New Solutions of Half-Space Contact Problems Using Potential Theory, Surface Elasticity and Strain Gradient Elasticity

Abstract

Size-dependent material responses observed at fine length scales are receiving growing attention due to the need in the modeling of very small sized mechanical structures. The conventional continuum theories do not suffice for accurate descriptions of the exact material behaviors in the fine-scale regime due to the lack of inherent material lengths. A number of new theories/models have been propounded so far to interpret such novel phenomena. In this dissertation a few enriched-continuum theories - the adhesive contact mechanics, surface elasticity and strain gradient elasticity - are employed to study the mechanical behaviors of a semi-infinite solid induced by the boundary forces.

A unified treatment of axisymmetric adhesive contact problems is developed using the harmonic functions. The generalized solution applies to the adhesive contact problems involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile, and it links existing solutions/models for axisymmetric non-adhesive and adhesive contact problems like the Hertz solution, Sneddon's solution, the JKR model, the DMT model and the M-D model.

The generalized Boussinesq and Flamant problems are examined in the context of the surface elasticity of Gurtin and Murdoch (1975, 1978), which treats the surface as a negligibly thin membrane with material properties differing from those of the bulk. Analytical solution is derived based on integral transforms and use of potential functions. The newly derived solution applies to the problems of an elastic half-space (half-plane as well) subjected to prescribed surface tractions with consideration of surface effects. The newly derived results exhibit substantial deviations from the classical predictions near the loading points and converge to the classical ones at a distance far away from those points. The size-dependency of material responses is clearly demonstrated and material hardening effects are predicted.

The half-space contact problems are also studied using the simplified strain gradient elasticity theory which incorporates material microstructural effects. The solution is obtained by taking advantage of the displacement functions of Mindlin (1964) and integral transforms. Significant discrepancy between the current and the classical solutions is seen to exist in the immediate vicinity of the loading area. The discontinuity and singularity exist in classical solution are removed, and the stress and displacement components change smoothly through the solid body.