Asymptotic models for elastic solids taking into account nonlocal boundary layers
The thesis is aimed at asymptotic analysis of the near-surface boundary layers in non-locally elastic solids. The dynamic response of a homogeneous half-space with a traction-free surface is analysed for a nonlocal exponential kernel. A typical wave-length is assumed to be much greater than the length scale associated with internal properties of the elastic medium. The dominant effect of the boundary layer is revealed. The leading order long-wave approximations are shown to coincide with the `local' problem for a half-space having a vertical inhomogeneity localised near the sur-face. An explicit correction to the classical boundary conditions on the surface of a `locally' elastic half-space is obtained by asymptotic analysis of the near-surface behaviour. The order of the derived correction exceeds that of the well-known correction to the governing differential equations of Eringen's model, e.g., see . The obtained refined boundary conditions enable evaluating the interior stress-strain solution outside a narrow boundary layer localised near the surface. As examples, the effect of nonlocal elastic phenomena on the Rayleigh wave speed and also a plane strain problem of a moving load on the surface of a half-space are studied. In addition, a thin layer with a traction-free upper face, subjected to prescribed displacements along its lower face, is investigated. Further, the 3D dynamic equations in nonlocal elasticity for a thin plate are considered, assuming the plate thickness to be much greater than a typical microscale size. The long-wave low-frequency approximations are obtained for both plate bending and extension. Boundary layers characteristic of nonlocal behaviour are revealed near the plate faces. It is established that taking into account the effect of the boundary layers results in first-order corrections to the bending and extensional stiffness in the classical 2D plate theory.
|Jun 1, 2018