Effect of residual interface stress on effective thermal expansion

Appl. Math. Mech. -Engl. Ed., 32(11), 1377–1388 (2011) DOI 10.1007/s10483-011-1508-9 c Shanghai University and Springer-Verlag Berlin Heidelberg 2011

Applied Mathematics and Mechanics (English Edition)

Effect of residual interface stress on effective thermal expansion coefficient of particle-filled thermoelastic nanocomposite∗ Ru-chao HUANG (),

Yong-qiang CHEN ()

(Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, P. R. China) (Communicated by Quan-shui ZHENG)

Abstract The surface/interface energy theory based on three configurations proposed by Huang et al. is used to study the effective properties of thermoelastic nanocomposites. The particular emphasis is placed on the discussion of the influence of the residual interface stress on the thermal expansion coefficient of a thermoelastic composite filled with nanoparticles. First, the thermo-elastic interface constitutive relations expressed in terms of the first Piola-Kirchhoff interface stress and the Lagrangian description of the generalized Young-Laplace equation are presented. Second, the Hashin’s composite sphere assemblage (CSA) is taken as the representative volume element (RVE), and the residual elastic field induced by the residual interface stress in this CSA at reference configuration is determined. Elastic deformations in the CSA from the reference configuration to the current configuration are calculated. From the above calculations, analytical expressions of the effective bulk modulus and the effective thermal expansion coefficient of thermoelastic composite are derived. It is shown that the residual interface stress has a significant effect on the thermal expansion properties of thermoelastic nanocomposites. Key words nanocomposite, thermal elastoplastic, effective thermal expansion, residual interface stress, size-dependent Chinese Library Classification O343 2010 Mathematics Subject Classification

1

74A40

Introduction

The classic micromechanics is an effective means for the study of the effective properties of heterogeneous materials. Much progress has been made in this field since 1980s, and the methodology and main results of the micromechanics have been summarized in several books and review articles such as Mura[1] , Nemat-Nasser and Hori[2] , Milton[3] , Torquato[4], Buryachenko[5], and Hu et al.[6] . The thermal properties of the composite have also received wide attention from many researchers. The effective thermal expansion coefficient of a twophase thermoelastic composite was given by Levin[7] αL = α0 + f (α1 − α0 ) : B 1 ,

(1)

∗ Received Jul. 26, 2011 / Revised Aug. 15, 2011 Project supported by the National Natural Science Foundation of China (Nos. 10602002 and 10932001) and the Major State Basic Research Development Program of China (973 Program) (No. 2010CB731503) Corresponding author Yong-qiang CHEN, Associate Professor, Ph. D., E-mail: [email protected]

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Ru-chao HUANG and Yong-qiang CHEN

where f is the volume fraction of the inclusion, α0 and α1 are the thermal expansion coefficient tensors of the matrix material and the inclusions, respectively, and B 1 is the fourth-order stress concentration tensor of the inclusions. If the matrix material and the inclusions are thermally isotropic, αr and αL in Eq. (1) can be written as αr = αr I

(r = 0, 1),

αL = αL I,

in which I is the second-order identity tensor. Moreover, for spherical inclusions, we have I : B 1 = Bm I, in which Bm =

K1 (3K0 + 4μ0 ) , (1 − f )K0 (3K1 + 4μ0 ) + f K1 (3K0 + 4μ0 )

K0 , μ0 , and K1 , μ1 are the bulk and shear moduli of the matrix and the inclusions, respectively. Thus, the effective thermal expansion coefficient in Eq. (1) can be expressed by αL =

K0 α0 (1 − f )(3K1 + 4μ0 ) + f K1 α1 (3K0 + 4μ0 ) . K0 (1 − f )(3K1 + 4μ0 ) + f K1 (3K0 + 4μ0 )

(2)

For a nanocomposite, due to the large ratio of the interface area between different constituents to the volume of the composite, the influence of the interface energy on the effective properties of the composite becomes prominent. Therefore, the surface/interface effect on the mechanical properties of nanocomposite has been a subject that has attracted considerable attention of researchers in recent years[8–15] . It is noted that Eq. (2) is derived under the condition that the displacement and the traction across the interface are continuous. In order to take the interface effect into account, the interface constitutive relations and the equilibrium conditions of the interface have to be employed. The interface effect on the effective thermal expansion coefficient of the thermoelastic nanocomposite has been discussed by Chen et al.[16] and Duan and Karihaloo[17]. However, in their works, the effect of the residual interface stress on the effective thermal expansion coefficient has not been considered, and in the following, it will be shown that this effect is quite important. As a matter of fact, the creation of an interface or a surface generally results in an interface/surface stress (which is referred to as the residual interface stress). Thus, there exists a surface or interface-induced stress field in the bulk even under no external loading. Based on this observation, a new surface/interface energy theory of multi-phase hyperelastic media was proposed by Sun et al.[18] , Huang and Wang[19] , Huang and Sun[20] , and Huang et al.[21] . The main contributions of Huang’s theory can be summarized as follows. (i) The concept of the fictitious stress-free configuration was first introduced by Sun et al.[18] and Huang and Wang[19] . This means that a complete analysis of the deformation of a multiphase medium with the interface energy effect should in general involve three configurations even for the infinitesimal deformation. These three configurations are the fictitious stressfree configuration κ∗ , the reference configuration κ0 with the residual interface stress-induced elastic field, and the current configuration κ. It is emphasized that the elastic energy within the bulk material should be calculated based on κ∗ , whereas the surface energy and the surface constitutive relations must be formulated based on the reference configuration κ0 . (ii) Hyperelastic constitutive relations of the interface in terms of the interface energy at the finite deformation were proposed in [19]. It can be considered as a generalization of the well-known Shuttleworth equation of the small deformation. It can be seen that even under the small deformation, the first and second Piola-Kirchhoff stresses of the interface and the Cauchy stress of the interface are different from one another[20,22] . (iii) A new energy functional was suggested. Based on the stationary condition of this energy functional, both the Eulerian and Lagrangian descriptions of the generalized YoungLaplace equations were derived [19,23] . The advantage of this new approach for constructing the

Effect of residual interface stress on effective thermal expansion coefficient

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Young-Laplace equations is that it can be used to more complex situations, for example, the derivation of the Young-Laplace equations of the micropolar composite[24] . (iv) Since we do not know the exact position of the deformed surface/interface in advance (for example, the curvature tensor of the interface), it is necessary to use the Lagrangian description of the Young-Laplace equations, as pointed out by Huang and Sun[20] . Moreover, as the first Piola-Kirchhoff surface stress appears in this description, it is natural to express the interface constitutive relations in terms of the first Piola-Kirchhoff interface stress. Thus, different from the results given by previous authors, the effect of the residual interface stress could also be important in the predictions of the mechanical properties of the nanosized structure and the nanocomposite (see [20] and [25]). This effect has already been demonstrated and confirmed by Park and Klein[26–27] . In this paper, the above surface/interface energy theory is used to study the effective properties of a thermoelastic composite filled with nanoparticles. The influence of the residual interface stress on the effective thermal expansion coefficient of the said composite is discussed. First, the thermoelastic constitutive relations of the surface/interface are presented. Then, by means of the three configurations concept, the deformations of Hashin’s composite sphere assemblage (CSA) with the residual interface stress are calculated. Based on the above calculations, the analytical expressions of the effective bulk modulus and the effective thermal expansion coefficient of the thermoelastic composite are derived. It is shown that, different from the results given by previous authors, the residual surface/interface stress does have a significant effect on the effective properties of the thermoelastic nanocomposite.

2

Basic equations of interface

There are two kinds of basic equations for the surface/interface. One is the constitutive relation of the surface/interface, and the other is the equilibrium equation of the interface (i.e., the Young-Laplace equation). Consider a smooth surface/interface A0 in the reference configuration κ0 . After the deformation, A0 becomes A. The covariant base vectors on the tangential planes of A0 and A are Aβ = r0,β and aβ = r0,β + u,β (β = 1, 2), respectively, where u is the displacement vector. The unit normal vectors of A0 and A are denoted by A3 and a3 , respectively. Then, the deformation gradient of the interface can be written as[23] Fs = Fs(in) + Fs(ou) , (in)

(3)

(ou)

where Fs = i0 + u∇0s and Fs = dβ A3 ⊗ Aβ correspond to the in-plane part and the out-plane part of the deformation gradient of the interface. u∇0s is the displacement gradient of the interface in the reference configuration and can be expressed by u∇0s = u0s ∇0s − un0 b0 = uλ0 β Aλ ⊗ Aβ − un0 b0 , (4) and dβ = uλ0 b0λβ + un0 ,β . b0λβ is the covariant component of the curvature tensor b0 on A0 [28] . i0 is the second-order identity tensor in the tangent plane of A0 . ∇0s is the gradient operator on the interface[28] . 1/2 By means of Cs = FsT · Fs and Us = Cs , the Lagrangian measure of the interface strain Es can be defined[19,23] . In the following, the surface/interface energy per unit area of A in the current configuration γ is assumed to be a function of the temperature θ and the interface strain Es , i.e., γ = γ(θ, Es ). Therefore, the surface/interface energy per unit area on A0 in the reference configuration is J2 γ, where J2 = det Us is the ratio between the area elements of dA and dA0 . Thus, the interface stress conjugate to Es can be written as Ts =

∂(J2 γ) . ∂Es

(5)

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In particular, Es can be chosen to be the Green strain of the interface, i.e., 1 (Cs − i0 ). 2

Es =

Then, the first and second interface Piola-Kirchhoff stresses can be expressed by[19,23] Ss = 2Fs ·

∂(J2 γ) ∂Cs

(6)

and

∂(J2 γ) , ∂Cs respectively, and the Cauchy stress of the interface is Ts(1) = 2

σs =

(7)

1 Fs · Ts(1) · FsT . J2

(8)

Obviously, Ss can also be decomposed into an in-plane part and an out-plane part, Ss = Ss(in) + Ss(ou) , where Ss(in) = 2Fs(in) ·

∂(J2 γ) , ∂Cs

∂(J2 γ) . ∂Cs

Ss(ou) = 2Fs(ou) ·

For the isotropic interface, we have γ = γ (θ, J1 , J2 ), where J1 = tr Us and J2 = det Us are the first and second invariants of Us . In the case that the deformation and the temperature change are small, the interface stress can be written as a linear function of the temperature[29–30] . If the higher-order small quantities are neglected, the series expansion of γ at the reference temperature θ0 can be given by (see [20] and [22]). (1)

γ = γ0 + γ1 (J1 − 2) + γ2 (J2 − 1) − γ0 (θ − θ0 ) 1 1 (2) + γ11 (J1 − 2)2 + γ22 (J2 − 1)2 − γ0 (θ − θ0 )2 2 2 (1)

(1)

+ γ12 (J1 − 2)(J2 − 1) − γ1 (J1 − 2)(θ − θ0 ) − γ2 (J2 − 1)(θ − θ0 ).

(9)

Therefore, Eq. (6) reduces to Ss = γ¯0 i0 + γ¯1 Es − γ¯0 ∇0s u + γ¯0 Fs(ou) , where γ¯0 = J2

∂γ ∂γ + J2 +γ , ∂J1 ∂J2

γ¯1 = J2

∂γ , ∂J1

Es =

(10)

1 (∇0s u + u∇0s ). 2

Substituting Eq. (9) into Eq. (10) leads to Ss = γ0∗ i0 − γˆ0 (θ − θ0 )i0 + (γ0∗ + γ1∗ )(tr Es )i0 + γ1 Es − γ0∗ ∇0s u + γ0∗ Fs(ou) ,

(11)

where γ0∗ = γ0 + γ1 + γ2 ,

γ1∗ = γ1 + 2γ2 + γ11 + 2γ12 + γ22 ,

(1)

(1)

(1)

γˆ0 = γ0 + γ1 + γ2 ,

and γ0∗ i0 is the residual interface stress. The second Piola-Kirchhoff stress of the interface and the Cauchy stress of the interface can also be obtained by substituting Eq. (9) into Eqs. (7) and


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(8). It can be seen that, due to existence of the residual interface stress γ0∗ , Ss in Eq. (11) and (1) Ts in Eq. (7) or σs in Eq. (8) are not the same. In order to compare the present result with those in the existing literature, the following notations are adopted: λ∗s = γ1∗ + γ0∗ ,

2μ∗s = γ1 − γ0∗ ,

Ks∗ = 2(λ∗s + μ∗s ),

and the thermal expansion coefficient of the interface αs is related to γˆ0 by γˆ0 = αs Ks∗ . In particular, if we neglecte the influence of the residual interface stress, the above notations reduce to λs = γ1∗ , 2μs = γ1 , Ks = 2(λs + μs ). Next, let us consider the equilibrium equations of the interface, i.e., the Young-Laplace equations. According to [23], the Lagrangian description of the Young-Laplace equations can be written as ⎧ 0 (in) (ou) ⎪ ⎨ A3 · S · A3 = −Ss : b0 − (A3 · Ss ) · ∇0s , (12) ⎪ ⎩ 0 (in) (ou) P0 · S · A3 = −Ss ∇0s + A3 · Ss · b0 , where S 0 is the first Piola-Kirchhoff stress relative to the reference configuration in the bulk material, S 0 is the discontinuity of S 0 across A0 , b0 is the curvature tensor of the interface in the reference configuration, and P0 = I − A3 ⊗ A3 , in which I is the second-order identity tensor in the three-dimensional (3D) space. It should be noted that S 0 in Eq. (12) depends not only on the deformation gradient F from the reference configuration κ0 to the current configuration κ, but also on the deformation gradient F ∗ from the fictitious stress-free configuration κ∗ to the reference configuration κ0 . The total deformation gradient is F = F · F ∗ , and the Green strains relative to κ0 and κ∗ can be written as[19,23] 1 E = (F T · F − I) (13) 2 and = 1 (F T · F − I) = F ∗T · E · F ∗ + E ∗ , (14) E 2 respectively, in which E ∗ = 12 (F ∗T · F ∗ − I) is the residual Green strain. It was emphasized by Huang and Wang[19] that the Helmholtz free energy in the bulk material should be calculated based on the stress-free configuration κ∗ , i.e., ψ = ψ(θ, E). Therefore, the first and second Piola-Kirchhoff stresses relative to the reference configuration κ0 can be written as[19,23] S 0 = F · T 0 = ρ0 F · F ∗ ·

∂ψ · F ∗T , ∂E

T 0 = ρ0 F ∗ ·

∂ψ · F ∗T , ∂E

(15)

where ρ0 is the mass density in the reference configuration. It is shown that from Eqs. (11) and (12), there is a residual stress field in the bulk material σ ∗ = S 0 F=I, . This residual stress field is induced by the residual interface stress Ss = γ0∗ i0 . θ=θ0

In the case that the deformation from κ0 to κ is small, the strain in the bulk material can be defined by 1 ε = (u∇0 + ∇0 u), (16) 2 where u is the displacement from κ0 to κ, and ∇0 denotes the gradient operator in the 3D space.

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Applying series expansion of T 0 in Eq. (15) with respect to ε and θ − θ0 at the reference temperature θ0 and F = I, we have T 0 = σ ∗ + L : (ε − α(θ − θ0 )), where L = ρ0

∂2ψ ∂E∂E

E=0

is a fourth-order elastic stiffness tensor and is dependent on F ∗ , i.e.,

L = L(F ∗ ). α is the thermal expansion tensor. Neglecting the higher-order small quantities in Eq. (15), we obtain S 0 = (I + u∇0 ) · (σ ∗ + L(F ∗ ) : ε − L(F ∗ ) : α(θ − θ0 )) . = σ ∗ + (u∇0 ) · σ ∗ + L(F ∗ ) : ε − L(F ∗ ) : α(θ − θ0 ).

(17)

If we further assume that the deformation from the stress-free configuration to the reference configuration is small, Eq. (14) can be approximately written as ε˜ = ε∗ + ε.

(18)

Hence, L(F ∗ ) can be regarded as a constant, and the second term on the right-hand side of Eq. (17) can be omitted, which gives S 0 = σ ∗ + L : (ε − α(θ − θ0 )) = σ ∗ + σ,

(19)

where L is the elastic stiffness tensor that is independent of F ∗ .

3

Effective properties of particle-filled thermoelastic composites

3.1 Displacement field in Hashin’s CSA Consider a particle-filled thermoelastic composite, in which both the matrix material and the particles are assumed to be isotropic and linear thermoelastic. Following the approach employed in [18], the Hashin’s CSA can be taken as the representative volume element (RVE). The radii of the particles and the composite sphere are a and b, respectively. Thus, the volume fraction of the particles is f = (a/b)3 . In the calculation of the effective thermal expansion coefficient, only the bulk modulus is involved. Hence, only the spherically symmetric boundary condition needs to be considered, and the displacement field in the CSA can be represented by the radial displacement ur in the spherical coordinates. Therefore, in the case of the small deformation and the uniform temperature change, the components of the strain and the stress can be written as εr = and

dur , dr

εθ = εφ =

ur , r

εrθ = εθφ = εφr = 0

⎧ 4μ ur 4μ dur ⎪ ⎪ + 2K − − 3KαΔθ, ⎪ σr = K + ⎪ 3 dr 3 r ⎪ ⎪ ⎪ ⎪ ⎨ 2μ ur 2μ dur + 2K + − 3KαΔθ, ⎪ σθ = σφ = K − ⎪ 3 dr 3 r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ τrθ = τθφ = τφr = 0,

(20)

(21)


1383

where Δθ = θ − θ0 is the temperature change, α is the linear thermal expansion coefficient, and K and μ are the bulk and shear moduli, respectively. It should be noted that their values in particles are different from those in the matrix. From the equilibrium equation r2

dur d2 ur + 2r − 2ur = 0, dr2 dr

(22)

we obtain the following general solution: ur = F r +

G . r2

(23)

The displacement and the stress in particles and in the matrix are u1r = F1 r,

1 σrr = 3K1 (F1 − α1 Δθ)

(24)

and u0r = F0 r +

G0 , r2

0 σrr = 3K0 F0 − 3K0 α0 Δθ −

4μ0 G0 , r3

(25)

respectively. In the above equations, the three constants F1 , F0 , and G0 need to be determined from the outer boundary condition at r = b and the interface conditions between the particle and the matrix at r = a. Obviously, for the above displacement field, the out-plane parts in Eqs. (3) and (11) are equal to zero, i.e., Fs(ou) = 0, Ss∗

Ss(ou) = γ0∗ Fs(ou) = 0.

The residual stress field σ ∗ in Eq. (19) can be determined from the residual interface stress = γ0∗ i0 in Eqs. (11) and (12) 2γ ∗ ∗ (26) σrr = 0, r=a a

in which the curvature tensor b0 = − a1 i0 is expressed in terms of the particle radius a at the reference configuration. By means of Eqs. (24) and (25), Eq. (26) can be rewritten as follows: (∗) 4μ0 G0 2γ ∗ (∗) (∗) ∗ σrr = 3K F − 3K F − = 0, 0 1 0 1 r=a 3 a a (∗)

(∗)

(27)

(∗)

in which F0 , G0 , and F1 are constants to be determined. In the reference configuration, the Hashin’s CSA is traction free at the outer boundary r = b. Therefore, (∗) μ0 G0 (∗) (0∗) σrr = 3K F − 4 = 0. (28) 0 0 r=b b3 From the condition that the displacement is continuous at the interface r = a, we have (∗)

(∗)

F1 a = F0 a + (∗)

(∗)

(∗)

Therefore, F0 , G0 , and F1 (∗)

F1

=−

(∗)

G0 . a2

(29)

can be determined from Eqs. (27), (28), and (29), for example,

2γ0∗ 4a3 μ0 + 3b3 K0 . 3a 4K0 μ0 (b3 − a3 ) + K1 (4a3 μ0 + 3b3 K0 )

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In particular, the residual displacement at the interface r = a and the outer boundary r = b in the reference configuration can be given by ⎧ (4a3 μ0 + 3b3 K0 )γ0∗ 2 (0∗) ⎪ ⎪ ⎪ ur r=a = − 3 4K μ (b3 − a3 ) + K (4a3 μ + 3b3 K ) , ⎪ ⎨ 0 0 1 0 0 (30) ⎪ ⎪ 2 ∗ ⎪ 2 b(4μ + 3K )γ a 0 0 0 ⎪ ⎩ u(0∗) . =− r r=b 3 4K0 μ0 (b3 − a3 ) + K1 (4a3 μ0 + 3b3 K0 ) 3.2

Effective bulk modulus and thermal expansion coefficient of thermoelastic composite In the following, we are only interested in the mechanical response of the composite from the reference configuration to the current configuration. Hence, S 0 and Ss in Eq. (12) can be replaced by ΔS 0 = S 0 − σ ∗ = σ and ΔSs = Ss − γ0∗ i0 , and by means of Eqs. (19) and (11), we have A3 · σ · A3 = −ΔSs(in) : b0 (31) or

2 2 σrr r=a = 2 (γ0∗ + 2γ1∗ + γ1 )ur r=a − γˆ0 Δθ. (32) a a In Eq. (32), the displacement ur is measured from the reference configuration. Substituting Eqs. (24) and (25) into the above equation leads to 3K0 F0 − 3K0 α0 Δθ − 4

μ0 G0 2 2 − 3K1 (F1 − α1 Δθ) = F1 (γ0∗ + 2γ1∗ + γ1 ) − γˆ0 Δθ. 3 a a a

(33)

The condition that the displacement is continuous at the interface r = a can be written as F1 a = F0 a +

G0 . a2

(34)

If a constant normal traction 13 Σm = 13 trΣ is applied to the outer boundary r = b, then we have μ0 G0 1 3K0 F0 − 3K0 α0 Δθ − 4 3 = Σm . (35) b 3 The above three unknown constants F0 , G0 , and F1 can be determined by Eqs. (33), (34), and (35). Therefore, the dilatational part of the strain of the CSA can be expressed in terms of the displacement at r = b u0r r=b 1 1 Em = tr E = , (36) 3 3 b where G0 u0r r=b = F0 b + 2 . (37) b (1)

(2)

Em in Eq. (36) can be decomposed into two parts: Em = Em + Em Δθ, in which (1) Em =

(−2f Ks∗ + 2Ks∗ + 4aμ0 + 3af K0 − 3af K1 + 3aK1 )Σm , 4f μ0 (2Ks∗ + 3aK1 ) + 3K0 (−4af μ0 + 2(Ks∗ + 2aμ0 ) + 3aK1 )

(38)

(2) Em =(24f γˆ0 μ0 + 18f γˆ0K0 + 36af μ0α1 K1 + 27af α1 K0 K1 + 9(1 − f )α0 K0 (2Ks∗ + 4aμ0

+ 3aK1 ))(4f μ0 (2Ks∗ + 3aK1 ) + 3K0 (−4af μ0 + 2Ks∗ + 4aμ0 + 3aK1 ))−1 .

(39)


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In the above equations, the notation Ks∗ = 2(λ∗s + μ∗s ) = γ0∗ + 2γ1∗ + γ1 has been used. The effective bulk modulus K and the effective thermal expansion coefficient α ¯ of the composite can be calculated from Eqs. (38) and (39). (1) (2) Since that the equations tr Σ = 3Ktr Em − 9K α ¯ Δθ or Σm = 3KEm + 3K(Em − 3α ¯ )Δθ are always valid for any Δθ, we have 1 Σm (40) K= (1) 3 Em and α ¯=

1 (2) E . 3 m

(41)

Substituting Eq. (38) into Eq. (40), we obtain 2K ∗ s + K1 − K0 f (4μ0 + 3K0 ) 3a . K = K0 + 2Ks∗ 4μ0 + 3f K0 + 3(1 − f ) K1 + 3a

(42)

This result has already been given in [20]. It should be mentioned that, in the above 2K ∗ expression, the residual interface stress is also included in 3as . If the effect of the residual interface stress γ0∗ is neglected, Ks∗ will reduce to Ks , and Eq. (42) will reduce to Eq. (33) in [16] or to Eq. (36) in [14]. The effective thermal expansion coefficient of the composite can be obtained by substituting Eq. (39) into Eq. (41)

2K ∗ 2αs Ks∗ K0 α0 (1 − f ) (3K1 + 4μ0 ) + a s + f K1 α1 + 3a (3K0 + 4μ0 ) α ¯= , (43)

2Ks∗ 2Ks∗ K0 (1 − f ) (3K1 + 4μ0 ) + a + f K1 + 3a (3K0 + 4μ0 ) in which γˆ0 is denoted by αs Ks∗ . It can be seen that the result given by [16] (Eq. (31)) or by [17] (Eq. (9)) is only a special case of the present paper, when the residual interface stress is neglected, i.e., Ks∗ is replaced by Ks . Obviously, if the interface effect is not considered at all, i.e., Ks∗ = 0, Eq. (43) will reduce to Eq. (2) given by Levin[7] . If neglecting the effect of γ0∗ , Ks∗ will reduce to Ks , and Eq. (43) will reduce to Eq. (31) in [16] or Eq. (9) in [17]. If totally neglecting the effect of the interface, Eq. (43) will reduce to the same result of Levin[7] , i.e., Eq. (43) will reduce to Eq. (2) by letting Ks∗ = 0.

4

Illustrative example

The influence of the residual surface/interface stress on the effective thermal expansion coefficient of the composite is illustrated as follows. Consider a thermoelastic composite containing spherical voids. The matrix material is polyethylene with its bulk modulus, shear modulus, and thermal expansion coefficient being Km = 3.33 × 109 Pa, μm = 0.345 × 109 Pa, and αm = 2.0 × 10−4 K−1[31] , respectively. According to [32], the value of γ0∗ is taken to be γ0∗ = 0.036 J/m2 . Since the experimental data of γ1 and γ1∗ are not available in the existing literature, here we assume that the values of γ1 and γ1∗ have the same order as γ0∗ and are chosen to be γ1 = γ1∗ = 12 γ0∗ , from which we have Ks∗ = 0.09 J/m2 and Ks = 0.054 J/m2 . Also, due to the absence of the experimental data for the thermal expansion coefficient of the interface αs , it is expediency to choose αs = 4αm and αs = 0.5αm for comparisons. The influence of the residual surface stress on the effective thermal expansion coefficient is depicted in Figs. 1–4. The effective thermal expansion coefficients given by the present paper and by the interface stress model (ISM)[16–17] are denoted by α ¯ and α ¯ ISM , respectively.

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In Figs. 1–4, the effective thermal coefficients α ¯ and α ¯ ISM [16–17] are normalized by α ¯ L , which was given by Levin. The variations of α/ ¯ α ¯ L and α ¯ ISM /α ¯ L with the void radius are shown in Figs. 1–2, where the thermal expansion coefficients of the surface are chosen to be αs = 4αm in Fig. 1 and αs = 0.5αm in Fig. 2. The relationship between the normalized effective thermal expansion coefficients and the void volume fraction is depicted in Figs. 3–4 with αs = 4αm in Fig. 3 and αs = 0.5αm in Fig. 4.

Fig. 1

Normalized effective thermal expan¯ ISM /α ¯L sion coefficients α ¯ /α ¯ L and α versus radius of voids (αs = 4αm )

Fig. 2

Normalized effective thermal expansion ¯ ISM /α ¯ L versus racoefficients α/ ¯ α ¯ L and α dius of voids (αs = 0.5αm )

Fig. 3

Normalized effective thermal expan¯ ISM /α ¯L sion coefficients α ¯ /α ¯ L and α versus volume fraction of voids (αs = 4αm )

Fig. 4

Normalized effective thermal expansion ¯ ISM /α ¯ L versus coefficients α/ ¯ α ¯ L and α volume fraction of voids (αs = 0.5αm )

From these figures, it can be seen that the effective thermal expansion coefficients predicted both in this paper and by previous authors are size-dependent. However, the influence of the residual surface/interface stress on the effective thermal expansion coefficient will be pronounced as the radius of the void decreases (or the void volume fraction increases), although the tendency of the variation of the effective thermal expansion coefficient may be different for αs = 4αm (¯ α /α ¯L > α ¯ISM /α ¯ L > 1) and for αs = 0.5αm (¯ α /α ¯L < α ¯ ISM /α ¯ L < 1).


5

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Concluding remarks

In this paper, the effective thermal expansion coefficient of a particle-filled nanocomposite is studied by using the three-configuration based surface/interface theory proposed by Huang et al. The analytical expression of the effective thermal expansion coefficient is derived, in which the residual interface stress is also included. First, two basic equations of the interface are presented. These equations are the thermoelastic surface/interface constitutive relation and the interface equilibrium equation. Second, based on the three configurations concept, the Hashin’s CSA model is utilized as the RVE, and an analytical expression of the effective thermal expansion coefficient of a thermoelastic composite is derived. In this derivation, the residual interface stress is also taken into account. It is shown that the residual surface/interface stress does have a significant effect on the effective thermal expansion properties, and the present result will degenerate to the one given by previous authors[16–17] if the residual surface/interface stress is neglected.

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