ISSN : 2250-0138 (Online) ISSN: 0976-2876(Print) VALUATION OF FFCTIV MATRIAL PROPRTIS OF RANDOMLY DISTRIBUTD CARBON NANOTUB COMPOSITS CONSIDRING INTRFAC FFCT HOSSINMAMANPUSH a, HOSSIN GOLSTANIAN b,c1 a MechanicalngineeringDepartement, UniversityofShahrekord, Shahrekord,Iran, b Faculty of ngineering, University of Shahrekord, Shahrekord, Iran c Nanotechnology Research Center, University of Shahrekord, Shahrekord, Iran, ABSTRACT The superlative mechanical properties of carbon nanotubes (CNTs) make them the filler material of choice for composite reinforcement. Substantial improvements in mechanical properties of polymers have been attained through the addition of small amounts of carbon nanotubes. The way in which CNTs are distributed inside the matrix can be divided into two main categories namely: aligned and randomly distributed. In this research the effect of both aligned and randomly distributed carbon nanotubes on the effective material properties of nanocomposite is investigated. In addition, effect of CNT/Matrix interface strength on the carbon nanotube reinforced polymer mechanical properties is considered. Theory of elasticity of anisotropic materials and finite element approach are used for the estimation of the effective material properties of randomly oriented carbon nanotube reinforced composites. Nanocomposite mechanical properties are evaluated using a square 3D nanoscale Representative Volume lement (RV). RVs consisting of aligned and randomly oriented CNTs are modeled to investigate the effects of nanotube dispersion on nanocomposite mechanical properties. The computational results indicated that mechanical properties of nanocomposite are remarkably dependent on the dispersion of the CNTs and also CNT/Matrix interface strength. KYWORDS:Nanocomposite, Carbon Nanotube, Mechanical Properties, CNT Dispersion, Randomly Distributed During the last decade carbon nanotubes (CNTs) have generated considerable interest in the scientific community because of their potential for large increase in strength and stiffness, when compared to ordinary carbon fiber reinforced polymer composites. The combination of extraordinary mechanical and physical properties makes CNTs prospective candidates for reinforcement of polymer matrix composite systems. Besides their extraordinary small size, CNTs are half as dense as aluminum, with a density of 1.33-1.4 g/cm 3. CNTs have Young's modulus of about 1 TPa and tensile strength 20 times that of steel alloys. Tensile strength of single walled carbon nanotubes (SWCNTs) is as high as 2 GPa, which is much higher than that of high strength steels (Guz et al., 2007, Collins and Avouris, 2000). The fact that the inclusion of even small amounts of CNTs coupled with suitable processing steps appears to significantly enhance mechanical properties of polymers have catapulted nanotube reinforced composites to being one of the most attractive areas of nanotechnology (Li and Chou, 2003, Chang and Gao, 2003, Krishnan et al., 1998, Yao and Lordi, 1998, Lu, 1997). Several fabrication and modeling issues need to be addressed to optimize the properties of such materials, including dispersion of the CNTs within the polymer, CNT polymer bonding and interaction, and nanotube orientation and alignment. ven though some nanotube composite materials have been characterized experimentally, the development of these materials can be extremely facilitated by using computational methods (Thosenson et al., 2005, Qian et al., 2002, Srivastava et al., 2003). * The approaches taken by previous researchers can be divided into two main categories: the atomistic approach and the continuum mechanics approach. The atomistic approach includes classical molecular dynamics(liu et al., 2004) and tight binding molecular dynamics (Nasdala and rnst, 2005). Although these approaches can be used to deal with any problem associated with molecular or atomic motions, their huge computational requirements bound their application to problems with small number of molecules or atoms (Nasdala and rnst, 2005). In particular, molecular dynamic simulation can predict the effective mechanical properties of nanocomposites on specific region. The continuum mechanics approach mainly involves classical continuum mechanics (Pipes and Hubert, 2002) and continuum shell modeling (Das and Wille, 2004).xperimental results, however, show large variations in nanotube strength data. Li and Chou (2003) determined shear modulus as a function of SWCNT diameter using the molecular structural mechanics approach. Xu and Sengupta (2005) used finite element method to investigate the interfacial stress transfer and possible stress singularities in a nanocomposite. Kulkarni et al. (2010) used numerical and experimental approaches to investigate the elastic response of a carbon nanotube fiber reinforced polymeric composite. Valavala and Odegard (2005) presented a review article discussing the major modeling tools available for predicting mechanical properties of nanostructured materials. Liu et al. (2005) developed a new continuum model of the CNT based composites for large scale analysis at the micro scale in order to characterize such composites. Chen et al. (2008) proposed a micromechanical method to predict size dependent plastic properties of composite materials. Chen and Liu(2004) evaluated effective mechanical properties of CNT based composites using a square Representative Volume lement (RV) based on the continuum mechanics using FM. Golestanian and Shojaie (2010) used finite element method to investigate the interface effects of CNT based polymer composites. Bogdanivich and Bradford (2010) produced 1 Corresponding Author
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS macroscopic textile preforms using CNTs and used them as reinforcements in makingnanocomposite tensile test samples. Golestanian and Matin (2011) determined the effects of CNT helical angle on mechanical properties of carbon nanotube reinforced polymer composites. Shady and Gowayed (2010) have discussed the impact of the nanotube curvature on the elastic properties of nanocomposites using the modified fiber model and the Mori Tanaka approach. It has been observed that there is very little change in the composite modulus by improving the dispersion for large orientation angles (Wang, 2008). Golestanian and Mamanpush (2014) investigated the effects of carbon nanotube orientation on the macroscopic stiffness of nanocomposites. Moreover, aligned or regularly dispersed CNTs are found to produce the largest improvements in nanocomposite tensile modulus, compared to randomly oriented carbon nanotubes (Joshi et al., 2010). There are many crucial issues that have yet to be examined such as the impacts of dispersion and interface strength of embedded nanotubes on the effective mechanical properties of the CNT reinforced polymer that provide the motivation for this current study. One of the most important problems in solid mechanics is the determination of effective elastic properties of a composite material made up of isotropic random distribution of elastic carbon nanotubes embedded in a continuous, isotropic and elastic matrix as shown in Fig. 1.Numerical models seem to be a convenient approach to describe the behavior of these materials, because there is no restriction on the geometry, material properties, number of phases in the composite, and on the size. The concept of unit cells or RVs has been applied to model CNT based nanocomposites at the nanoscale level (Lakhnitskii, 1981). In order to achieve the best predictions of a nanocomposite material behavior by computational means, three dimensional numerical simulation of an RV is most suitable (Joshi et al., 2012). Figure 1. Procedure of homogenization method applicable to randomly distributed CNTreinforced polymer. In the present work several computational experiments are presented to know the influence of different dispersion of CNTs in a nanocomposite. The specific parameters investigated in this research are the effects of randomly distributed CNTs and CNT/matrix interface strength on the mechanical properties of nanocomposites. To achieve our goals, FA models consisting of several CNTs are created. Relations are derived based on the elasticity theory to extract the effective material constants from numerical models of the square RVs under specified load cases. RVs consisting of uniformly distributed and randomly oriented CNTs are modeled to investigate the effects of nanotube dispersion on nanocomposite mechanical properties. Further, the effects of interface strengthon nanocomposite properties are determined by modeling an elastic intermediate layer at the CNT/matrix interface. Two typesof interface material behaviors are considered, namely:perfect bondingandthe elastic interface.ultimately, the results of the cases of randomly oriented and uniformly distributednanotube reinforced polymer composites are compared at a volume fraction of 1%. The analysisand resultsof this investigation are presented in the following sections. ANALYSIS The homogenized effective elastic constants of the composite are obtained through finite element analysis of a periodic cubic RV of volume L 3 as shown in Fig. 2 consisting of uniformly distributed and randomly distributed carbon nanotubes. Several RVs reinforced with: (i) The 4 4 array of CNTs aligned in the z direction, shown in Fig. 3(a) and (ii) sixteen randomly distributed nanotubes, shown in Fig. 3(b) are modeled. In all cases, both perfect bonding and elastic interface conditions are modeled. lasticity solutions can be obtained under certain load cases. The RV reinforced with randomly oriented CNTs are not symmetric and are considered as an anisotropic body.in these cases, generalized Hooke's law is used to determine material constants. The analysis approachand the corresponding formulations are represented in the following subsections. -133-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS Figure 2. The RV used to model the CNT based polymer composite. Figure 3. (a) Distribution of the uniformly oriented CNTs in the RV, (b) Distribution of the randomly oriented CNTs in the RV (In both cases matrix removed for a better demonstration of the CNTs in the RV). The meshed geometry of the 3-D RV is shown in Fig. 4(a). In Figs. 4(b) and 4(c) the meshed CNTs are shown with the matrix removed for better observation. One layer of elements is found to be sufficient for meshing the CNTs in this FA model after performing a grid study. Figure 4. (a) A 3-D FM model of the square RV, (b) A 3-D FM model of the randomly oriented CNTs, (c) A 3-D FM model of the uniformly distributed CNTs (in cases b and c only nanotubes are shown for a better demonstration). Generalized Hooke s law To derive the relations for extracting the equivalent material constants, a homogenized elasticity model of the square RV as shown in Fig. 2 is considered. The analysis approach is presented in this section. Generalized Hooke s law is valid for every continuous medium, regardless of its physical properties. Hooke s law explains the relations between strain and stress components in the body. The generalized Hooke s law is represented in quation (1). In general the number of elastic constants is 36. 1 υ yx υ zx η yz, x η zx, x η xy, x υ xy 1 υ zy η yz, y η zx, y η xy, y ε x yy yy yy yy yy yy σ x ε y υ υ yz 1 η yz, z η zx, z η xy, z σ y ε z zz zz zz zz zz zz σ z = γ yz η x, yz η y, yz η z, yz 1 µ zx, yz µ xy, yz τ yz γ G yz G yz G yz G yz G yz G yz τ γ xy η x, zx η y, zx η z, zx µ yz, zx 1 µ xy, zx τ xy G G G G G G η x,xy η y, xy η z, xy µ yz, xy µ zx, xy 1 G xy G xy G xy G xy G xy G xy In this equation, x, y and z are Young s moduli in directions x, y, and z. G xy, G, and G yz are the shear moduli for planes parallel to the coordinate planes. The constants η yz,x,, η xy,z characterize extension in the directions of the coordinate axes produced by shearing stresses acting in the coordinate -134- (1)
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS planes. They are termed the mutual influence coefficients of the first kind. Finally, η x,yz,, η z,xy, are called the mutual influence coefficients of the second kind. These coefficients express shears in the coordinate planes due to normal stresses acting in the directions of the coordinate axes (Lakhnitskii, 1981). These constants are usually non zero for an anisotropic elastic body and are zero for an isotropic body. Analysis of the RV reinforced with aligned nanotubes The cubic RV used to model the nanocomposite reinforced with aligned nanotubes is shown in Fig. 2. This RV is symmetric; therefore it has five independent material constants, namely; x = y, z, υ xy,υ, and G.In this investigation four of the five material constants are determined and G is not considered. The RV reinforced with the aligned nanotubes is transversely isotropic. The general 3D strain stress relations relating the normal stresses and strains for a transversely isotropic material can be written as (Liu et al., 2005): 1 υ yx υ zx ε x σ x υ xy 1 υ zy ε y = σ y yy yy yy ε z σ z υ υ yz 1 zz zz zz To determine the four unknown material constants ( x, z, υ xy and υ zx ), four equations are needed. These equations are obtained from loading the RV in different directions as illustrated in Figs. 5 and 6. These loading cases and formulations for each case are presented in the following sections. (2) Figure 5. The RV under an axial elongation, L, in the z direction. In the case of randomly distributed CNTs, an axial distributed load is applied in the z direction. Figure 6. The RV under a transverse distributed load in the y direction. Cubic RV under an axial elongation In this loading case, shown in Fig. 5, the RV is placed under an arbitrary axial elongation L in the z direction. The stress and strain components on the lateral surface are given by (Bogdanivich, 2010); L x σ = σ = 0, ε =, ε = along x = ± a x y z L x a y ε = along y = ± a y a Where, a is the change in one half of the cross section length, a, under the elongation L in the z-direction. Integrating and averaging the third equation in (2) on the plane z = L/2, we obtain; σave L z= = σave (3) εz L Where,σ ave is the average value of stress in the z direction, given by; 1 σ ave = σ z ( x, y, L / 2) dxdy (4) A A Where, A is the RV cross sectional area. The value of σ ave is evaluated by averaging stresses in the z direction over all elements in the RV cross section at L/2. Using one of the -135-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS relations in quation (2), together with the value of z found from quation (3), along x = ±a we have (Liu et al., 2005); ε = υ zx x z σ z = υ L zx L = a a Thus, we can obtain an expression for the Poisson s ratio as follows; a L υzx= / a L quations (3) and (6) can be applied to estimate the effective Young s modulus z and Poisson s ratio υ zx ( = υ zy ), once the contraction a and the average stress, ơ ave, in this loading case are obtained. Cubic RV under a lateral uniform load In this load case, the square RV is loaded with a uniformly distributed tensile load, p, in one of the lateral directions, for example the y direction. The RV is constrained in the z direction so that the plane strain condition is maintained in order to simulate the interactions between the nanotube and matrix materials existing in the z direction. Thus, the 3D strain stress relations for normal components in quation (2) are reduced to (Golestanian and Matin, 2011); 1 υ 2 2 zx υxy υ zx ε x x z x z σ x ε = y υxy 2 1 2 σ υ y zx υ zx x z x z For the corresponding elasticity model, we have the following results for the normal stress and strain components at a point on the lateral surfaces; x σ = 0, σ = p, ε = along x = ± a x y x a y ε = along y = ± a y a Where, xand y are the changes of dimensions in the x and y directions, respectively. Applying the first equation in (7) for points along x = ±a, and the second equation in (7) for points along y = ±a together with the above conditions, we obtain; (5) (6) (7) Solving these two equations gives the effective Young s modulus and Poisson s ratio in the transverse direction, x y plane, to be; 1 x= y= y υ 2 (9) + zx pa z x υ 2 + zx pa υ z (10) xy= y υ 2 + zx pa z The results of axial elongation loading case are used in quations (9) and (10) for z and υ zx. Once the changes in dimensions, xand y, are determined for the square RV from a finite element analysis, x ( = y ) and υ xy can be computed from quations (9) and (10), respectively. Analysis of RV reinforced with randomly distributed CNTs The RV reinforced with randomly distributed CNTs is not symmetric and is considered as an anisotropic body. In this case, to extract material constants of the nanocomposite, the first three relations in quation (1) are used. Thus, we have (Liu et al., 2005); 1 υ yx υ η yz, x η zx, x η zx xy, x σ x σ y ε x υ xy 1 υ zy η yz, y η zx, y η xy, y σ z ε y = yy yy yy yy yy τ yy yz ε z υ υ yz 1 η τ yz, z η η zx, z xy, z xy zz zz zz zz zz τ zz In this paper in case of randomly distributed CNTs, the nine independent material properties were calculated namely (three moduli of elasticity and six Poisson s ratios) for the anisotropic nanocomposite. Shear moduli are not considered in this paper.the shear moduli can be calculated using torsional loadings applied to the RV in different directions. To determine the nine unknown material constants, nine independent equations are needed. (11) υxy υ 2 zx x ε = + p = x x z a 1 υ 2 zx y ε = p = y x z a (8) -136-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS Figure 7. The RV of the randomly distributed CNT reinforced polymer under a transverse distributed load in the x direction. Three different loading cases have been devised to provide these equations. In these loading cases, illustrated in Figs. 5-7, coefficients η ij,k are set equal to zero since τ ij = 0. Then, quation (11) reduces to (Golestanian and Matin, 2011): Table 1. Comparison of results for the case with m = 100 GPa. Investigator z/ m x / m y/ m Current investigation 1.0451 0.9971 0.9971 Chen and Liu (2004) 1.0391 0.9342 0.9342 1 υ yx υ zx ε x σ x υ xy 1 υ zy ε y = σ y yy yy yy ε z σ z υ υ yz 1 zz zz zz (12) In case of the nanocomposite reinforced with randomly oriented CNTs, three distributed loading cases were applied. ach of these three loading cases consisted of a distributed load applied in one of the coordinate directions. That is, three different models were created and in each one the distributed load was applied in one coordinate direction. Initially, the RV shown in Fig. 2 is loaded by a tensile stress along the x axis. With this loading, Young s modulus can be calculated using the first relation in quation (12). Using a similar method, for loadings in the y and z directions, and using the second and the third relations in quation (12), Young s moduli yy and zz, are determined. In quation (12), strain components in different directions are calculated using; ε = u L Where u is the average value of displacement of nodes on the considered plane and L is the initial length of RV in the corresponding direction. For example, to calculateεz, the RV is cut at an arbitrary location perpendicular to the CNT in the composite region. Then, the displacements in the z direction of nodes located on the cross section are averaged. Finally, the calculated average value is divided by the initial width of the RV in the z direction. Verification of the modeling aproach In this section, the results of the current investigation are compared with those presented by Chen and Liu (2004), to validate the FA models. This comparison is presented in Table 1. For this comparison a short CNT inside the square RV is modeled as shown in Fig. 8. Chen and Liu used a modulus of elasticity of 100 GPa for the matrix in their models and assumed perfect bonding at the CNT/matrix interface. The dimensions of the RV and CNT are kept the same as those in Chen and Liu s model. In both cases CNT volume fraction is 1.62%. Thus the comparison is made with our results at the same conditions. Note that the results of this investigation are in good agreement with the results presented by Chen and Liu. (13) Figure 8. Quarter of the RV used to model the CNT based polymer composite. RSULTS AND DISCUSSION As explained above, two types of models were created and analyzed. In the first set, perfect bonding condition is applied in both of the aligned and randomly distributed CNT reinforced polymer composites. In the models with perfect bonding, the conditions of no slip are imposed at the CNT/matrix interface. Material properties of constituents are listed in Table 2. In the second set of models, a thin layer of an elastic material was modeled surrounding the CNT. In all cases, several types of matrix materials with different moduli of elasticity are considered. Matrix modulus is varied from 3.2 to 100 GPa to investigate the effects of matrix strength on nanocomposite mechanical properties. The RV is a 100 100 100 nm cube. The nanotube dimensions are: length = 50 nm, inner radius = 4.6 nm and outer radius = 5.0 nm. The CNT volume fraction in all models is 1%. Table 2. Mechanical properties of nanocomposite constituents. Material (GPa) Matrix 3.2-100, Variable 0.3 CNT 1000 0.3-137-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS Aligned CNT-reinforced nanocomposite results In this section, the results of aligned CNT reinforced nanocomposite with perfect bonding interface are presented. The effective material properties for this case are listed in Table 3.The variation of nanocomposite longitudinal modulus to matrix modulus, z/m, with matrix modulus is shown in Fig. 9 Table 3. Mechanical properties of uniformly distributed CNT nanocomposite. m z/ m x/ m= y/ m υ zx= υ zy υ xy 3.2 1.5101 0.9181 0.4452 0.2341 10 1.4372 0.9214 0.4205 0.2479 20 1.3601 0.9222 0.3657 0.2541 50 1.1540 0.9292 0.3414 0.2768 70 1.1314 0.9311 0.3191 0.2819 100 1.0326 0.9344 0.3024 0.2902 1.6 1.5 1.4 z / m 1.3 1.2 1.1 1 0 20 40 60 80 100 m (GPa) Figure 9. Variation of z/ m with matrix modulus for the uniformly distributed CNT-reinforced polymer. Note in Table 3 and Fig. 9 that the strengthening effect of the CNTs in the longitudinal direction decreases sharply as the matrix modulus increases. A 51 percent increase is observed for the matrix with modulus of 3.2 GPa, with the addition of only 1% nanotube to the matrix. For the matrix with a modulus of 100 GPa, the increase in matrix modulus is only 3.26 percent in the longitudinal direction. These results indicate a higher efficiency of the nanotube in raising nanocomposite effective modulus when the difference between constituent moduli is large. The ratio of nanocomposite transverse modulus to matrix modulus, x / m, is below unity in all cases. These results suggest that the aligned nanotubes do not contribute to nanocomposite strength in the transverse directions. In fact, CNTs act as cavities in the matrix relative to the transverse directions and weaken the matrix in those directions. Randomly distributed CNT-reinforced nanocomposite results In this section, the results of nanocomposite reinforced with randomly distributed nanotubes in the RV are presented. The results of these models are listed in Tables 4 and 5. -138-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS Table 4. Computed Young's moduli of randomly distributed CNT-reinforced nanocomposite. m x/ m y/ m z/ m 100 90 70 50 30 20 10 1.021 1.037 1.045 1.050 1.108 1.132 1.168 3.2 1.289 Note in Table 4 that, the ratio of nanocomposite modulus to matrix modulus in the longitudinal direction, z/m, is equal to 1.310 in this case for the matrix with modulus of 3.2 GPa. This is about 20% lower than the uniformly distributed CNT reinforced nanocomposite longitudinal modulus. A comparison of the results with those obtained for the uniform distribution of straight CNTs indicates that strengthening is much more homogeneous in the randomly distributed CNT case. That is, the ratios of nanocomposite longitudinal and transverse moduli to matrix modulus are close in this case (at equal mvalues). These results suggest that the random distribution of the CNTs 1.027 1.038 1.049 1.070 1.133 1.146 1.178 1.030 1.041 1.055 1.093 1.121 1.185 1.231 1.293 1.310 reinforces the matrix in all three directions. In fact, in case of transverse moduli, the portions of CNTs that are oriented along the x and y directions contribute to the nanocomposite moduli in those directions. The variation of nanocomposite longitudinal modulus to matrix modulus, z/m, with matrix modulus is shown in Fig. 10. Note that the ratio of nanocomposite longitudinal modulus to matrix modulus decreases as the matrix modulus increases in this case as well. Note in Table 5 that the six Poisson's ratios have close values for all matrix types. This also suggests a more homogeneousnanocomposit Table 5. Poisson's ratios of randomly distributed CNT reinforced nanocomposite m xy yx yz zx zy 100 0.2981 0.2981 0.2978 0.2937 0.2927 0.2982 90 0.2925 0.2978 0.2928 0.2989 0.2975 0.2942 70 0.2926 0.2926 0.2981 0.2908 0.2912 0.2998 50 0.2847 0.2858 0.2898 0.2871 0.2849 0.2851 30 0.2953 0.2883 0.2988 0.2892 0.2941 0.2953 20 0.2903 0.2844 0.2985 0.2906 0.2945 0.2991 10 0.2983 0.2933 0.2974 0.2992 0.2905 0.2930 3.2 0.2888 0.2858 0.2905 0.2933 0.2912 0.2950 z / m 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1 0 10 20 30 40 50 60 70 80 90 100 m (GPa) Figure 10. Variation of z / m with matrix modulus for randomly oriented CNT reinforced nanocomposite. -139-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS ffects of CNT/matrix interface strength Next, models are developed to simulate imperfect bonding at CNT/matrix interface. In order to determine the effects of interface strength on nanocomposite moduli, a thin layer of an elastic material with 0.142 nm thickness, equal to the bond length, was modeled as the CNT/Matrix interface. Six types of interface materials with moduli of elasticity varying from 10 to 100 GPa are considered. The matrix modulus was kept constant at 100 GPa in these models. The schematic of the model is shown in Fig. 11. Figure 11. Schematic of the elastic interface model. A part of each component is shown. Nanocomposite reinforced with uniformly distributed CNTs In this section, the results of interface strength on the mechanical properties of nanocomposite reinforced with a 4 4 array of nanotubes with a uniform distribution are presented. The results of this investigation are presented in Table 6 and Fig. 12. Note that the ratio of the nanocomposite longitudinal modulus to matrix modulus increases as the interface strength increases. Note also that, increasing the interface modulus from 10 to 100 GPa results in an increase in nanocomposites longitudinal modulus by almost 1.2 percent. These results indicate that higher interface strength results in a more efficient load transfer between the CNT and matrix. For a weak interface, interface = 10 GPa, nanotube strengthening results in 2% increase in effective modulus. As the interface modulus increases, the strengthening effect of the nanotube also increases. This trend continues until the interface modulus reaches that of the perfect bonding case. This is the point where interface modulus is equal to the matrix modulus, 100 GPa. The point corresponding to the perfect bonding case is also shown in Fig. 12. In addition, the interface strength does not have a significant effect on nanocomposite transverse moduli. This is expected since the main strengthening effect of CNTs is in the longitudinal direction. Table 6. The effects of elastic interface strength on moduli of elasticity for uniformly distributed CNT reinforced matrix. interface(gpa) z/ m x/ m y/ m 10 1.0204 0.9659 0.9662 20 1.0241 0.9630 0.9654 40 1.0292 0.9624 0.9614 60 1.0310 0.9587 0.9585 80 1.0317 0.9572 0.9576 100 1.0324 0.9547 0.9551 Variation of z / m with interface strength is shown in Fig. 12. Note also that the results of the elastic interface models are in excellent agreement with the perfect bonding results for interface =100GPa. -140-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS Perfect Bonding Figure 12. Variation of z/ m with interface modulus for the uniformly distributed CNT reinforced polymer. Nanocomposite reinforced with randomly distributed CNTs Next, models were developed to investigate the effects of CNT/matrix interface strength on the mechanical properties of randomly oriented CNT reinforced nanocomposite. The results of this investigation are presented in Table 7 and Fig. 13. Note that the ratio of the nanocomposite longitudinal modulus to matrix modulus increases as the interface strength increases in this case as well. Note also that, increasing the interface modulus from 10 to 100 GPa results in an increase in nanocomposites modulus in all directions by almost 1 percent. Table 7. The effects of elastic interface strength on moduli of elasticity for randomly distributed CNT reinforced matrix. interface(gpa) z/ m x/ m y/ m 10 1.010 1.018 1.020 20 1.012 1.019 1.021 40 1.016 1.021 1.024 60 1.018 1.023 1.026 80 1.019 1.025 1.027 100 1.020 1.026 1.029 The variation of the ratio of nanocomposite longitudinal modulus to matrix modulus with interface strength is shown in Fig. 13. As the interface modulus reaches matrix modulus ( interface = 100 GPa), z/m value almost reaches the perfect bonding case. The difference is about 1% at this point. In addition, in this case, the interface strength has a significant effect on nanocomposite transverse moduli.these results indicate the importance of obtaining a good bonding between the matrix and the nanotube to take advantage of the nanotube high modulus in manufacturing nanocomposites. Figure 13. Variation of z/ m with interface modulus for the randomly distributed CNT-reinforced polymer. -141-
HOSSIN MAMANPUSH AND HOSSIN GOLSTANIAN : VALUATION OF FFCTIV MATRIAL PROPRTIS Conclutions In this article, the effects of random distribution of CNTs and interface strength on the mechanical properties of nanocomposites are investigated. The elasticity theory for the anisotropic body and the continuum simulation are used to determine mechanical properties of straight CNT reinforced polymers. Two different cases were considered namely: uniformly distributed and randomly distributed CNTs. Also, the effects of matrix modulus on nanotube strengthening efficiency were investigated for each nanocomposite. In addition, for both aligned and random CNT reinforced nanocomposites models were developed to investigate the effects of interface strength on nanocomposite mechanical properties. The following conclusions can be drawn from the results of this investigation. Nanocomposites reinforced with uniformly distributed nanotubes had a higher longitudinal modulus compared to the nanocomposite reinforced with randomly distributed nanotubes. A comparison of the results of randomly oriented with those obtained for the uniform distribution of CNTs indicates that strengthening is much more homogeneous in the randomly distributed CNT case. The elastic interface results on aligned CNT reinforced nanocomposite indicate that the nanocomposite longitudinal modulus increases with increasing the interface strength. In case of the randomly distributed CNT nanocomposite, increasing the interface strength resulted in an increase in nanocomposite modulus in all three directions. Thus, to use the full capacity of the CNTs reinforcing effect, we need to enhance the bonding between the CNT and matrix. Finally, the strengthening effect of the CNTs reduces as the matrix modulus increases, in all cases. This result suggests that CNTs are more efficient in strengthening lower moduli matrices. RFRNCS Guz, I. A, Rodger, A. A, Guzand, A. N, Rushchitsky, J. J. (2007): Developing the mechanical models for manomaterials,compos. A38: 1234-1250. Collins, P. G and Avouris, P. (2000): Nanotubes for electronics. Scientific American, December Li, C. Y and Chou, T. W. (2003): A structural mechanics approach for the analysis of carbon nanotubes, Int. J. Solids. Struct.40; 2487 2499. Chang, T and Gao, H. (2003): Size-Dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model,j. Mesh. Phys. Solids.51: 1059-1074. Krishnan, A, Dujardin,, bbesen, T. W, et al. (1998): Young s modulus of single-walled nanotubes,phys. Rev. B 58: 14013 14019. Yao, N and Lordi.(1998): Young s modulus of single-walled carbon nanotubes,j. Appl. Phys. 84; 1939 1943. Lu, J. P. (1997): lastic properties of carbon nanotubes and nanoropes,phys. Rev. Lett.79; 1297 1300. Thostenson,. T, Li, C and Chou, T. W. (2005): Nanocomposites in context,compos. Sci. Technol.65; 491 516. Qian, D, Wagner, G. J, Liu W. K, et al. (2002): Mechanics of carbon nanotubes,appl. Mech, Rev55; 495 530. Srivastava, D, Wei, C and Cho, K. (2003): Nanomechanics of carbon nanotubes and composites. Appl. Mech. Rev. 56; 215 230. Liu, B, Huang, Y, Jiang, H, et al. (2004): The atomic-scale finite element method, Computer. Meth. Appl. Mech. ng. 193; 1849-1864. Nasdala, L and rnst, G. (2005): Development of a 4-node finite element for the computation of nano-structured materials, Comput. Mater. Sci.33; 443-458. Pipes, R. B and Hubert, P. (2002): Helical carbon nanotube arrays: Mechanical properties,compos. Sci, Technol,62; 419-428. Das, P. S and Wille, L. T. (2004): Atomistic and continuum studies of carbon nanotubes under pressure,comput. Mater. Sci.24; 159-162. Xu, L. R and Sengupta, S. (2005): Interfacial stress transfer and property mismatch in discontinuous nanofiber/nanotube composite materials,j. Nanosci. Nanotechnol.5 (4); 620 626. Kulkarni, M, Carnahan, D, Kulkarni, K, et al. (2010): lastic response of a carbon nanotube fiber reinforced composite: a numerical and experimental study,compos. B41; 414-421. Valavala, P. K and Odegard, G. M. (2005) :Modeling techniques for determination of mechanical properties of polymer nanocomposites,rev. Adv. Mater. Sci. 9; 34 44. Liu, Y, Nishimura, N and Otani, Y. (2005): Large-scale modeling of carbon nanotube composites by a fast multipole boundary element method,comput. Mater. Sci. 34; 173 187. Chen, H, Liu, X and Gengkai, H. (2998): Overall plasticity of micropolar composites with interface effect, Mech. Mater. 40; 721 728. Chen, X. L and Liu, Y. J. (2004): Square representative volume elements for evaluating the effective material properties of carbon nanotube-based composites,comput. Mater. Sci. 29; 1 11. Golestanian, H and Shojaie, M. (2010): Numerical characterization of CNT-based polymer composites considering interface effects,comput. Mater. Sci.50; 731 736. Bogdanivich, A. and Bradford, P. D,(2010): Carbon nanotube yarn and 3-D braid composites. Part I: Tensile testing and mechanical properties analysis,compos. A 41; 230-237. Golestanian, H and Matin Ghahfarokhi, Z. (2011): ffects of nanotube helical angle on mechanical properties of carbon nanotube reinforced polymer composites, Comput. Mater. Sci.50; 3171 3177. Golestanian, H and Mamanpush, H. (2014): Investigating the effects of carbon nanotube orientation on the macroscopic stiffness of nanocomposites, Int. J. Curr. Li. Sci. 4(4); 1168-1174. Shady,, Gowayed, Y. (2010):ffect of nanotube geometry on the elastic properties of nanocomposites, Compos. Sci. Technol. 70; 1476 1481. Wang, S. (2008) Role of structure and morphology in the elastic modulus of carbon nanotube composites,j. Mater. Sci.43 (17); 5837 5844. Joshi, U. A, Sharma Satish, C and Harsha, S. P. (2010): Influence of dispersion and alignment of nanotubes on the strength and elasticity of carbon nanotubes reinforced composites,j. Nanotechnol. ng. Med.2 (4); 417-426. Lekhnitskii, S. G. (1981): Theory of elasticity of an anisotropic body, nglish Translation, Translated From the Revised Russian dition, Moscow: MIR Publishers. Joshi, U. A, Sharma, S. C and Harsha, S. P. (2012): ffect of carbon nanotube orientation on the mechanical properties of nanocomposites, Compos. Part B43; 2063 2071. -142-