Tensile Properties of Thermoplastic-Laminated Composites Based on a Polypropylene Matrix Reinforced with Continuous Twaron Fibers J. L. MENA-TUN, P. I. GONZALEZ-CHI Centro de Investigación Científica de Yucatán, A. C. Calle 43 No. 130, Colonia Chuburná de Hidalgo, C. P. 97200, Mérida, Yucatán, Mexico A. DÍAZ-DÍAZ Centro de Investigación en Materiales Avanzados, S. C. Miguel de Cervantes 120, Complejo Industrial Chihuahua, C. P. 31109, Chihuahua, Chihuahua, Mexico Received: October 14, 2011 Accepted: August 25, 2012 ABSTRACT: The present paper focuses on a semiempirical macroscopic approach for the prediction of the tensile properties (modulus and strength) of thermoplastic laminates based on polypropylene (PP) reinforced with Twaron fibers. The influence of fiber content on the orthotropic stiffnesses and strengths of the unidirectional composite was experimentally determined; then, these orthotropic properties were the input data for the prediction, by means of the classical lamination theory, of the tensile effective properties of several PP/Twaron laminates. Good agreement was found between the predicted and experimental tensile data of the laminates. A previous publication focused on the modeling of the nonlinear behavior of PP/Twaron laminates; consequently, the Correspondence to: P. I. Gonzalez-Chi; e-mail: ivan@cicy.mx Advances in Polymer Technology, Vol. 32, No. S1, E749 E759 (2013) C 2012 Wiley Periodicals, Inc.
results from the present work proved that the whole mechanical response of thermoplastic laminates PP/Twaron can be simulated by means of macroscopic models originally developed for thermosetting composites. C 2012 Wiley Periodicals, Inc. Adv Polym Techn 32: E749 E759, 2013; View this article online at wileyonlinelibrary.com. DOI 10.1002/adv.21318 KEY WORDS: Thermoplastics Composites, Mechanical properties, Modeling, Tension, Introduction Laminated composites have traditionally been manufactured using thermosetting polymers because these are liquid and consequently easy to combine with fibers; they also have good compatibility with the fibers and cure at room temperature. All these factors contribute to an economical fabrication method; nevertheless, owing to international environmental concerns, there is an increasing interest in the development and use of thermoplastic composites for structural applications because of their recyclability. 1 Moreover, thermoplastic-laminated composites offer several advantages over conventional thermosetting composites, such as improved environmental resistance, increased interlaminar toughness, and enhanced damage tolerance. 2 Characterizing the mechanical behavior of thermoplastic composites is an integral part of developing these new materials. Fiber-reinforced composites can be studied at the micro- and macrolevel. The micromechanical analysis is aimed to provide an understanding of the mechanical behavior of a unidirectional composite in terms of the fiber and matrix properties; in this sense, several models of varying degrees of sophistication have been proposed to simulate the microstructure of a composite to predict its elastic properties. Surveys about such models as well as their developments have been given, among others, by Chamis and Sendeckyj, 3 Hashin, 4, 5 McCullough, 6 and Halpin 7 ; almost all these models were developed based on unidirectional fiberreinforced composites, giving good predictions in general for their linear elastic properties. However, it is difficult to generalize these models to predict other properties such as the tensile strength and the failure behavior of the composite. Failure models for composites are difficult to formulate and prone to error because composite materials exhibit various and complex failure behavior. 8 Macromechanics is the approach used to predict the overall response of laminated composites based on the average apparent properties of single orthotropic laminas. The classical lamination theory 9 is a widely accepted approach to predict laminate behavior based on assumptions of plane stress. Some work has been done for modeling the nonlinear stress strain response of polypropylene (PP)/Twaron thermoplastic laminates subjected to in-plane tension. 10 The present study is aimed at establishing a semiempirical methodology for the macromechanical prediction of the elastic properties and strength of PP/Twaron laminates. The first part of the study was to establish the experimental variation of the elastic properties and strength in the orthotropic axes of the unidirectional composites as a function of the fiber volume fraction. The experimental results were compared with the predictions from the micromechanical model of mechanics of materials to see how well these predictions worked for the thermoplastic composite; the experimental orthotropic properties from the unidirectional composite were the input data for the macroscopic modeling, and this modeling consisted of simulating the elastic modulus and tensile strength of several PP/Twaron laminates using the classical lamination theory in combination with the maximum stress failure criterion. Mathematical Formulation The laminates analyzed in the present work were subjected to a tensile load in the x direction. Each layer of the laminate was considered as a unidirectional composite (Fig. 1) and modeled as an orthotropic material with axes along the fibers, normal to the fibers at the laminate plane, and normal to the plane of the laminate. These orthotropic axes are designated as 1, 2, and 3, respectively; the orientation of a ply in a laminate is given by the angle E750 Advances in Polymer Technology DOI 10.1002/adv
E, G, ν, andv are the Young s modulus, the shear modulus, the Poisson s ratio, and the volume fraction, respectively. Subscripts f, m, 1, 2, and12refer to the fiber, the matrix, and unidirectional composite orthotropic axes. For the orthotropic strengths of the unidirectional composite, the mechanics of materials approach 12, 13 gives X T 1 = σ ult f V f + σ m V m (2a) FIGURE 1. Unidirectional composite. X T 2 = σ ult m [ ( 4Vf 1 π ) 1 ] 2 (2b) θ between the reference x axis and the orthotropic 1-axis, measured in a counterclockwise direction on the x y plane. MICROMECHANICAL ANALYSIS OF THE UNIDIRECTIONAL COMPOSITE The simplest micromechanical approach used for the analysis of a unidirectional composite is the model of mechanics of materials. This model assumes the fibers to be homogeneous, linearly elastic, perfectly aligned, and of uniform cross section. The matrix is assumed to be homogeneous, linearly elastic, and isotropic. The fiber/matrix adhesion is assumed to be perfect. Based on these assumptions, the elastic moduli of the unidirectional composite are given by Eqs. (1a) (1d) 11, 12 : E 1 = E f V f + E m V m (1a) [ ( ) 1 ( ( G m 4Vf 2 4Vf S = G 12 + 1 G f π π ) 1 )] 2 γm ult (2c) where X and S are axial and shear strengths, respectively; σ and γ are axial stress and shear strain, respectively. Superscripts T and ult refer to tensile and ultimate strength, respectively. LAMINATE BEHAVIOR (MACROMECHANICAL ANALYSIS) The classical lamination theory predicts the overall mechanical performance of a laminate as a function of the mechanical properties and the stacking sequence of its individual layers. This theory is based on assumptions of plane stress. Simple relations can be derived by means of this theory for the effective engineering stiffnesses of a laminate 11 E 2 = E f E m E m V f + E f V m (1b) E x = 1 ha 11 (3a) G 12 = G f G m V f G m + V m G f (1c) E y = 1 ha 22 (3b) ν 12 = ν f V f + ν m V m (1d) G xy = 1 ha 66 (3c) where E m E m = 1 νm 2 ν xy = a 12 a 11 (3d) where h is the laminate thickness and a ij are the elements of the laminate compliance matrix. Subscripts x, y, andxy refer to the laminate global axes. Advances in Polymer Technology DOI 10.1002/adv E751
TABLE I Mechanical Properties of Twaron Fiber and PP Matrix Property Fiber Matrix Young s modulus 113.9 GPa 1.56 GPa Shear modulus 42.2 GPa 0.57 GPa Poisson s ratio 0.35 0.38 Ultimate stress 3.01 GPa 25.5 MPa Ultimate strain 3.01% 304% The simplest and consequently the most widely used failure theory reported in the literature, to predict the laminate strength, is the maximum stress failure criterion. 12 This criterion establishes that a laminate will fail if the strength of the unidirectional composite (orthotropic coordinates) is reached at any ply, 12 namely, if σ 1 X T 1 or σ 2 X T 2 or τ 12 S (4) where τ is shear stress. Methodology MATERIALS The thermoplastic laminates were molded from PP (Valtec from Indelpro, Altamira, Mexico) reinforced with continuous aramid fibers (Twaron 2200 from Teijin, Amhem, Netherlands). PP was supplied in the form of pellets, and its specific weight was 0.9 g/cm 3. The Twaron fiber was supplied in the form of filament yarn with a monofilament diameter of 12 μm and specific weight of 1.45 g/cm 3. Table I presents the average mechanical properties of these constituent materials. IMPREGNATION OF THE FIBERS The PP was ground, and the Twaron was impregnated with the matrix using the dry powder impregnation method in a continuous impregnation line designed and constructed to conduct the present project. This impregnation line is currently under the patenting process and basically consists of a series of rubber rods, which guide the fiber yarn through the three main impregnation steps: The filaments are separated by a constant flow of air, and then they are guided through the impregnation chamber where the powder particles of the matrix stick to the fibers; the supply rate of powder matrix to this chamber controls the fiber volume fraction of the final composite. Finally, the impregnated fibers go through a heating tunnel in which the conditions of time and temperature permit the powder matrix to partially melt and adhere to the fibers, becoming a preform of the composite. The preform is collected at room temperature in an end bobbin before the molding process. MOLDING OF THE COMPOSITES The preform was cut with ceramic scissors and fitted into the mold, stacking each layer of the laminates with a lineal density of 2 preforms/cm. The laminates (250 250 mm) were compression molded at 185 C for 40 min with no pressure and then for 20 min at 130 kpa. The mold was then cooled to 80 C, and the composite was removed from the mold. Table II presents the composites molded to TABLE II Laminates Prepared to Characterize the Composite s Orthotropic Properties Study Laminate Fiber Volume Fraction (%) Number of Samples Variationofthe longitudinal properties and Poisson s ratio with the fiber content Variation of the transverse properties with the fiber content Variation of the in-plane shear properties with the fiber content [0] 8 11 8 [0] 16 17 8 [0] 24 27 8 [0] 32 35 8 [90] 4 5 8 [90] 8 10 8 [90] 16 16 8 [90] 24 28 8 [0 4 /90 8 /0 4 ] 10 5 [0 8 /90 16 /0 8 ] 17 5 [0 12 /90 24 /0 12 ] 25 5 E752 Advances in Polymer Technology DOI 10.1002/adv
TABLE III Laminates Experimentally Characterized for Modeling 14, 15 of Its Effective Properties Stacking Fiber Volume Laminate Sequence Fraction (%) TENSILE PROPERTIES OF THERMOPLASTIC-LAMINATED COMPOSITES 1 [45] 6.8 2 [90/±45/0] S 7.0 3 [90/±45/0] S 6.4 4 [±45] 5.8 5 [±45] S 7. 7 6 [±45] 2S 7.0 7 [45] 5 8.4 8 [±43] 2S 9.0 9 [±45/0/±45/0/±45] 6.1 10 [±45/0/±45/0/±45] 2 7.0 11 [±45/0/±45/0/±45] 3 9.2 12 [±45] 2S 9.8 13 [±45] 6S 11.7 14 [(0/±45/0) S ] 3 11.0 FIGURE 2. Geometry and dimensions of the tensile test specimens according to ASTM D-3039. characterize the variation of the orthotropic properties of PP/Twaron in a function of its fiber volume fraction. The experimental orthotropic properties of PP/Twaron were the input data to model the effective modulus and strength of several laminates with the stacking sequence and fiber volume fractions presented in Table III. The classical lamination theory and the maximum stress failure criterion were used for the macroscopic analysis; the laminate effective properties predicted by this approach were compared with experimental data 14, 15 of these laminates. TEST METHODS Specimen Geometry The specimens for the mechanical characterization were cut from each laminate with a band saw, and the edges were polished to eliminate any irregularity. Figure 2 shows the specimen geometry and dimensions according to ASTM D-3039 for the tensile samples (for all laminates from Tables II and III, except [0 4 /90 8 /0 4 ], [0 8 /90 16 /0 8 ], and [0 12 /90 24 /0 12 ]); a 0/90 strain gauge was cemented on the center of specimens [0] 8,[0] 16,and[0] 24 to measure the Poisson s ratio. Figure 3 shows the specimen geometry, according to ASTM D-5379, for the Iosipescu shear test of laminates [0 4 /90 8 /0 4 ], [0 8 /90 16 /0 8 ], and FIGURE 3. Geometry and dimensions of the Iosipescu shear test specimens. [0 12 /90 24 /0 12 ]; a rosette strain gauge was cemented on the center of these specimens to measure the shear strain. Tensile Tests Tensile tests were performed according to ASTM D-3039 in a Shimadzu universal testing machine (model AG-1). The crosshead speed was set to get a strain rate of 8.33 10 3 min 1 ; an extensometer was fitted to each specimen to measure the tensile strain. The tensile load and strain were simultaneously monitored to estimate tensile strength and Young s modulus of the laminate specimens. Shear Tests The shear properties of PP/Twaron composites were performed using the Iosipescu shear tests according with ASTM D-5379 in a Shimadzu universal testing machine (model AG-1). The crosshead speed was set to get a shear strain rate of 8.33 10 2 min 1 ; the data of the shear load and strain were simultaneously monitored to estimate the shear strength and the shear modulus of the composites. Advances in Polymer Technology DOI 10.1002/adv E753
FIGURE 4. Longitudinal Young s modulus vs. fiber volume fraction of the unidirectional PP/Twaron composites. FIGURE 5. Transverse modulus of the unidirectional PP/Twaron composite vs. fiber volume fraction. Results ORTHOTROPIC PROPERTIES OF THE UNIDIRECTIONAL COMPOSITE Longitudinal Young s Modulus Figure 4 presents the longitudinal Young s modulus of composites [0] 8,[0] 16,and[0] 24 with a fiber volume fraction of 11%, 17%, and 27%, respectively. The solid line represents the theoretical longitudinal modulus of the composite given by the model of mechanics of materials (Eq. (1a)) in which the Young s moduli of matrix and fiber are those given in Table I. The experimental Young s modulus coincides with the theoretical prediction, which is in accordance with reports in the literature for most of the composite materials. The fitting of the experimental data with the model has a coefficient of determination R 2 = 0.97. Transverse Young s Modulus Figure 5 shows the transverse Young s modulus of composites [90] 8, [90] 16, and [90] 24, which have a fiber volume fraction of 10%, 16%, and 28%, respectively. The solid line represents the theoretical transverse modulus of the composite (Eq. (1b)) and fits the experimental results with good approximation. The fitting of the experimental data with the model has a coefficient of determination R 2 = 0.76. FIGURE 6. In-plane shear modulus of the PP/Twaron composite vs. fiber volume fraction. In-Plane Shear Modulus Figure 6 shows the Iosipescu results for the inplane shear modulus G 12 of composites [0 4 /90 8 /0 4 ], [0 8 /90 16 /0 8 ], and [0 12 /90 24 /0 12 ] with fiber volume fractions of 10%, 17%, and 25%, respectively. The theoretical shear modulus in Fig. 6 was obtained from Eq. (1c) and Table I. The experimental shear modulus is higher than the predicted one; in accordance with the literature, 11 the mechanics of materials underestimates the shear modulus of most composite materials. Taking into account that the composite shear modulus must coincide with the matrix modulus for a fiber volume fraction of zero, an exponential function G 12 = 0.57 e 1.49V f with a coefficient of determination R 2 = 0.70 was fitted to the experimental data to estimate the in-plane shear modulus of the E754 Advances in Polymer Technology DOI 10.1002/adv
FIGURE 7. Major Poisson s ratio of the unidirectional PP/Twaron composite vs. fiber volume fraction. FIGURE 8. Tensile strength of the PP/Twaron composite vs. fiber volume fraction. PP/Twaron composite. This fitting also intends to keep a qualitative agreement with the model curve. Poisson s Ratio The theoretical Poisson s ratio given by the model of mechanics of materials (Eq. (1d); Table I) is compared to the experimental Poisson s ratio of the composites (Fig. 7). It has been found in a previous work 16 on PP/Twaron that a fiber weight fraction of 0.5 (corresponding to a fiber volume fraction of 0.4) has no enough PP matrix to fully wet the fibers; consequently, the resulting composite has dry zones that affect its mechanical properties. Taking into account that the Poisson s ratio of the composite must coincide with the Poisson s ratio of the matrix for zero fiber volume fraction, a polynomial function ν 12 = 2.45V 2 f 0.05V f + 0.38 with a coefficient of determination R 2 = 0.98 was fitted to the experimental data to describe the variation of the composite Poisson s ratio in a function of the fiber content. In a practical measurement, a strain gauge would not be able to measure the transverse contraction in a composite with a large number of dry zones in which the matrix and the fibers are not bonded together; consequently, the composite Poisson s ratio tends to be zero for a fiber volume fraction of 0.38, as shown in Fig. 7. The development of a micromechanical model for the Poisson s ratio of PP/Twaron is beyond the scope of the present study; an accurate model should consider the combined effect of several factors such as the fiber matrix bond strength, fiber arrangement in the matrix, fiber alignment, fiber flaw distribution (including statistical variations of flaw severity), dry zones distribution, and residual stresses originated during the manufacturing process. Longitudinal Strength Figure 8 shows the longitudinal strength of [0] 8, [0] 16,and[0] 32 composites. The theoretical curve in Fig. 8 was obtained using Eq. (2a) in which the fiber tensile strength is 3.01 GPa (Table I), and the tensile stress of the matrix was ignored since it is far lower than the fiber tensile strength. Experimental results showed that the longitudinal strength of specimens with a low fiber volume fraction is close to the theoretical prediction; however, the specimens with a higher fiber volume fraction are under the predicted behavior. Stress concentrations in areas where neighboring fibers come into close contact with each other may be leading to a premature failure of this composite. The molding process of this thermoplastic composite does not guarantee a homogeneous distribution of the Twaron fibers in the PP matrix, namely the fibers are somewhat randomly placed rather than being packed in a regular array; as the fiber volume fraction increases, the fibers are more likely to touch each other instead of being surrounded by the matrix material. The longitudinal strength of PP/Twaron can be estimated by means of the polynomial equation, X T 1 = 3322V2 f + 3191V f + 5, which fits the experimental data with a coefficient of determination R 2 = 0.99. Advances in Polymer Technology DOI 10.1002/adv E755
FIGURE 9. Transverse strength of the PP/Twaron composites vs. fiber volume fraction. Transverse Tensile Strength Figure 9 presents the tensile strength of composites [90] 4, [90] 8, [90] 16, and [90] 24 with fiber volume fractions of 5%, 10%, 16%, and 28%, respectively. The model curve was constructed with Eq. (2b) and Table I. In a composite with a perfect fiber/matrix bond, the interfacial strength is higher than the matrix strength; consequently, the transverse fracture will take place through the matrix. The matrix strength is the upper boundary for the transverse strength of a unidirectional composite. On the other hand, the model curve given by Eq. (2b) for the transverse strength of the unidirectional composite constitutes a lower boundary for this property since this model considers the fibers as cylindrical holes in the matrix; this model could be used even for composites with no interfacial bonding. Nevertheless, the transverse strength of the PP/Twaron composite is lower than the model s prediction. An exponential function X2 T = 25.56 e 6.975V f with a coefficient of correlation R 2 = 0.99 was fitted to the experimental data in Fig. 9 to describe the transverse strength of the PP/Twaron composite. The low transverse strength of the PP/Twaron composites may be attributed to the combination of two key factors: a random packing of the fibers in the matrix and a low fiber/matrix quality. The model for the transverse strength of the composite assumes that the fibers are distributed in the matrix in a square array and assumes that the applied load is entirely supported by the matrix; as the compression molding of this thermoplastic composite does not control the distribution of the fibers, the composite is likely to fail in zones with the greater presence of fibers where the matrix thickness is minimum. Figure 10 shows the thickness of a couple of fractured composites; the heterogeneous distribution of fibers is more evident in Fig. 10a because of the low fiber content, but both Figs. 10(a) and 10(b) show some single fibers separated from the composite, which may indicate poor wetting of fibers or poor interfacial adhesion. The interface quality between the Twaron fiber and the PP has been studied before 17 ; it has been found that their interfacial strength is relatively low FIGURE 10. Thickness of (a) [90] 8 and (b) [90] 16 composites. E756 Advances in Polymer Technology DOI 10.1002/adv
FIGURE 11. SEM image of matrix fracture in a [±45] 2S PP/Twaron laminate. compared with the interfacial strength of Twaron with other matrices such as polyethylene terephthalate and polyethylene. PP is an inert material with no functional groups that can react with the surface of the Twaron fiber to produce a chemical bond. Figure 11 shows a scanning electron microscopy (SEM) image of a typical fracture of laminates [±45] 2S. It is possible to notice fiber completely separated from the matrix, which evidences the low fiber/matrix adhesion in the composite. In-Plane Shear Strength The Iosipescu shear tests could not be completed to fracture, as the test fixture collided with the specimens; consequently, Fig. 12 shows the initial parts of the typical shear stress strain curves of composites [0 4 /90 8 /0 4 ], [0 8 /90 16 /0 8 ], and [0 12 /90 24 /0 12 ]with fiber volume fractions of 10%, 17%, and 25%, respectively. The specimens showed that a high ultimate shear strain was causing the change in the fiber orientation; consequently, the shear stress state is no longer in the direction 1 2 of a unidirectional composite. Thus, the shear strength must be reported (according to ASTM D-5379) for a shear strain of 5%. Based on the experimental data, the orthotropic properties of the unidirectional composite PP/Twaron can be defined in terms of the fiber volume fraction by the following set of expressions: FIGURE 12. Typical Iosipescu shear stress strain curves from PP/Twaron composites; shear strength was defined at a shear strain of 5%. E 2 = E f E m E m V f + E f V m G 12 = 0.57 e 1.49V f ν 12 = 2.45V 2 f 0.05V f + 0.38 X T 1 = 3322V2 f + 3191V f + 5 (5b) (5c) (5d) (5e) E 1 = E f V f + E m V m (5a) X T 2 = 25.56e 6.975V f (5f) Advances in Polymer Technology DOI 10.1002/adv E757
FIGURE 13. Experimental and simulated modulus E x of the PP/Twaron laminates from Table III. FIGURE 14. Experimental and simulated failure strength of the PP/Twaron laminates from Table III. γ ult 12 = 0.05 (5g) where γ12 ult is the failure shear strain of the unidirectional composite material. LAMINATES EFFECTIVE PROPERTIES Equations (5a) (5d) were used to estimate the orthotropic elastic properties of unidirectional composites with fiber volume fractions corresponding to the laminates presented in Table III; the estimated values were the input data to the classical lamination theory, which was used to compute the axial modulus E x (Eq. (3a)) of the laminates. A comparison between predicted and experimental axial modulus is presented in Fig. 13. Good agreement can be observed. The orthotropic strengths of the unidirectional composites were estimated using Eqs. (5e) (5g), and the laminates axial tensile strength was predicted using the maximum stress failure criterion (Eq. (4)). Figure 14 presents the comparison between predicted and experimental data. Experimental data are well approximated by the theoretical predictions. It can be concluded that classical lamination theory in combination with the maximum stress failure criterion is a suitable tool to predict the tensile properties of PP/Twaron laminates. As the modeling of the nonlinear stress strain response of PP/Twaron laminates has been reported elsewhere, 10 present results prove that the whole tensile properties of thermoplastic PP/Twaron laminates can be simulated by means of available macromechanical approaches developed for thermoset composites. Conclusions The tensile performance of the PP/Twaron thermoplastic composite was studied at both micromechanical and macromechanical levels. The comparison of experimental results with predictions from the micromechanical model of mechanics of materials, demonstrated that the elastic moduli of the thermoplastic composite show a similar behavior to those commonly reported in the literature for thermoset composites; however, the composite Poisson s ratio showed a particular behavior. The longitudinal tensile strength of the composite is accurately predicted by the model only to a certain fiber volume fraction of approximately 17%; at higher volume fractions, the longitudinal strength is affected negatively. Moreover, the transverse tensile strength of the composite is lower than the theoretical prediction. The low strength of the composite can be attributed to both poor fiber/matrix adhesion and stress concentrations generated in the composite by a random distribution of the fibers in the matrix, especially when the amount of fiber is high. The orthotropic properties of PP/Twaron were defined in terms of the fiber volume fraction by mathematical equations derived from the fitting of the experimental data from the unidirectional composites. At the macromechanical level, the results from several laminates with different stacking sequences showed that the classical lamination theory in combination with the maximum stress failure criterion E758 Advances in Polymer Technology DOI 10.1002/adv
is the right tool for predicting effective engineering properties and tensile strength of PP/Twaron laminates. The present article complements a previous report in which the nonlinear tensile response of PP/Twaron laminates was studied. 10 The present results are useful to conclude that the whole tensile mechanical response of thermoplastic laminates PP/Twaron can be simulated by means of available macromechanical approaches developed for thermoset composites. Acknowledgments The authors would like to thank Teijin Aramid for kindly providing the Twaron fiber. References 1. Kemmochi, K.; Takayanagi, H.; Nagasawa, C.; Takahashi, J.; Hayashi, R. Adv Perform Mater 1995, 2, 385 394. 2. Oya, N.; Hamada, H. Composites, Part A 1997, 28, 823 832. 3. Chamis, C. C.; Sendeckyj, G. P. J Compos Mater 1968, 2 332 358. 4. Hashin, Z. J Appl Mech 1979, 46, 543 550. 5. Hashin, Z. J Appl Mech 1983, 50, 481 505. 6. McCullough, R. L. In Micromechanical Materials Modeling (Delaware Composites Design Encyclopedia, Vol. 2), Whitney, J. M., McCullough, R. L., Eds.; Technomic: Lancaster, PA, 1990; pp. 49 142. 7. Halpin, J. C. Primer on Composite Materials Analysis, 2nd ed.; Technomic: Lancaster, PA, 1992; pp. 153 192. 8. Echaabi, J.; Trochu, F.; Gauvin, R. Polym Compos 1996, 17, 786 798. 9. Ashton, J. E.; Whitney, J. M. Theory of Laminated Plates, Technomic: Stamford, CT., 1970. 10. Mena-Tun, J. L.; Díaz-Díaz A.; Gonzalez-Chi P. I. Compos Struct 2011, 93, 2808 2816. 11. Daniel, I. M.; Ishai, O. Engineering Mechanics of Composite Materials; Oxford University Press: New York, 1994; Chap. 4. 12. Kaw, A. K. Mechanics of Composite Materials, 2nd ed.; CRC Press: Boca Raton, FL, 1975; chap. 3. 13. Mingsze-Chan, M. M.Sc. Thesis, University of Toronto, Canada, 1998. 14. Soberanis-Monforte, G. A. M.S. Thesis, Yucatan Center for Scientific Research, Mexico, 2006. 15. Mena-Tun, J. L. M.S. Thesis, Yucatan Center for Scientific Research, Mexico, 2007. 16. Gonzalez-Chi, P. I.; Ramos-Torres, W. Rev Mex Ing Quim 2007, 6, 51 58. 17. Gonzalez-Chi, P. I.; Mena-Tun, J. L.; Carrillo-Baeza, J. G. Int J Polym Mater 2002, 51, 497 509. Advances in Polymer Technology DOI 10.1002/adv E759