Prediction of the macroscopic behavior of multi-layered elastic composites

. In this work, elastic fiber reinforced composite materials consisting of several layers are considered. Each layer is assumed to be a unidirectional fiber reinforced composite. The interface between layers is assumed to be perfect, which means the displacement and the traction through interfaces are continuous. To predict the overall behavior of the considered composite a two step homogenization procedure is considered. In the first step, effective properties of fiber reinforced layers are predicted based on the Mori-Tanaka micromechanical model and in the second one, a general matrix method is constructed to obtain the macroscopic properties of multi-layered composites. Analytical expressions of effective properties, for reinforced composites constituted with arbitrary N layers, are derived. The combination of the Mori-Tanaka method with the general matrix method allowed obtaining the effective properties of reinforced elastic multi-layered composites.


Introduction
Composites materials are widely used in many engineering applications and industrial fields due to their unique properties compared to normal materials: light weight, resistance to damage, thermal resistance etc. The choice of the constituents' materials and their geometries allows one to deign composites with specific properties. Several methods are developed to compute effective properties taking into account the microscopic behavior and the morphological description of components.
Multilayered composites are very used especially in space craft production and many researchers work to develop mathematical models to analyze and predict the behavior of multilayered composites. Kim [1] developed a matrix method to predict the effective behavior of anisotropic multilayered composites. Kim and Rokhlin [2] determined elastic constants of anisotropic multilayered composites using acoustic microscopy measurements. Wang et al. [3] extended the matrix method to investigate the macroscopic behavior of multilayered functionally graded multiferroic composites.
In this paper, a two steps homogenization method is used to estimate the effective behavior of multilayered composites where each layer is reinforced with short aligned fibers. First, the elastic moduli are determined for the multilayered composite without considering reinforcements. Then, the Mori-Tanaka mean field method is used to estimate the effective behavior of each reinforced lamina (layer). The obtained elastic moduli are injected into the derived expressions of multilayered composites effective properties.

Macroscopic properties of multilayered elastic composites
The linear elastic behavior of a homogeneous multilayered material is described by where ij  and kl  are constituents of the stress and strain tensors and ijkl C are the elastic constants.
The following vector notations are considered: , , n n nn nt t t tn nn with subscripts n and t refer respectively to out of plane and in-plane stresses and strains. These quantities are explicitly expressed by Equation (1) is reformulated in the following matrix vector-matrix form: , n nn n nt t t tn n tt t where nn C , nt C , tn C and tt C are the generalized Voigt matrices given by with , , where T denotes matrix transpose. Now, an elastic multi-layered composite of N homogeneous layers of elastic materials is considered (see Fig. 1). In the following, the super-script ( ) k is associated to the k th layer where ( 1, 2,..., k N  ). Then, the constitutive behavior of the th k elastic layer, is given by It is assumed that the multilayered composite is subjected to a uniform state of strain, even though the real composite may be subjected to arbitrary loading. Based on this assumption, one can write: Taking the average of ( ) k n  over the volume of the composite and using Eqs. (7) and (8), one obtains: represents the volume fraction of the th k layer. The compact form of Eq. (9), is given by Averaging ( ) k t  over the volume of the composite, we then obtain which can be explicitly, after some mathematical operation, written as The condensed form of is given by The effective properties of the considered multilayered composite is fully calculated by Eq. (11-12) and (16-17). In the next section, the Mori-Tanaka approach which further allows considering reinforcements in each layer will be considered.

Mori-Tanaka method for multi-layered composites
The Mori-Tanaka method is very popular among the existing micromechanics methods due to its effectiveness: it is easy to implement, there is no need to iterative procedures, and is known to give good estimation of effective properties of the considered composites (see references [4][5][6][7] for more readings). The Mori-Tanaka is used here to predict the behavior of reinforced layers. Each layer is constituted of a matrix reinforced with aligned fibers (inclusions) in direction (2 The obtained effective behavior allows one to numerically predict the effective elastic properties of the reinforced layer for a wide range of reinforcement's geometry. In this work, reinforcements are considered to be fibers aligned in the 2 x direction.

Numerical results
Numerical results are presented for a composite constituted of three layers. The first layer and the third one are considered to be made up of an epoxy material reinforced with carbon fibers aligned in the 2 x direction. The second layer is made up of an Epoxy material. The thicknesses of layers 1, 2 and 3 are .
. The elastic coefficients of the used materials are listed in Table 1. Reinforcements in the modeling is taken into account as ellipsoids of half axes a, b and c that are aligned with the global coordinates. So, to obtain fibers in the x 2 direction ellipsoids half axes are given the following values: 1, 1000, 1 a b c    . Table 1. Material properties of the used materials constituting the multi-layered composite.

Parameters Carbon fiber Epoxy matrix
Young's modulus (GPa) 227 3 Poisson's ratio 0.30 0.40 As a first step, the Mori-Tanaka method is used to estimate the effective properties of the first and the third layers. Figs. 3a, 4a and 5a present the predicted elastic constants with respect to the volume fraction of the epoxy material.      Then, in the second step, prediction of the behavior of the considered multi-layered composite is performed based on the homogenization procedure combining the Mori-Tanaka method with the matrix one. Figs. 3b, 4b and 5b present the elastic coefficients of the multilayered composite with respect to the volume fraction of epoxy material in layers 1 and 3. When the volume fraction equals 0 layers one and three are constituted of carbon material and when the volume fraction equals 1, layers one and three become epoxy.

Conclusions
In this work, a homogenization procedure combining the Mori-Tanaka mean field approach for each reinforced layer with the matrix homogenization method for multilayered composites was developed. It allows the estimation of multi-layered reinforced composites effective properties. Numerical results were presented for a three layers reinforced composite. The first and third layers are constituted with an epoxy matrix reinforced with carbon fibers and the second layer is considered to be epoxy. The numerical results were presented as effective properties for the reinforced layers using the Mori Tanaka model and for the whole three layers reinforced composite using the combined homogenization procedure.