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UBC Theses and Dissertations

Surface loading and rigid indentation of an elastic layer with surface energy effects Zhao, Xujun


With the growing interest in nanotechnology, it is becoming important to understand the nanoscale mechanics to achieve successful design and fabrication of nanoscale devices. However, the classical continuum theory is not directly applicable to the analysis of nanoscale domains due to size-dependent behavior of nanostructures. Since the surface-to-volume ratio of a nanoscale domain is relatively high compared to that of a macro-scale domain, the energy associated with atoms at or near a free surface is different from that of atoms in the bulk. The effect of surface free energy therefore needs to be considered. Ultra-thin film/substrate systems, which are encountered in applications involving nanocoatings, nanotribology and material characterization based on nano-indentation, may be analyzed using modified continuum elasticity theory incorporating surface energy effects. This thesis presents a set of analytical solutions for elastic field of a layer of nanoscale thickness bonded to a rigid base under surface loading and indentation by a rigid body. Surface energy effects are accounted for by using Gurtin-Murdoch elasticity theory. Fourier and Hankel integral transforms are used to solve the two- and three- dimensional boundary-value problems involving non-classical boundary conditions associated with the generalized Young-Laplace equation. In the case of a two-dimensional semi-infinite medium, the solutions can be expressed in closed form. The elastic field is found to depend on the layer thickness and surface elastic constants, and the influence of surface energy is shown to be more significant under a horizontal load than under a vertical load. A characteristic length scale related to the surface material properties can be identified for the present class of problems. The solution for the indentation problem is considered for flat, conical and spherical rigid indenters. The mixed boundary-value problem corresponding to a rigid indenter is formulated in terms of a dual integral equation system that is solved by using numerical quadrature. Selected numerical results are presented to show the influence of the indenter shape, surface properties and size-dependency of response.

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