On the effect of stiffness/softness and morphology of interphase phase on the effective elastic properties of three-phase composite material

A BSTRACT . In the present study, Composite material consisting of an elastic homogeneous isotropic matrix in which are embedded coated elastic isotropic inclusions, widely used in many applications is investigated by a homogenization approach coupled to the finite element method. A finite element model is proposed to predict the Young’s and shear modulus of the three-phase composite containing spherical inclusions surrounded by a spherical or ellipsoid interphase layer. Three cases of particle volume fractions and interphase were considered with the addition of two interphase morphologies. Young’s modulus of the interphase region was varied from soft to hard than the matrix properties. We note the interphase morphology and properties play an important role in the elastic properties of composite with increasing the volume fraction of inclusions and interphase. The results were compared to the first-order bounds Voigt and Reuss, and the mean-field homogenization techniques. A sensitive study of the effect of mesh density on the results of the von Mises stresses and elastic properties has been made. the unit cell approach is proposed to the estimation of effective elastic properties of three-phase coated spherical inclusion using homogenization techniques and finite elements method for different volume fractions and different interphase morphology. Two different representative volume elements were created and finite element analysis using kinematic uniform boundary conditions was performed. The effective elastic constants have been calculated with the finite element model. The numerical results demonstrate that the developed FEM approach is very useful and give good and efficient results for the analysis of unit cell models of composite, with the presence of the interphases.


INTRODUCTION
or the industrial need in material characteristics, properties and strength of these materials in their mission can be ameliorated by mastering the design of components properties (improve building energy efficiency [1][2][3][4][5][6]. Various techniques are used to give to the composite a great performance by choice of the inclusions shapes, the inclusions properties or the numbers of inclusions [7][8][9][10]. Another method is by the creation of interphase between the matrix and other phases. These interphases are formed by a desired or natural chemical reaction between the matrix and particles or the use of protective coatings during manufacturing. The incorporation of an interphase between the constituents of composite affects it elastic properties. Although small in thickness, interphases affect the overall mechanical properties of the particle reinforced composites. The effective elastic properties of the composite are related to the constituent's elastic properties and their volume fraction. Numerous experimental, analytical or computational techniques like finite element method were adopted to study the effects of various parameters like the morphology, properties of inclusions on the composite material [11][12][13][14][15][16][17][18]. This interphases phenomena influence was the object of many works and various micromechanical techniques were proposed in the literature. A simple example of the material where interphase is important is the concrete with the creation of the interfacial transition zone (ITZ) which exists in cement paste near sand particles and aggregates. Concrete must be considered as a three-phase composite: (1) cement paste, (2) ITZ, and (3) aggregates, see [6,19]. For the Ductile Cast Irons, fatigue crack propagation is strongly influenced by the matrix and by the presence of graphite inclusions. In the work of [20], the role played by the graphite inclusions and interphase during the fatigue crack propagation was investigated by means of Light Optical Microscope observations of transversal sections of metallographically prepared fatigue fracture surfaces. Because of their potential, core/shell composites have been the subject of numerous experimental and theoretical investigations as the use of core/shell nanoparticles in different fields such as building materials for energy efficiency, as well as in biomedical and biological fields, for the treatment of tumors [21,22] and medical imaging. Composites with a metal matrix reinforced with particles find their uses for their strength /rigidity ratio in a wide range of products such as the aerospace, weapons, automotive industries [23]. CNT (carbon nanotube) is characterized by a surrounding region which has a significant influence on the mechanical properties of the composite [24]. For this, the properties of the latter must be taken into account in the modeling of these materials. Recent research has explored the possibility of having complex composites with controllable particle sizes and morphologies. The properties thus obtained allow the use of these composites in solar energy, photothermal and electronic fields, [25]. For the micromechanical model, Voigt and Reuss proposed a general expression for all the composites with an upper and lower bounds for the elastic modulus taking into account the volume fractions of phases and their elastic properties [26,27]. The beginner to investigate the effects of a thin interphase on local fields and the resulting effective properties of composite made of coated ellipsoidal inclusion were Walpole and Cherkaoui et al. [28,11], by the self-consistent estimate developed a general expression for the prediction of effective parameters of a particle-filled composite with thin interphase while Aboutajeddine and Neale [29] proposed a general form of elastic properties taken into account the Mori-Tanaka method. In other papers, [30,31], the thermo-elastic behavior of matrix reinforced with long continuous coated fiber was studied. Herve and Zaoui in [32,33] generalized the model used for the representation of the three-phase model to a model composed of n-layered spherical inclusion. In [34], Lipinski et al. proposed the use of the model proposed by Hervé and Zaoui's work for the n-layered spherical inclusion morphology to study the n-layered ellipsoidal inclusion configurations. In Berger et al. [35], the effective thermo-mechanical properties of three-phase composites (one of the three-phase is considered an interphase) for different configurations were estimated using a unit cell. Alexander et al. studied the prediction of effective thermal conductivity of spherical random distributed core-shell particles distributed in a continuous matrix [36]. In Cabeza et al.- [6] and Benjamin, [12], the effective mechanical properties phase change materials PCM used in building material composed of three-phase were predicted using finite element technique and elastic deformation. Benjamin in [12,13] examined the effects of microencapsulated phase change materials (PCMs)on the thermal deformation behavior of cement-based composites. In Böhm, [37] material reinforced by coated spheres is investigated by some analytical techniques and the use of periodic homogenization. In all the preceding work, composite material is taken like a periodic array of repetitive unit cells. In this paper, we present a computational micromechanical investigation of the effect of interphase on the mechanical properties and strength of materials. We study a few material configurations (properties, morphology, and volume fraction) using the finite element method to explore how the interface morphology and properties affect the material behavior. The F objective of the current paper is the determination of the effective elastic properties and the study of the effects of interphase on these properties.

ANALYSIS
Unit cell and phases properties n our case, composite material is taken like a periodic array of repetitive unit cells, constructed with uniform distribution of the same morphology for the reinforcing phase. The micro-mechanical approach chooses this unit cell as the representative volume element (RVE) for the composites. The length of the unit cell is set to 0.1 mm. The proposed RVE containing the interphase phase, surrounded by the matrix and the spherical inclusion located at the center is schematically represented in Fig. 1, see Amraei, [38]. The coated inclusions are formed by spherical inclusions surrounded by the interphase. For the morphology of the interphase, the spherical and ellipsoid form is adopted. The volume of a hollow sphere (interphase) with radius of the overall sphere r 1 and the radius of the hollowed region r 2 can be determined as: For the ellipsoid interphase we have: Tab. 1 shows the cases of RVE in this study. In this study, i V , i P and i r represents respectively the volume of inclusions, the volume fraction and the radius of the i th phase. P is the volume fraction of the matrix phase. Three cases of volume fraction of the interface are considered in this work. Firstly the volume of interphase is equal to the volume of inclusion, secondly is half and finally the volume of interphase is one on three of inclusion volume. Three different volume fractions of matrix are considered  95%, 90% P and 70% . In this paper,     All the components are considered isotropic, this means that their combination in the unit cell generates a composite with isotropic behavior. We note that interphase is located between the matrix and inclusion, its elastic properties are affected by the elastic properties of the two phases. The Young's modulus and the Poisson's ratio of all phases are presented in Tab. 2.

Governing equations
Linear elastic constitutive relationships for an isotropic material are given by [39].
where     , , , 1,2,3 i j k l  For homogeneous and isotropic materials the tensor C is given by: with  and  are the Lamé parameters and   denote a Kronecker delta. The elastic moduli are expressed by: This constitutive law can be determined based on the detailed fields in the selected unit cell through an "averaging" procedure. First, the differential equilibrium equation in any component is expressed as (Hjelmstad, 2005): Second, the strain-displacement relation in any component is given by: where    , , T u u v w is the displacement vector. Finally, the constitutive law for each constituent is given by: (16)-(18), results in governing equations expressed solely in terms of the displacement field. These equations are referred to as Navier's equations, which, for the present case, are given by: where  I and  I are the Lamé parameters for each component.  It is important to note that continuous/welded contact between the inclusion, interphase and the matrix implied displacement continuity across their interfaces. The stress and strain components were not uniform throughout the heterogeneous structure, and thus were volume-averaged to obtain the average stresses. Using FEM the volume-averaged stresses and strains can be calculated as: where V is the volume of the RVE. The averages are then treated as the effective stress and strain fields in the homogenized RVE. The relations between  ij and  ij determine the "effective" constitutive law.

Boundary conditions
In order to fully define the problem at hand, boundary conditions must also be prescribed for the unit-cell domain shown in Fig. 2. To do so, six boundary conditions are required, three tensile and three shear loads, which were selected in order to model the elastic deformation of a computational domain. The approach proposed in [35] is used in this work. For instance, to predict the unknown effective elastic coefficients of stiffness matrix we impose the boundary conditions in such away is not equal to zero and all other strains are zero. The remaining that the macroscopic strain coefficients can be determined in a similar way.

RESULTS AND DISCUSSIONS
he finite element method is a powerful tool to solve the governing equations over the RVE domain that was discretized with satisfactorily mesh using the boundary conditions given in the previous subsection. Volume averaged stresses and strains were used for the computation of the effective Young's modulus eff E and the effective shear modulus eff μ from Eqns. (10) and (11). For the verification of the exactitude of results, a case of homogeneous composite was studied by putting the same properties for the three-component composite (inclusion, interphase and matrix) with each component. The numerically predicted effective properties Young's modulus and shear modulus were found to be identical to the values of E and mu of each component.

Convergence of the results with regard to mesh size
First, mesh convergence was verified by analysing the relative difference in the effective Young's modulus and shear modulus between different meshes. We note that the curves of von Mises stress in the unit cell in all the different meshes tested here (coarse, coarser and extra coarse) have the same appearance in addition they overlap almost completely with negligible differences, see Fig. 3 . The elastic properties and von Mises stress are independent from the mesh size chosen or computation.

Comparison with analytical bounds and estimation
The Voigt and Reuss models give too rough a framework for estimating the effective modules. Voigt's model assumes constant deformation. This model leads to an upper bound for the tensor of the modules of the effective medium. The expression is given by: For the lower bound, Reuss uses the constant stress. The expression is given by: T Figure 3: von Misses stress and shear ratio for the case of 10 % as a function of unit cell length for different mesh density ellipsoid interphase morphology (case 10% case 2) Mean-field (MF) homogenization techniques [40,41] describe the composite rheological behavior based on the average stress and strain tensors on phase level and composite level. In the MF approach, the problem of a composite with coated inclusions can be seen as a three-phase composite. Model's prediction in this research is based on the first order Mori-Tanaka homogenization (MTH) technique. The inclusions are first homogenized with their coatings (first level of homogenization), and the resulting material, which can be seen as an "effective inclusion", is then homogenized with the matrix (second level of homogenization), see Fig. 4.

Figure 4: Multi-level homogenization
The MFH approach is a fast (requires no generation of RVE model and meshing) and efficient way to forecast the effective elastic properties of linear elastic composites. The mean-field stress ( σ ) and strain (  ) in each phase i and associated parameters are given in [42]. Figs. 5, 6 and 7 plots the ratio of Young's modulus and shear modulus of the composite, to the matrix Young's modulus and shear modulus, as a function of interphase Young's modulus for two different interphase morphology spherical and ellipsoids morphologies. We note that the lower bound of Reuss gives very good results compared to the finite elements computation results for the case of prediction of shear modulus. The results are independent of the kinds of interphases morphology. The Mean-field (MF) homogenization technique give a good agreement with the finite elements estimation of the Young's modulus but it presents a big difference in the case of shear modulus equal to 25% in the case of spherical interphase morphology. For the case of ellipsoid interphase morphology, it is equal to 15%.    with the morphology of the interphase. For the contrast greather than 3, the curves are nearly linear and stable. But for the contrast less than 3, we note a big gain in elastic properties. It is also observed that for the big volume fractions of inclusions and interphase, the Young's modulus and shape affect considerably the elastic properties of composite materials, see figures. For the cases of soft interphase, the amelioration for the case of 5% in Young's and shear modulus is about 5% compared to the amelioration in other cases study of 10 and 30% are about 10 and 35% respectively. We also note a divergence of results between the cases studied especially for the volume fractions 30% but with less effect for the volume fractions 5 and 10%.  For the cases of stiff interphase, the amelioration for the case of 5% in Young's and shear modulus is about 8% compared to the amelioration in other cases study of 10 and 30% are about 16 and 60% respectively.

Effect of interphase morphology
In this section, we discuss the influence of morphological parameters namely interphase shape and volume fraction of this phase on the effective Young's and shear modulus of the three phases material. With the variation of volume of interphase from 1, 1/2 and 1/3 times the inclusion volume, we note here the increase of the elastic properties (bulk and shear modulus) in all the cases for spherical and ellipsoid interphase. In all the cases, for the soft interphase compared to the matrix stiffness, for all the volumes fractions and morphologies, the elastic properties are independent. For the stiff interphase compared to the matrix stiffness, the 1 times for all the volumes fraction and morphology presents better properties compared to the others volume 1/2 and 1/3 times, see Fig. 8, 9, 10, 11, 12, 13 and 14.    5,5] they exceed the stress found in the matrix. We note that the value of Von Mises stress in the two cases of ellipsoid and spherical interphase morphologies nearly present the same value with a small growth for the ellipsoid interphase. In the case where the interphase Young's modulus is equal to matrix Young's modulus, Fig. 17  For the ellipsoid interphase, we note a difference between the von Mises results in the two cross-section planes (ZX and ZY), see Fig.18. It is a logical conclusion because of the shape of the ellipse which has three rays (r, r, R) according to the three dimensions. It is very important to know the location of the maximum stress, for that many simulations were conducted. A lot of parameters are tested in this work by varying the interphase Young's modulus from 0.1 to 10 [MPa]. Fig.18 plots the stress for the case where we have a stiff interphase, that means the interphase Young's modulus is equal to 10 times the matrix Young's modulus (E_interphase = 10 [MPa]). In this case, the maximum stress is located in the region of interphase area for all the morphologies. Fig. 19 plots the stress for the case of interphase Young's modulus is equal to the matrix Young's modulus (E_interphase = 1 [MPa]). In this case, the maximum stress is located in the region of inclusion inside the interphase area for all the morphologies.  ). For the soft interphase, the maximum stress is located in the region around the outdoor interphase for all the morphologies. It should be pointed out here that it is according to the direction (planes XY, ZY, XZ) and the type of loading, the stresses follow the direction.

CONCLUSION
new model based on the unit cell approach is proposed to the estimation of effective elastic properties of threephase coated spherical inclusion using homogenization techniques and finite elements method for different volume fractions and different interphase morphology. Two different representative volume elements were created and finite element analysis using kinematic uniform boundary conditions was performed. The effective elastic constants have been calculated with the finite element model. The numerical results demonstrate that the developed FEM approach is very useful and give good and efficient results for the analysis of unit cell models of composite, with the presence of the interphases.

A
We observed that for the big volume fractions of inclusions and interphase, the Young's modulus and interphase shape affect considerably the elastic properties of composite materials. With the variation of volume of interphase from 1, 1/2 and 1/3 times the inclusion volume, we note here the increase of the elastic properties (bulk and shear modulus) in all the cases for spherical and ellipsoid interphase. We note here that the value of Von Mises stress in the two cases of ellipsoid and spherical interphase morphologies nearly present the same value with a small growth for the ellipsoid interphase. The maximum of Von Mises stress are observed in the interphase area. This value is located for the case with interphase young modulus equal to 8 and 10 [MPa] for the ellipsoid and spherical interphase morphology. The present work has laid down a foundation for further applications of micro-mechanical model analysis for problems, such as an investigation of stress field in the interphase for the study of inelastic behavior in this area. In addition, this model offers opportunities to study the behavior of the interphase in temperatures fields.