Theory of Elasticity at the Nanoscale

We have shown in a series of recent papers that the classical theory of elasticity can be extended to the nanoscale by supplementing the equations of elasticity for the bulk material with the generalized Young-Laplace equations of surface elasticity. This review article shows how this has been done in order to capture the often unusual mechanical and physical properties of nanostructured particulate and porous materials. It begins with a description of the generalized Young-Laplace equations. It then generalizes the classical Eshelby formalism for nano-inhomogeneities; the Eshelby tensor now depends on the size of the inhomogeneity and the location of the material point in it. Then the stress concentration factor of a spherical nanovoid is calculated, as well as the strain fields in quantum dots (QDs)with multi-shell structures and in alloyed QDs induced by the mismatch in the lattice constants of the atomic species. This is followed by a generalization of the micromechanical framework for determining the effective elastic properties and effective coefficients of thermal expansion of heterogeneous solids containing nano-inhomogeneities. It is shown, for example, that the elastic constants of nanochannel-array materials with a large surface area can be made to exceed those of the nonporous matrices through pore surface modification or coating. Finally, the scaling laws governing the properties of nanostructured materials are derived. The underlying cause of the size dependence of these properties at the nanoscale is the competition between surface and bulk energies. These laws provide a yardstick for checking the accuracy of experimentally measured or numerically computed properties of nanostructured materials over a broad size range and can thus help replace repeated and exhaustive testing by one or a few tests.