Carbon nanotubes (CNTs) demonstrate unusually high stiffness, strength and resilience, and are ideal reinforcing materials for polymer-based nanocomposites. Van der Waals interactions between the nanotubes often result in formation of nanotube bundles and/or ropes. Determination of mechanical properties of the nano bundle and nano rope-based nanocomposites is difficult due to the complicated geometries of the reinforcement structures. However, to better utilize nanocomposites, it is crucial to determine mechanical properties of nanocomposites in their real form as much as possible. In this paper, the elasticity theory for anisotropic body and continuum modeling are used to determine effective mechanical properties of bundle and nanorope-based nanocomposites. Numerical models are developed using a Representative Volume Element (RVE) consisting of nanoropes and nanobundles made up of different numbers of nanotubes to investigate the effect of nanorope geometry on nanocomposite mechanical properties. Models of several RVEs are developed with nanoropes consisting of five, seven, nine, and thirteen CNTs. CNT volume fraction is kept constant in all models for a valid comparison. Also, the effect of matrix modulus on the strengthening efficiency of CNTs is investigated. The results indicate that nanocomposite longitudinal modulus decreases with increasing the number of CNTs in the nanorope. Keywords: Nanorope; Nanobundles; Anisotropy; Nanocomposite; Mechanical properties Evolutions Mech Eng Copyright © Seyed Hossein Mamanpush 2/9 How to cite this article: Seyed H M, Zohre MG,Hossein G. Numerical Modeling of Nano Bundle and Nano Rope-Reinforced Polymer Mechanical Properties. Evolutions Mech Eng . 1(4). EME.000518.2018.Volume 1 - Issue - 4ing angle increases, the kink progresses along the nanotube and its shape changes in both longitudinal and circumferential directions. In many cases SWCNT bundles naturally assume a twisted form, which is similar to the wire ropes at a continuum scale [22]. It is well known from continuum mechanics analysis of other types of wire or fiber forms that twisting the wires or “weaving” the fibers can lead to a cable or rope that has a much better load transfer mechanism in tension than a straight bundle [23]. Ren et al. [24] used molecular mechanics simulation and tensile loading to investigate fatigue failure mechanism of SWCNT bundles embedded in a polymeric matrix. They found that SWCNTs within the rope do not break at the same time or at the same location. Vodenitcharova investigated the inter-tubular van der Waals interactions that produce the initial cross-sectional distortion of single-walled carbon nanotubes during a bundle formation [25]. These investigators determined the distributions of the van der Waals force by combining the analysis of molecular dynamics with continuum mechanics. Walters et al. [26]. obtained yield strength of 45 ±7 GPa for nanotube ropes. Pipes & Hubert [20,27] used textile mechanics and elasticity methods to evaluate the effective elastic properties of twisted carbon nanotube arrays. The study by Qian et al. [28] showed that twisting could significantly decrease the required contact length for a complete load transfer between nanorope and matrix. Ashrafi & Hubert [29] used finite element analysis and strain energy approaches to investigate the elastic properties of SWCNT arrays and their composites [29]. They predicted properties of twisted SWCNT nano-arrays. These investigators studied the effect of nano-array volume fraction and aspect ratio for dilute polymer composite systems using conventional micromechanics. Their results indicated that the axial Young’s modulus of a nanotube array decreases dramatically even for small twist angles. As stated above, nanotubes seldom exist as straight and individual in the matrix and usually form bundles or ropes consisting of several CNTs twisted together. Figure 1 shows an SEM image of a nanorope inside a matrix in which the roping/ twisting of the nanotube is obvious. As can be seen from the presented literature review, much research has been performed on dispersed (individual) nanotube-reinforced nanocomposites. Few researchers have addressed the bundling and roping effect of CNTs on nanocomposite mechanical properties. In this research, the elasticity theories for anisotropic bodies are used to evaluate effective mechanical properties of nanobundle and nanorope-reinforced polymers. To achieve our goals, several models are developed and analyzed in ABAQUS finite element software. Representative Volume Elements (RVE) reinforced with nanobundle and nanoropes consisting of different numbers of SWCNTs are modeled to investigate the influence of reinforcement characteristics on nanocomposite effective mechanical properties. The effect of matrix modulus on the strengthening efficiency of CNTs is also investigated. The analysis and results of this investigation are presented in the following sections. Figure1: SEM image of a carbon nanorope [13]. Analysis of the RVE Reinforced with Nanobundle Figure2: The RVE used to simulate the CNT bundle-reinforced polymer. Evolutions Mech Eng Copyright © Seyed Hossein Mamanpush 3/9 How to cite this article: Seyed H M, Zohre MG,Hossein G. Numerical Modeling of Nano Bundle and Nano Rope-Reinforced Polymer Mechanical Properties. Evolutions Mech Eng . 1(4). EME.000518.2018.Volume 1 - Issue - 4To derive the relations for extracting nanocomposite equivalent material constants, a homogenized elasticity model of the square RVE is considered as shown in Figure 2. The CNTs in the bundle are straight and parallel to each other and are oriented in the z-direction along the RVE. The RVE is 266nm long with a square cross section of 43.28 nm in length on each side. The CNTs are 236 nm long and 5.44 nm in diameter. The nanobundle is placed in the middle of the RVE. Nanotube volume fraction in the RVE is kept constant at 7.7% in all models for a better comparison of the results. The directions of the coordinate axes are also shown in this figure. Elasticity solutions can be obtained under certain load cases. The elasticity model has five independent material constants (Ex= Ey, Ez, Gxy, υxy, and υzx). In this paper, nanocomposite Young’s moduli are determined. The general 3-D strain-stress relation relating the normal stresses and strains for a transversely isotropic material can be written as [30]. Equation (1) is the generalized Hook’s law for an orthotropic body. To determine Ex, Ez, we also need to determine Poisson’s ratios υxy, υzx as will be shown in follow Section. Thus, four equations will be needed in this case [30]. Two loading cases have been devised to provide four such equations based on the elasticity theory, as explained in the following sections. Square RVE under an axial elongation ΔL In this case, shown in Figure 3a, let L denote the length of simulated RVE in the z-direction and the cross-sectional dimension be denoted by letter a. An elongation ΔL is imposed on the RVE in the z-direction. Due to Poisson’s ratio effect, changes of cross sectional dimensions equal to Δx and Δy are created in the x- and y-directions, respectively. Stress and strain components on the lateral surface are as follows [30] 0, , ,xy z xLxalong x aLaσσ ε ε∆∆= = = = =±Integrating and averaging the third equation in (1) on the plane z = L/2, we obtain ave Z ave zLELσσε∆= = (2) Where ơave is the averaged value of stress in the z-direction, given by ()21, , /2ave Ax y L dxdyAσσ=∫ (3) In equation (3), A is the RVE cross sectional area. The value of σave is evaluated by averaging stress in the z-direction on the plane z = L/2 in the FEA model. Using the first relation in equation (1), and the relation between stress and strain, along x = ±a, we have zx x z zx zLa E Laυε συ∆∆= −= −= (4) Thus, we can obtain an expression for the Poisson’s ratio as follows zxaL aLυ∆∆  =−     (5) Equations (2) and (5) can be applied to estimate the effective Young’s modulus Ez and Poisson’s ratio υzx (= υzy), once the contraction Δa and the average stress, ơave, in case (a) are obtained from FEA results. Square RVE under a lateral uniform load In this load case, shown in Figure 3b, the RVE is placed under a lateral uniform load, p, in the y-direction. The RVE 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, since ez=0 for plane strain case, equation (1) reduces to 22 221 1xy zx zx xx xz x z yy xy zx zx x z xzEE E E E E EEυυυ εσ εσυυυ− −− = − −−  (6) Figure3: (a) RVE under axial distributed load p. and (b) RVE under lateral distributed load p. In this load case, the values of stress and strain components at a point on the lateral surfaces are given by , ,, ,xy x yxp along x aa yalong y aaσσ ε ε∆= = = =± ∆= =± Appling the first and second equations in (6) for points along x=±a and y= ±a, together with the above conditions, we obtain 2 21xy zx x xz zx y xzxpEE a ypEE aυυε υε ∆= −+ =   ∆= −− =  (7) Evolutions Mech Eng Copyright © Seyed Hossein Mamanpush 4/9 How to cite this article: Seyed H M, Zohre MG,Hossein G. Numerical Modeling of Nano Bundle and Nano Rope-Reinforced Polymer Mechanical Properties. Evolutions Mech Eng . 1(4). EME.000518.2018.Volume 1 - Issue - 4Solving these two equations gives the effective Young’s modulus and Poisson’s ratio in the transverse direction as follows 21 xy zx zEEy pa Eυ= =∆+ (8) 2 2zx z xy zx zx pa E y pa Eυ υυ∆+ =∆+ (9) The results of axial elongation loading case are used in equations (8) and (9) for Ez and υzx. Once the changes in dimensions, Δx and Δy, are determined for the square RVE from a finite element analysis, Ex=Ey and υxy can be computed from equations (8) and (9), respectively [30]. Analysis of the RVE reinforced with nanorope In this section, the models used to determine effective mechanical properties of nanorope-based nanocomposites are presented. Four RVEs, reinforced with nanoropes consisting of five, seven, nine, and thirteen nanotubes are modeled. CNT volume fraction in the RVEs is kept at 7.7% in all cases. Figure 4 shows the RVE used to model the nanorope-based nanocomposite in this investigation. The RVE consists of a nanorope and the surrounding matrix. Several assumptions are made in the analysis. First, the nanorope is perfectly bonded to the matrix. Second, the individual SWCNTs in the nanorope have uniform diameters and equal lengths. Third, there is no direct interaction between the adjacent CNTs in the nanorope. And finally, the matrix and the constituent nanotubes in the nanorope are homogeneous materials. Approach In this section, the approach used to extract relations for evaluating the effective material constants of the square RVEs are established based on elasticity theory for anisotropic bodies. The generalized Hooke’s law based on the elasticity theory for anisotropic body is given by equation (10) [31]. ,, , ,,, ,, , , ,, ,1 1 1 1yx yz x zx y xy z zx xx xx xx xx xx xx xy zy yz y zx y xy y yy yy yy yy yy yyx yz yz z xy z zx z y xz zz zz zz zz zz zz z yz x yz y yz z yz zx yz xz yz yz yz yz y xyEEE E E E EEE E E E EEE E E E GGG G Gυ ηηηυ υ υη η η ε υη η η ευ ε γηηη µ γ γ−− −−  −−=  , ,, , ,, , ,, , ,1 1x y z yz xy yz xz z yz xyy zx yz zx xy zx x zx z zx xz xz xz xz xz xz x xy y xy z xy yz xy zx xy xy xy xy xy xy xyG GGG G G G GGG G G Gσ σ σ τµ τ τ ηµ µηη ηηηµ µ                         (10) In equation (10), Ex, Ey, and Ez are Young’s moduli in directions x, y, and z. Gxy, Gxz, and Gyz are the shear moduli for planes parallel to the coordinate planes. In addition υij are Poisson’s ratios characterizing the strain in jth direction due to the applied stress in ith direction. The coefficients μzx,yz,…, μzx,xy characterize shears in planes parallel to the coordinate planes produced by shearing stresses acting in other planes parallel to the coordinate planes. These coefficients are called Choentsov’s coefficients. The constants ηyz,x, …, ηxy,z characterize extension in the directions of the coordinate axes produced by shearing stresses acting in the coordinate planes. They are termed the mutual influence coefficients of the first kind. Finally, ηx,yz, …, ηz,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 [31]. These constants are usually non-zero for an anisotropic elastic body and are zero for an isotropic body. Since the nanorope is an anisotropic body, the RVE containing the nanorope is also anisotropic. In the general case of anisotropy each strain component is a linear function of all six stress components. Then, the RVE has nine independent material constants. In order to determine these constants, nine independent equations are needed. These equations are obtained by loading the RVE in the x-, y-, and z-directions by arbitrary distributed stresses separately and using the first three relationships in equation (10). Using this approach, we can obtain the nine needed equations to determine the nanocomposite effective mechanical properties. Equation (10) reduces to equation (1) for this case since no shearing stresses are applied to the RVE [32]. Initially, the RVE shown in Figure 4 is loaded by a tensile stress along the x-axis. Then, using the first relation in equation (1), Young’s modulus Ex can be calculated. Using a similar method for loadings in the y- and z-directions and using the second and the third relations in equation (1), Young’s moduli Ey and Ez, are determined. In equation (1), strain components in different directions are calculated using Figure4: The RVE used to model the polymer nanocomposite reinforced with nanorope. u Lε= Where, u is the average value of displacement of nodes on the considered plane and L is the initial length of the RVE in the corresponding direction. The strains needed in equations are extracted from the FEM results. The FEA models and analysis results are described in the following sections. Evolutions Mech Eng Copyright © Seyed Hossein Mamanpush 5/9 How to cite this article: Seyed H M, Zohre MG,Hossein G. Numerical Modeling of Nano Bundle and Nano Rope-Reinforced Polymer Mechanical Properties. Evolutions Mech Eng . 1(4). EME.000518.2018.Volume 1 - Issue - 4Continuum modeling Four nanocomposites reinforced with nanoropes consisting of five, seven, nine, and thirteen SWCNTs were modeled. These models were analyzed to investigate the effects of nanorope characteristics on nanocomposite effective mechanical properties. Simulated nanorope and its cross section are shown in Figure 5 for the nanorope consisting of seven SWCNTs. In this nanorope, a straight SWCNT is placed at the center of the nanorope and six helical SWCNTs surround the central SWCNT . The other three nanoropes are similar in shape and one straight CNT is in the middle surrounded by the other helical CNTs. The dimensions of the RVE are 43.28×43.28×266nm and the length of the nanorope inside the RVE is 236nm. Diameter of each nanotube in the nanoropes is 5.44nm and helical angle of the surrounding nanotubes in the nanoropes is 16 degrees. Material properties of the constituents are taken from