A model was developed to predict the effects of wall friction upon the elastic modulus of powder beds. The model combined a Janssen stress distribution with non-linear contact mechanics. Overall bed properties were predicted by integrating strain over the bed depth and relating this to applied stress to find the apparent spring constant of the bed. This model was compared to the experimental data of Yanagida et al., for particulate materials subject to low-magnitude vibration. These data cover a range of particulate materials in test cells ranging from 0.0345 to 0.15 m in diameter. The model gave excellent qualitative correspondence with the experimental data, showing the following points: • for shallow beds, first peak frequencies are independent of cell diameter and only dependent upon cell depth, • for intermediate bed depths, first peak frequency is dependent upon both bed depth and cell diameter, • there are indications that, for deeper beds, first peak frequency may be independent of bed depth and only dependent upon cell diameter. Quantitative comparisons were made with the experimental data by three methods, with two or three fitted parameters from the model. One method gave good quantitative agreement with physically realistic values of the fitted parameters. This method determined an elastic parameter from shallow bed data, where first peak frequency is proportional to bed depth to the power −5/6. Two stress distribution constants were then determined by a least squares of error method on the data.
Matchett, A. J., & Yanagida, T. (2003). Elastic modulus of powder beds - the effects of wall friction: a model compared to experimental data. Powder Technology, 137(3), 148-158. https://doi.org/10.1016/j.powtec.2003.08.049