Determination of composite properties in the presence of fiber deboning and matrix damage by using the method of cell

In this study, the elastic properties of composite materials are investigated, considering the effects of separation of fiber-matrix joint boundary and matrix failure. In this method, by assuming periodic microstructure and using a linear approximation of the displacement field by applying continuity and equilibrium conditions, the composite fiber composite relation is determined. The effect of separation is assumed by introducing tangential and normal scalar parameters in the equations by assuming the displacement field jump at the common boundary. In order to express the effect of matrix micro-cracks, the fracture mechanics framework of continuous environments was used and the microcracks parallel to the fibers, perpendicular to the fibers and in the thickness direction with scalar parameters were expressed. At the end of the effect of these parameters the results are presented in graphs. The results show that the presence of defective joint at the joint boundary and the matrix microcomponents reduce the hardness of the composite and thus it’s bearing load, which is more significant at the defective joint state. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 6 September 2020 doi:10.20944/preprints202009.0143.v1 © 2020 by the author(s). Distributed under a Creative Commons CC BY license.


Introduction
The use of composite materials has increased for half a century due to their unique properties such as strength, hardness, lightness and low cost. Also, the combination of different materials to achieve new properties gives designers a wide variety of applications. The industrial and commercial applications of composites are very diverse, some of which include aerospace, automotive, sports, mining and infrastructure. In addition, fiber-reinforced polymer composites in electronic components (circuit boards), construction (floor beams), sofas and mannequins (seat springs), power industries (transformer compartments), oil industry (oil rig design) and suction tubes, the medical industry (implants, bone plates, and dentures), power transmission shafts, and oxygen tanks.
Determining the effective mechanical properties of composite materials using the properties of components (matrices and fibers) has been of interest to researchers. In general, various factors such as the properties and volume percentages of the constituents and the joint boundary properties affect the composite properties. In general, various factors such as the properties and volume percentages of the constituents and the joint boundary properties affect the composite properties. The most important failure modes of the micro scale are matrix micro-crack, fiber-matrix joint separation, and fiber breakage [1].
In recent years, various models have been used to study the behavior of composites. According to the Voigt and Reuss model, the stiffness and softness properties of the mixing law are calculated and obtain the upper and lower strength limits [2,3]. Eshelby in 1957 presented a general solution for inclusions in the matrix and using their presented model the shear value and shear modulus was calculated [4]. In 1973, Mori and Tanaka [5] proposed the average strain of several inclusions in the matrix of the strain concentration tensors and then the bulk and shear modulus of the composite by his investigations are calculated. In the self-consistent method, the properties of two-phase composites are obtained by dissolving an inclusion in the infinite matrix [6]. Another valid model to predict the elastic modulus of composites is the Halpin-Tsai model [7], originally developed for fiber-reinforced composites based on strains in a variety of composites [8][9][10][11]. These methods suffer from high computational volume and are not suitable for larger-scale complex geometries, despite the advantages of local field determination. In 1989, Aboudi introduced the cell method for determining the behavior of composite materials, in which the composite is considered periodic and consists of duplicate single cells [12]. This method, while simply providing a closed solution, predicts the elastic properties and can influence the influence of the joint boundary and matrix failure mechanisms on the properties in its flexible formulation. This method is a good candidate for multi-dimensional analysis and practical and industrial applications.
In the present study, the elastic properties of the composite were calculated using the cell method and the effect of the joint defect and matrix failure was investigated. First, the micromechanical formulation of the cell method is presented, the tangential and normal boundary separation parameters of the joints involved in the relationships, and also by applying the fracture mechanics of continuous environments, the influence of the matrix breakdown density on the reduction of the composite stiffness in the stiffness relation is applied. At the end of the properties diagrams are plotted in terms of fiber volume and different values of breakdown parameters.   Duplicate unit cells with 4 sub cells [12] The analysis of a duplicate single cell includes the stress vector coherence conditions and the displacement field at the common boundary between the sub cells as well as the neighboring unit cells and the static equilibrium conditions. One of the advantages of the cell method is the ability to calculate the properties of composites whose constituents exhibit nonlinear (corrosive or inelastic) behavior. The unit cell in this method consists of 4 square / rectangular sub-cells, one of which is fiber and the rest is matrix. So this method gives an approximation of the mechanical properties of the composite because the fiber is actually circular. However, in practical applications this approximation is in good agreement with the experimental data. The general and local coordinates and geometrical dimensions of the fiber and matrix sub-cells are shown in Fig. 1. In order to obtain the average behavior of the composite, a first-order theory is used to extend the displacement field in terms of the distance from the center of the substrates,

Micromechanical Basis of the Cell Method
x . Expansion field of displacement for each cell [12]: The following equations are extracted by using the algebraic adaptation equation of displacement and applying the continuity condition at the joint boundary of the sub-cells. (1) and (2) and applying the condition The mean strain volume is obtained from the following equation: The cell volume is equal to hl V = . By writing the equation comprising each sub-cell, the stress-strain relation is determined. Assuming the transverse isotropic behavior of the matrix and the fiber, we can write the equation of the sub-cells as follows: is the stiffness matrix component. The volume average stress is obtained from the following relation: After calculating the micro-variables in terms of geometrical parameters and general strains and using the volume average of the stress and strain tensors, one can write the general composite relationship of the unidirectional composite [12]: is the share of the thermal strain and B the hardness matrix that is: The values of ij b are presented in detail in [12]. In this study, thermal deformation is ignored.

Separation of common border
The interface between fibers and matrices in composite materials is very important because the However, the computation of the mid-phase properties is complicated, so for practical applications one can use the Jones [13] model, in which a small thickness flexible joint boundary is considered to model the fiber-matrix joint boundary separation. The scalar parameters t R and n R are expressed. In this method, the displacement field at the common boundary has a jump that is conditioned on the continuity of the common boundary. To express the separation phenomenon in the present micromechanical formulation, relation (2) is given as follows: represents the normal and tangential separation parameters. By repeating the process of the previous section, these two parameters appear in the hard elements. In the

2.2-Micro-crack matrix
Another common failure mechanism in composites is micro-crack matrix, which can increase the number of larger cracks and lead the composite to the final fracture stage. The presence of micro-cracks reduces the stiffness and composite strength, which can be explained by the failure mechanism of continuum mechanics. In this method Kachanov [14] introduced a failure variable that manifests itself as a decrease in the cross-section of the material. The effective cross-section of A is obtained by subtracting the levels occupied by the breakdowns of A . To do this, Kachanov [14] introduced a bulk element that is large enough on a macro scale to accommodate multiple failures, yet small enough to be considered a material point. The effective cross section is equal to: It should be noted that the generalization of the scalar parameter D to the tensor form is also used to calculate the stiffness reduction in 3D and with different failure modes. In composite materials, micro-cracks can be perpendicular to the material axis, that is, parallel to the fibers, perpendicular to the fibers and in the direction of thickness. So assuming three parameters of failure, the deteriorated properties can be expressed as follows [16]:

Results
In this section, using the above-mentioned equations for a single-layer composite with glass / epoxy, boron / aluminum systems, the elastic properties and their changes are calculated in terms of the separation parameters and the matrix breakdown density. The material properties of fiber and matrix are listed in Table 1.  Figure 3. It is then plotted for glass / epoxy and boron / aluminum systems with a volumetric modulus of 30% and a transverse and axial shear modulus according to the tangent separation parameter of for different volumetric ratios. As shown in Figures 4 and 5, the increase in tangential separation reduces the properties, which is more significant for the A G axial shear modulus as expected, because increasing t R means that the boundary is slippery, so the fibers under load the 13 12 ,  shear cannot completely transfer the load from the matrix to the fibers. It can also be concluded that the effect of the incomplete joint boundary on the properties is independent of the fiber and matrix systems and the higher hardness constituents behave similarly.      and decreases with increasing modulus density. According to the figure it can be concluded that in the worst case the transverse modulus reduction is less than 1 GPa because the transverse loads are parallel to the micro-crack surfaces. The greater impact of failure at higher volumetric percentages may be due to the interaction between the stress concentration of the fibers and the microtubules. As shown in Fig. 9, the longitudinal modulus decreases with increasing breakdown density at high volumetric percentages because as the composite reinforcement percentage increases, the fibers withstand higher hardness than the load matrix.

Conclusion
In this study, the elastic properties of composite materials were determined using the cell method. In particular, the effect of incomplete connection between Leaf-Matrix common boundaries was introduced by introducing scalar parameters in the formulation of cell method. In the following, using the framework of mechanical failure of continuous environments, the effect of matrix micro-cracks on reducing the hardness of the composite was also discussed. The results of the properties change for the glass and boron / aluminum epoxy system show that the tangential and normal separation, respectively, significantly reduces the axial shear modulus and the transverse modulus of a composite. Changes in breakdown density were then investigated and it was observed that increasing the breakdown density