Computation of Effective Elastic Properties Using a Three-Dimensional Semi-Analytical Approach for Transversely Isotropic Nanocomposites  
[PERSON]  
1 Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias, Carr. al Lago de Guadalupe Km. 3.5, State of Mexico 52926, Mexico; [EMAIL_ADDRESS]  
[PERSON]  
2 Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, Universidad Nacional Autonoma de Mexico, Apartado Postal 20-126, Alcaldia Alvaro Obregon, Mexico City 01000, Mexico; [EMAIL_ADDRESS]  
[PERSON]  
3 Facultad de Matematicas y Computacion, Universidad de la Habana, San Lazaro y L, Vedado, La Habana 10400, Cuba; [EMAIL_ADDRESS]  
[PERSON]  
1 Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias, Carr. al Lago de Guadalupe Km. 3.5, State of Mexico 52926, Mexico; [EMAIL_ADDRESS]  
Footnote 1: email: {mori,tjose,  
Homogenization techniques have always been a useful tool to describe the structure-property relationship of composite materials [17; 18; 19; 20; 21]. For example, [PERSON] et al. [22] determined the effective properties of elastic FRCs with square and hexagonal fiber distributions by means of the asymptotic homogenization method (AHM) for bone mechanics applications. [PERSON] et al. [23] calculated the effective properties of elastic FRCs with transverse isotropic constituents and parallelogram-like periodic unit cells using the AHM. [PERSON] et al. [24] predicted the effective elastic properties of polymer nanocomposites using a novel numerical implementation of AHM. [PERSON] et al. [25] proposed a two-step homogenization approach to analyze the mechanical behavior of polymeric nanocomposites based on multiscale homogenization and the clustering density concept. In the literature, different homogenization implementations and computer modeling techniques to predict the effective properties of nanocomposites can be found; see for instance [26; 27; 28]. They represent a rewarding advantage through validations by comparing different models. In addition, [PERSON] et al. [29] proposed a semi-analytical method to find the effective coefficients of elastic FRCs with aligned inclusions and transversely isotropic constituents. An extension of this 2D semi-analytical method for coupled field problems was reported by [PERSON] et al. [30]. [PERSON] et al. [31] developed a semi-analytical approach based on FEM linked with the homogenization method to solve the inverse problem to determine microscale distributions. [PERSON] et al. [32] presented a new asymptotic expansion homogenization model to study the thermodynamic properties of 3D composites. In their work, the mathematical treatment takes into account the benefits of 2D homogenization and periodicity in the in-plane direction. The mechanical behaviors of three-dimensional short fiber composites, including interface debonding, were studied using an extended finite-element method (XFEM) [33]. [PERSON] et al. [34; 35] studied the effect of carbon nanotube (CNT) inclusions on the effective elastic properties of composites by numerical simulation techniques, using finite elements coupled with the Mori-Tanaka model. A computational approach based on FEM was implemented to predict the elastic behavior of a hybrid nanocomposite reinforced by CNTs and nano-clay platelets in an epoxy system [36]. A review of FEM applications in the analysis of composite materials was reported in [37]. More recently, many research works have reported on the field of nanocomposites reinforced by CNT inclusions; for example, [PERSON] et al. [38] analyzed the effect of CNT additives on the mechanical behavior of three-phase polymer-matrix composites by experimental techniques. [PERSON] et al. [39] proposed a multi-scale method to obtain the mechanical properties of interfacial regions of SWNT/epoxy nanocomposites. [PERSON] et al. [40] estimated the elastic properties of laminated composites reinforced by CNT using a numerical homogenization approach, and comparisons with experimental data were reported.