Oscillatory singularity for bending of a partially clamped nanoplate with consideration of surface effect

Abstract

This article studies a partially clamped flexible ultrathin elastic nanoplate with surface effect. An emphasis is to analyze the singular elastic behavior near the tips of partially clamped boundary. A mixed boundary value problem associated with a partially clamped nanoplate having parabolic deflection is presented and solved. Using the Kirchhoff plate theory incorporating the Gurtin–Murdoch surface elasticity, some basic equations for nanoplates with surface elasticity are established. With the aid of the Fourier transform and the superposition principle, two coupled singular integral equations with Cauchy kernel are derived during bending of the nanoplate and then converted to a singular integral equation with two Cauchy kernels. The exact solution to the singular integral equation with two Cauchy kernels is determined and the oscillatory singular elastic fields near the tips of the clamped boundary are given in elementary functions. The bending moments and the normal stress components in the vicinity of the tips simultaneously have the oscillator singular behavior like r−3/2−iɛ, r being the distance from the tip and ɛ relying on the material properties. Besides, the effective shear force possesses an r−5/2−iɛ singular behavior. The influences of the geometrical parameters and material properties of nanoplates on the oscillatory singular behavior are presented graphically. When surface effect is neglected, the solution is applicable to classical thin plates with partially clamped boundary.