THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Thermal Elastic Buckling o plates made o carbon nanotube reinorced polymer composite materials J. Naar Dastgerdi,, S. Marzban 3, G. Marquis, Applied mechanics department, Aalto University, Espoo, Finland, Department o Mechanical Engineering, Isahan University o Technology, Isahan, Iran, National Iran Oil Petrol Distribution Company (N. I. O. P. D. C), Tehran, Iran. * Corresponding author (Jairan.naardastgerdi@aalto.i) Keywords: Polymer composite, Carbon nanotubes, Thermal buckling Introduction The high strength and stiness o carbon nanotubes have generated enormous interest in the scientiic community in recent years [- 3]. One o the areas has been the applicability o carbon nanotubes as a reinorcing constituent [4-4]. In act, the unique atomic structure, high aspect ratio, light weight, etraordinary mechanical properties [5], and thermal conductivity make SWNT a potentially very attractive material to be used in polymer/carbon nanotube composite application. Carbon nanotubes (CNTs) have radii on the order o nanometers and lengths on the order o micrometers resulting in large aspect ratios beneicial to their use in composites. As the mechanical properties o composites depend directly upon the embedded iber mechanical behavior, replacing conventional carbon ibers with CNTs can potentially improve composite properties, such as tensile strength and elastic modulus and can be a very promising candidate as the ideal reinorcing ibers or advanced composites. Composites o carbon nanotubes dispersed in metallic or polymeric matries have attracted a considerable attention recently [6, 7]. Most studies on carbon nanotube reinorced composites (CNTRCs) have ocused on their material properties [8-3]. Several investigations have shown that the addition o a small percentage o nanotubes (~5% by weight) in a matri may considerably increase the composite s mechanical, electrical and thermal properties [9-3]. The eects o CNT dispersion and orientation [4], deormation mechanisms [5, 6], interacial bonding [7, 8] on mechanical properties o CNT reinorced composites have been investigated eperimentally. Wuite and Adali ound that the stiness o CNTRC beams can be improved signiicantly by the homogeneous dispersion o a small percentage o CNTs [9]. Vodenitcharova and Zhang studied the pure bending and bending-induced local buckling o CNTRC beams [30]. Formica et al. [3] presented the vibration behavior o CNTRC plates by employing an equivalent continuum model based on the Mori Tanaka approach. They ound that the improvement achieves a maimum when the carbon nanotubes are uniormly aligned with the loading direction. Instabilities in CNTRCs have also been o substantial interest and many eperiments have observed buckling [3-34]. An increase in the utilization o nanocomposites is due to their physical properties which can be improved simultaneously. Physical properties such as low density, low thermal epansion, high resistance interering with thermal shock and high strength in high temperatures attract researcher s attention. Evidently, such composites are o paramount interest in aeronautic and astronautic technology, automobile and many other modern industries. High speed air crat structures are subjected not only to aerodynamic loading, but also to aerodynamic heating. The temperature rise may buckle the plate and ehaust the load caring capacity; thereore, elastic buckling o the composite plate reinorced CNTs corresponding to the practical application in air crat and car industries is too prominent. Motivated by these considerations, the present work ocuses attention on the thermal elastic buckling o composite plates reinorced
with carbon nanotubes subjected to in- plane uniorm temperature. Due to this aim, we derive the governing equations o this plate and ind a relation or critical load obtained by use o the principle o minimum potential energy, and then we study the eects o carbon nanotube volume raction, radius o nanotubes, plate aspect ratio and the orientation angle o nanotubes in polymer matri composite. Equations Total potential energy o the plate, due to the internal strain and the surace traction can be epressed as total i i R () st V U dr T u d Where U is deined as composite strain total energy, R is volume o elastic body, T i is ith component o surace traction, u i is ith component o displacement, and St is a portion o plate surace on which surace traction is eerted. The composite strain energy U total is composed o strain energy in the matri and CNTs, and the cohesive energy in CNT/matri interaces can be written as in the ollowing [37] Utotal Ucohsive U : dv da () int Where U cohesive is the cohesive energy in the surace o the nanotube and polymer matri and U is the strain energy in the matri and nanotubes. Considering the irst stage o the cohesive law proposed by Tan et al. [37], the cohesive energy or CNT/ matri interace is obtained as below Kn J U dv (3) cohesive R R Where,, R are respectively radius o nanotube, carbon nanotube volume raction, and equivalent stress o the composite plate [37]. K n is the linear modulus o the interace [38]. J is a constant coeicient can be written as in the ollowing 9 0.03 9 n J (0.3 kn0 )( ) 0.5k 0 3 R S (4) In Eq. (), U in Cartesian coordinate system is deined as ollow U ij ij y y z z z z yz yz y y (5) According to the principle o minimum potential energy, it necessary that stresses replace with strains and then strains transorm into displacements by use o strain displacement relations. Thereore the Eq. () is rewritten in matri orm and takes the ollowing orm T Kn J U CdV dv total (6) V The constitutive relating the stress and the strain can be written as 3 0 0 6 C C C C C C C 0 0 C 3 6 y y z C3 C3 C33 0 0 C 36 z yz 0 0 0 C44 C45 0 yz 0 0 0 C C 0 z z 54 55 y y C6 C6 C36 0 0 C 66 R R (7) Where C ij are transormed elastic coeicients in the global coordinate system. Based on the classical plate theory and eliminating the shear deormation the constitutive relating the stress and the strain is as ollow C C C 6 y C C C 6 y (8) y C6 C 6 C 66 y The components o transormed elastic matri are given as ollow C C cos ( C C )sin cos C sin 4 4 66 C ( C C 4 C )sin cos C(sin cos ) 4 4 66 C C sin ( C C )sin cos C cos 4 4 66 C ( C C C )sin cos ( C C C )sin cos 3 3 6 66 66 3 3 C ( C C C )sin cos ( C C C )sin cos 6 66 66 (9) 4 4 C66 ( C C C C66)sin cos C66(sin cos ) Where is the angle between orientation o ibers and the aial o the composite plate. The composite plate is assumed to be orthotropic
C6 C6 0. The other parameters are calculated as ollow E E C, C (0) E E C, C66 G In the above ormulas, E, E and G are the longitudinal, transverse and shear moduli o the composite with the uniaial straight ibers. is the longitudinal transverse Poisson s ratio. To simpliy the analysis we will calculate these parameters using the rule o miture [35] EEm( ) E, E Em E GmG E E m G, Gm, G G G ( ) ( ) ( ) m m E m( ), E () Where E m, E, G m, G and m, are Young s moduli and Poisson s ratios o the matri and the ibers. is the volume ractions o the ibers. The relations between the strain and the displacement in the classical plate theory are deined as u0 w z v0 w y z () y y y u0 v0 w z y y Where u 0 and 0 are the displacements o the mid-plane in and y directions, respectively, which are assumed to be zero because there is no coupling between the inplane and the out o plane displacements. wy (, ) denotes the displacement in z direction, i.e. the lateral delection o the composite plate. Substituting Eq. () into Eq. (6) yields: U total w z C h a b h 0 0 w w z C w w z C6 y w w z C6 y y w w z C 66 z C y y kn J z w 9 y w w w y y (3) Where a, b, and h are the length, the width, and the thickness o the rectangular composite plate respectively. The work done by the surace traction is as ollow a b u0 w v0 w Tuds i i N 0 0 Ny st y y u0 u0 ww (4) Ny ddy y y Where N and N y are the resultant orces in the and y directions, respectively and N y denotes the shear orce. The composite plate is subjected to in- plane uniorm temperature; thereore, N y is assumed to be zero. N and N y are calculated as ollow y N C C h T N C C h T y y (5), are respectively longitudinal and y transverse thermal coeicient, respectively. Substituting Eqs. (3) and (4) in Eq. () yields the ollowing simpliied epression or the total potential energy subjected to in- plane uniorm temperature
a b w w w V D 0 0 D y D w w D y y 66 w w D6 y w w D6 y y kn J w w w w dd 54R y y a b w C 0 0 C y Th ddy a b w C 0 0 C y Th ddy y (6) D ij is the bending stiness matri whose elements are deined as D ij h h Cij z dz (7) In this work, the plate is orthotropic, thus, D, D 0. 6 6. Solution methodology According to the levy solution [36], it is important to ind suitable unction or the lateral delection. It is assumed that the lateral delection can be written as the ollowing separate unction o and y variables: w(, y) ( ) g( y) (8) The ollowing boundary conditions are assumed or all edged simply supported plate situation 0 0 y b a 0 y b w0 0 a y 0 0 a y b (9) The solution unction or buckling response is proposed as n my w, y Amn sin sin (0) n m a b The assumed displacement ield Eqs. (0) completely satisies the boundary conditions speciied in Eqs. (9). The substitution o the assumed solution unction Eqs. (0) and its derivatives into Eqs. (6) provides a relationship as in the ollowing 4 Kn J n ab 54R a 4 n m V D A 4 Kn J m ab D 54R b 4 n m mn mn Kn J m n ab 66 D D A 54R b a 4 n C Cy Th a ab 4 A Amn n m n ab y mn a 4 n m C C Th A mn n m () For the plate to be in equilibrium, the total energy should be stationary V 0 () Amn Hence, solving Eq. () yield the ollowing buckling load 4 4 Kn J n m n m D D 54R a b a b Tcr n CCy hc Cy h a mn mn D 4D66 b a b a n CCy hccy h a (3) Another case considered in this work is an asymmetric (clamped supported at one edge and simple support on remaining edges) boundary condition as in the ollowing 0 0 y b w0 0 a y 0 0 a y b (4) w 0 b 0 y b The assumed displacement ield which completely satisies the boundary conditions speciied in Eqs. (5) is proposed as in the ollowing
3 3 4 n w, y An b y3by y sin (5) n a The substitution o Eqs. (5) and its derivatives into Eqs. (6) and use o the principle o minimum potential energy lead to a relationship or critical load in the asymmetric boundary condition case as below 4 4 Kn j Kj n 3 D 4 D 4 54 R a 54 R b Tcr 4 C C yh C C y h a Knj 4 5 DD66 54 R ab 4 C C yh C C y h a 3 m 4 5 n0.5 m (6) 3 Result and discussion In this work, the thermal buckling o a composite plate reinorced by CNTs subjected to in plane uniorm temperature is investigated by the analytical method. Based on a relation or critical thermal buckling load obtained by use o the principle o minimum potential energy, the eects o carbon nanotubes volume raction, radius o nanotubes R, plate aspect ratio a/ b and the orientation angle o nanotubes in polymer matri composite on the thermal buckling behavior o plates made o carbon nanotube-reinorced polymer composite materials are studied or two dierent kinds o boundary conditions. The composite plate is composed o polyethylene as the matri with the Young s modulus and the Poisson's ratio o Em 0.98Gpa, and m 0.33, respectively. The CNTs are modeled as long, transversely isotropic ibers. The material properties o SWCNTs are E 350Gpa and 0.6. Fig. shows the eect o the CNTs volume raction on the thermal buckling behavior o plates reinorced with SWCNTs or armchair carbon nanotubes (6, 6), the orientation angle o nanotubes 45 o, and two symmetric and asymmetric boundary conditions. It can be seen that the plate with volume raction 0% has the highest critical thermal buckling load in comparison with volume ratio o %, 5%. For constant aspect ratio and CNTs volume raction, the highest critical thermal buckling load occurs or asymmetric boundary condition. Fig. shows the eect o CNT size on the critical load aspect ratio curve. The orientation angle o nanotubes and the volume raction is 45 o and 0%, respectively. Three dierent armchair CNTs, (8, 8), (, ) and (6, 6), are studied, and the corresponding radii are R.5, 0.83 and 0.4 nm, respectively. Small CNTs clearly give stronger reinorcing eect than large CNTs because, at a ied CNT volume raction, there are more small CNTs than large ones, and thereore there eist more interaces. This observation o strong reinorcing eect or small CNTs also holds ater CNTs are deboned rom the matri. According to this act, armchair carbon nanotubes (6, 6) are the best choice in comparison with armchair carbon nanotubes (8, 8) and (, ). Fig. 3 shows the eect o the orientation angle o nanotubes on the thermal buckling behavior o a plate reinorced carbon nanotubes or dierent boundary conditions. The volume raction o armchair (6, 6) carbon nanotubes is 0%. From presented results, there is an optimum value or orientation angle o nanotubes. This value is 46 o or symmetric boundary conditions and 4 o or asymmetric boundary conditions. In Fig. 4 the critical buckling load is plotted with respect to aspect ratio or dierent number o hal- waves in the X direction in the case that the orientation angle o nanotubes and the volume raction is 45 o and 0%, respectively, or armchair carbon nanotubes (6, 6). There is a minimum in Fig.4. By increasing the aspect ratio o the composite plate, the critical thermal buckling load decreases at the beginning and attains a minimum value and then it increases. The rate o critical load's variations beore reaching the minimum value is higher
than the net step at which the critical load increases. 4 Conclusions In this study, the thermal buckling o a composite plate reinorced by CNTs subjected to in plane uniorm temperature is investigated by the analytical method. Based on a relation or critical thermal buckling load obtained by use o the principle o minimum potential energy, the eects o carbon nanotubes volume raction, radius o nanotubes, the orientation angle o nanotubes, and the plate aspect ratio in polymer matri composite are presented. Two dierent symmetric (simple supported at all edges) and asymmetric (clamped supported at one edge and simple support on remaining edges) boundary conditions have been considered. The results show that the plate with volume raction 0% has the highest critical thermal buckling load in comparison with volume ratio o %,5%. For constant aspect ratio and CNTs volume raction, the highest critical thermal buckling load occurs or asymmetric boundary condition. Armchair carbon nanotubes (6, 6) are the best choice in comparison with armchair carbon nanotubes (8, 8) and (, ). Small CNTs clearly give stronger reinorcing eect than large CNTs. From presented results, there is an optimum value or orientation angle o nanotubes. This value is 46 o or symmetric boundary conditions and 4 o or asymmetric boundary conditions. By increasing the aspect ratio o the composite plate, the critical thermal buckling load decreases at the beginning and attains a minimum value and then it increases. The rate o critical load's variations beore reaching the minimum value is higher than the net step at which the critical load increases. (a) (b) Fig.. The critical load aspect ratio relation o a carbon nanotube reinorced polyethylene matri composite. The volume raction o armchair (6, 6) carbon nanotubes is 0%, 5% and %. The orientation angle o nanotubes is 45 o. (a) Symmetric boundary condition and (b) asymmetric boundary condition
(a) (a) (b) (b) Fig.. The critical load aspect ratio relation o a carbon nanotube reinorced polyethylene matri composite plate. The orientation angle o nanotubes and the volume raction is 45 o and 0%, respectively, or three dierent armchair carbon nanotubes (8, 8), (, ) and (6, 6). (a) Symmetric boundary condition and (b) asymmetric boundary condition Fig. 3. The critical load orientation angle o nanotubes relation o a carbon nanotube reinorced polyethylene matri composite plate. The volume raction is 0% or armchair carbon nanotubes (6, 6). (a) Symmetric boundary condition and (b) asymmetric boundary condition
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