Analysis Of Nonlinear Static And Dynamic Behaviors Of Micron/Nanon-Scaled Structures

Owing to small size and special processing technology, mechanical properties of nano/microstructures are very different from that in the nano/micro-scale range . Therefore , thoroughly understanding and accurate characterization of mechanical properties of nano/microstructures is critical for optimal design and reliability of MEMS devices and products. Nowadays, many nano/micro-scale experiments have verified that mechanical properties of some materials at the micron or nano scale are size dependent. No material length scale parameter is introduced into the constitutive relation of conventional theory, so it cannot characterize and interpret size effect on the mechanical properties of materials and structures at the micron scale.Strain gradient theories, which include effects of strain gradient tensor and possess material length scale parameter, can characterize and interpret size effect on the mechanical properties of micro-structures at the micron scale. Besides the classical elastic theory incorporating with surface energies can characterize and interpret size effect on the mechanical properties of nano-structures at the nano scale.A size-dependent beam theory considering the effects of surface energies at nano-scaled thickness has been developed for beams. Unlike the classical beam theory, the transverse normal stress is preserved in order to retain the balance of surface stresses. Based on this size-dependent beam theory the governing equation of the nanobeam is derived where the geometry nonlinearity is also taken into account and the Galerkin method is applied to simplify this problem into a reduced-order equation. The influence of the surface effect on the free vibration is discussed. Then a powerful technology, AEM is applied to solve this complex problem. The influence of the surface effect on the static and dynamic responses, pull-in voltage and pull-in time is also discussed. The numerical results clearly show the size-dependent elastic responses of nano-scaled elements. The surface energies affect the results through the presence of some intrinsic length scales.A new beam model is developed for viscoelastic microbeam based on a modified couple stress model which contains only one material length scale parameter. The governing equations of equilibrium together with initial conditions and boundary conditions are obtained by a combination of the basic equations of modified couple stress theory and Hamilton's principle. This new beam model is then used for an electrically actuated microbeam-based MEMS structure. The dynamic and quasi-static governing equations of an electrically actuated viscoelastic microbeam are firstly given where the axial force created by the midplane stretching effect is also considered. Galerkin method is used to solve the above equation and this method is also validated by the finite element method (FEM) when our model is reduced into an elastic case. The numerical results show that the instantaneous pull-in voltage, durable pull-in voltage and pull-in delay time predicted by this newly developed model is larger (longer) than that predicted by the classical beam model. A comparison between the quasi-static model results and the dynamic model results is also given.A large-deflection model has been proposed to study the dynamic behaviors of microscale elastic thin plate. It is an extension of Von Kármán's geometrically nonlinear theory for elastic plates by adopting Kirchhoff's kinetic hypothesis. Moreover, the modified model is considered the modified couple stress theory. The governing equations are derived by using Hamilton's principle. Based on this new plate model, numerical results are represented for the effects of nonlinearity and material length scale parameter on the free vibration. For the free vibration of a microscale plate, the incremental harmonic balanced method is used and it is found that the nonlinear frequencies are much higher than the linear ones. This finding is very useful for high-performance microresonators in which one of the basic requirements is to match the frequencies of signals of interest.A new couple stress theory based inflated viscoelastic Timoshenko beam model is developed for the analysis of the mechanical properties of the yeast. The governing equations of equilibrium are obtained by a combination of the basic equations of modified couple stress theory and the principle of minimum total potential energy where the effect of the change in the enclosed volume is also considered. The concepts of instantaneous critical buckling force and durable critical buckling force are set up. The numerical results show that the instantaneous and delay buckling critical forces predicated by the Timoshenko beam model are both smaller than those predicated by the most-used Euler beam model. We also introduced the initial value theorem and the final value theorem to determine the instantaneous and delay buckling critical forces directly. The effects of the internal material length scale parameter and the turgor pressure on them are also discussed. After the yeast buckled, the effects of the internal material length scale parameter and turgor pressure on the buckling amplitude are also given. Here the final value theorem is also utilized to determine the steady buckling amplitude. We present a viscoelastic shell model for the study of the yeast where the effect of the turgor pressure is also considered. The governing equations are based on the first order shear deformation theory with a Flügge shell model. The critical axial compressive force in the phase space is obtained by using the Laplace transformation, and the Bellman numerical inversion method is then applied to the analytical result to obtain the corresponding numerical results in the physical phase. The concepts of instantaneous critical buckling force, durable critical buckling force and delay buckling are set up firstly. And finally the effects of the transverse shear deformation and the turgor pressure on the buckling phenomena are given.Lastly, based on a modified couple stress theory, a new Timoshenko beam model is established to address the size effect of MTs. The bending equation and the buckling equation are derived from the minimum total potential energy principle. The results got from present model show that length-dependence of MTs is related not only to the shear effect but also to the size effect, and the size effect is coupled in the shear effect, which means that the phenomenon of length-dependence will disappear when the shear effect is neglected. Moreover, when very long MTs are considered, the persistence lengths are related to the internal material length scale parameter, which is different from the conclusions got from the classical and previous nonlocal beam models. The effect of the internal material length scale parameter on the buckling wavelength, the buckling growth rate and the buckling amplitude of the MTs is also discussed in this paper, and a comparison between present and previous results is presented.