Nanoscale problems for an elastic semi-space subjected to an axisymmetric loading


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Due to technological advancements, problems at nanoscale have gained significant interest in recent years. Material properties at nanoscale exhibit size-dependent behavior which can be attributed to the influence of surfaces and interfaces. Correspondingly, problems at nanoscale have to take into account surface energy. Here, we consider two problems involving an isotropic half-space subjected to either nanoscale contact or nanoscale patch loading. In the first problem, we consider an isotropic half-space subjected to nanoscale contact with a rigid punch. The surface energy in the Steigmann-Ogden form is used to model the half-space while linear elasticity is used to model the bulk of the material. The nanoindentation problem is solved using Boussinesq’s displacement potentials and Hankel integral transforms. The problem is reduced to a single integral equation, the character of which is studied, and a numerical method of solution to the corresponding integral equation using Gauss-Chebyshev quadrature is presented. In the second problem, we consider another problem at nanoscale in which an isotropic half-space is subjected to a nanoscale patch load. The surface effects are considered on a circular subset of the surface by assuming the Steigmann-Ogden model on the subset, while the remainder of the surface is assumed to not have surface effects. Using techniques presented previously in the work, the problem is reduced to a system of singular integro-differential equations with boundary conditions. Solutions corresponding to the dominant systems with Cauchy kernels are explored. A numerical method of solution of the system of integral equations using approximation by Chebyshev polynomials and Gauss-Chebyshev quadrature for computation of the integrals is presented and results are compared to those available the literature.



Axisymmetric, Surface energy, Singular integral equation, Frictionless, Nanoscale loading

Graduation Month



Doctor of Philosophy


Department of Mathematics

Major Professor

Anna Zemlyanova