# Free vibration and buckling analysis of beams with a modified couple stress theory, by J. V. Araujo dos Santos and J. N. Reddy

A model based on a modified couple stress theory for the free vibration and buckling analyses of beams is presented. The model also incorporates the Poisson's effect and allows the analysis of Timoshenko beams with any arbitrary end boundary condition. The natural frequencies and buckling loads are computed using the Ritz method. Para-metric studies show that, while the natural frequencies and the buckling loads increase monotonically with the increase of the material length scale, they present a minimum in certain values of the Poisson's ratio. A study relating the classical elasticity and the couple stress strain energies is also presented. By establishing this relation, explicit for-mulas to obtain the natural frequencies and buckling loads, in which the couple stress and Poisson's effects are accounted for, in terms of the buckling loads of the classical elasticity are found. These formulas, which are valid when the shear strain and stress are zero, allow an expedite computation of natural frequencies and buckling loads of beams with couple stress and Poisson's effect.

### Very good paper

Many thanks for your very good paper.

I am a PhD student who seeking for a PhD scholarship in your
research field. I have many international publications related to your work.
Moreover, i am now working in the same field: nano continuum mechanics of
solids.

Now, I am preparing some papers in which the classical
nd
modified to be consistent with the unique behavior of nano materials. The
fundamental equations and relations for the couple stress theory are derived to
illustrate the microstructural effects on nano materials. Moreover, the effect
of the material surface free energy is incorporated into the modified continuum
theory. Then the modified continuum theories are used to investigate the size-dependent
models for nano structures (Nano Mindlin and Kirchhoff plates) for functionally
graded nano materials. Moreover, a comparison between the modified couple
stress theory proposed by yang et al. 2002 and the consistent couple
stress theory proposed by
Hadjesfandiari and Dargush 2011 is performed to show the effect of the symmetric
couple-stress tensor (Yang) and the skew-symmetric couple-stress tensor

Consequently, if you have an available PhD position
for me, I am looking to hear from you. I attached to you my CV.

M. Shaat

Best regards,
-------------------------------------------------------
Mohammed Ibrahim Shaat (M.Shaat)
Research Assistant
Mechanical Engineering Department, Zagazig
University, Zagazig 44511, Egypt

E-Mail: ShaatScience@yahoo.com
Tel: +20 0122 2232469