Effects Of Surface Stress On The Deformation Of Micro-and Nano-sized Cantilever Plates

Cantilevers made of thin film materials have been widely inmicroelectromechanicalsystems (MEMS) and nanoelectromechanicalsystems (NEMS). Whenthe thicknesses of thin film materials reachmicron or nanometers, surface stress (surfaceelasticity)can have significant influence on theirmechanical properties. Cantilever bendingtests and the resonant tests have been commonly applied to characterize the mechanicalproperties of thin film materials. For example, microcantilever bending tests have applied tocharacterize Young’s modulus, yield strength, fracture toughness and residual surface stressof thin film materials, microcantilever resonant tests have been used to determine elasticmodulus of thin films. Microcantilever bending tests and resonant tests used in themeasurement of the mechanical properties of thin film materials are usually based oncantilever beam formulations from the mechanics of materials approach. In most applications,the thickness, width and length dimensions of microcantileversare more closer to the scope ofclassical plate theory than theclassical beam theory. In order to more accurately determine themechanical properties of thin films, analytical solutions of the deflections of cantilevermicro-plates are more desirable.In this study, the bending and in-plane deformations of a cantilever rectangular plate dueto residual surface stresses applied on its upper and bottom surfaces are solved analyticallybased on the superposition of bisection Fourier cosine series and low-order polynomials. Notethat the bisection Fourier cosine series can satisfy the bi-harmonic differential equation. Thedeflection formulation for the middle point of the free end can be used to accurately determinethe residual surface stress in thin film materials. Based on the Gurtin-Murdoch surfaceelasticity theory, the effect of surface elasticity on the bending stiffness of thin films withthickness at nanometer level has been investigated. Analytical solution has been obtained forthe deflection of a cantilever rectangular nano-plate subjected a concentrated force applied atthe middle point of its free end.In the investigation of the bending of a rectangular cantilever plate induced by residualsurface stress, solutions of the deflection, slope and curvature at the middle point of the free end are presented for the aspect ratio ranging from0.1to10and for different Poisson’s ratios.Analytical solutions based on Fourier cosine series have been compared with simpleanalytical formulations such as the Stoney’s equation and numerical results based on finiteelement analysis. It has been revealed that, though the Poisson’s ratio has important effects onthe deflection, slope and curvature of the free end, the aspect ratio of the rectangular plate hasmore pronounced effect on them. When the aspect ratio is less than one, the clamped end hassignificant on the bending deformation of the plate, on the other hand, for a plate with largeaspect ratio (e.g. larger than3), the influence of the clamped end on the deflection of the plateis small. In most practical cases, the aspect ratio is greater than one andthe maximumpercentage errors of Stoney’s equation for the deflection, slope and curvature at the middlepoint of its free end are16%,16%and10%, respectively. Based on the present Fourier cosineseries with the first two leading terms, the maximum percentage errors of the deflection, slopeand curvature of the free end are reduced to3%,2%and3%respectively.For the problem of the in-plane deformation of a cantilever rectangular plate induced byresidual surface stresses at its upper and bottom surfaces, none zero in-plane stress resultantscan be introduced into the region of the plate adjacent to the clamped end. When the aspectratio is less than one, none zero stress resultants are distributed across the majority portion ofthe plate, and the residual surface stress has important effect on the in-plane stiffness of thecantilever plate. On the other hand, when the aspect ratio is greater than one, none zero stressresultants are localized to the region close to the clamped end, and the residual surface stresshas little influence on the in-plane stiffness of the cantilever plate. In addition, the Poisson’sratio have also influence on the magnitude and distribution of none zero stress resultants ofthe cantilever rectangular plate.When the thicknesses of thin-film materials reach the nano-meter level, the surfaceelastic constants of thin films have pronounced effect on the bending stiffness of nano platesmade of thin films. The bending stiffness becomes size-dependent, the bending stiffness caneither increase or decrease with the thickness of a nano plate depending on the elastic stiffnessof the upper and bottom surfaces of the nano plate. The deflection of a cantilever rectangular nano plate due to a concentrated force applied at the middle point of its free end has beenobtained by using bi-direction Fourier cosine series. The numerical solutions for thedeflection of the middle point of the free end are compared with those obtained by finiteelement analysis and they show good agreement. When the aspect ratio is less than one, theaccuracy of the deflection based on simple cantilever beam formulation is poor. When theaspect ratio is greater than one and less than two, the maximum relative error of the simplecantilever beam formulation is within11%. When the aspect ratio is greater than two, themaximum relative error of the simple cantilever beam formulation is reduced to7%. Based onthe present Fourier cosine series with the first three leading terms, the maximum percentageerror of the deflection of the free end is reduced to2%, showing significant improvement overthan the simple cantilever beam formulation.