Achieving remarkable mechanical properties for very low levels of reinforcement (<5%), nanocomposites are nowadays a new class of materials with high potential. Nanocomposites are obtained by using polymer-based [1, 2], metal-based [3] or ceramic-based [4] matrix and nanosized particles as the reinforcement. The reinforcing material can be made up of particles (e.g., minerals), sheets (e.g., exfoliated clay stacks) or fibers (e.g., carbon nanotubes). The matrix material properties are significantly affected in the vicinity of the reinforcement. Thanks to the stiffening aspect of the interface region, these composites often exhibit improved mechanical and physical (electrical and thermal conductivity, optical properties, dielectric properties, heat resistance) properties compared to the conventional composites reinforced with micron-sized particles. The origin of this size effect is revealed by the definition of the stress vector discontinuity condition at the interface between two different phases [5, 6]. For instance, this situation is classicaly managed when dealing with the poroelastic behavior of unsaturated porous media. Indeed, the interfacial domain separating the liquid and the gaseous phases introduces a discontinuity of the normal component of the stress vector. The latter is controlled by the surface tension *γ* according to the Young-Laplace equation:

$$\underset{\_}{n}\cdot \u301a\sigma \u301b\cdot \underset{\_}{n}={p}_{g}\text{-}{p}_{\ell}={\gamma}^{\ell g}trb\text{\hspace{1em}(1)}$$(1)

where **b**=-**grad** *n* is the surface curvature tensor associated to the interface morphology.

When solid/solid interfaces are involved, the interface equilibrium condition yields the following equation, which may be seen as a generalized Young-Laplace equation [7]:

$$\u301a\sigma \u301b\cdot \underset{\_}{n}+\mathrm{(}{\sigma}^{s}:b\mathrm{)}\underset{\_}{n}+{\text{div}}^{s}{\sigma}^{s}=0\text{\hspace{1em}(2)}$$(2)

where **σ**^{s} is the surface stress tensor defined on the interface separating two phases. The normal discontinuity condition is taken into account through the term (**σ**^{s}:**b**) *n*. Remarkably, defining the surface stress tensor as **σ**^{s}=*γ*^{lg}**1** allows one to retrieve (1).

The surface divergence operator involved in (2) accounts for the tangent discontinuity of the stress vector. It is defined as:

$${\text{div}}^{s}\text{\hspace{0.17em}}{\sigma}^{s}=\nabla {\sigma}^{s}:{1}_{T}\text{\hspace{1em}(3)}$$(3)

where **1**_{T}=**1**-*n*⊗*n* is the second-order identity tensor of the plane *T* tangent to the interface at the considered point, whereas ∇ is the gradient operator.

Clearly enough, the use of (2) requires clarifying the definition of the surface stress tensor **σ**^{s}. Following [8] and [9], the surface stress tensor is linearly related to the surface strain tensor according to:

$${\sigma}^{s}={\u2102}^{s}:{\epsilon}^{s}=2{\mu}_{s}{\epsilon}^{s}+{\lambda}_{s}\mathrm{(}tr\text{\hspace{0.17em}}{\epsilon}^{s}\mathrm{)}1\text{\hspace{1em}(4)}$$(4)

where *μ*^{s} and *κ*^{s}=*λ*^{s}+*μ*^{s} represent the shear and the bulk surface moduli.

From an energy point of view, this additional surface contribution modifies the elastic energy stored in the inclusion phase. One part of this energy $$\mathrm{(}{W}_{\text{V}}^{\text{e}}\mathrm{)}$$ is defined as the integration over Ω_{I} of the elastic energy volume density, the other part $$\mathrm{(}{W}_{\text{S}}^{\text{e}}\mathrm{)}$$ is defined as the integration over its external surface of a surface elastic energy:

$${W}_{I}^{e}=\frac{1}{2}{\displaystyle {\int}_{{\Omega}_{\text{I}}}\sigma :\epsilon \text{\hspace{0.17em}}dV}=\underset{={W}_{V}^{e}}{\underbrace{\frac{1}{2}{\displaystyle {\int}_{{\Omega}_{I}}{\epsilon}_{I}:{\u2102}^{I}{\epsilon}_{I}\text{\hspace{0.17em}}dV}}}+\underset{={W}_{S}^{\text{e}}}{\underbrace{\frac{1}{2}{\displaystyle {\int}_{\partial I}{\epsilon}^{s}:{\u2102}^{s}{\epsilon}^{\text{s}}dS}}}\text{\hspace{1em}(5)}$$(5)

The comparison of the order of magnitude of each of these two terms allows one to understand the increasing contribution of surface energy with decreasing inclusion characteristic size. Let us consider a homogeneous spherical inclusion (domain Ω_{I}) of radius *R*, undergoing a uniform strain on its external boundary (∂Ω_{I}). Surface and volume inclusion strains are then both equal to this uniform strain, so that the surface-to-volume elastic energy ratio reads:

$$\frac{{W}_{S}^{e}}{{W}_{V}^{e}}=\frac{{M}_{s}}{{M}_{v}}\frac{\mathrm{|}\partial {\Omega}_{I}\mathrm{|}}{\mathrm{|}{\Omega}_{I}\mathrm{|}}=\frac{{M}_{s}}{{M}_{v}}\frac{3}{R}\text{\hspace{1em}(6)}$$(6)

where *M*_{v} (resp. *M*_{s}) symbolically represents the volume (resp. surface) elastic moduli contribution. Considering finite values of elastic moduli for both volume and surface behaviors, (6) clearly shows that surface energy has an increasing contribution for decreasing values of the inclusion characteristic size. By contrast, when the characteristic size of the inclusion is large enough, the elastic energy is essentially stored in its bulk volume. This is the situation usually addressed by micromechanics approaches. Considering surface effects in the framework of usual micromechanics developments is precisely the starting point of recent efforts of the scientific community toward the better understanding of nanocomposite behavior [8, 10, 11]). For instance, estimates (Mori-Tanaka type [12] or generalized self-consistent types) are now available for the elastic moduli of nanocomposites reinforced by spherical nanosized particles. As expected, these estimates show a dependence on the size of the nanoparticles.

The concept of interface is a two-dimensional (2D) idealization of the transition zone between two phases, which may be thought of as a thin 3D layer of infinitesimal thickness *h*. The surface stress tensor then appears as:

$${\sigma}^{s}=\underset{h\to 0}{{\displaystyle lim}}{\displaystyle {\int}_{0}^{h}{\sigma}^{m}dz}\text{\hspace{1em}(7)}$$(7)

The mathematical singularity of the membrane stress concept lies in the fact that the integration over a layer whose thickness tends toward 0 yields a non-vanishing limit. As opposed to the bidimensional concept of membrane stress, **σ**^{m} is a 3D stress field. The equivalence of the interface surface and the thin elastic layer is demonstrated in [9, 13]. It was shown that it is possible to introduce a 3D stiffness tensor ℂ^{m} accounting for the interface represented by an elastic layer of small, but finite, thickness *h*, made up of an isotropic, linearly elastic material with Young’s modulus *E*^{m} and Poisson’s ratio *ν*^{m}. Then the 2D and 3D representations are asymptotically equivalent when *h*→0^{+}. Finally, the 3D corresponding stiffness tensor reads:

$${\u2102}^{m}=3{\kappa}^{m}\mathbb{J}+2{\mu}^{m}\mathbb{K}\text{\hspace{1em}(8)}$$(8)

where J and K are the spherical and deviatoric fourth-order projectors. The 3D bulk and shear modulus *κ*^{m} and *μ*^{m} of the equivalent elastic layer are deduced from their 2D (surface) counterparts (4) by:

$${\kappa}^{m}=\frac{4{\kappa}^{s}{\mu}^{s}}{33{\mu}^{s}-{\kappa}^{s}\mathrm{)}h}\mathrm{;}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mu}^{m}=\frac{{\mu}^{s}}{h}\text{\hspace{1em}(9)}$$(9)

Taking advantage of this 2D-3D equivalence, the elastic surface energy in (5) may be advantageously replaced by the 3D contribution of an elastic layer (domain Ω_{m}) of infinitely small thickness *h* according to:

$${W}_{S}^{e}=\frac{1}{2}{\displaystyle {\int}_{\partial {\Omega}_{I}}{\epsilon}^{s}:{\u2102}^{s}:{\epsilon}^{s}dS\approx \underset{h\to {0}^{+}}{\mathrm{lim}}}\frac{1}{2}{\displaystyle {\int}_{{\Omega}_{m}}{\epsilon}^{m}:{\u2102}^{m}:{\epsilon}^{m}dV}\text{\hspace{1em}(10)}$$(10)

so that (5) may equivalently be written as:

$${W}_{I}^{e}=\frac{1}{2}{\displaystyle {\int}_{{\Omega}_{I}}\epsilon {}_{I}:{\u2102}^{I}:\epsilon {}_{I}\text{\hspace{0.17em}}dV+\underset{h\to {0}^{+}}{\mathrm{lim}}}\frac{1}{2}{\displaystyle {\int}_{{\Omega}_{m}}{\epsilon}^{m}:{\u2102}^{m}:{\epsilon}^{m}\text{\hspace{0.17em}}dV}\text{\hspace{1em}(11)}$$(11)

On a micromechanics point of view, the 3D behavior of nanocomposites requires a specific morphological definition of the nanosized particles embedded in the matrix phase. These particles should be modeled as ellipsoidal inhomogeneous inclusions: a (assumed) homogeneous core made up of the particle material itself and a (assumed) homogeneous shell made up of an elastic medium with ‘large’ elastic moduli in the sense defined in (9) with *h*→0.

The case of spherical inclusions has already been adressed in the literature [8, 9, 11, 13]. Because of their difficult mathematical treatment, ellipsoidal inclusions have received less attention. It is the purpose of the present paper to propose a strategy that allows managing the ellipsoidal morphology of the inhomogeneous inclusion nanophase. The idea consists in assuming that it is possible to define an equivalent inclusion that may replace the inclusion including interface effects. This idea has already been used in a previous paper in the context of displacement discontinuity conditions between inclusion and matrix phases[14–16].

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