Study On Two Types Of Scale Effects Of Vibration Characteristics Of Nanosclae Beams And Plates

With the rapid development and wide application of nanomaterials,more and more researchers pay attention to nanomechanical research.However,under the micro and nanoscale scale,the size of the structure is quite close to the size of the material molecule so that the scale effect can not be ignored.However,the scale effect can not be described very well by classical elastic theories and it is particularly important to develop new theories to study the mechanical properties of microscale materials.In this paper,the bending,buckling and vibration characteristics of one dimensional nanobeam to two-dimensional thin nanoplate are studied respectively based on the nonlocal strain gradient theory proposed by Lim,et al,which combine two modern relatively complete theories: nonlocal theory and strain gradient theory.In order to be more adaptive to the development and application of modern nanomaterials,the vibration characteristics of the piezoelectric nanoscale under the combined action of thermal,electrical and mechanical loads is also studied.Main contents are as follow:First,the basic contents of nonlocal strain gradient theory is briefly introduced,and the stress relation between the theory and classical theory is given.Meanwhile,the transformation relation between the theory and the nonlocal theory and strain gradient theory is extended.For one dimensional nanobeam,a theoretical model of Timoshenko nanobeam is established and its governing equation is derived by Hamiltonian variational principle.The governing equation is solved by the separation variable method to get the analytic solutions and the numerical example analysis is carried out.The results show that the increase of the nonlocal parameter will increase the flexural deflection of the nanobeam and decrease the critical buckling force and natural frequency.However,the increase of the material characteristic parameter and ratio of length to height will reduce the flexural deflection and increase the critical buckling force and natural frequency.The nonlocal effect will weaken the equivalent stiffness of beams,while the strain gradient effect will enhance the equivalent stiffness of beams.When the ratio of length to height increases to a certain value,the effect on the flexural deflection,critical buckling force and natural vibration frequency of the beamdecreases obviously,or even it might not exist at all.Then,from one-dimensional nanobeam to two-dimensional thin nanoplate model,the series expression of the stress under the nonlocal strain gradient theory is obtained by using iterative method.The bending differential equation and governing equation are obtained by displacement method and Hamiltonian principle respectively.The analytic solutions are evaluated through Navier method and the numerical value is plugged in to analyze the results.The results show that the nonlocal effect increases the maximum bending deflection of the nanoplate and reduces the natural frequency,while the strain gradient effect is the opposite.The increase of the half wave number and the thickness of the plate will also improve the natural frequency.The increase of the two-dimensional size of the nanoplate will make the maximum bending deflection approach the value of classical theory and reduce its natural frequency.In addition,the two kinds of scale effects only have a significant effect on the maximum bending deflection and natural frequency of the nanoplate under small scale.Finally,a theoretical model is established for a piezoelectric thin nanoplate under the joint action of thermal,electrical and mechanical loads.The vibration control equation is derived using the Hamilton principle and the analytic solution of its dimensionless form is gained by Navier method.Through specific example analysis,the results show that the nonlocal effect will weaken the equivalent stiffness of the plate and reduce the first natural frequency,while the strain gradient effect is the opposite.The two kinds of scale effect have a more significant effect on the higher order natural frequency.The increase of the length-width ratio of the piezoelectric nanoplate increases the first natural frequency,while the ratio of length to thickness is the opposite.With the enhancement of the nonlocal effect,the effect of length-width ratio and ratio of length to thickness on the first-order natural frequency becomes smaller and smaller.On the contrary,the two ratios will have more and more influence on the first-order natural frequency with the enhancement of the strain gradient effect.When the axial tension or negative voltage applied on the upper and lower surface of piezoelectric nanoplates increase,the first three-order vibration frequencies will increase.Conversely,the increase of axial pressure and positive voltage will reduce the first three-order vibration frequencies.Because PZT-4 is not a thermal sensitive material,the increase of temperature difference will only reduce slightly the first three-order vibrationfrequencies of the PZT-4 piezoelectric nanoplate,which is almost negligible.