The dynamic indentation of a finite elastic plate

The Shore hardness test has been widely used and has become the standard for characterizing materials in the rubber and plastics industries. The obtained hardness scale values, ASTM or IRHD, are often related to the static elastic properties of rubber like-materials. Recently, progress on the problem of obtaining dynamic viscoelastic properties has been made through the use of an oscillating spherical indentor and modeling the measured input impedance by the radiation impedance of a radiating sphere. When the sphere radius is small, such that ka<<l for longitudinal waves, then the measured shear modulus and loss factor is in good agreement with that found using other dynamic methods such as the DMTA. This model is appropriate only for relatively soft materials or very thick hard materials where the thickness of the material can be ignored. The object of the current work is to extend the methodology to layers of finite thickness. The problem is studied under the assumptions of linear lasticity and the condition that there is perfect adhesion between the indentor and the elastic layer. The method of solution that is adopted will be described and as such it constitutes the first serious attempt to solve an elastodynamic boundary value problem of this type.