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Science and Engineering of Composite Materials

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Volume 22, Issue 5

Issues

An energy-based equivalent inclusion for the determination of nanocomposite behavior

Yamen Maalej
  • Corresponding author
  • Mechanical Engineering, National Engineering School of Tunis (ENIT), University of Tunis El Manar, BP 37, Le Belvedere, 1002, Tunis, Tunisia; and U2MP, ENIS, University of Sfax, Sfax
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Luc Dormieux
  • Institut Navier, Ecole des Ponts ParisTech, 6 et 8, Avenue Blaise Pascal, Cité Descartes, Champs sur Marne, F-77455 Marne-la-Vallée Cedex 2, France
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/ Eric Lemarchand
  • Institut Navier, Ecole des Ponts ParisTech, 6 et 8, Avenue Blaise Pascal, Cité Descartes, Champs sur Marne, F-77455 Marne-la-Vallée Cedex 2, France
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Published Online: 2014-04-11 | DOI: https://doi.org/10.1515/secm-2013-0154

Abstract

Achieving remarkable mechanical properties for very low levels of reinforcement, nanocomposites are nowadays a new class of materials with high potential. Thanks to the stiffening aspect of the interface region between the matrix and the inclusions, nanocomposites often exhibit improved mechanical and physical properties compared to the conventional composites reinforced with micron-sized particles. This paper presents a strategy based on the introduction of an equivalent inclusion phase that allows managing the ellipsoidal morphology of inhomogeneous inclusion nanophase in the determination of estimates for the homogenized elastic moduli. Application to clay-polymer nanocomposites is presented in the framework of the dilute scheme. The latter gives theoretical estimates in good agreement with experimental results reported in the literature.

Keywords: ellipsoidal inclusions; interface clay platelets; micromechanics; nanocomposites

1 Introduction

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:

n_σn_=pg-p=γgtrb (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]:

σn_+(σs:b)n_+divsσs=0 (2)(2)

where σs is the surface stress tensor defined on the interface separating two phases. The normal discontinuity condition is taken into account through the term (σs:b) n. Remarkably, defining the surface stress tensor as σs=γlg1 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:

divsσs=σs:1T (3)(3)

where 1T=1-nn 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 σs. Following [8] and [9], the surface stress tensor is linearly related to the surface strain tensor according to:

σs=s:εs=2μsεs+λs(trεs)1 (4)(4)

where μs and κs=λs+μs 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 (WVe) is defined as the integration over ΩI of the elastic energy volume density, the other part (WSe) is defined as the integration over its external surface of a surface elastic energy:

WIe=12ΩIσ:εdV=12ΩIεI:IεIdV=WVe+12Iεs:sεsdS=WSe (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 ΩI) of radius R, undergoing a uniform strain on its external boundary (∂ΩI). Surface and volume inclusion strains are then both equal to this uniform strain, so that the surface-to-volume elastic energy ratio reads:

WSeWVe=MsMv|ΩI||ΩI|=MsMv3R (6)(6)

where Mv (resp. Ms) 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:

σs=limh00hσmdz (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, σm 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 ℂm 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 Em and Poisson’s ratio νm. Then the 2D and 3D representations are asymptotically equivalent when h→0+. Finally, the 3D corresponding stiffness tensor reads:

m=3κmJ+2μmK (8)(8)

where J and K are the spherical and deviatoric fourth-order projectors. The 3D bulk and shear modulus κm and μm of the equivalent elastic layer are deduced from their 2D (surface) counterparts (4) by:

κm=4κsμs33μsκs)h;    μm=μsh (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 Ωm) of infinitely small thickness h according to:

WSe=12ΩIεs:s:εsdSlimh0+12Ωmεm:m:εmdV (10)(10)

so that (5) may equivalently be written as:

WIe=12ΩIε:II:εIdV+limh0+12Ωmεm:m:εmdV (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].

2 Equivalent inclusion model

In this section, we present an energy-based strategy that will allow us to estimate the effective moduli of nanocomposites reinforced by ellipsoidal inclusions. The main idea consists in replacing the inhomogeneous particle (inclusion+‘3D’ membrane) by an equivalent inclusion of identical shape. This equivalent inclusion behavior is characterized by a stiffness tensor ℂeq in such a way that the elastic energy stored in this equivalent inclusion is equal to the elastic energy stored in the inhomogeneous inclusion under the same loading condition on its external boundary.

2.1 Geometrical and material description

The geometrical shape is an oblate spheroid with equatorial radii a, polar radii c (aspect ratio X=c/a) and revolution axis ez. The equation of this geometrical shape in a (x, y, z) Cartesian coordinate system reads:

x2a2+y2a2+z2c2=1 (12)(12)

Let n be the external normal vector to the surface:

n=2xa2ex+2ya2ey+2zc2ez (13)(13)

Using the spherical coordinate system (er, eθ, eφ), the ellipsoid can be parametrized by:

{x=rsinθsinϕy=rsinθcosϕz=rcosθ (14)(14)

Substituting (14) in (12) yields the following parametric equation r(θ):

r(θ)=1sin2θa2+cos2θc2=aXsin2θX2+cos2θ (15)(15)

which allows one to reconsider (13) as:

n=er+r2(θ)sinθcosθ(1a2-1c2)eθr2(θ)sin2θa4+cos2θc4 (16)(16)

The external surface acts as a membrane of constant infinitesimal thickness h occupying the geometrical space between r=r-(θ) and r=r+(θ) (Figure 1).

Geometrical description.
Figure 1

Geometrical description.

Recalling that the interfacial domain is assumed infinitesimal (h<<1), we may write:

hcosδ(r+(θ)-r(θ)) (17)(17)

where δ is the angle between the normal n and er:

cosδ=ner=1r2(θ)sin2θa4+cos2θc4 (18)(18)

2.2 Elastic energy at the membrane

In the framework of micromechanics approaches, the inhomogeneous inclusion may be interpreted as a morphologically representative pattern [17] whose analytical treatment is a complicated task. In order to overcome this difficulty, we develop hereafter an energy-based strategy allowing us to replace the inhomogeneous inclusion by an equivalent homogeneous one. It is therefore necessary to identify the elastic moduli of the equivalent inclusion so that the elastic energy stored in this equivalent inclusion is equal to the elastic energy stored in the inhomogeneous inclusion under the same loading condition on its external boundary, defined by a uniform macroscopic deformation load ℂ (Figure 2). The stiffness tensor of the equivalent inclusion is denoted by ℂeq in the sequel.

Equivalent inclusion concept.
Figure 2

Equivalent inclusion concept.

Using the revolution symmetry around ez, the field of displacement in the membrane is sought in the form:

ξ(r,θ,ϕ)=ξr(r,θ,ϕ)er+ξθ(r,θ,ϕ)eθ (19)(19)

which should comply with the following boundary conditions:

ξ(r(θ),θ,ϕ)=Er(θ)er (20)(20)

Within the framework of a variational approach, the displacement ξr and ξθ are sought in the subspace of the kinematic admissible displacement field defined by:

i=r,θ        ξi(r,θ,ϕ)=ei.Erer(1+αif(r)) (21)(21)

for any scalar function f(r) meeting the condition f(r(θ))=0. αr and αθ in (21) are two Lagrangian multipliers that will be optimized by taking advantage of the potential energy minimization principle.

The elastic strain in the membrane domain e(r,θ,ϕ)=12(ξ+Tξ) generates a density of elastic energy:

wem=12e(r,θ,ϕ):m:e(r,θ,ϕ) (22)(22)

where the 3D stiffness tensor ℂm characterizes the elastic behavior of the membrane layer according to (8) and (9). The total strain energy Wem in the membrane domain Ωm, calculated in spherical coordinate system and using the density of elastic energy (22), reads:

Wem=Ωmwem(r,θ,ϕ)dV=02π0πrr+wem(r,θ,ϕ)r2sinθdrdθdϕ (23)(23)

Recalling that the interfacial domain is assumed infinitesimal, (23) may be well approximated by:

Wem=02π0πwem(r(θ),θ,ϕ)r2(θ)(r+(θ)-r-(θ))sinθdθdϕ (24)(24)

Substituting (17) and (18) in (24), the total membrane strain energy Wem may be approximated by:

Wem=h02π0πr4(θ)wem(r,θ,ϕ)sin2θa4+cos2θc4sinθdθdϕ (25)(25)

Optimized values of parameters αr and αθ are then obtained in order to minimize the total membrane energy (25). The minimization problem to solve reads:

αiopt(E)/Wemαi|αi=αiopt=0    for   i=r,θ (26)(26)

2.3 Total elastic energy

The macroscopic elastic energy of the equivalent inclusion stored by the uniform strain ℂ applied on its external boundary yields:

Weeq=12|ΩI|E:eq:E (27)(27)

where |ΩI|=43πa2c is the volume of the equivalent inclusion.

Similarly, the elastic energy of the inhomogeneous inclusion is defined in (11). Still, considering the same uniform strain boundary condition ξd=E·z (11) yields:

Weinc=12|ΩI|E:I:E+Wem(αropt(E),αθopt(E)) (28)(28)

where Wem(αropt,αθopt) is derived from (25) and (26).

Eventually, taking advantage of the energy consistency between the elastic energy stored in the inhomogeneous inclusion (28) and the one stored in the equivalent inclusion (of identical shape) (27), it is possible to identify the elastic moduli of the equivalent inclusion that are mobilized by the uniform strain 𝔼 applied on the external boundary:

12|ΩI|E:eqI:E=Wem(αropt(E),αθopt(E)) (29)(29)

The next sections are devoted to the derivation of the previous methodology to two classes of inclusion geometry: the spherical inclusion and the oblate spheroidal inclusion characterized by an aspect ratio X→0.

3 Application to a spherical inclusion geometry

Let us first consider the case of a spherical inhomogeneity of radius R. This is equivalent to considering the case X=1 (a=c) in the previous reasoning. The parametric equation (15) also yields r(θ)=R, so that the elastic energy in the 3D membrane, defined in (25), takes the following form:

WemhR202π0πwem(R,θ,ϕ)sinθdθdϕ (30)(30)

Due to the spherical symmetry of this problem, the equivalent inclusion behavior is isotropic and fully characterized by two coefficients keq and μeq. Two different homogeneous strain definitions have to be considered in order to determine these coefficients.

3.1 Hydrostatic deformation: E∞=E 1

Considering a hydrostatic deformation E=E1, the elastic energy stored in the equivalent inclusion (27) reads:

We,1eq=|ΩI|keqE2 (31)(31)

The displacement field in the membrane (19) reduces to as ξ(r, θ, φ)=ξr(r, θ, φ)er with:

ξr(r,θ,ϕ)=Erer(1+αrf(r)) (32)(32)

The optimization procedure (26) provides the unique optimal parameter αr,1opt, which yields, whatever the choice for function f(r), the following estimate of the equivalent bulk modulus according to the energy equivalence principle (29):

keqkI+12κmμm4μm+3κm(ha) (33)(33)

or equivalently, using (9):

keqkI+4κs3R (34)(34)

3.2 Deviatoric deformation: E=E(e1e1- e2e2)

Considering a uniform loading defined by a deviatoric strain E=E(e1e1-e2e2) applied on the spherical inhomogeneity, the elastic energy stored in the equivalent inclusion (27) reads:

We,2eq=2|ΩI|μeqE2 (35)(35)

The displacement field in the membrane is defined by (19) and (21). The optimization procedure (26) provides optimal parameters αr,2opt and αθ,2opt. Used in (29), these optimal values yield, whatever the choice for function f(r), the following estimate of the equivalent elastic shear modulus μeq:

μeqμI+μm(8μm+9κm)5(4μm+3κm)(ha) (36)(36)

or equivalently, using (9):

μeqμI+6μs+κs5R (37)(37)

Concerning the order of magnitude of the equivalent inclusion elastic moduli, Eqs. (34) and (37) clearly exhibit the increasing contribution of the surface (elastic) energy for decreasing values of the inclusion radius R. The latter defines the characteristic size of the spherical inclusion phase. In practice, incorporating a spherical inclusion having a radius of the order of a few nanometers, likely to significantly increase elastic moduli of nanocomposites, may appear as a complicated technical task. Many studies have therefore turned into natural materials defined at the nanoscale by solid-phase inclusions of high surface area. Clay materials, and more particularly smectites, meet this requirement (among others) and have been widely used as inclusion phase for nanocomposites. Depending on their level of deflocculation (exfoliation), the nanoscopic solid phase of these natural materials may be represented by ellipsoidal inclusions whose aspect ratio may be very small (1/100) for complete deflocculation. The next section takes advantage of the energy equivalence strategy in order to propose a micromechanical model to the nanocomposite behavior. The ‘oblate spheroid’ morphology with the additional surface energy contribution accounts for a montmorillonite (MMT) clay platelets nano-inclusion and is introduced in a polymer matrix. The validation is performed by using reference experimental study.

4 Application to a polymer matrix reinforced by MMT clay platelets nanocomposite

The present section is devoted to the determination of estimates for the macroscopic elastic moduli of nanocomposites. These estimates are derived in the framework of Eshelby-based micromechanics approaches [18, 19]. The very low level of reinforcement usually encountered suggests resorting to the well-known dilute scheme. The nanocomposite here is a polymeric matrix reinforced by parallel clay platelets. The latter are modeled as a non-interacting oblate spheroid inclusion embedded in a continuous polymeric matrix. These ellipsoids are characterized by the same aspect ratio X=c/a→0 (2a=clay particle diameter in the e1/e2 plane; 2c=clay particle thickness in the e3 direction; particle volume ΩI=43πXa3) and the same orientation that coincides with the direction of the axis of rotational symmetry parallel to the sample axis e3 (Figure 3).

Parallel clay platelets reinforcement.
Figure 3

Parallel clay platelets reinforcement.

The nanocomposite is therefore a two-phase medium made up of:

  • a polymer matrix phase, characterized by an isotropic elastic tensor ℂ0.

  • an inclusion phase modeled as a flat oblate spheroid inhomogeneity. The core material is the inclusionary material (clay platelets here). Its behavior is assumed transversely isotropic (isotropic in the e1/e2 plane). The thin shell material is isotropic with elastic moduli defined in (8) and (9).

Following the results of the previous section, the inclusion phase is replaced by a transversely isotropic equivalent inclusion with stiffness tensor ℂeq. The homogenized behavior itself is expected transversely isotropic and the elasticity tensor is characterized by five independent coefficients. However, the reference experimental results that are used to validate the developed model concerns Young’s modulus as the nano-reinforcements are randomly distributed. In the sequel; only the moduli in the principal direction E11=E22 and E33 are estimated for the considered polymeric matrix reinforced by parallel clay platelets.

4.1 Dilute scheme

Due to the low level of reinforcement (≤5%), all the effective elastic moduli of this transversely isotropic nanocomposite are here estimated in an explicit analytical way by using the so-called dilute scheme approximation.

A direct application of usual micromechanics relations allows us to define the homogenized elastic moduli according to [19]:

hom=0+fδ0eq:Aeq    with    δ0eq=eq0 (38)(38)

where 𝔸eq is the equivalent inclusion strain concentration tensor. In the case of a dilute scheme approximation, the latter reads:

Aeq=[I+0:δ0eq]1 (39)(39)

As discussed in the previous section, we will restrict our analysis to the determination of estimates for the longitudinal compliance modulus 1/E33 and the transversal compliance modulus 1/E11=1/E22. As a consequence, there is no need to fully determine ℂeq, but only the coefficients involved in the determination of these estimates. In the present case, up to four coefficients, or groups of coefficients, have to be identified. According to Section 2, this requires three (resp. four) appropriate uniform strain tensors E to be considered in order to estimate 1/E33 (resp. 1/E11=1/E22). Details of the procedure are developed in the Appendix.

5 Results and comments

Two experimental studies are used to validate the developed model. The first concerns the experimental results obtained by Cauvin et al. [20], which is devoted to the mechanical behavior of a polypropylene (PP) reinforced by randomly distributed clay nanoplatelets (MMT). The elastic properties of the PP matrix are E=0.9 GPa and v=0.4, and of the MMT inclusion are E=170 GPa and v=0.25. The second concerns the experimental results presented by Luo and Daniel [21] devoted to polymer/clay nanocomposites consisting of epoxy matrix filled with randomly distributed silicate clay particles where the elastic properties of the epoxy matrix are E=2.05 GPa and v=0.35; the clay particles have the same characteristics as those used by Cauvin et al. [20].

Figures 4 and 5 provide a comparison of the reported experimental results with the developed model. The results of the model using spherical inclusion (black) show a significant underestimation of the nanocomposite elastic modulus even taking account of the interface effects. For the flat oblate spheroid inclusion, the longitudinal modulus E11=E22 (red) represents the behavior obtained for a macro inclusion distribution oriented in the loading direction. This corresponds to the configuration where all MMT particles are oriented in the same direction, which gives, thanks to the interface properties and the great length of particles, an upper limit of the macroscopic behavior in this direction. By contrast, the transversal modulus E33 (green) represents the case where only the thickness of the platelets contributes to the macroscopic stiffness. Given that this thickness is very low, the supply of MMT platelet stiffness in this direction represents a lower bound of the macroscopic behavior. The dashed curve represents an average of the two modules for each fraction of inclusion.

Comparison with experimental results from Cauvin et al. [20].
Figure 4

Comparison with experimental results from Cauvin et al. [20].

Comparison with experimental results from Luo et al. [21].
Figure 5

Comparison with experimental results from Luo et al. [21].

The developed model based on the use of a flat oblate spheroid inclusion clearly shows a good agreement with the two experimental results. The strong effect of the nano-inclusion on the variation of Young’s modulus as a function of the MMT volume fraction (<5%) is well predicted.

It is worth noting that the the aspect ratio X was numerically calibrated to fit experimental results; its direct determination needs microstructural observations, which are not provided. We also note that the exfoliation process of these materials is not under control, and in the absence of microstructural observations, it is difficult to predict if all the platelets of the MMT aggregates are separated and dispersed randomly within the matrix (complete exfoliation) or partially separated and emerging platelet aggregates (partial exfoliation). This particular point needs further research based on the developed model and choosing an adequate homogenization scheme taking into account the microstructure morphology. In the case of complete exfoliation, the macroscopic behavior is expected to be isotropic. For an ongoing research, an extension to spatial distribution of the flat oblate spheroid inclusion should be taken into account in the developed model.

Acknowledgments

This research is conducted through a collaboration between the Ecole des Ponts-ParisTech and the National Engineering School of Sfax. This collaboration is financed by the French Institute of Tunisia according to the program “Séjour Scientifique de Haut Niveau” (SSHN). The authors gratefully acknowledge the support of these institutions.

Appendix

In this section, the elastic moduli of the transversely isotropic equivalent inclusion are computed using various appropriate uniform strain tensors E.

Hydrostatic deformation: E=E1

Considering a hydrostatic deformation E=E1, the elastic energy stored in the equivalent inclusion (27) reads:

Weeq,1=|ΩI|C1eqE2 (40)(40)

where:

C1β=(1:β:1)/2=C1111β+C1122β+2C1133β+C3333β2 (41)(41)

The displacement field in the membrane (19) reduces to ξ(r, θ, φ)=ξr(r, θ, φ)er. The optimization procedure (26) leads to the sole optimal value αropt,1:

αropt,1=24κX23μκ+O(X3) (42)(42)

which tends toward 0 when the aspect ratio X→0. Substituting (42) in (29) yields the following estimate of the elastic coefficient C1eq:

C1eqC1I+9μmκm(3μmκm)c (43)(43)

Biaxial deformation: E=E(e1e1+e2e2)

Similarly, the elastic energy stored in the equivalent inclusion (27) now reads:

Weeq,2=|ΩI|C2eqE2 (44)(44)

where (1p=e1e1+e2e2):

C2β=(1p:β:1p)/2=C1111β+C1122β (45)(45)

The optimization procedure (26) yields the optimal values:

{αropt,2=32μmμm2-12κmμm+3κm23μm-κmX2+O(X3)0αθopt,2=32μm(κm-μm)+O(X3) (46)(46)

Substituting (46) in (29) yields the following estimate of the elastic coefficient C2eq:

C2eqC2I+3(μm2+10κmμm-3κm2)4(3μm-κm)c (47)(47)

Deformation: E=E(e3e3)

The elastic energy stored in the equivalent inclusion (27) reads:

Weeq,3=|ΩI|C3eqE2 (48)(48)

where:

C3β=C3333β/2 (49)(49)

The optimization procedure (26) yields the optimal values:

{αropt,3=3κ-5μ2(3μ-κ)+O(X3)αθopt,3=32+O(X3) (50)(50)

Substituting (50) in (29) yields the following estimate of the elastic coefficient C3eq:

C3eqC3I+3μm243μm-κm)c (51)(51)

Deformation: E∞=E(e1e1)

The elastic energy stored in the equivalent inclusion (27) reads:

Weeq,4=|ΩI|C4eqE2 (52)(52)

where:

C4β=C1111β/2 (53)(53)

The optimization procedure (26) yields the optimal values:

{αropt,4=32-12κμ+3κ2+μ2m(3μ-κ)X2+O(X3)0αθopt,4=32κ-μμ+O(X3) (54)(54)

Substituting (54) in (29) yields the following estimate of the elastic coefficient C4eq:

C4eqC4I+3π1615μm2+26κmμm-9κm2(3μm-κm)c (55)(55)

References

About the article

Corresponding author: Yamen Maalej, Mechanical Engineering, National Engineering School of Tunis (ENIT), University of Tunis El Manar, BP 37, Le Belvedere, 1002, Tunis, Tunisia; and U2MP, ENIS, University of Sfax, Sfax, e-mail:


Received: 2013-07-25

Accepted: 2014-03-01

Published Online: 2014-04-11

Published in Print: 2015-09-01


Citation Information: Science and Engineering of Composite Materials, Volume 22, Issue 5, Pages 475–483, ISSN (Online) 2191-0359, ISSN (Print) 0792-1233, DOI: https://doi.org/10.1515/secm-2013-0154.

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