Chirality and small scale effects on embedded thermo elastic carbon nanotube conveying fluid

This paper presents the mechanical buckling properties of fluid conveying thermo elastic embedded single walled carbon nanotube (SWCNT) with small scale and chirality effect. The analytical formulation is developed based on Eringen’s non local elasticity theory. The nonlocal form of governing equations that contains partial differential equations for (SWCNT) single walled carbon nanotube is derived by considering thermal and chirality effect. The analytical solution is obtained by using Euler–Bernoulli beam theory. The equivalent Young’s modulus and shear modulus of chirality SWCNT is derived. The computed non dimensional wave frequency, phase velocity and group velocity are presented in the form of dispersion curves and the physical characteristic are studied.


Introduction
Nano materials has extensive mechanical, chemical and electronic properties and it has been developed recently because of the fact that it is applied in all fields of science, engineering and technology. The interaction between Carbon Nanotubes (CNTs) and polymer matrix have gained great attention due to their vital mechanical properties. The CNT resting on a polymeric matrix will create a remarkable volume fraction in the interfacial layer between CNT and the bulk polymer matrix. Therefore, a thorough understanding of the dynamical and mechanical behaviours of Nano sized structures and their interacting medium is of importance in the analysisand design of Nano or micro structures such as micro-and Nano-electromechanical systems (MEMS and NEMS).
Erignen. [1][2][3] introduced the new nonlocal continuum field theories and discussed their applicability in CNT models. Aydogdu [4] revealed the common features of presented papers on Eringen's nonlocal elasticity which are related to the investigation of nanobeams nanowires and nanotubes. Semmah et al. [5] have verified the thermal buckling analysis of a zigzag SWCNT via nonlocal Timoshenko beam theory. They verified the effect of the nonlocal parameter, the ratio of the length to the diameter, rotary inertia and the chirality of SWCNT on the thermal buckling properties. Naceri et al. [6] explained the thermal effect on the vibration characteristics of armchair SWCNTs via nonlocal Levinson beam theory. Theoretical formulations carried out by including the small scale effect, the temperature change and the chirality of armchair carbon nanotube. Ebrahimi, and Dabbagh. [7] checked the magnetic field effects on thermally affected propagation of acoustical waves in rotary double nanobeam systems. Zhang et al. [8] developed a method for transverse vibrations of an elastic beam under distributed transverse pressure on basis of the Bernoulli Euler beam theory and thermal elasticity. Ansari et al. [9] discussed CNT's embedded on polymer matrix is getting important structural idea among researchers. Besseghier et al. [10] developed the vibration analysis of DWCNTs embedded in elastic support.PradhanandPhadika [11] investigated the vibration of single layered and multi-layered graphene sheets embedded in polymer matrix and has been carried out with continuum models. Bensattalah et al. [12] have used nonlocal Timoshenko beam theory to study the vibration of SWCNTs embedded in an elastic medium, considering the thermal and chirality effects. Popov and Doren [13] studied the Young's and shear moduli of various SWNT's and they estimated the analytical formulas derived within a lattice-dynamical model for nanotubes. Our study is mainly concerned with the non local thermo elastic waves in a fluid conveying single walled carbon nanotube resting on polymer matrix incorporated with chirality and small scale effect. We have researched the basic equations based on the Eringen's non local elasticity theory. The governing equations having the partial differential equations for single walled carbon nanotube is derived by considering thermal effect along with the nonlocal parameters. The computed non dimensional wave frequency, phase velocity and group velocity are studied.

Atomic configuration of CNT
Carbon nanotubes are constructed via graphene sheet with  T vector.

Eringen Nonlocal Theory of beam The motion equation of a material is reads from Eringen [2] as follows
is the stress tensor at ' x in the body, this stress and displacement equation is The kernel function which will add nonlocal effect in the relation is represented by is an external characteristic length of the system, hence the non-local modulus has form Where 0 e is a material constant which has to be calculated for each material differently and " , x x  " is the Euclidian distance. Then, "Eq. (2)" can be simplifies to partial differential equation as follows

Formulation of SWCNT with nonlocal relations
The partial differential equation which governs the elastic waves in nanotube conveying fluid via thermal force using Simsek [16] can be expressed as, () x  shows the interactive force between nanotube and polymer matrix. A is the cross section of CNT and c m is a mass of nanotube per unit length. Shear force S on nanotube cross section is defined in the following equilibrium equation where T is the temperature change . The force for unit length due to plug flow fluid is taken as  (10) here where xx  is the nonlocal axial stress defined by nonlocal continuum theory. The constitutive equation of a homogeneous isotropic elastic solid for one-dimensional nanotube is considered as where x is the axial coordinate, xx  is the axial strain,   a e 0 is a nonlocal parameter which represents the impact of nonlocal scale effect on the structure. a is an internal characteristic length and E is young modulus. The nonlocal relations in "Eq. (12)" can be written with temperature environment as follows In the context of Euler -Bernoulli beam model, the axial strain xx  for small deflection is defined as and The simplified form of above equation via presuure and thermal environment is derived as The pressure of the CNT due to polymer matrix in unit axial length is reads from Winkler-type model by Yoon [17][18]

Ultrasonic wave form
In order to analysis the elastic wave charectertic of SWCNT, a harmonic solution of displacement ) , ( t x w is reads from Eringen [3] and Narender [19] The phase velocity is read as

Results and illustrations
In this paper thermo elastic wave in a fluid conveying single walled carbon nanotube resting on polymer matrix incorporated with chirality and small scale impact is studied. From lee [20] the SWCNT have a young modulus E=1 Tpa, thickness to be 0.35 nm and mass density as 2.3 g/cm3 , the mass density of water is 1000 kg/m3,

Conclusions
Effect of zigzag and chirality on the dynamic variation of thermal carbon nanotube embedded in polymer matrix is dealt in this study via nonlocal Euler-Bernoulli beam equations. We reached the following conclusions  The frequency, group velocity and bending moment are highly dependent on temperature, scale effect and chirality impact.  In room temperature, frequency ratios ) , ( t x w varies in linear manner with temperature  and is more responsive to the changes of chirality.  The rate of increases of the frequency ratio is lower for higher values of nonlocal parameter.  The scale effect shows the maximum energy whenever the elastic medium is absent.