Small oscillations of a rigid sphere on an elastic half space: a theoretical analysis

Abstract

Harmonic motion is a significant topic in undergraduate lectures for university physics students. A typical example, presented in lectures around the globe, is the motion of a mass at the end of a spring. It should be emphasized that the harmonic motion in the aforementioned case is just an approximation for small displacements. However, the value of this example for teaching purposes is undeniable, if used properly. Its significance is connected to the adoption of an analytical way of thinking by the students that can help them to not only memorize isolated fragments but to really understand the topic in depth. As a result, the students should be able to apply a similar analysis to new and unsolved problems and explore unknown physical systems using basic theoretical tools. To move in the right direction, a new example is presented in this paper: the motion of a rigid sphere on an elastic half space. This phenomenon is extremely complicated. However, in this paper it is shown that for a specific range of values, this motion can be approximated to harmonic motion. This fact will significantly help students to learn how to resolve unknown problems and tasks using their knowledge. In addition, the analysis provided by this paper can be used as an alternative and simpler solution for data processing in nanoindentation experiments. As a result, it can be used as a valuable tool not only for undergraduate physics students but also for young researchers in the fields of nanotechnology and materials science.

Revisiting the theory behind AFM indentation procedures. Exploring the physical significance of fundamental equations

“…Firstly, the sample that is being indented should be flat, homogeneous and isotropic and the indenter should be significantly smaller compared to the sample's dimensions (in other words the sample should behave as an elastic half space. An elastic half space is an elastic material that extends infinitely in all directions including the depth, with the surface at the top as boundary [39]). Secondly, the contact between the sample and the indenter should be adhesionless and frictionless.…”

Exploring the non-linear oscillation of a rigid sphere on an elastic half-space

“…The aforementioned motion is described by a second-order nonlinear ordinary differential equation that presents significant complexity [32]. In a previous publication, it was shown that for small displacements, the sphere's motion can be considered as harmonic [32]. However, in this paper, cases with significant displacement range will be explored.…”

“…The problem to be studied in this paper is the nonlinear oscillation of a rigid sphere on an elastic half-space. The aforementioned motion is described by a second-order nonlinear ordinary differential equation that presents significant complexity [32]. In a previous publication, it was shown that for small displacements, the sphere's motion can be considered as harmonic [32].…”

An estimation for the effective force transfer medium in radial loading of the cylindrical and spherical geometries

“…Recently, Zhupanska and Ulikto [26] used a similar approach to solve the contact of a rigid cylinder indenting an elastic half-space, both for non-slipping and slipping conditions with a finite friction coefficient. The oscillatory behavior of the sphere in the half-space has been analyzed by Kontomaris et al [27], which has described an analytical solution in the range of . .…”