The atomistic and the continuum theories of crystal elasticity

Abstract

The lattice theory of crystal elasticity is developed with special reference to the basic postulates of the lattice models, the role of initial stresses and the validity of the rotational invariance condition. The elastic behaviour of the infinite lattice is investigated theoretically by the method of homogeneous deformation and by the method of long waves. The exact generalization of the classical theory, viz., the existence of surface couples, is deduced from these considerations. It has also been proved that the presence of volume couples does not alter the structure of the classical theory. A complete correspondence between the lattice and the continuum approaches is exhibited. Some subtle considerations, regarding the extension of the continuum concepts of strains and stresses to atomic lattices, and the clear distinction between the initial stress and the natural stress problems are also pointed out.