Measurement of granular contact forces using frequency scanning interferometry
The propagation of stress within a granular material has been studied for many years, but only recently have models and theories focused on the micromechanical (single grain) level. Experiments at this level are still rather limited in number. For this reason, a system using optical techniques has been developed. The substrate on which the granular bed is assembled is a double layer elastic substrate with high modulus epoxy constituting the top layer and silicone rubber as the bottom layer. In between the two layers, gold is coated which acts as a reflective film. To design the substrate, a Finite Element Analysis package called LUSAS was used. By performing a non-linear contact analysis, the design of the substrate was optimised so as to give a linear response, high stiffness, deflection in the measurable range, and negligible cross-talk between neighbouring grains. Fabrication and inspection techniques were developed to enable samples to be manufactured to this design. The deformation of the gold interface layer is measured using interferometry. The interferometer utilised a frequency tunable laser which acts both as the light source and the phase shifting device. The optical arrangement is based on the Fizeau set-up. This has removed several problems such as multiple reflections and sensitivity to vibration that occurred when using a Mach-Zehnder configuration. A fifteen-frame phase shifting algorithm, was developed based on a Hanning window, which allows the phase difference map to be obtained. This is then unwrapped in order to obtain the indentation profile. The deflection profile is then converted to a single indentation depth value by fitting a Lorentzian curve to the measured data. Calibration of the substrate is carried out by loading at 9 different locations simultaneously. Spatial and temporal variations of the calibration constants are found to be of order 10-15%. Results are presents showing contact force distributions under both piles of sand and under face-centred cubic arrangements of stainless steel balls. Reasonable agreement was obtained in the latter case with both the expected mean force and the probability density function predicted by the so-called 'q' model. The experimental techniques are able to measure small displacements down to a few nanometers. To the best of my knowledge these experiments are the first to employ the interferometer method in attempting to measure the contact force distribution at the base of a granular bed.