Advances in Applied Mechanics (eBook)

4 Micromechanical Framework for Nano-inhomogeneities with Interface Stress

The interface stress contributes to the effective moduli of composites in two ways. First, it affects the average stress (strain) in each inhomogeneity, and this effect can be taken into account by the use of the so-called stress (strain) concentration tensor in the inhomogeneity (cf. Section 2.2). Second, the discontinuities in the traction across the interface directly participate in the calculation of the volume average of the strain or stress, and this effect can be taken into account by use of the stress (strain) concentration tensor at the interface. The effective moduli of solids with interface stress effect can be modelled with the well-known micromechanical schemes, provided these are suitably modified to account for the discontinuity in the tractions across the interfaces. For example, Benveniste (1985) has given the general framework with consideration of the displacement discontinuity, and Hashin (1991) has used this framework to predict the effective moduli of composites with linear spring interfaces (displacement discontinuity). In view of the importance of the surface/interface stress effect at the nanoscale, a framework has been proposed by Duan et al. (2005b) to include stress discontinuity in order to take into account the surface/interface stress effect.

Many micromechanical schemes have been successfully used for obtaining effective elastic constants of heterogeneous solids. For a comprehensive exposition, one can refer to the monographs of Aboudi (1991), Nemat-Nasser and Hori (1999), Milton (2002), and Torquato (2002). For the prediction of the effective properties of nonlinear composites, one can refer to the works of Ponte Castañeda and Suquet (1998) and Willis (2000). For the sake of simplicity but without loss of the physical essence of the surface/interface stress effect, we shall use three schemes to predict the effective elastic constants of solids containing nanoinhomogeneities with the surface/interface stress effect described by Eqs. (2.2) and (2.6). The three schemes are Hashin’s composite sphere assemblage model (CSA; Hashin, 1962), the Mori–Tanaka method (MTM; Mori and Tanaka, 1973), and the generalized self-consistent method (GSCM; Christensen and Lo, 1979).

Consider a representative volume element (RVE) consisting of a two-phase medium occupying a volume V with external boundary S, and let VI and Vm denote the volumes of the two phases ΩI and Ωm, respectively. The interface stress effect is taken into account at the interface Γ with outward unit normal n between ΩI and Ωm. The heterogeneous material is assumed to be statistically homogeneous with the inhomogeneity moduli CI (compliance tensor DI) and matrix moduli Cm (compliance tensor Dm). fI and (1 – fI) denote the volume fractions of the inhomogeneity and matrix, respectively. To define the effective elastic moduli of a composite, we use the usual concept of homogeneous boundary conditions imposed on a RVE. As in the work of Benveniste and Miloh (2001), we define the average strain ¯ and average stress ¯ as follows:

¯=12V∫SN⊗u+u⊗NdS,

¯=1V∫Sσ⋅N⊗xdS,

where N is the outward unit normal vector to S, and x is the position vector. In the presence of interface stress effect (stress discontinuity), the average strain and average stress are

¯=1−fIε¯m+fIε¯I,

¯=1−fIσ¯m+fIσ¯I+fIVI∫Γσ⋅n⊗xdΓ,

where ¯k and ¯k (k = I, m) denote volume averages of the strain and stress over the respective phases in the RVE. [σ] = σI – σm. As usual, the effective elastic moduli of the composite can then be determined by subjecting the external surface S to homogeneous displacement or traction boundary conditions, defined as

S=ε0⋅x,

S=σ0⋅N,

where ε0 and σ0 are constant strain and stress tensors, respectively.

In the following, we will first derive formulas relating the average stress (strain) in the inhomogeneities and at the interface to the applied stress (strain) under both types of boundary condition (4.5) and (4.6). These formulas are needed to calculate the effective moduli of the composite according to the dilute concentration approximation and GSCM schemes. It is easy to derive formulas relating the average stress (strain) in the inhomogeneities and at the interface to the average stress (strain) in the matrix, again under both types of boundary condition (4.5) and (4.6). These formulas are needed in MTM.

Under homogeneous displacement boundary conditions Eq. (4.5), define a strain concentration tensor R in the inhomogeneity and a strain concentration tensor T at the interface such that

¯I=R:ε0,1VI∫Γσ⋅n⊗xdΓ=Cm:T:ε0

From Eqs. (4.3)–(4.5) and (4.7), the effective stiffness tensor ¯ of the composite is given by

¯=Cm+fICI−Cm:R+fICm:T.

Under homogeneous traction boundary conditions Eq. (4.6), define two stress concentration tensors U (in the inhomogeneity) and W (at the interface) by the relations

¯I=U:σ0,1VI∫Γσ⋅n⊗xdΓ=W:σ0.

Then the effective compliance tensor ¯ of the composite is given by

¯=Sm+fISI−Sm:U−fISm:W.

Eqs. (4.8) and (4.10) can be used to calculate the effective moduli of composites by using the dilute concentration approximation and GSCM once R, T, U, and W have been obtained. If the inhomogeneity and matrix are both isotropic, and the composite is macroscopically isotropic, then R and T in Eq. (4.7) can be expressed as

=R1J1+R2J2,T=T1J1+T2J2

in which

1=13I2⊗I2,J2=−13I2⊗I2+I4s

with I(4s) the fourth-order symmetric identity tensor. R1, R2, T1, and T2 are four scalars to be determined from the adopted micromechanical scheme. Then Eq. (4.8) decouples into

¯=κm+fIκI−κmR1+κmT1,

¯=μm+fIμI−μmR2+μmT2.

Let us now relate the average stress (strain) in the inhomogeneity and the average stress difference at the interface to the average stress (strain) in the matrix, following the Mori–Tanaka procedure (Benveniste, 1987). Under the homogeneous displacement boundary conditions Eq. (4.5), define two strain concentration tensors M (in the inhomogeneity) and H (at the interface) by the relations

¯I=M:ε¯m,1VI∫Γσ⋅n⊗xdΓ=Cm:H:ε¯m,

where ¯m=I4s+fM−I4s−1:ε0. Then the effective stiffness tensor can be obtained from

¯=Cm+fICI−Cm:M+Cm:H:I4s+fIM−I4s−1.

Likewise, under the homogeneous traction boundary conditions Eq. (4.6), the effective compliance tensor can be obtained from

¯=Sm+fISI−Sm:P+Sm:Q:I4s+fIP+Q−I4s−1.

where P and Q are stress concentration tensors in the inhomogeneity and at the interface, respectively. They are defined by

¯I=P:σ¯m,1VI∫Γσ⋅n⊗xdΓ=Q:σ¯m,

where ¯m=I4s+fIP+Q−I4s−1:σ0. Eqs. (4.16) and (4.17) can be used to...