Singular elastic field induced by a rigid line inclusion in a thin nanoplate with surface elasticity

Abstract

This paper analyzes the mechanical behavior of a Kirchhoff thin nanoplate with surface stresses containing a thickness-through rigid line inclusion. Based on the bulk and surface elasticity, a rigid line inclusion problem of a prescribed displacement is reduced to an associated mixed boundary value problem. By using the Fourier transform, the problem is converted to dual integral equations, then to a weakly singular integral equation with the logarithmic kernel. An exact solution for a rigid line inclusion with an off-nanoplate rigid rotation is obtained and full elastic fields including the deflection, bending moments, and effective shear forces at any position of the nanoplate are determined explicitly in terms of the elementary functions. The singularity coefficients of the bending moments and effective shear forces near the tip of the rigid line inclusion are evaluated. The obtained results show that the bending moments and the stress components exhibit an r−3/2 singularity near the rigid line inclusion tip, where r is the distance from the rigid line inclusion tip. The effective shear forces have an r−5/2 singularity. The influence of surface elasticity on the singularity coefficients is examined.