Self-amplification of solid friction in interleaved assemblies (1508.03290v2)

Abstract: It is nearly impossible to separate two interleaved phonebooks when held by their spines. A full understanding of this astonishing demonstration of solid friction in complex assemblies has remained elusive. In this Letter, we report on experiments with controlled booklets and show that the force required increases sharply with the number of sheets. A model captures the effect of the number of sheets, their thickness and the overlapping distance. Furthermore, the data collapse onto a self-similar master curve with one dimensionless amplification parameter. In addition to solving a long-standing familiar enigma, this model system provides a framework with which one can accurately measure friction forces and coefficients at low loads, and that has relevance to complex assemblies from the macro to the nanoscale.

• The paper demonstrates that friction scales nonlinearly, with a tenfold increase in sheets producing a four orders of magnitude rise in traction force.
• It introduces a theoretical model based on Amontons-Coulomb laws and an ordinary differential equation to relate interleaving with frictional amplification.
• The findings offer practical insights for advanced material design and micro-device engineering by controlling friction under varied loads.

Self-Amplification of Solid Friction in Interleaved Assemblies

The research paper titled "Self-Amplification of Solid Friction in Interleaved Assemblies" presents a detailed paper of frictional forces in interleaved assemblies, with a specific focus on the well-known phenomenon of the interleaved phonebook. This classic problem, where two interleaved phonebooks are nearly inseparable without significant force, is explored in depth through controlled experiments that analyze the factors affecting the required separation force, such as the number of sheets, their thickness, and the overlap distance.

Experimental Setup and Observations

The authors conducted experiments using identically constructed books made up of interleaved sheets of paper, held together by clamps and subjected to tensile tests. Force measurements were taken to determine the traction force necessary to separate the books. The paper reports several key observations:

• Nonlinear Force Scaling: The force needed to separate the books does not increase linearly with the number of interleaved sheets. Instead, a tenfold increase in the number of sheets results in a four orders of magnitude increase in the required force.
• Reproducibility and Environmental Factors: Despite potential variations in humidity (30-60%), the experiments showed consistent results, highlighting the robustness of their setup against common environmental fluctuations.

Theoretical Model and Analysis

The paper derives a theoretical model capturing the essence of the observed phenomena, utilizing established Amontons-Coulomb (AC) laws of friction. AC laws dictate that frictional force is independent of contact area but proportional to the applied load during sliding. The authors introduce a dimensionless amplification parameter that condenses their observations into a self-similar master curve. This equation relates the traction force to frictional amplification, solely dependent on the amplification parameter.

• Equation Derivation: The model employs a simple ordinary differential equation (ODE) that describes the transformation of traction forces into resisting normal loads within the interleaved sheets. This ODE predicts a proportionate increase in traction with increased interleaving, a factor verified by their experimental results.
• Effective Coefficient of Kinetic Friction: The model assumes an average coefficient of kinetic friction for each setup. Interestingly, the research indicates that the coefficient of friction varies inversely with load, consistent with previously reported material behavior, where adhesion forces increase friction under decreased normal loads.

Implications and Future Research

The implications of the research extend beyond a mere resolution of a longstanding demonstrative trick. Understanding the self-amplification of friction in such assemblies holds potential for manipulating friction in practical applications, from micro-devices to industrial engineering solutions involving layered structures. The insights into load-induced friction changes are particularly relevant for developing advanced materials and understanding tribological behaviors at micro and nano scales.

The paper opens pathways for future research to explore friction's characteristics at smaller scales or explore potential variations in materials and environmental conditions. It also suggests possible applications of similar frictional principles in new domains, such as biometric sensing or robotic manipulation, where controlled friction is crucial.

In conclusion, the paper elucidates the underlying mechanics of friction in interleaved assemblies through both experimental and theoretical lenses, presenting a comprehensive account that leverages fundamental tribological principles to address complex frictional phenomena in a structured manner.