| With the rapid development of MEMS(micro-electro-mechanical systems)and micro/nano technology,elastic structures have been their important building blocks,and how to accurately design these structures is a hot topic hitherto.At micro and nanoscale,with the significant decrease of the sizes of elastic structures,they have much bigger surface energy to volume ratios,thus rendering them very strong surface effect and size effect.Aiming to study the elastic thin plates at micro and nanoscale,the Gurtin’s theory of surface elasticity has been introduced in this work to analyze the static deformation and vibration of these plates,where the surface elasticity and residual surface stress have been considered.The effects of plate shape(rectangular plate and circular plate),loading types and boundary conditions on the static deformation of plates have been thoroughly investigated.We have also studied the effect of residual surface stress on the natural frequency and natural model of the nanoplate.In the modeling process,firstly,according to the Young-Laplace equation,the transverse load induced by the residual surface stress is considered.Besides this feature,the surface elasticity effect of the thin plate at micro and nanoscale is also discussed,where the corrected expression of the bending stiffness is used.The rectangular plate and circular plate are both investigated in the present work.Firstly,the displacement solution of the circular plate is studied in the polar coordinate system.For the clamped circular thin plate,the displacement solution expressed in terms of the Bessel function is given by considering its transversely uniform load.The analytical solutions are obtained in the light of the positive or negative residual surface stress.In succession,based on the infinitesimal deformation theory,the differential equation of the rectangular thin plate with nanoscaled thickness is built in consideration of surface effects.The analytical solutions of the deflection of a plate with all edges simply supported and uniform load,with all edges simply supported and bending moment distributing its two pairs of edges and with two edges simply supported and the others clamped,through series expansion,have been presented.The results show that,the residual surface stress has strengthened the bending stiffness of the thin plate,thus making it more difficult to deform,and its deflection has decreased.With the decrease of the residual surface stress,and especially when it is zero,the result can be compared with that of the classical elasticity theory.This indicates that the positive residual surface stress makes the plate“stiffer”,and the negative one makes it “softer”.Another issue of this study is the vibration analysis of the circular micro and nanoplate.Firstly,based upon Gurtin’s theory of surface elasticity,the surface elasticity and residual surface stress are both considered,and thus the mathematical model of a thin plate in vibration at micro and nano-scale with clamped boundary condition is developed.Subsequently,the natural frequency and natural model of the plate are solved.The effects of the residual surface stress on the first four order natural frequency and vibration mode have all been presented.The results manefist that the first order natural frequencies all decrease with the increase of the residual surface stress parameter.Moreover,the residual surface stress effect takes more important effect on the first natural frequency than others,and as the vibration order increases,the effect becomes weaker.The first four order vibration modes for various surface effect parameters are also given in the study.As a consequence,the influence of the vibration mode is similar to that of the natural frequency;and the influence of the surface effects is greater on the first main mode than those of other modes.These results are beneficial to fully understand the surface effects of materials and structures at micro and nanoscale,and can also provide some inspirations for the design of micro-sensors,MEMS and smart materials. |
