Abstract
In this chapter, a study on the displacement field equations of nonlocal elasticity is developed for mechanical analyses of nano structures. Considering the small scale effect, the three dimensional equations of nonlocal elasticity are obtained. At first, three decoupled equations in terms of displacement components and three decoupled equations in terms of rotation components are obtained. These equations are also invariant with respect to the coordinate system. In order to solve a nonlocal elasticity problem based on the presented formulation, one of the three equations in terms of displacement components and corresponding rotation equation should be solved independently. Using some relations, the other two displacement components can be obtained in terms of the mentioned displacement and rotation component. In an illustrative example, these equations associated with simply supported boundary conditions are solved for a nano-plate using the Fourier series technique. In addition, the results are compared with the first order and third order shear deformation theories. It is seen that the natural frequencies of the nonlocal plate theories are not as accurate as the results in classical plate theories.
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References
Ozer, T.: Symmetry group classification for two-dimensional elastodynamics problems in nonlocal elasticity. Int. J. Eng. Sci. 41, 2193–2211 (2003)
Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int. J. Eng. Sci. 10, 233–248 (1972)
Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, 4703–4710 (1983)
Wang, Q., Liew, K.M.: Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys. Lett. A 363, 236–242 (2007)
Reddy, J.N.: Nonlocal theories for bending, buckling and vibration of beams. Int. J. Eng. Sci. 45, 288–307 (2007)
Peddieson, J., Buchanan, G.R., McNitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 41, 305–312 (2003)
Kitipornchai, S., He, X.Q., Liew, K.M.: Continuum model for the vibration of multilayered graphene sheets. Phys. Rev. B 72, 075443–075449 (2005)
Pradhan, S.C., Phadikar, J.K.: Nonlocal elasticity theory for vibration of nanoplates. J. Sound Vib. 325, 206–223 (2009)
Aghababaei, R., Reddy, J.N.: Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J. Sound Vib. 326, 277–289 (2009)
Plevako, V.P.: On the theory of elasticity of inhomogeneous media. J. Appl. Math. Mech. 35, 806–813 (1971)
Levinson, M.: Free vibrations of a simply supported, rectangular plate: an exact elasticity solution. J. Sound Vib. 98, 289–298 (1985)
Charalambopoulos, A., Dassions, G., Fotiadis, D.I., Massalas, C.V.: Note on eigenvector solutions of the Navier equation on cylindrical coordinates. Acta Mech. 134, 115–119 (1999)
Poullikkas A, Karageorghis A, Georgiou G The method of fundamental solutions in three dimensional elastostatics. International conference on parallel processing and applied mathematics, vol. 2328, pp. 747–755. Leczow, Poland (2001)
Ozer, T.: The solution of Navier equations of classical elasticity using Lie symmetry groups. Mech. Res. Commun. 30, 193–201 (2003)
Saidi, A.R., Atashipour, S.R., Jomehzadeh, E.: Reformulation of Navier equations for solving three-dimensional elasticity problems with applications to thick plate analysis. Acta Mech. 208, 227–235 (2009)
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The authors wish to acknowledge the financial support from National Elite Foundation of Iran.
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Jomehzadeh, E., Saidi, A.R. (2012). A Study on the Displacement Field of Nonlocal Elasticity for Mechanical Analysis of Nano Structures. In: Öchsner, A., da Silva, L., Altenbach, H. (eds) Materials with Complex Behaviour II. Advanced Structured Materials, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22700-4_21
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DOI: https://doi.org/10.1007/978-3-642-22700-4_21
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