 (or ζ2) and η1 (or η2) introduced earlier by Milton and Torquato [6–9] A11α = A 22 α = −A 12 α = 2 3 v1v2ζα , α = 1, 2; B11α = B 22 α = −B 12 α = 3 10 v1v2ηα + 1 2 v1v2ζα . (41) The bounds (26) and (30) for the models at ranges of v2, with k1 = 1, µ1 = 0.3, k2 = 20, µ2 = 10, together with Hashin-Shtrikman bounds are projected in Figs. 2b, 3b. 282 Vu Lam Dong, Pham Duc Chinh (a) 0.1 0.2 0.3 0.4 0.5 0.6 0 1 2 3 4 5 v 2 µ HS HARD (b) Fig. 2. Hashin-Strikman bounds (HS) and the bounds (HARD) on the elastic shear modulus of the random suspension of equisized hard spheres. (a) A random suspension of equisized hard spheres; (b) The bounds (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2 4 6 8 10 v 2 µ HS OVERLA (b) Fig. 3. Hashin-Strikman bounds (HS) and the bounds (OVERLA) on the elastic shear modulus of the random suspension of equisized overlapping spheres. (a) A random suspension of equisized overlapping spheres; (b) The bounds 4. CONCLUSION In this paper the authors have constructed three-point correlation bounds on the effective shear elastic modulus of statistically isotropic N -component materials from min- imum energy principles, using multi-free-parameter trial fields. The bounds are specified Construction of bounds on the effective shear modulus of isotropic multicomponent materials 283 to the practical class of symmetric cell materials and random suspensions of equisized spheres, with numerical illustrations. The trial polarization fields (6), (29) used in this paper depend on 2N − 2 free pa- rameters [i.e., 2N parameter aα, bα restricted by 2 constraints (10), (11)], hence are more general than the Hashin-Shtrikman ones used [3–5], which contain just 2 free parameters. Therefore the new bounds are more restricting in the cases N ≥ 3. We remind the particu- lar example of three-phase double-coated-sphere composite [1], where the parameters Aβγα have been determined analytically, our new bounds converge to the exact effective bulk modulus, while the old bounds in [3, 5] do not. Note also that the trial fields (6), (29) for the shear modulus containing 2N −2 free parameters are also more sophisticated than the respective trial fields for the bulk modulus in [1] containing just N − 1 free parameters. ACKNOWLEDGEMENT The authors thank the financial support of Vietnam’s Nafosted, project No 107.02- 2013.20. REFERENCES [1] Pham Duc Chinh, Vu Lam Dong, Three-point correlation bounds on the effective bulk modulus of isotropic multicomponent materials,Vietnam Journal of Mechanics, 34, (2012), pp. 67–77. [2] Hashin Z., Shtrikman S., A variational approach to the theory of the elastic behaviour of multiphase materials, J.Mech.Phys. Solids, 11, (1963), pp. 127–140. [3] Pham D.C., Bounds on the effective shear modulus of multiphase materials, Int.J, Engng. Sci, 31, (1993), pp. 11–17. [4] Pham D.C., Bounds for the effective conductivity and elastic moduli of fully-disordered multi- component materials, Arch. Rational Mech. Anal, 127, (1994), pp. 191–198. [5] Pham D.C.,Bounds for the effective properties of isotropic composite and poly-crystals, D. Sci. Thesis , Hanoi, (1996). [6] Milton G.W., The theory of Composites, Cambridge University Press, (2001). [7] Torquato S.,Random heterogeneous media, New York, Springer, (2002). [8] Pham D.C., Torquato S., Strong-contrast expansions and approximations for the effective con- ductivity of isotropic multiphase composites, J. Appl, Phys, 94, (2003), pp. 6591–6602. [9] Pham D.C., Three-point interpolation approximation for the macroscopic properties of isotropic two-component materials, Philos, Mag, 87, (2007), pp. 3531–3544. Received July 19, 2013 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 4, 2013 CONTENTS Pages 1. Nguyen Manh Cuong, Tran Ich Thinh, Ta Thi Hien, Dinh Gia Ninh, Free vibration of thick composite plates on non-homogeneous elastic foundations by dynamic stiffness method. 257 2. Vu Lam Dong, Pham Duc Chinh, Construction of bounds on the effective shear modulus of isotropic multicomponent materials. 275 3. Dao Van Dung, Nguyen Thi Nga, Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory. 285 4. N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: II. The numerical analysis. 299 5. Tran Van Lien, Trinh Anh Hao, Determination of mode shapes of a multiple cracked beam element and its application for free vibration analysis of a multi- span continuous beam. 313 6. Phan Anh Tuan, Pham Thi Thanh Huong, Vu Duy Quang, A method of skin frictional resistant reduction by creating small bubbles at bottom of ships. 325 7. Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong, An effective algorithm for reliability-based optimization of stiffened Mindlin plate. 335