Small Scale Effect on the Vibration of Orthotropic Plates Embedded in an Elastic Medium and Under Biaxial In-plane Pre-load Via Nonlocal Elasticity Theory
محورهای موضوعی : Engineering
M Mohammadi
1
(Department of Engineering, Ahvaz branch, Islamic Azad university)
M Goodarzi
2
(Department of Engineering, Ahvaz branch, Islamic Azad university)
M Ghayour
3
(Department of Mechanical Engineering, Isfahan University of Technology)
S Alivand
4
(Department of Engineering, Ahvaz branch, Islamic Azad university)
کلید واژه: Vibration, Pasternak foundation, Nonlocal elasticity theory, Biaxial in-plane pre-load, Orthotropic Nanoplates,
چکیده مقاله :
In this study, the free vibration behavior of orthotropic rectangular graphene sheet embedded in an elastic medium under biaxial pre-load is studied. Using the nonlocal elasticity theory, the governing equation is derived for single-layered graphene sheets (SLGS). Differential quadrature method (DQM) has been used to solve the governing equations for various boundary conditions. To verify the accuracy of the present results, a Navier’s approach is also developed. DQM results are successfully verified with those of the Navier’s approach. The results are subsequently compared with valid result reported in the literature. The effects of the small scale, pre-load, Winkler and Pasternak foundations and material properties on natural frequencies are investigated. The results are shown that with the decrease of in-plane pre-loads the curves isotropic and orthotropic non-dimensional frequency in approaches close to each other.
[1] Iijima S., 1991, Helical microtubules of graphitic carbon, Nature 354: 56–58.
[2] Wang J., 2005, Carbon-Nanotube based electrochemical biosensors: A Review, Electro Analysis 17: 7–14.
[3] Kinaret J.M., Nord T., Viefers S., 2003, A carbon-nanotube-based nanorelay, Physics Letters A 82: 1287–1289.
[4] Reddy C.D., Rajendran S., Liew K.M., 2006, Equilibrium configuration and continuum elastic properties of finite sized grapheme, Nanotechnology 17: 864–870.
[5] Sakhaeepour A., Ahmadian M., Vafai A., 2007, Applications of single-layered graphene sheets as mass sensors and atomistic dust detectors, Solid State Communications 145: 168-172.
[6] Zhang Y.Q., Liu G.R., Han X., 2006, Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure, Physics Letters A 349: 370–376.
[7] Aydogdu M., 2009, Axial vibration of the nanorods with the nonlocal continuum rod model, Physica E 41: 861–864.
[8] Filiz S., Aydogdu M., 2010, Axial vibration of carbon nanotube heterojunctions using nonlocal elasticity, Computational Materials Science 49: 619-627.
[9] Murmu T., Adhikari S., 2011, Nonlocal vibration of carbon nanotubes with attached buckyballs at tip, Mechanics Research Communications 38: 62–67.
[10] Malekzadeh P., Setoodeh A.R, Alibeygibeni A., 2011,Small scale effect on the thermal buckling of orthotropic arbitrary straight-sided quadrilateral nanoplates embedded in an elastic medium, Composite Structures 93: 2083-2089.
[11] Khademolhosseini F., Rajapakse R.K.N.D., Nojeh A., 2010, Torsional buckling of carbon nanotubes based on nonlocal elasticity shell models, Computational Materials Science 48: 736–742.
[12] Eringen A.C., 1983, On differential of non-local elasticity and solutions of screw dislocation and surface waves, Journal Applied Physic 54: 4703–4710.
[13] Pradhan S.C., Murmu T., 2009, Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics, Computational Materials Science 47: 268–274.
[14] Pradhan S.C., 2009, Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory, Physics Letters A 373: 4182–4188.
[15] Murmu, T. Pradhan, S.C., 2009, Vibration analysis of nano-single-layered graphene sheets embedded in elastic medium based on nonlocal elasticity theory, Journal Applied Physic 105: 064319.
[16] Pradhan S.C., Phadikar J.K., 2009, Small scale effect on vibration of embedded multi layered graphene sheets based on nonlocal continuum models, Physics Letters A 373: 1062–1069.
[17] Aghababaei R., Reddy J.N., 2009, Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates, Journal of Sound and Vibration 326: 277-289.
[18] Pradhan S.C., Murmu T., 2009, Small scale effect on the buckling of single-layered graphene sheets under biaxial compression via nonlocal continuum mechanics, Computational Materials Science 47: 268–274.
[19] Murmu T., Pradhan S.C., 2009, Buckling of biaxially compressed orthotropic plates at small scales, Mechanics Research Communications 36: 933–938.
[20] Murmu T., Pradhan S.C., 2009, Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity, Journal Applied Physic 106: 104301.
[21] Chen Y., Lee J.D., Eskandarian A., 2004, Atomistic viewpoint of the applicability of microcontinuum theories, International Journal Solids Structure 41: 2085-2097.
[22] Bert CW., Malik M., 1996, Differential quadrature method in computational mechanics: a review, Applied Mechanic Review 49: 1–27.
[23] Farajpour A., Shahidi A.R., Mohammadi M., Mahzoon M., 2012, Buckling of orthotropic micro/nanoscale plates under linearly varying in-plane load via nonlocal continuum mechanics, Composite Structures 94: 1605–1615.
[24] Danesh M., Farajpour A., Mohammadi M., 2012, Axial vibration analysis of a tapered nanorod based on nonlocal elasticity theory and differential quadrature method, Mechanics Research Communications 39: 23– 27.
[25] Shu C, Richards Be.,1992, Application of generalized differential quadrature to solve two-dimensional incompressible Navier Stokes equations, International Journal for Numerical Methods in Fluids 15: 791–798.
[26] Shu C., 2000, Differential Quadrature and its Application in Engineering, Springer, Great Britain.
[27] Wang Q., Wang C.M., 2007, The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes, Nanotechnology 18: 075702.
[28] Duan W.H., Wang C.M., 2007, Exact solutions for axisymmetric bending of micro/ nanoscale circular plates based on nonlocal plate theory, Nanotechnology 18: 385704.
[29] Liew K., He M., X Q., Kitipornchai S., 2006, Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix, Acta Material 54: 4229-4236.
