Structural Mechanics Analysis Of The Theoretical Beam Of Hyperbolic Shear Deformation With Size Effect

Micro-and nano-sized micro-beams and thin plates often exhibit size effects.Due to the small size of the device,its mechanical properties and deformation behavior are closely related to the size effects.Mechanical models which based on classical continuum mechanics that do not contain any material length scale parameters cannot capture these effects.Therefore,it is of great significance to strengthen the analysis and study of the size effect of microstructure deformation behavior.Based on the modified couple stress theory and surface elasticity theory,a new hyperbolic shear deformation beam model is proposed for static bending,elastic buckling and free vibration analysis of micro-scale uniform beam and functionally graded beam.The theoretical model can directly obtain the transverse shear stress by using the constitutive relation.It satisfies the stress-free boundary condition of the top and bottom of the beam,and avoids the need of the shear correction factor.The model makes the symmetry of the couple stress tensor by introducing an additional equilibrium condition of the even moment,which contains only one material scale parameter and enable the new model to describe the scale effects associated with the microstructure and surface energy.According to the generalized Young-Laplace equation,the stress continuity conditions of the inner and outer layers of the beam are established.A single variable field can describe the displacement mode of the beam.By considering the thickness of the surface layer and the continuous stress conditions of the surface layer in the displacement field,the new model can more accurately predict the scale effects related to micro-scale and surface energy.The governing equations of the beam and the associated boundary conditions are derived by the Hamilton principle.The differential quadrature method(DQEM)is used to study the mechanical response of beams under different boundary conditions.The numerical solution is compared with the analytical solution.In this paper,the effects of surface tension,material length scale parameter,beam thickness and functional gradient parameters on the mechanical response of the beam are analyzed.It is found that when the beam thickness is small,the difference between the bending stiffness,buckling load and natural frequency predicted by the two models is very significant snd decreases with the increase of the beam thickness,which proves the existence of the scale effect.When the microstructure correlation and surface energy effects are not considered,the model reduces into a classical Euler beam model.