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Nanorods and nanotubes are widely used as biosensors and reinforced fibers. Surface effects should be considered due to their nano-meter order dimension in at least one direction. This paper studies vibration and stability of a nanocantilever carrying mass under non-conservative loading. Special emphasis is placed on determination of flutter loads and resonant frequencies of a nanocantilever-mass system subjected to a generalized follower force. An exact characteristic equation is derived. For various concrete cases, the load–frequency interaction curves are displayed and the influences of surface elasticity and residual surface tension are analyzed for both conservative and non-conservative forces. The resonant frequencies are calculated for bending vibration of a sensor with attached mass. Flutter loads are obtained under a generalized follower force. Beck’s column with the surface effects is a special case of the present. Surface influences from residual surface tension and surface elasticity on the resonant frequencies and critical loads are examined. It is found that there exists the lowest flutter load independent of the tip mass through which all force–frequency interaction curves passes at a given frequency for any generalized follower force in possible direction. [ABSTRACT FROM AUTHOR]

• 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. [ABSTRACT FROM AUTHOR]

We consider the torsional deformation of an infinite three-dimensional isotropic elastic solid weakened by a penny-shaped crack whose boundary is enhanced by the incorporation of surface elasticity. Furthermore, we assume the presence of a Dugdale-type plastic zone around the crack front. The ensuing elastoplastic problem gives rise to a non-standard mixed boundary value problem which is then reduced to a hypersingular integral equation upon application of the Hankel transform. The hypersingular integral equation is subsequently solved numerically using orthogonal expansions revealing the effects of surface elasticity on the size of the plastic zone and crack-face torsional displacements. Our results demonstrate the effect of surface elasticity on the width of the plastic zone thus providing a significant shielding effect while increasing the load-bearing capacity of the solid in the vicinity of the defect. Our conclusions will be of particular interest when the presence of surface elasticity along a crack boundary is used in enhanced continuum-based models to study fracture at micro- and nano-scales, for example, in the prediction of fracture and failure in micro- and nano-structures. [ABSTRACT FROM AUTHOR]

We analyze the transient response of a linearly elastic solid weakened by a nanoscale crack (nanocrack) when the solid is subjected to anti-plane impact loading. We account for the nanoscale by incorporating surface elasticity into the model of deformation. The problem is formulated as a nonclassical mixed initial–boundary value problem. Most significant is the fact that both surface stress and surface mass inertia are included in the boundary condition on the crack surface. By applying Laplace and Fourier transform methods, we reduce the associated boundary value problem in the Laplace transform domain to a singular integro-differential equation. The latter is solved numerically leading to the determination of the corresponding stress intensity factors in the (Laplace) transformed domain. The Gaver–Stehfest algorithm is then applied to perform a numerical inversion of the Laplace transform from which we obtain dynamic stress intensity factors in the time domain. Numerical results characterizing the transient response of the crack-tip field are presented with particular emphasis on the influence of surface elasticity on the dynamic fracture parameters. • Transient response of the crack-tip field with surface effects. • Dynamic stress intensity factors of a nanoscale mode-III crack subjected to sudden impact. • Surface elasticity alters the overshoot magnitude, does not change its occurrence time. • Crack surface mass inertia affects the overshoot magnitude and occurrence time. [ABSTRACT FROM AUTHOR]

The bending fracture of an ultra-thin plate containing a thickness-through crack with consideration of surface elasticity is studied in this paper. Based on the Kirchhoff plate theory incorporating surface elasticity, a governing equation is derived for static bending of nanoplates with surface elasticity. The fracture problem of an infinite isotropic elastic nanoplate with a thickness-through crack is presented and solved when the plate is subjected to uniform bending moment, twisting moment, and out-of-plane tearing load, respectively. The Fourier integral transform is applied to solve associated mixed boundary value problems. A hypersingular integral equation is deduced and an exact solution is determined for each case. Complete elastic fields at any position of the nanoplate are given explicitly in terms of elementary functions and the expressions for the intensity factors of stress, shear force, and bending/twisting moment at the crack tips are obtained in closed form. The intensity factors of stress and moment exhibit a square-root singularity, while the effective shear force intensity factors have an r - 3 / 2 singularity near the crack tips, r being the distance from the crack tip. Results are presented graphically to show that the stress intensity factors (SIFs) are related to the bulk and surface material properties. When neglecting surface elasticity, the obtained size-dependent SIFs reduce to the well-known classical counterparts. [ABSTRACT FROM AUTHOR]

• Full elastic field induced by a rigid line inclusion in a nanoplate is obtained. • Influence of surface elasticity on the elastic field is analyzed. • Bending moment near the inclusion tip exhibits a power-law singularity of -3/2 order. • Effective shear force near the inclusion tip exhibits a power-law singularity of -5/2 order. [Display omitted] 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. [ABSTRACT FROM AUTHOR]