On the microscopic foundations of elasticityI. Goldhirsch1 and C. Goldenberg2
1 Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
2 School of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel
(Received 18 March 2002 and Received in final form 29 May 2002 / Published online: 17 December 2002)
The modeling of the elastic properties of disordered or nanoscale solids requires the foundations of the theory of elasticity to be revisited, as one explores scales at which this theory may no longer hold. The only cases for which microscopically based derivations of elasticity are documented are (nearly) uniformly strained lattices. A microscopic approach to elasticity is proposed. As a first step, microscopically exact expressions for the displacement, strain and stress fields are derived. Conditions under which linear elastic constitutive relations hold are studied theoretically and numerically. It turns out that standard continuum elasticity is not self-evident, and applies only above certain spatial scales, which depend on details of the considered system and boundary conditions. Possible relevance to granular materials is briefly discussed.
46.25.Cc - Static elasticity: Theoretical studies.
61.43.-j - Disordered solids.
62.25.+g - Mechanical properties of nanoscale materials.
83.80.Fg - Granular solids.
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag 2002