This study deals with bounds for the effective elastic moduli of granular materials in terms of micromechanical parameters. The case considered is that of two-dimensional isotropic assemblies of non-rotating particles with bonded contacts and a linear elastic contact constitutive relation. Based on variational principles, rigorous upper and lower bounds are obtained for the elastic moduli. To this end, compatible and equilibrated fields are constructed from local characteristics, based on approximate equilibrium and compatibility, respectively. Results of discrete element simulations are used to compare the obtained bounds with the actual moduli. This comparison shows that the actual moduli are narrowly bracketed by these bounds. The corresponding fields of relative displacement and force at the contacts are analysed, showing fairly close agreement with those obtained from the discrete element simulations.
- Variational principles
- Granular materials