Nonlocal theories for bending, buckling and vibration of beams

Abstract Various available beam theories, including the Euler–Bernoulli, Timoshenko, Reddy, and Levinson beam theories, are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived, and variational statements in terms of the generalized displacements are presented. Analytical solutions of bending, vibration and buckling are presented using the nonlocal theories to bring out the effect of the nonlocal behavior on deflections, buckling loads, and natural frequencies. The theoretical development as well as numerical solutions presented herein should serve as references for nonlocal theories of beams, plates, and shells.

[1]  H. P. Lee,et al.  Dynamic properties of flexural beams using a nonlocal elasticity model , 2006 .

[2]  Vijay K. Varadan,et al.  Vibration of carbon nanotubes studied using nonlocal continuum mechanics , 2006 .

[3]  J. Reddy.  Theory and Analysis of Elastic Plates and Shells , 2006 .

[4]  C. Wang,et al.  Buckling analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko beam theory , 2006 .

[5]  A. Eringen.  On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves , 1983 .

[6]  John Peddieson,et al.  Application of nonlocal continuum models to nanotechnology , 2003 .

[7]  L. Sudak,et al.  Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics , 2003 .

[8]  A. C. Eringen,et al.  Nonlocal polar elastic continua , 1972 .

[9]  Mingtian Xu,et al.  Free transverse vibrations of nano-to-micron scale beams , 2006, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[10]  J. N. Reddy,et al.  Energy principles and variational methods in applied mechanics , 2002 .

[11]  Quan Wang,et al.  Wave propagation in carbon nanotubes via nonlocal continuum mechanics , 2005 .

[12]  A. Eringen,et al.  On nonlocal elasticity , 1972 .

[13]  A. Eringen,et al.  Nonlocal Continuum Field Theories , 2002 .

[14]  C. Wang,et al.  Deflection relationships between classical and third-order plate theories , 1998 .

[15]  Gui-Rong Liu,et al.  Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression , 2004 .

[16]  Gui-Rong Liu,et al.  Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity , 2005 .

[17]  M. Levinson,et al.  A new rectangular beam theory , 1981 .

[18]  J. Reddy,et al.  A higher order beam finite element for bending and vibration problems , 1988 .

[19]  J. Reddy.  A Simple Higher-Order Theory for Laminated Composite Plates , 1984 .