Self-Amplification of Solid Friction in Interleaved Assemblies

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 selfsimilar master curve with one dimensionless amplification parameter. In addition to solving a longstanding 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.Many of us are familiar with a classical demonstration of the strength of friction: take two phonebooks, interleave their sheets and try to separate them by pulling on their spines. This demonstration has been carried out spectacularly by attempting to pull the books apart with people, trucks, lifting a car [1], and even two military tanks [2], only to fail and suggest that the inner friction between these sheets prevails. The simple explanation often given is that gravity provides the normal force that generates the tangential friction, but this hypothesis is easily proven to be wrong as there is no discernible difference between such an experiment carried out in the vertical or horizontal direction. In this Letter, we study the force needed to separate two books as a function of the number of sheets, the thickness of the sheets, and the interleaving distance. In particular, we show that the force required to separate the books increases abruptly with the number of sheets. The strength of the system is due to the operator: the person, car, truck, or tank, amplifies any small friction arising from the normal force acting on the boundaries of the stack. We present a simple model that captures all the data into a self-similar master curve. The model depends on one single dimensionless amplification parameter, and thus gives insight into the mechanisms at play in this deceivingly complex system. In addition to solving a long-standing familiar enigma related to the classical problem of friction, this model system provides a framework within which one can accurately measure friction forces and coefficients at low loads, and opens the way to the technologically-relevant engineering of friction in complex assemblies from the macro to the nanoscale.The first-known systematic studies of friction were carried out five centuries ago by da Vinci [3, 4] who discov- * frederic.restagno@u-psud.fr ered basic rules which were later confirmed by Amontons [3]. In particular, these laws establish that the friction force is independent of the contact area and proportional to the applied load during sliding, the proportionality constant being the coefficient of kinetic friction. Coulomb rediscovered these laws and further determined...

Qual é a expressão correta para o trabalho realizado pela força de atrito cinético?

“…A expressão entre parênteses, em (14), é igual a M v cm,cm , e como v cm,cm = 0, segue a igualdade (13) − o que conclui nossa demonstração da igualdade (6), partindo da igualdade (4).…”

“…Não posso deixar de comentar que o tema "atrito" ainda envolve muita investigação científica, e em um nível de sofisticação bastante superior às modelagens que foram discutidas neste artigo. Como apontam Alarcón e colaboradores [4], esforços em micro e nanoescalas e na biologia, durante as últimas décadas, têm resultado no ressurgimento de atividades na área de tribologia. 11 Se for de seu interesse, veja, por exemplo, as referências 5 a 15, na publicação de Alarcón e colaboradores, e artigos relacionados.…”

“…Tais trabalhos envolvem tratamentos teóricos avançados (que vão muito além da fenomenologia), microscopia de força atômica e investigações com nanotubos de carbono e folhas de grafeno, entre outras coisas. No entanto, como comentam Alarcón e colaboradores [4], é notável que embora fenômenos complexos não-lineares possam ser observados em escalas micrométricas, é frequentemente o caso que em escalas de maior comprimento as chamadas leis de Amontons-Coulomb simples (leis AC) com as quais estamos tão familiarizados (como f c = µ c f n , incluindo a independência do valor de f c em relação à velocidade relativa entre as superfícies em contato) são válidas. Nas modelagens apresentadas neste trabalho, nos restringimos a corpos de dimensões macroscópicas, com a suposição da validade das leis AC, e fizemos uso da mecânica newtoniana, apenas.…”

Emergent Strain Stiffening in Interlocked Granular Chains

“…Performing indentation experiments, we observe an exponential growth of the resistance force with the product of the indentation depth by the square root of the number of beads per chain. The first factor is reminiscent of the selfamplification of friction in a capstan [17], as well as in interleaved books [18,19], where the increase in resistance is created by, and proportional to, the force exerted by the operator. The second factor points towards the central role played by inter-chain topological constraints, as in polymer physics [20].…”

“…Previous studies about self-amplified friction in complex assemblies have shown that the frictional resistance essentially grows exponentially with the relevant control parameter, e.g. the angle in a capstan [17], or the Hercules number [18,19] in interleaved books. This general behaviour is intimately rooted in the fact that the increase in the force exerted by the operator is proportional to that force itself and to an effective coefficient of friction induced by the specific geometry.…”