Anomalous Enhancement of Yield Strength due to Static Friction

Abstract

Friction is fundamental to mechanical stability across scales, from geological faults and architectural structures to granular materials and animal feet. We study the mechanical stability of a minimal friction-stabilized structure composed of three cylindrical particles arranged in a triangular stack on a floor under gravity. We analyze the yield force, defined as the threshold compressive force applied quasi-statically from above at which the structure collapses due to sliding at the floor contact. Using singular perturbation analysis, we derive an expression which quantitatively predicts the yield force as a function of the static friction coefficient and a small dimensionless parameter $ε$ characterizing elastic deformation.

Figures (8)

• Figure 1: (a) Schematic of the system: three frictional cylinders, with a cylinder-floor friction coefficient $\mu$ are stacked under gravity and compressed from above by an external force $f$ applied to the top cylinder. (b) Contact forces $a,b,c,d$, and $e$ act between the cylinders and the floor. The $x$ and $y$ axes indicate the horizontal (along the floor) and vertical (normal to the floor) directions, respectively. The unit normal and tangential vectors are defined at the contact point between cylinders.
• Figure 2: (a) Dependence of the yield force on the friction coefficient $\mu$. Comparison among the rigid-body result (purple curve), DEM simulations (symbols), and perturbative solutions (solid curves) for $k_n r / mg= 10^2$, $10^3$, and $10^4$. (b) Scaling of the yield force with respect to the dimensionless stiffness $k_n r / mg$ across the transition point, shown for $\mu = 0.25$, $\mu_c$, and $0.33$. Data symbols represent DEM simulation results, and solid curves indicate perturbative solutions.
• Figure 3: Displacement of the bottom cylinders under an applied force $f$. The bottom cylinders move vertically by a displacement $z$.
• Figure 4: Displacement variables under an applied force $f$. Cylinder 1 undergoes vertical displacement $\delta_{11}$, while cylinder 2 displaces vertically by $\delta_{21}$, horizontally by $\delta_{22}$, and rotates by an angle $\theta$.
• Figure S1: Preparation protocol for the initial state. The stacking procedure consists of three steps: (1) inward displacement of the walls by a distance $A$; (2) release of the top cylinder at a constant velocity $v_{\mathrm{fall}}$; and (3) retraction of the walls at a constant velocity $v_{\mathrm{wall}}$.