Extending the Boussinesq model for impacts in granular media
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Extending the Boussinesq model for impacts in granular media Francisco Martínez1 · María P. Urrea1 · Claudia M. Gonzalez1 · Germán Varas2 Received: 27 April 2020 / Accepted: 1 October 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Granular impact phenomena have implications in many areas. A fundamental problem deals with the mechanical response of a granular medium under external loads, such as those derived from the impact of a rigid object on the top of the medium. Here, the load changes its magnitude and intensity during the impact, being also applied in much shorter intervals than those of a static load, where the Boussinesq model is usually valid. This dynamic has been poorly addressed in the literature, a void to which this study aims to contribute, where we measure the pressure distribution transmitted at the bottom of a uniform, dry granular layer under the action of gravitational impacts of a steel sphere of fixed diameter. Exploring different bed thicknesses and drop heights, it is found that the structure of this distribution follows a similar form to the Boussinesq model, which was proposed initially for static conditions. This surprising result opens up several questions and future research challenges that could help validate or refute this model in other scenarios. Keywords Granular impact · Mechanical response · Pressure distribution
1 Introduction Impact phenomenon has acquired a great interest in the last decades, due to its implications in different disciplines. In the field of geology, the study of this topic has allowed to extract information about Earth’s data [1]. In soil mechanics, impacts are typically involved in compaction processes performed for the improvement of soil quality, its stability, and bearing capacity [2]. From the physical point of view, the study of impacts has allowed getting insight about properties of frictional resistance of granular media, their rheological * Francisco Martínez [email protected] María P. Urrea [email protected] Claudia M. Gonzalez [email protected] Germán Varas [email protected] https://fis.ucv.cl/gvaras/ 1
Escuela de Ingeniería Civil, Pontificia Universidad Católica de Valparaíso (PUCV), Av. Brasil 2147, Valparaiso, Valparaiso, Chile
Instituto de Física, Pontificia Universidad Católica de Valparaíso (PUCV), Avenida Universidad 330, Valparaiso, Chile
2
behavior, and the mechanical response issue from the application of external loads. However, very few studies have addressed this topic. Boussinesq’s model was a pioneer attempt to propose a general law describing the mechanical response in an inner point (r, z) (Fig. 1b) of a semi-infinite, homogeneous, linear and isotropic elastic medium under a single point-force in a static case,
3F 1 P(r, z) = 2𝜋 z2
[( ) ]−5∕2 2 r F +1 = 2 IB (r, z), z z
(1)
where F is a point-force applied at the top, and IB is the Boussinesq stress coefficient dependent on the coordinates of the stress point [3, 4]. Different experi
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