Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate

Transcription

1 Buckling Behavior o 3D Randoly Oriented CNT Reinorced Nanocoposite Plate A. K. Srivastava 1*, D. Kuar 1* Research scholar, Mechanical Departent, MNIT Jaipur, INDIA, e-ail: ashish.eech@gail.co Asst. Pro., Mechanical Dapartent, MNIT Jaipur, INDIA, e-ail: dkuar.ech@nit.ac.in Abstract: In this paper, the inluence o carbon nanotube (CNT) in a etal atrix on the buckling behavior o the resulting nanocoposite plate is studied. The Young's odulii o CNT reinorced nanocoposite are predicted irst using the representative volue eleent (RVE) ethod. CNTs are considered to have the periodic spatial distribution in the RVE and their 3D orientations are randoized using the livelink between COMSOL Multiphysics and the MATLAB. Thereater, the eect o CNT reinorceent on the buckling behavior o siply-supported nanocoposite plate is predicted using the plate physics o COMSOL Multiphysics. The procedure to calculate the Young's odulus is veriied by coparing the results o the single CNT reinorced RVE with the analytical solution obtained by the rule o ixtures. Buckling strength o the nanocoposite plate is also copared with the classical plate theory (CPT) based analytical result. It is concluded that the enhanced stiness properties o the CNT-reinorced nanocoposite resulted in the increase o buckling strength o the nanocoposite plate. Keywords: Carbon nanotube, Representative volue eleent, Elastic odulii, Buckling strength 1. Introduction Carbon nanotubes (CNTs) discovered in 1991 by Iijia [1] are the rontier research area because o their extraordinary strength, resilience, stiness and low density properties [-4]. CNTs have attracted the attention o researchers ro alost all ields, because o their superior electrical, theral and echanical properties, CNTs are potentially great reinorcing eleent in dierent atrix aterials such as polyers, ceraics and etals, to enhance the overall properties o the resulting coposite aterial. These CNTs are sp bonded 3D structure o ullerene aily, ade o one ato thick rolled sheet which are stronger than the sp 3 bonded alkanes and diaonds that provide the unique properties. The buckling strength o these resulting nanocoposite plates are also studied by ew authors using analytical solution based on irst order shear deoration theory (FSDT) [5] and approxiate solution based on eleent-ree kp-ritz ethod [6]. The ai o present paper is to predict the elastic properties o randoly oriented and periodically distributed CNT reinorced nanocoposite using 3-D nano scale representative volue eleent (RVE) based on continuu echanics approach. Subsequently, the eect o CNT reinorceent on the buckling strength o the nanocoposite plate is also predicted. Perect bonding is assued between the nanoiller (i.e. CNT) and the atrix aterial. Strength o aterials based rule o ixtures and analytical orula or buckling strength o the plate based on classical plate theory (CPT) are used to validate, respectively, the axial Young's odulus o the CNT reinorced nanocoposite aterial and the buckling strength o nanocoposite plate obtained ro COMSOL Multiphysics.. Representative volue eleent Multi-scale ethods are required to deal with the wide length and tie scales o nanocoposites that vary ro nano to acro level, by integrating the olecular dynaics and continuu echanics approaches. This has posed any challenges to all the researchers in the area. The concept o representative volue eleent, originally used or iber reinorced coposites [7], has also been carried orward in the characterization o nano scale aterials. Liu and Chen [8] proposed three types o nanoscale RVE, naely cylindrical, square and hexagonal. It was shown that cylindrical RVE provide good prediction o elastic properties o CNT nanocoposite in the case o axisyetric loading, but at the sae tie cylindrical RVEs tends to overestiate the volue raction o nanoillers in a atrix aterial [9]. Square and hexagonal RVEs are used when nanoillers are distributed in a atrix aterial in square and hexagonal arrays, respectively. In the present Excerpt ro the Proceedings o the 015 COMSOL Conerence in Pune

2 study CNTs are considered to be oriented randoly and distributed periodically, thereore a square (cubical) RVE is considered to study the eect o CNT's reinorceent in a etal atrix on the buckling behavior o nanocoposite plate. 3. Analytical approach Strength o aterials based rule o ixtures (ROM), based on Reuss odel, is a very good ethod or estiating the elastic odulus in axial direction, and because o its siplicity and accuracy, the ROM has been requently used by any researchers [8-1] to validate their results. ROM doesn't predicts good results or transverse Young's odulii because it does not take into the account the geoetry and diensions o the reinorceent but considers only the volue raction o reinorceent. Under the assuption o perect bonding between the reinorceent and the atrix, ollowing relationships describes the ROM, that are used to calculate the stiness properties o the RVE. E V E (1 V ) E (1) 1 E E E () E V 1 G 1 V E G G G V G (1 V ) (3) V (1 V ) (4) 1 Volue raction o CNT in a atrix aterial is calculated as: V RVE i LCNT i L RVE VCNT n r0 r (5) V a nr where r i and r o are the inner and outer radii o the CNTs; a denotes the side o square RVE; L CNT and L RVE represent the lengths o the CNT and RVE, respectively, and; n is the nuber o CNTs. A single CNT reinorced RVE is odeled (i.e. n =1), to predict the inite eleent analysis (FEA) based results or elastic properties o the resulting nanocoposite, that are urther validated with the corresponding results obtained by applying ROM, as speciied in Eqs. (1-4). At the sae tie, the buckling strength o the plate ade o nanocoposite aterial is calculated using COMSOL Multiphysics and that is validated with the corresponding results obtained ro the analytical orula based on the classical plate theory (CPT). For all edges siply supported, the critical buckling load o a plate is given as: 4 D cr (6) b 3 Eh where, D 11 where, b and h represent side and thickness o the square plate, respectively; E and ν are the Young's odulus and Poisson's ratio o the plate aterial, respectively. 4. Validation For the study purpose Magnesiu (Mg) is considered as the atrix aterial and that is reinorced with 1% o CNT to ake a nancoposite aterial. Elastic properties o the reinorceent (i.e., CNT) and the atrix aterials are given below: For atrix: Young's odulus E = 45 GPa, Poisson's ratio, ν = 0.3. For Carbon nanotube: Young's odulus E =1000 GPa, Poisson's ratio, ν = 0.3. Diensions o the CNT are taken as: Outer radius, r o =3.6 n, Inner radius, r i = 3. n, Length, L CNT =50 n. For a given volue raction and L CNT = L RVE, the side o the RVE (ade o a single CNT reinorced in the atrix aterial, as shown in Fig. 1) a can be calculated using Eq. (5). The inite eleent eshing o RVE using tetrahedron eleents is perored using physics controlled eature o COMSOL MultiPhysics. The elastic properties o the nanocoposite are obtained by inite eleent analysis o the RVE subjected to dierent periodic displaceent boundary conditions, as discussed in Re. [7]. In the current study, perect bonding is assued between the reinorceent and the atrix to sipliy the analysis. The eective elastic constants o the nanocoposite are extracted ro the ratios o their respective voluetric averages o the stress and the strain. The results o coparison between the present study (carried out on a single CNT-reinorced RVE) and the ROM are tabulated in Table 1. A good agreeent between the values o E 1 and ν 1 obtained ro the FEA analysis and those obtained ro ROM can be observed in Table 1. But as expected, there is deviation between the Excerpt ro the Proceedings o the 015 COMSOL Conerence in Pune

3 values o E, E 3 and G 1 obtained in the present investigation ro that calculated ro the ROM. Figure 1 Single CNT RVE or the v = To calculate the rando orientation o each CNT, rando nubers are generated using randi coand o MATLAB. The odeled cubical RVE (having L RVE = a and L CNT = 50n) o the CNT-Mg nanocoposite (having 1 % volue raction o CNT reinorceent) with periodic distribution and rando orientation o n CNTs is shown in Fig. (). Using the above validated procedure in Section 4, a coplete characterization o this RVE or elastic properties is carried out. Subsequently, the buckling strength o the siply-supported plate ade o the above characterized CNT-Mg nanocoposite aterial is studied using plate physics o COMSOL Multiphysics. The boundary conditions o a siply supported plate used in the study are shown in Fig. (3). Side o the square plate is taken as, b = 0. and its thickness is taken as Obtained results are copared with that obtained ro Eq.(6). 6. Results and Discussion Figure. RVE consisting o randoly oriented and periodically distributed CNT or v = Table 1. Coparison o Elastic constants o CNT nanocoposite obtained in the present study and the ROM Elastic Constants FEA results ROM E E =E G 1 = G ν Present Study In the present study, a coplete characterization o the nanocoposite (ade o the sae CNT- Mg aterials cobination) with periodically distributed and randoly oriented n CNTs is carried out. For this study, CNTs are given spatial periodic distribution and 3D rando orientation in the atrix using livelink between the COMSOL Multiphysics and the MATLAB. The eective elastic properties o the CNT-Mg nanocoposite are listed in Table. There is approxiately 7 % increent in the E x, E y and E z noral elastic odulii relative to that o pure atrix aterial. Further, shear odulii G xz, G yz and G xy are iproved by 8 % as copared to the odulus o pure atrix aterial. Since, the obtained values o the nanocoposite iic the isotropic behavior o actual randoly oriented CNT nanocoposite, thus an average values o extensional odulus, shear odulus and Poisson's ratio are considered in the present work to study the buckling behavior o nanocoposite plate. Evaluated average values o Elastic constants are given below: E = GPa, G = GPa and ν = Buckling strength o pure Mg plate and CNT-Mg nanocoposite plate obtained ro CPT based analytical solution using Eq. (6) and FEM solution using COMSOL are listed in Table 3. First our ode shapes o the CNT-Mg nanocoposite plate are shown in Fig. (4). The deviation ound by the two ethod can be attributed to the act that CPT doesn't consider the eect o shear deoration through the thickness o the plate whereas plate physics section o COMSOL MultiPhysics considers the Excerpt ro the Proceedings o the 015 COMSOL Conerence in Pune

4 sae and works on Mindlin-Reissner theory o plates which is an extension o CPT. As copared to pure Mg plate, approxiately 7 % increent in the buckling strength o the nanocoposite plate is obtained which can be urther iproved by increasing the percentage o CNT. Thereore, the enhanced stiness properties o nanocoposite have resulted in the enhanced buckling strength o nanocoposite plate. Figure 3. Boundary conditions or the all edges siply supported plate 7. Conclusion Eective elastic constants o randoly oriented and periodically distributed CNT reinorced Mg nanocoposite are predicted using cubical representative volue eleent subjected to dierent periodic boundary conditions. The obtained odel shows the isotropic behavior that can iic the actually developed isotropic nanocoposites. Finite eleent ethod is used to peror the analysis using COMSOL Multiphysics sotware. The procedure is validated by coparing FEA result or the axial odulus with the corresponding analytical result obtained ro echanics o solid based rule o ixtures (ROM). Coplete characterization in ters o various elastic odulii o CNT nanocoposite is done. The eect o CNT reinorceent on the buckling behavior o a siply supported square plate is also predicted using the plate physics section o COMSOL Multophysics. Based on the present study, it can be concluded that reinorceent o CNTs enhances the stiness properties o the etal atrix which in turn increases the buckling strength o the plate. Table Eective elastic properties o randoly oriented CNT-Mg nanocoposite E x E y E z G xy G xz G yz ν xy ν xz ν yz Table 3 Coparison o buckling strengths o pure Mg and CNT-Mg nanocoposite (having 1 % reinorceents) obtained using the COMSOL Multiphysics and the analytical orula based on CPT. Plate Material COMSOL Multiphysics (KN/) Analytical results based on CPT (KN/) Mg CNT-Mg Nanocoposite Excerpt ro the Proceedings o the 015 COMSOL Conerence in Pune

5 Figure 4. (a-d) First our ode shapes o CNT-Mg nanocoposites plate having 1% CNT reinorceent with a/h = Reerences [1] Iijia, S. (1991). Helical Microtubules o Graphitic Carbon. Nature, 354, [] Sears, A., & Batra, R. C. (004). Macroscopic properties o carbon nanotubes ro olecular-echanics siulations. Physical Review B, 69, [3] Laurent, C., Flahaut, E., & Peigney, A. (010). The weight and density o carbon nanotubes versus the nuber o walls and diaeter. Carbon, 48, [4] Ruo, R. S., Qian, D., & Liu, W. K. (003). Mechanical properties o carbon nanotubes: theoretical predictions and experiental easureents. Coptes Rendus Physique, 4, [5] Jaari Mehrabadi S, Sobhani Aragh B, Khoshkhahesh V, Taherpour a. Mechanical buckling o nanocoposite rectangular plate reinorced by aligned and straight single-walled carbon nanotubes (01). Copos Part B Eng, 43, [6] Lei ZX, Liew KM, Yu JL (013). Buckling analysis o unctionally graded carbon nanotube-reinorced coposite plates using the eleent-ree kp-ritz ethod. Copos Struct, 98, [7] Sun CT, Vaidya RS. Prediction o coposite properties ro a representative volue eleent(1996). Copos Sci Technol, 56, [8] Liu, Y., & Chen, X. (003). Evaluations o the eective aterial properties o carbon nanotube-based coposites using a nanoscale representative volue eleent. Mechanics o Materials, 35, [9] Chen, X. L., & Liu, Y. J. (004). Square representative volue eleents or Excerpt ro the Proceedings o the 015 COMSOL Conerence in Pune

6 evaluating the eective aterial properties o carbon nanotube-based coposites. Coputational Materials Science, 9, [10] Joshi, P., & Upadhyay, S. H. (014). Evaluation o elastic properties o ulti walled carbon nanotube reinorced coposite. Coputational Materials Science, 81, [11] Joshi, P., & Upadhyay, S. H.(014). Eect o interphase on elastic behavior o ultiwalled carbon nanotube reinorced coposite. Coputational Materials Science, 87, [1] Joshi, U. A., Joshi, P., Harsha, S. P., & Shara, S. C. (010). Evaluation o the Mechanical Properties o CNT Based Coposites Using Hexagonal RVE. Journal o Nanotechnology in Engineering and Medicine, 1, 1 7. Excerpt ro the Proceedings o the 015 COMSOL Conerence in Pune