1 Introduction

In recent years, great progress has been achieved with regard to the manufacturing techniques and application of nanocomposites. For a nanostructured material whose characteristic size is in the nanometer range, the interface effects become prominent because of the high ratio of the interfacial area to the volume. Therefore, it is important to evaluate the effects of the interface on the effective mechanical properties of nanocomposites.

The micromechanical methods can be utilized to predict the effective properties of nanocomposites that account for the interface effects. There are two approaches to study the interface effects. One is to include the interface/interphase directly into the micromechanical method (referred to as the DM in the following)^[1-7]. The other is to introduce the equivalent inclusion (EI) in the micromechanics-based method (referred to as the EIM). The latter approach can be applied to classical micromechanics with perfect bonding. Therefore, this approach is more convenient in practical calculations.

The concept of equivalent fiber has been introduced by Hashin^[8] to obtain the effective properties of unidirectional fiber composites with imperfect interface or fiber coating. The properties of the equivalent fiber that consists of the fiber and interphase were first calculated, and then the effective properties of the composite were derived by replacing the inclusion in the generalized self-consistent method (GSCM) with the equivalent fiber. A two-step homogenization technique was also adopted by Dai et al.^[9]. The inclusion and the coating were homogenized first as a homogeneous "composite inclusion". Then, the "composite inclusion" and the matrix were homogenized as the homogenous composite medium to obtain the effective properties of this composite. A new kind of differential effective medium theory that maps each inclusion particle, surrounded by a shell of another phase, onto an effective particle of uniform elastic moduli was put forward by Garboczi and Berryman^[10] to study the properties of the concrete. To address the composites with inhomogeneous interphases, in the second part of a two-part paper^[11-12], the idea of Garboczi and Berryman^[10] was adopted by Zhong et al.^[12], and a differential scheme was proposed to calculate the elastic moduli of the effective inclusion. A replacement technique that is similar to the differential scheme employed by Zhong et al.^[12] was used by Shen and Li^[13] to convert a fiber/sphere with its interphase into a homogeneous fiber/sphere to study the effective moduli of composites with inhomogeneous interphases. The replacement technique was also employed by Lombardo^[14] to obtain the effective thermal expansion coefficient of composite materials that contain inhomogeneous interphases. A modified differential method to replace the inclusion and the inhomogeneous interphase layer with an equivalent homogeneous inclusion was implemented by Sevostianov and Kachanov^[15] to study the effects of interfaces on the overall elastic and conductive properties of composites. An energy equivalency condition was utilized by Duan et al.^[16] to calculate the elastic constants of equivalent particles or fibers that are employed to replace the inhomogeneities with interface effects. In the paper of Nguyen and Pham^[17], the two-phase (coated) compound inclusions were substituted by equivalent one-phase inclusions using dilute solution results, and then the available effective medium approximations and bounds with the EIs were applied to obtain the elastic moduli of two-dimensional suspensions of compound inclusions. Through analyzing the Helmholtz free energy of the inclusion together with its interface, the thermoelastic properties of the EI were obtained by Chen et al.^[18] and utilized to derive the size-dependent effective specific heats of nanocomposites.

The concept of EI can also be applied to study the effect of interfacial debonding. A micromechanical damage model for two-phase functionally graded materials was developed by Paulino et al.^[19], in which the effect of interfacial debonding was considered by replacing the debonded isotropic particles with perfectly bonded orthotropic particles in terms of the elastic equivalency. The same idea was also employed by Liu et al.^[20] to model the elastoplastic behavior of particle-reinforced metal-matrix composites with particle-matrix interfacial debonding.

In summary, in the equivalent inclusion method (EIM), the properties of an EI, i.e., the inclusion together with its interface/interphase, are derived first, and then the effective properties of the composites can be obtained by replacing the inclusion in the classical micromechanical models with the EI. Because the properties of composites without consideration of interface effects can be derived easily in the classical micromechanical methods, it is more convenient to obtain those properties with interface effects by using the EIM than the DM.

In this paper, the EI and its applicability for predicting the effective properties of nanocomposites incorporating the interface energy are discussed. The elastic properties of the EI are usually obtained by embedding a single inclusion with its interface into an infinite matrix material and by applying the linear relationship between the volume averages of the stress and strain of the EI under certain boundary conditions^[21-23]. However, there arises a question as to whether the above mentioned EI can be applied universally in different micromechanical methods. To the best of the authors' knowledge, this question has been rarely discussed in the literature, and thus is investigated in this study.

This paper is organized as follows. In Section 2, the elastic properties of a particulate EI and a fibrous EI are derived by embedding a single inclusion with its interface into an infinite matrix. In Section 3, the effective elastic properties of the nanocomposites that take into account the interface are obtained by directly using the DM. In Section 4, the question as to whether the defined EI is universal for different micromechanical methods is discussed by comparing the effective elastic properties of nanocomposites obtained by means of the EIM that is based on different micromechanical methods with those obtained from the DM. It is found that some elastic properties of the EI are related only to the properties of the inclusion and the interface, whereas others (i.e., two shear moduli) are related to the properties of the matrix. The former properties of the EI can be applied to both the classical Mori-Tanaka method (MTM) and the GSCM. However, the latter properties of the EI can be utilized only in the MTM. In Section 5, two new kinds of EIs are proposed to obtain the two effective shear moduli based on the GSCM. Finally, closing remarks are given in Section 6.

2 Elastic constants of EI

In order to obtain the elastic properties of the EI, a single inclusion that includes the interface is embedded into an infinite matrix^[21-23]. Then, far-field boundary conditions are applied to calculate the volume averages of the stress and strain of the EI. The elastic properties of the EI are obtained by applying the linear elastic relationship between the volume averages of the stress and strain.

2.1 Basic equations for interface

In this study, the interface energy theory proposed in Refs. [21]-[24] is used to describe the interface between the inclusion and the matrix. This interface energy theory provides two kinds of basic equations for the interface, i.e., the interface constitutive equation and the interface equilibrium equation. In this paper, only small deformation is considered, and thus, these two basic equations in the infinitesimal deformation approximations are presented.

The interface constitutive equation is expressed by the first Piola-Kirchhoff stress as^[22]

(1)

where i₀ is the second-rank identity tensor in the tangent plane of the interface in the reference configuration, is the displacement of the interface from the reference configuration to the current configuration with A₁ and A₂ being the covariant base vectors on the interface and A₃ being the unit normal base vector to the interface, E_s is the Green strain tensor of the interface, ∇_0s is the interface gradient operator in the reference configuration, in which is the curvature tensor of the interface in the reference configuration with A^λ and A^α being the contra-variant base vectors of the interface, is the out-of-plane term of the deformation gradient of the interface, γ₀^* is the residual interface stress, and λ_s and μ_s are the Lamé constants of the interface.

The interface equilibrium equation based on the Lagrangian description can be expressed by^[22]

(2)

where and I is the second-rank identity tensor in the three-dimensional space, represents the discontinuity value of a physical quantity, S⁰ is the first Piola-Kirchhoff stress in the bulk material relative to the reference configuration, in which σ^* is the residual stress in the bulk material that is induced by the residual interface stress, and σ is the Cauchy stress, and and ΔS_s⁽ⁱⁿ⁾ and ΔS_s^(ou) are the in-plane and out-of-plane terms of ΔS_s, respectively, i.e., and

2.2 Elastic constants of spherical particulate EI

The matrix and the spherical particulate inclusion are assumed to be isotropic materials. K_p, μ_p, and υ_p are denoted as the bulk modulus, shear modulus, and Poisson's ratio of the particulate inclusion, respectively, K_m, μ_m, and υ_m are the elastic constants of the matrix, and K_*, μ_*, and υ_* are the elastic constants of the spherical particulate EI. The radius of the particulate inclusion is denoted as a.

The elastic properties of the particulate EI are obtained by embedding a single particulate inclusion with its interface into an infinite matrix as mentioned above. The bulk modulus K_* and the shear modulus μ_* of the particulate EI have been given in the literature^[21-22], the expressions of which are

(3)

(4)

where

2.3 Elastic constants of cylindrical fibrous EI

The matrix and the fibrous inclusion are assumed to be isotropic materials. Thus, the fibrous EI (the fibrous inclusion together with its interface) is transversely isotropic. There are five independent elastic coefficients for transversely isotropic materials. If x₁ is the axis of symmetry, the stress-strain relationship can be expressed in Hill's notations^[25], i.e., k, n, m, p, l,

(5)

For isotropic materials, the Hill's notations degenerate into

(6)

where λ and μ are the Lamé constants. In the following, for convenience, the elastic coefficients of the isotropic fibrous inclusion and the matrix are also denoted by using the Hill's notations as (k_f, n_f, m_f, p_f, l_f) and (k_m, n_m, m_m, p_m, l_m), respectively, and these Hill's notations can be obtained by Eq. (6) using the isotropic elastic constants. The cross-section radius of the fibrous inclusion is denoted as a₁. The elastic coefficients of the fibrous EI are denoted by (k_*, n_*, l_*, p_*, m_*).

The elastic coefficients of the fibrous EI are derived in two steps by embedding a single fibrous inclusion with its interface into an infinite matrix. In the first step, far-field boundary conditions are imposed and the displacements in the fibrous inclusion and matrix are obtained. To derive the axisymmetric moduli k_*, n_*, and l_*, the following far-field axisymmetric boundary condition is applied:

(7)

where z is in the axisymmetric direction of the fibrous inclusion in a cylindrical coordinate system. The displacements in the fibrous inclusion and the matrix can be given in the following form^[26]:

(8)

where j=f, m represents the fibrous inclusion and the matrix, respectively, and F_j and G_j are constants to be determined. To derive p_*, the far-field anti-plane shear boundary condition is applied,

(9)

The displacements in the fibrous inclusion and the matrix are^[3]

(10)

with

(11)

where M_j and N_j are constants to be determined. To derive m_*, the in-plane far-field shear boundary condition is applied,

(12)

The displacements in the fibrous inclusion and the matrix are^[3]

(13)

with

(14)

where and A_j, B_j, C_j, and D_j are constants to be determined. The undetermined constants in Eqs. (8), (11), and (14) can be obtained under four conditions: (ⅰ) the displacement at the origin r=0 cannot be singular; (ⅱ) the boundary conditions, Eqs. (7), (9), and (12); (ⅲ) the displacement continuity condition at the interface r=a₁; and (ⅳ) the stress discontinuity condition at the interface, Eq. (2).

In the second step, the volume averages of the stress and strain of the fibrous EI are calculated as

(15)

where is the volume of the fibrous inclusion. It should be noted that the stress in the integral of the first equation in Eq. (15) should be that of the matrix at r=a₁ so that the interface effects can be included. The elastic tensor of the EI, i.e., L_*, satisfies the relationship which gives

(16)

(17)

(18)

3 Effective elastic properties of nanocomposites including interface derived directly using micromechanical methods

Section 2 describes how the elastic properties of a spherical particulate EI and a cylindrical fibrous EI are obtained. The effective properties of the nanocomposites can be obtained from the EIM, i.e., by replacing the inclusion in the classical micromechanical model with the EI. The aim of this study is to investigate whether the EI obtained in Section 2 is universal for different micromechanical methods. To this end, an `inclusion-matrix-effective medium' (IMEM) model is suggested to calculate directly the effective elastic properties of nanocomposites considering interface effects. This method is referred to more simply as the direct method (DM) in this paper. The universality of the EI is investigated by comparing the results obtained from the EIM that is based on different classical micromechanical methods with those obtained from the DM.

3.1 Effective elastic properties of particle-reinforced nanocomposites considering interface effects

Consider a particulate IMEM model, in which a composite sphere with the interface (a spherical particle with its interface surrounded by the matrix) is embedded in an infinite effective medium (see Fig. 1). The radius of the composite sphere is b, and the volume fraction of the inclusion can be expressed as c=(a/b)³. The interface energy effect becomes prominent when the size of the inclusion is in the nano scale. Therefore, to consider the interface effects for nanocomposites, the interface energy theory proposed in Refs. [21]-[24] as introduced in Subsection 2.1 is used to describe the interface between the particulate inclusion and matrix. Thus, the stress at the interface r=a is discontinuous, and the stress discontinuity condition is given by Eq. (2). In addition, the displacement at the interface r=a and both the displacement and the stress at the interface r=b are continuous. To derive the effective elastic properties of such a particle-reinforced nanocomposite (PRNC), far-field boundary conditions are applied to the particulate IMEM model, and the volume averages of the strain and stress of the composite sphere, are calculated. The effective elastic properties are given by the relationship where L is the effective elastic tensor of the composite.

To derive the effective bulk modulus of a PRNC, a spherically symmetric remote strain is imposed at infinity in the particulate IMEM model,

(19)

where E_m is the trace of the imposed strain tensor E^∞, and I is the second-rank identity tensor. According to the Lurie solution^[27], the displacements in the inclusion, the matrix, and the effective medium can be written as

(20)

where A₀^(j) and D₀^(j) are constants to be determined, and j=p, m, e represents the particulate inclusion, the matrix, and the effective medium, respectively. To avoid singularity at the origin, D₀^(p) should vanish, and according to the boundary condition, i.e., Eq. (19), The other constants and D₀^(e) can be determined by the interface conditions at r=a and r=b. With all the constants A₀^(j) and D₀^(j) determined, the volume averages of the stress and strain of the composite sphere can be calculated and the effective bulk modulus is obtained,

(21)

To derive the effective shear modulus of the PRNC, a pure deviatoric remote strain is imposed at infinity in the particulate IMEM model,

(22)

where E_e is the equivalent strain of the imposed strain tensor, i.e., and e₁, e₂, and e₃ are the base vectors of the Cartesian coordinate system. According to the Lurie solution^[27], the displacements and stresses in the inclusion, the matrix, and the effective medium can be given in a spherical coordinate system,

(23)

where and D₂^(j) are constants to be determined, and μ_e and υ_e are the shear modulus and Poisson's ratio of the effective medium, respectively. To avoid singularity at the origin, C₂^(p) and D₂^(p) should vanish, and according to the boundary condition, i.e., Eq. (22), A₂^(e) =0 and B₂^(e)= The other constants can be determined using the interface conditions at r=a and r=b. Then, the volume averages of the stress and strain of the composite sphere, can be calculated, respectively, and the relationship gives the effective shear modulus,

(24)

Appendix A provides the expressions of ξ₁, ξ₂, ξ₃, and ξ₄.

It should be noted that when the properties of the effective medium are set to be the same as those of the nanocomposite, Eqs. (21) and (24) actually correspond to the effective elastic moduli predicted by the GSCM. When the properties of the effective medium are set to be the same as those of the matrix, Eqs. (21) and (24) correspond to the effective elastic moduli predicted by the MTM^{[9, 28]}.

3.2 Effective elastic properties of fiber-reinforced nanocomposites considering interface effects

The effective elastic properties of a fiber-reinforced nanocomposite (FRNC) are derived by using a fibrous IMEM model, where a composite cylinder with the interface (a cylindrical fiber with its interface surrounded by the matrix) is embedded into an infinite effective medium. The radius of the composite cylinder is b₁, and the volume fraction of the fibrous inclusion is c=(a₁/b₁) ². In the following, the interface energy theory proposed in Refs. [21]-[24] introduced in Subsection 2.1 is used to describe the interface between the fibrous inclusion and matrix. The stress at the interface r=a₁ is discontinuous, and this stress discontinuity condition is given by Eq. (2). In addition, the displacement at the interface r=a₁ and both the displacement and the stress at the interface r=b₁ are continuous. To derive the effective elastic properties of an FRNC, the boundary conditions, Eqs. (7), (9), and (12), are applied to the fibrous IMEM model, and the displacement and stress analysis procedure is similar to that given in Subsection 2.3. The volume averages of the stress and strain of the composite cylinder, and in the fibrous IMEM model are calculated, respectively, and then the effective elastic properties are given by the relationship where L is the effective elastic tensor of the composite. The effective elastic properties of the FRNC are expressed by

(25)

(26)

(27)

Appendix B provides the expressions of ξ₅, ξ₆, ξ₇, and ξ₈ in Eq. (27).

It should be noted that when the properties of the effective medium are set to be as those of the nanocomposite, Eqs. (25), (26), and (27) correspond to the effective moduli predicted by the GSCM. When the properties of the effective medium are set to be as those of the matrix, Eqs. (25), (26), and (27) then correspond to the effective moduli predicted by the MTM^{[9, 28]}.

4 Discussion of universality of EI

For a certain micromechanical method, if the effective elastic properties of nanocomposites obtained by the EIM are the same as those derived by the DM, the EI can be applied to this micromechanical method. Therefore, whether or not the EI obtained in Section 2 is universal for different micromechanical methods can be investigated by comparing the effective elastic properties of nanocomposites derived from the EIM with those obtained from the DM.

Expressions of the effective elastic properties of the PRNC and FRNC without considering interface effects and predicted by the GSCM and the MTM can be found in the literature^{[26, 29]}. By using the EIM, i.e., replacing the inclusion in these expressions by the EI, the effective elastic properties of nanocomposites that consider interface effects can be obtained. For comparison, the results obtained from the EIM that is based on the classical MTM are denoted by K_{MT_E}, μ_{MT_E}, k_{MT_E}, n_{MT_E}, l_{MT_E}, p_{MT_E}, and m_{MT_E}, whereas the results obtained from the EIM that is based on the classical GSCM are denoted by K_{G_E}, μ_{G_E}, k_{G_E}, n_{G_E}, l_{G_E}, p_{G_E}, and m_{G_E}. The results obtained by directly using the MTM that includes the interface, as described in Section 3, are denoted by K_MT, μ_MT, k_MT, n_MT, l_MT, p_MT, and m_MT, whereas the results obtained by directly using the GSCM that includes the interface, as described in Section 3, are denoted by K_G, μ_G, k_G, n_G, l_G, p_G, and m_G.

From the obtained results, we have

(28)

Equation (28) shows that the effective bulk modulus K of a PRNC, and the four effective elastic coefficients k, n, l, and p of an FRNC obtained by the EIM are equivalent to those derived from the DM, if the GSCM and the MTM are used. Moreover, the effective properties K, k, n, l, and p derived from the GSCM are the same as those derived from the MTM.

However, for the effective shear modulus μ of a PRNC and the effective transverse shear modulus m of an FRNC, we have

(29)

Equation (29) shows that μ and m obtained from the EIM are equivalent to those derived by the DM if the MTM is employed. However, μ and m obtained by the EIM are not equivalent to those derived from the DM, if the GSCM is utilized. Also, please note that the effective properties μ and m derived from the GSCM are different from those derived from the MTM.

An example is given in the following to show the difference between μ_{G_E} and μ_G, and that between m_{G_E} and m_G. Both the matrix and the inclusion (i.e., the particle or fiber) are assumed to be isotropic. The bulk moduli of the matrix and the inclusion are assumed to satisfy K_i/K_m =10, and Poisson's ratios are υ_i =υ_m =0.3. The properties of the interface are assumed to be for the particle (or for the fiber) and 2μ_s =λ_s =0.1γ₀^*. The effective shear modulus of the PRNC and the effective transverse shear modulus of the FRNC, derived by using the classical GSCM without consideration of the interface, are denoted by μ_{G_C} and m_{G_C}, respectively. The normalized results, μ_{G_E}/μ_{G_C}, μ_G /μ_{G_C}, m_{G_E} /m_{G_C}, and m_G /m_{G_C}, are shown in Figs. 2 and 3. These figures show that μ_{G_E} and m_{G_E} are not equivalent to μ_G and m_G, respectively. However, in this example, the differences between μ_{G_E} and μ_G, and m_{G_E} and m_G are small and thus can be neglected. For example, even the volume fraction of the inclusion reaches 50%, the relative difference between μ_{G_E} and μ_G, and that between m_{G_E} and m_G are only 0.7% and 0.4%, respectively.

Looking back on the results obtained in Section 2, it can be found that the bulk modulus K_* of the particulate EI in Eq. (3), and the four elastic moduli of the fibrous EI, k_*, n_*, l_* in Eq. (16) and p_* in Eq. (17), are related only to the properties of the inclusion and the interface, whereas the shear modulus μ_* of the particulate EI in Eq. (3) and the transverse shear modulus m_* of the fibrous EI in Eq. (18) are related not only to the properties of the inclusion and the interface, but also to the properties of the matrix. From the above discussion, it can be concluded that when the elastic properties of the EI are related only to the properties of the inclusion and the interface, the EI can be applied to both the MTM and the GSCM to obtain the effective elastic properties of nanocomposites considering interface effects.

However, if the elastic properties of the EI are related also to the properties of the matrix, the EI is not universal for different micromechanical methods. The shear moduli of the EI, i.e., Eqs. (4) and (18), can be applied only to the MTM. The effective shear moduli of the nanocomposites obtained from the EIM that is based on the classical GSCM with the EI are not equivalent to those obtained directly from the GSCM. However, the numerical example suggests that the relative difference between the results obtained by these two methods is small. To obtain results more close to those obtained directly from the GSCM, two new kinds of EIs are proposed based on the GSCM to derive the effective shear modulus of a PRNC and the effective transverse shear modulus of an FRNC.

5 New EIs based on GSCM

Two new kinds of EIs are proposed based on the GSCM. The first new EI is defined similar to that described in Section 2, except that the model used here is the IMEM model rather than a single inclusion with the interface embedded in an infinite matrix. The properties of the EI are related to the effective medium, of which the properties are unknown. By using a suggested expression of the properties of the effective medium in this paper, the properties of the first new EI are obtained. Compared with the EI defined in Section 2, the shear moduli of the nanocomposites obtained from the EIM with this first new kind of EI are more close to those obtained by the DM that is based on the GSCM. To make the shear moduli of the nanocomposites obtained by the EIM exactly the same as those obtained by the DM that is based on the GSCM, a second new kind of EI is proposed. By using the EIM that is based on the GSCM, the shear moduli of the nanocomposites can be obtained by solving a quadratic equation in which the coefficients are functions of the properties of the EI. The shear moduli of the nanocomposites in the quadratic equation are set to be the same as those obtained by the DM that is based on the GSCM, and then we can rearrange the quadratic equation in terms of the properties of the EI. For the particulate EI, this equation is a cubic equation expressed in terms of the shear modulus of the EI. Instead of solving directly the cubic equation, it is first reduced to a quadratic equation expressed in terms of the shear modulus by assuming Poisson's ratio of the EI to be a known value, for example, that of the inclusion. After solving the obtained quadratic equation, the assumed Poisson's ratio and the corresponding shear modulus are considered to be the properties of the second new kind of particulate EI. For the fibrous EI, a quadratic equation of the transverse shear modulus of the EI can be obtained, whose solution is the property of the second new kind of fibrous EI.

5.1 First new EI

The IMEM model discussed in Section 3 is used to derive the properties of the first new EI. By imposing a certain boundary condition on the IMEM model, the volume averages of the stress and strain of the inclusion together with the interface can be derived, which are denoted as and , respectively. The elastic tensor of the first new EI is denoted as , and the elastic properties of the new EI can be derived from the relationship It is found that the bulk modulus K_*' of the new particulate EI is the same as Eq. (3), and the elastic moduli k_*', n_*', l_*', and p_*' of the new fibrous EI are the same as those derived in Subsection 2.3, i.e., Eqs. (16) and (17). The shear modulus of the new particulate EI is

(30)

where β_i (i=1, 2, …, 6) are presented in Appendix C. The transverse shear modulus of the new fibrous EI is

(31)

where χ_i(i=1, 2, …, 6) are presented in Appendix C.

It is noted that μ_*' and m_*' are related to μ_e and m_e, respectively. However, μ_e and m_e are unknown. In the case where the interface effects are not considered, it is common to use the Voight and Reuss bounds to estimate the properties of the effective medium^{[9, 28]}. Therefore, to obtain explicit expressions of μ_*' and m_*', the Voight and Reuss bounds (Eqs. (28) and (29)) are used, in which the properties of the matrix and the properties of EI derived by embedding a single inclusion with the interface into the infinite matrix are applied to estimate the properties of the effective medium when the interface effects are considered,

(32)

(33)

In Eqs. (32) and (33), μ_* and m_* are the properties of the EI that are derived by embedding a single inclusion into an infinite matrix, i.e., Eqs. (4) and (18), respectively. The new EIs obtained by substituting Eq. (32) into Eqs. (30) and (31) are denoted by μ_*V' and m_*V', respectively. The new EIs obtained by substituting Eq. (33) into Eqs. (30) and (31) are denoted by μ_*R' and m_*R', respectively. It is found that the effective shear moduli obtained from the EIM with μ_*V' and m_*V' are closer than those obtained from the EIM with μ_*R' and m_*R' to the results obtained by the DM based on the GSCM (see Fig. 4 and Fig. 5 in the following). Therefore, Eq. (32) is applied to estimate β_i(μ_e) and χ_i(m_e) in Eqs. (30) and (31) to obtain the explicit expressions for μ_*' and m_*'.

5.2 Second new EI

The properties of the second new EI are derived by making the effective properties of the nanocomposites obtained from the EIM that is based on the GSCM be equal to those obtained by the DM. The elastic tensor of the second new EI is denoted by L_*''. As discussed in Section 4, it is clear that the bulk modulus K_*'' of the new particulate EI is the same as that in Eq. (3), and the elastic moduli k_*'', n_*'', l_*'', and p_*'' of this new fibrous EI are the same as those derived in Subsection 2.3, i.e., Eqs. (16) and (17). The procedure to derive the shear modulus μ_*'' of the particulate EI and the transverse shear modulus m_*'' of the fibrous EI is as follows.

In the GSCM, μ=μ_e, and thus Eq. (24) can be rearranged into a quadratic equation in terms of μ_e,

(34)

where and are presented in Appendix A. The effective shear modulus of a PRNC can be obtained as

(35)

By using the EIM based on the GSCM, another quadratic equation for μ_e can be obtained,

(36)

where A₁, B₁, and C₁ are related to the shear modulus μ_*'' and Poisson's ratio υ_*'' of the EI. Assume that μ_e in Eq. (36) is the same as in Eq. (35). Then, Eq. (36) can be rearranged into an equation for μ_*'',

(37)

where A₂, B₂, and C₂ are given in Appendix D. It should be noted that A₂, B₂, and C₂ are related to υ_*''. Because and K_*'' is equal to K_* in Eq. (3), Eq. (37) is actually a cubic equation of μ_*'' and is difficult to solve. To avoid solving this cubic equation, υ_*'' is assumed to be a known value, and is set to be, for example, the value of υ_p. Then, μ_*'' can be derived as

(38)

It also should be noted that the bulk modulus of the EI determined using υ_p and μ_*'' in Eq. (38) is not equal to K_*'' or K_* in Eq. (3), which means that the EI for the effective shear modulus of a PRNC is different from the EI defined for the effective bulk modulus of an FRNC.

Similarly, m_*'' can be derived as

(39)

where A₅, B₅, and C₅ are given in Appendix D. m_e in the expressions of A₃, B₃, and C₃ is

(40)

where and

Comparisons between the results of the two shear moduli derived by the EIM together with the two new kinds of EIs and the results of the shear moduli derived by the DM that is based on the GSCM, are presented in Fig. 4 and Fig. 5. In these figures, μ_{G_E} and m_{G_E} are the results derived from the EIM with the EIs defined in Subsection 2. μ_{G_EV1} and m_{G_EV1} are the results derived from the EIM with the first new kind of EI defined in Subsection 5.1, where μ_e and m_e are estimated by Eq. (32). μ_{G_ER1} and m_{G_ER1} are the results derived from the EIM with the first new kind of EI defined in Subsection 5.1, where μ_e and m_e are estimated by Eq. (33). and μ_{G_E2} and m_{G_E2} are the results derived from the EIM with the second new EI defined in Subsection 5.2. The properties of the matrix, the inclusion, and the interface in these examples are the same as those in the example given in Section 4.

Figures 4 and 5 indicate that the results obtained from the EIM with the first new kind of EI defined in Subsection 5.1, where μ_e and m_e are estimated by Eq. (32), are much closer to the results derived from the DM that is based on the GSCM than those obtained from the EIM with the EI defined in Section 2. For example, when the volume fraction of the inclusion is 0.5, the relative errors between μ_{G_E} and μ_G, and m_{G_E} and m_G, are 0.7% and 0.4%, respectively, whereas the relative errors between μ_{G_EV1} and μ_G, and m_{G_EV1} and m_G, are 0.2% and 0.08%, respectively. The first new kind of EI defined in Subsection 5.1 with μ_e and m_e estimated by Eq. (33) is not as good as that with μ_e and m_e estimated by Eq. (32). Moreover, the results obtained from the EIM with the second new kind of EI defined in Subsection 5.2 are the same as those derived from the DM that is based on the GSCM.

6 Concluding remarks

The interface effects on the mechanical properties of nanocomposites can be taken into account conveniently by applying an EI to the classical micromechanical methods. The properties of the EI, i.e., the inclusion together with the interface, usually are obtained by embedding a single inclusion with its interface in an infinite matrix^[21-23]. This paper focuses on investigating whether the EI can be applied universally for different micromechanical methods. From this study, the following conclusions are drawn.

(ⅰ) Some of the elastic properties of the EI are related only to the properties of the inclusion and the interface, i.e., the bulk modulus K_* of a particulate EI, and the elastic moduli k_*, l_*, n_*, and p_* of a fibrous EI. These properties can be applied to both the MTM and the GSCM to obtain the corresponding effective elastic properties of nanocomposites that consider interface effects, i.e., the effective bulk modulus of a PRNC K, and the effective elastic moduli k, l, n, p of an FRNC.

(ⅱ) However, some other elastic properties of the EI are also related to the properties of the matrix, i.e., the shear modulus μ_* of a particulate EI, and the transverse shear modulus m_* of a fibrous EI. These properties can be applied only in the MTM to obtain the corresponding effective elastic properties of nanocomposites, i.e., the effective shear modulus μ of a PRNC, and the effective transverse shear modulus m of an FRNC. Therefore, such an EI is not universal for different micromechanical methods.

(ⅲ) Although μ_* and m_* cannot be applied to the GSCM, μ and m obtained by using the EI in the classical GSCM are close to those obtained directly using the GSCM that includes the interface, as shown in the numerical examples presented in this paper. Therefore, the EI can still be regarded as applicable in practice to obtain μ and m that are based on the GSCM.

(ⅳ) Two new kinds of EIs are proposed based on the GSCM to derive μ and m. Compared with the results obtained by using a typically defined EI, the results of μ and m obtained by applying the first new kind of EI in the classical GSCM are much closer to those obtained directly by the GSCM that includes the interface. The results of μ and m obtained by applying the second new kind of EI are exactly the same as those obtained directly from the GSCM that includes the interface.

In this study, both the matrix and inclusion of the composites are considered to be isotropic, and the above conclusions are drawn for these composites. For composites whose constituents are anisotropic materials (such as cubic and tetragonal materials), similar conclusions may be made, i.e., if the properties of the defined EI are only related to those of the inclusion and the interface, the EI is universal for different micromechanical methods. Otherwise, if the properties of the defined EI are also related to those of the matrix material, the EI can only be applied for some specific micromechanical method. The universality of EI for composites with anisotropic constituents will be studied in our future work.

Appendix A

Expressions of ξ₁, ξ₂, ξ₃, and ξ₄ in Eq. (24) are

where

Appendix B

Expressions of ξ₅, ξ₆, ξ₇, and ξ₈ in Eq. (27) are

where

and

Appendix C

Expressions of β_i (i=1, 2, …, 6) in Eq. (30) are

where

Expressions of χ_i (i=1, 2, …, 6) in Eq. (31) are

Appendix D

Expressions of A₂, B₂, and C₂ in Eq. (37) are

where

Expressions of A₅, B₅, and C₅ in Eq. (39) are