Granular drag and lift force on a flexible fiber

TL;DR

This work investigates how a flexible fiber experiences drag and lift while traversing a static granular bed using discrete element simulations. It identifies a steady-state regime after large displacement and introduces two geometric scales, $\\lambda$ and $R_g$, along with a single elastogranular parameter $\\chi$ that governs the deformation and force response. The authors derive scaling laws for drag, $F_d = C_d\\rho g h \\lambda d$, with a size-corrected coefficient, and lift, $F_l = C_l\\rho g h \\dfrac{\\lambda^2}{R_g} d$, with $C_l = 2$, demonstrating data collapses onto master curves across fiber geometries. Ultimately, all forces and shapes collapse onto $\\chi$-controlled master curves, offering a compact predictive framework for elastogranular interactions and guiding future 3D extensions and experimental validation.

Abstract

In this work, we investigate the forces acting on a flexible fiber dragged through a granular bed. Using discrete element simulations, we observe that, after a sufficiently large displacement, the system reaches a steady state in which both the fiber's shape and the forces acting on it become, on average, constant. Under these conditions, we identify two characteristic lengths that describe the fiber's shape and propose unique scaling laws for the drag and lift forces, valid across a wide range of fiber flexibilities, from highly deformable to nearly rigid, based on these lengths. We highlight that the fiber-grains interaction is governed by a single dimensionless elastogranular parameter, defined as the ratio of the fiber's elastic properties to the granular pressure. Finally, we demonstrate that both the forces and the characteristic lengths can be expressed solely as functions of this dimensionless parameter. Our findings offer a fundamental insight into the behavior of a flexible fiber interacting with a granular medium.

Figures (9)

• Figure 1: (a) Typical geometry of the numerical setup. The granular bed has dimensions $L_x\times L_y$. The fiber underformed length is $L_f$. The fiber is dragged inside the bed with a constant velocity $v$ along the $x$-direction. (b) Scheme of the deformed fiber with indication of the projections on the drag $S_x$ and orthogonal-to-drag $S_y$ directions, the diameter $d_f$ of the fiber and the distance between two nodal particles $l_f$. (c) Sketch of the configuration with a top pressure (top) and a thicker granular layer (bottom).
• Figure 2: (a) Instantaneous drag $F_d$ and lift $F_l$ force on the fiber as a function of the displacement $\Delta x$ (fiber length $L_f=20.5d$ and elastogranular parameter $\chi\approx0.2$ as defined in Eq. \ref{['eq_:chiParam']}). The shaded region indicates the region of interest (ROI) where steady state values are computed. At the top of the plot we show the evolution of the shape of the fiber at some steps of the drag test (displacement step of $2.5d$). (b) Typical steady shape of a fiber of length $L_f=20.5d$ for different values of the bending stiffness (the associated elastogranular parameter $\chi$ is reported under each image). Images are a superimposition of instantaneous geometrical configurations of the fiber in the range $40d \leq \Delta x \leq 60d$ (displacement step of $0.05d$).
• Figure 3: (a) Drag force as a function of the characteristic length $\lambda$. Data collapse on the dashed line which corresponds to Eq. \ref{['eq:eq_fd']}. Inset: Comparison with data obtained by changing: top pressure $P_{top}=30$-$75 \rho g d$ ($\square$), burial depth $h=47d$ ($\triangleleft$), gravity $2g$ ($\lozenge$), drag velocity $v=0.1$-$1\sqrt{gd}$ ($\triangleright$). (b) Drag coefficient $C_d$ as a function of the ratio $\lambda/d$ (data are fitted by Eq. \ref{['eq:eq_cd']}, $C^{\infty}_d=3.8$, $\alpha_0=2.6$, $\tilde{\lambda}=2$). The diamond markers ($\blacklozenge$) indicates the drag coefficient obtained for a disks of different diameter rescale by the ratio $C^{\infty}_d/C^{\infty}_{d,disk}\approx2.1$. Inset: Effect of the friction coefficient $\mu_c$ on the drag coefficient ($\mu_c=0.1,~0.2,~0.4,~0.6$). The legend applies to both figures.
• Figure 4: Lift force as a function of $\lambda^2/R_g$. Data collapse on the dashed line which corresponds to Eq. \ref{['eq:eq_fl']} (with $C_l=2$). Inset: Comparison with data obtained changing the main system parameters. Markers are the same as those used in the inset of Fig \ref{['fig:fig2']}a.
• Figure 5: (a) Characteristic length $\lambda$ as a function of the elastogranular parameter $\chi$. Dashed line corresponds to Eq. \ref{['eq:eq_lamChi']} ($\alpha_{\lambda}=0.85$, $\chi_{_{\lambda}}=0.17$). Three steady-shapes typical of the elastogranular regimes are displayed inside the plot. The circular filled area around the fibers has radius $R_g$ and is centered on the center of mass of the deformed fiber. (b) Gyration radius $R_g$ as a function of $\chi$. Dashed line corresponds to by Eq. \ref{['eq:eq_RcChi']} ($\chi_{_{R_g}}=0.13$) Inset: Ratio $\lambda^2/R_g$ as a function of the rigidity parameter $\chi$. Markers are the same as those used in the inset of Fig \ref{['fig:fig2']}a for all the figures.