A continuum model of drag and lift forces for inclined planes dragged through granular beds
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ORIGINAL PAPER
A continuum model of drag and lift forces for inclined planes dragged through granular beds Hong Guo1,2 · Jiangtao Fu1 · Rui Guo1 · Qian Xu1 · Hong Jiang1 · Nengyuan Chen2 Received: 9 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Both drag and lift forces impact an inclined plane when it is dragged through a granular bed. In this paper, the following results have been obtained: the drag and lift forces grow with the velocity of motion; when the immersion depth is constant, the inclination angle has no effect on drag force, however, the lift force increases linearly with this inclination angle. In order to describe this physical process macroscopically, a continuum wedge model based on the Coulomb model is established to predict drag and lift forces. Particularly, the dynamic friction angle in the assumed shear band is predicted as a function of both inclined angle and moving velocity.
* Hong Guo [email protected] 1
School of Civil Engineering and Architecture, Shaanxi University of Technology, Hanzhong 723001, China
China Electronic Research Institute of Engineering Investigations and Design, Xi’an 710054, China
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Vol.:(0123456789)
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Page 2 of 10
H. Guo et al.
Graphic abstract
Keywords Granular flow · Drag and lift forces · Discrete element method
1 Introduction Much of the matter on the surface of the earth is granular, such as sand, soil, snow etc. Granular materials can be found in geotechnical, industrial and agricultural processes. Activities such as mixing, plowing, skiing, and excavation all involve motion through a granular material [1]. Drag force is the main focus of rigid bodies dragged through granular media. We have already investigated the drag force on vertical plates [2], however, an inclined plate also experiences a lift force when it is dragged through a granular material, making the mechanical properties more complicated. Figure 1a shows a vertical plate, of unit width and immersion depth Z , moving horizontally with a small displacement x through granular matter that has volumetric density 𝜌∗ . A wedge is created that slides along the shear band with a dip angle θ, which resembles a mass sliding along an inclined
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plane as shown in Fig. 1b. In addition to the drag force Fd , the forces acting on the wedge are gravity, mg , the normal force, N , and the frictional resistance, 𝜇N , see Fig. 1c. The pushed wall is a inclined plate with a width of L,as shown in Fig. 1d. Figure 1e shows all the forces applied on the wedge, whereFl is the lift force applying on the wedge. The drag force applied on the vertical plate [2] can be obtained by extending Coulomb’s model, in which a passive yielding force can be determined through limit equilibrium, and can be described as
Fd = mg
sin 𝜃 + 𝜇 cos 𝜃 cos 𝜃 − 𝜇 sin 𝜃
(1)
where 𝜃 is the angle of inclination, 𝜇 is the frictional coefficient and m = 21 𝜌∗ Z 2 L∕ tan 𝜃 is the mass of the wedge [3]. When passive yielding occurs within the granular matter in front of the adva
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