Singular elastic field induced by a rigid line adhering to a micro/nanoscale plate during bending

• Published in: Applied Mathematical Modelling Vol. 108; pp. 567 - 584
• Main Authors: Hu, Zhen-Liang, Yang, Ying, Li, Xian-Fang
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.08.2022; Elsevier BV
• Subjects: Bending moments; Boundary value problems; Elastic analysis; Elasticity; Exact elastic fields; Integral transforms; Parabolic displacement; Plate theory; Rigid line; Shear forces; Singular integral equations; Singularities; Singularity coefficients; Surface elasticity; Thin plates; Rigid line; Exact elastic fields; Surface elasticity; Parabolic displacement; Singularity coefficients
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2022.04.004

•Singular elastic field induced by a rigid line in a flexural nanoplate.•The effective shear forces at the parabolic rigid line tips admit an r−3/2 singularity.•The bending moments at the parabolic rigid line tips have an inverse square-root singularity.•Surface effects cause nanoplates to become stiffer than classic plates. In this work, the elastic analysis of a micro/nanoscale plate with a stiffer ribbon adhering to it is studied. The stiffer ribbon is understood as a rigid line. The problem is solved by superposition and the singular elastic field induced by a rigid line is concerned when the plate is during bending. The singular elastic field of a micro/nanoscale elastic thin plate with surface elasticity is determined for a parabolic rigid line. The theoretical analysis is based on a mixed boundary value problem that is solved through the Fourier integral transform technique. According to the classical Kirchhoff thin plate theory incorporating surface elasticity, some basic equations are established. The mixed boundary value problem is reduced to a singular integral equation of the first kind with Cauchy kernel. Explicit expressions for the moments and effective shear forces as well as the surface and bulk stress components in any position of the whole nanoplate are obtained. The singularity coefficients of the moment and effective shear force at the tip of the rigid line are determined in closed form. From the obtained results, we find that the normal stress components for bulk and surface phases and bending moments at the rigid line tips have a usual inverse square-root singularity, but the effective shear forces behave like r−3/2, r being the distance from the tip of the rigid line or admit an r−3/2 singularity. The numerical results are displayed graphically and show that the singularity coefficients are heavily dependent on the surface and bulk material constants.