Micro-origin of Macro-strength: Friction

Abstract: This paper presents an analytical study about the behavior of arbitrary shaped and sized non-cohesive two-dimensional granular materials. Several mechanical properties and relations are unraveled by connecting micro and macro scales in an explicit fashion that, at the same time, provides the basis of an analytical-theoretical framework for the development of new multi-scale techniques. Furthermore, the work herein presented is based on three main ideas that are developed and connected progressively; namely, the obtention of explicit expressions that enable us to relate micro-scale parameters, such as contact forces and fabric, to stress as a macro (continuum) physical property. Then, with these powerful tools, physical connections and relations between the mentioned micro-parameters and macro-constitutive parameters, in specific, Mohr-Coulomb's mobilized internal friction angle, are established. Finally, a non-linear optimization problem, which includes physical constraints at the contact point level, is proposed and solved in order to find the limit (maximum) internal friction angle in terms of the aforementioned micro-parameters. Thus, throughout this theoretical study, some important features about strength, anisotropy, contact buckling, and non-uniqueness of systems of contact forces are extracted, allowing us to have a deeper insight, as well as, a better understanding of the mechanical behavior of such complex-to-model materials.