Chen, Y., Lee, J.D. and Eskandarian, A. (2002) “Applicability analysis of continuum theories from viewpoint of phonon dispersion relations”, US 14th Congress of Theoretical and Applied Mechanics, June 22-27, Blacksburg, Virginia, USA.

Abstract: The dynamics of atoms in crystals - lattice dynamics - are basic to many fields of study in the solid state, and has been characterized by phonon dispersion relations, which is an essential feature representing the atomic structure and atomic motions in crystal. Phonon dispersion relations can be calculated from atomic force by the discrete atomic model. The analog of atomic force is the constitutive relations in the continuum theory, which serve to distinguish one material from another. It follows that the phonon dispersion relations can be also calculated through the constitutive relations of a material by a continuum model if the model can manifest the dynamics of atoms in crystals, and hence provides a bridge to link the interatomic force to the constitutive relations. This also provides a chance for the continuum field theories to be examined on the region of applicability and validity from the viewpoint of phonon dispersion relations.

The phonon dispersion relations based on the classical continuum theory, Eringen’s micromorphic continuum theory and micropolar theory, Mindlin’s microstructure elastic theory, Mindilin and Toupin’s couple stress theory, Cosserat’s theory, strain gradient theories and nonlocal theories have been presented. The results have been compared and analyzed. The applicability of each continnu field theory has been discussed. It is seen that (1) the classical continuum theory is only valid for very long wave-length problems, (2) the couple stress theory, strain grdient theories and nonlocal theories do not stem from the consideration of microstructure and micromotion. On the other hand, both the atomic structure and the dynamics of atoms have been manifested in micromorphic theory and micropolar theory, which therefore dramatically extend the application region of continuum theories to the microscopic length and time scales.