Asymptotic Model of the Long-Wavelength Vibrations of an Ultrathin Beam-Strip Taking into Account Surface Effects

The work is concerned with the derivation of asymptotically consistent equations governing the long-wavelength vibrations of an ultrathin elastic beam-strip taking into account surface effects within the framework of the Gurtin–Murdoch theory of surface elasticity. Two-dimensional equations of motion of an elastic isotropic medium are used as the initial ones. In the general case, the beam-strip is under the action of variable unsteady surface forces. The presence of residual shear stresses is assumed on the face surfaces. The ratio of the strip thickness to the characteristic length of bending deformation is considered as a small parameter. Within the framework of the Gurtin–Murdoch theory, two cases are considered, providing for the presence of large (a) residual stresses on the faces and (b) the effects of surface inertia. Using the method of asymptotic integration over the transverse direction, the relations for displacements and stresses in an ultra-thin beam-strip are obtained, and equivalent one-dimensional Timoshenko-type equations taking into account surface effects are derived. As an example, the free vibrations of a simply supported beam taking into account surface effects are considered.