Small-scale Effect Of A Graphene Sheet Based On Nonlocal Theory

The graphene sheet shows the broad prospects in a few of fields and attracts the attentions of some researchers because of its high mechanical strength,the powerful capacity of energy storage as well as the good catalytic effect.The solution for the problem of nanometer material mechanics by the classical continuum mechanics will no longer be accurate due to the surface and small-scale effect.Fortunately,the nonlocal elastic theory given by Eringen can make up for the defect of classical continuum mechanics.It not only takes into account the nonlocal effects of continuous medium but also retains the advantages of the classical continuum mechanics.Objects that always have a certain characteristic length(distance or size)are made up of the child object such as particles(atoms,molecules,etc.),the external load also has the characteristic length or time(such as the distribution of the external load with smooth area,wavelength,frequency,etc.).When the internal and external characteristic lengths are almost similar,the classical continuum theory will no longer be appropriate.In this case the classical continuum theory of the second and third assumption have to be given up and incorporating the nonlocal continuum theory into it.Nonlocal continuum theory is not a microscopic theory,and it is still a phenomenological method,but it considers the effect caused by microscopic properties.This makes it possible to build a bridge between phenomenological theory and atomic or molecular theory.The new theory can be used to explain or solve some mechanics problem beyond of the classical continuum theory..Nonlocal continuum mechanics may be employed to research the mechanics problems of nonlocal elastic solids and nonlocal fluid.So the small scale effect of nanomaterials can be investigated based on nonlocal theory.This thesis consists of the three parts(1)Small-scale effect on the static deflection of a clamped graphene sheet and influence of the helical angle of the clamped graphene sheet on its static deflection are investigated.Static equilibrium equations of the graphene sheet are formulated based on the concept of nonlocal elastic theory.Galerkin method is used to obtain the classical and the nonlocal static deflection from Static equilibrium equations,respectively.The numerical results show that the static deflection and small-scale effect of a clamped graphene sheet is affected by its small size and helical angle.Small-scale effect will decrease with increase of the length and width of the graphene sheet,and small-scale effect will disappear when the length and the width of graphene sheet are both larger than 200 um.(2)Small scale parameter and bending stiffness of graphene sheet are regarded as the interval variables,vibration equation of a simply supported graphene sheet with uncertainty is established based on nonlocal theory.Trigonometric function series solution and the modified interval analysis method are employed to obtain the upper and lower bounds of response of the simply supported graphene sheet.the response uncertainty level of the graphene sheet with the different dimensions are investigated.The numerical results show that when theuncertainty level of the small scale parameter and bending stiffness remain unchanged,the uncertainty level of the response will decrease with increase of the graphene sheet dimensions,and a small uncertainty level of the small scale parameter can lead to very large uncertainty level of the response before the small-scale effect disappears.In addition,we find that the bending stiffness can lead to smaller uncertain level of the response than its own when the dimensions of the graphene sheet increases.