Axisymmetric indentation of an elastic thin plate by a rigid sphere revisited

Abstract

A simple analytical model based on a Kerr‐type differential relation is proposed to study the indentation problem of a circular elastic thin plate indented by a rigid sphere. Unlike some existing methods which matched two solutions inside and outside the contact zone, the present method is based on a simple differential relation between contact pressure and the normal deflection of the pressured surface of elastic plate, which holds both inside and outside the contact zone and makes it possible to analyze both zones by a single governing differential equation. Explicit relations between the contact‐zone radius, indentation displacement and indentation load are derived for different boundary conditions. The proposed model is validated by good agreement between the predicted results and some known data available in existing literature. In particular, the present model confirms that the indenter will lose contact with the plate around the center of the plate as the indentation load exceeds a certain critical value, and then the contact zone becomes to an annular zone. Detailed analytical and numerical results beyond existing literature are demonstrated for the annular contact zone regarding the critical contact‐zone radius, the width of annular contact zone and the associated contact pressure distribution.

Adhesion of an elastic sphere on a tensioned membrane

“…Jennings et al [24] and Wan et al [25]). On the other hand, an equilibrium state characterized by full contact between the sphere and the membrane inside the circular zone ( a ≥ r ) does not exist when W = 0 but D > 0, consistent with the known fact that indentation of a rigid sphere on an elastic thin plate with non-zero bending rigidity ( D > 0) and vanishing thickness will cause a circular annular contact zone, rather than a solid circular contact zone (see Li et al [26]).…”

“…0, consistent with the known fact that indentation of a rigid sphere on an elastic thin plate with non-zero bending rigidity (D . 0) and vanishing thickness will cause a circular annular contact zone, rather than a solid circular contact zone (see Li et al [26]).…”

“…Apparently, Equation (A-13) reduces to (26) when the tension force T is much larger than the adhesion energy W.…”

Analytical Study of a Circular Thin Plate Contacting with an Elastic Sphere

“…Consequently, geometrical nonlinearity has to be considered, and detailed analysis need to be performed. Many researchers have performed theoretical, numerical, and experimental studies on the behavior of thin plates subjected to impulsive loadings [9][10][11][12][13][14][15][16][17][18][19]. This research primarily focused on the behavior of thin plates subjected to impulsive loadings.…”

A Kerr‐type elastic foundation model for the buckling analysis of a beam bonded on an elastic layer

“…[36] To obtain a precise critical buckling force theoretically, it is essential to find a suitable elastic foundation model which is able to accurately describe the mechanical response of the elastic layer. Therefore, the Kerr-type model, [22] which has been employed to study the indentation [37][38][39][40] and the wrinkling [41] of an elastic film, is developed in this work. A major advantage of the Kerrtype model over the classical linear elastic theory, is that the Kerr-type model provides a simple linear differential relation between the pressure and the normal deflection on the surface of the elastic layer, and pays little attention to the complicated stress distribution inside the layer.…”