## 5 Conclusions  
In this work, the estimation of the effective elastic properties of periodic reinforced nanocomposites is developed. The nanocomposites are defined with isotropic or transversely isotropic inclusions embedded in an isotropic matrix. Different inclusions in the form of discs, ellipsoidals, spheres, carbon nanotubes and carbon nanowires are also considered. Two configurations of inclusions are taken into account: aligned and random. A three-dimensional semi-analytical approach (SAFEM-3D) is implemented, using a combination of the asymptotic homogenization method and the finite element method. The asymptotic homogenization method provides the local problems and the effective coefficients, whereas the finite element method gives the solution to the local problem by using the minimum potential energy.  
The results have been compared with theoretical models validating the efficiency and accuracy of the SAFEM-3D for the analysis of the composites. Based on the numerical results, we have concluded that fiber-reinforced composites with aligned nanotube inclusions show a higher effective longitudinal Young's modulus than random nanotube arrays. Random carbon nanotube arrays show that the nanotube length has a significant effect on the composite elastic properties in the axial direction. Furthermore, according to the values of the computed relative errors, it is concluded that the SAFEM-3D values have a relative error less than 2.2% when compared with the other reported theoretical models; see Tables 4 and 5. These results show the efficiency and accuracy of the SAFEM-3D model. For the case of long fibers, a slight difference is observed between SAFEM-3D and AHM (or SAFEM-2D) values when the inclusion volume fraction increases; see Table 3. This error increment is not attributed to the accuracy of SAFEM-3D; it is rather a result of the fact that the long fibers are not considered in the model.  
A comparison with experimental data for CNW nanocomposites is reported and a good agreement is obtained. An increase in the CNW weight content enhances the magnitude of \(E_{11}^{*}\). This effect is described by the SAFEM-3D results for different orientations of inclusions. This is also observed in the experiment when the CNW weight content is 1% and 3%; however, when the CNW weight content is equal to 5%, a decrease in the property is observed. From the physical point of view, we consider that this last result is not consistent with the expected behavior; thus, we assume that it is a consequence of the composite's manufacture.  
An important advantage of the SAFEM-3D approach is the possibility to design any kind of geometry reinforcement for any type of material, including isotropic and transversely isotropic nanocomposites in three-dimensional configurations for a huge range of random angle values.  
Conceptualization and methodology, J.A.O. and R.R.-R.; software and validation, M.T., Y.E.-A. and J.A.O.; formal analysis, investigation and writing--original draft preparation, J.A.O., R.R.-R., Y.E.-A. and M.T.; writing--review and editing, supervision J.A.O., R.R.-R. and Y.E.-A. All authors have read and agreed to the published version of the manuscript.  
This research was sponsored by the Consejo Nacional de Ciencia y Tecnologia (CONACYT) under grant number 412024 and the Instituto Tecnologico y de Estudios Superiores de Monterrey (ITESM).  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
[PERSON] would like to thank CONACYT for scholarship funding under Grant No. 412024 and the Instituto Tecnologico y de Estudios Superiores de Monterrey (ITESM). [PERSON] gratefully acknowledges the Program of Postdoctoral Scholarships of DGAPA from UNAM, Mexico. [PERSON] would like to thank the Department of Mathematics and Mechanics at IIMAS, UNAM.