This paper revisits continua with microstructure from a geometric point of view. We model a structured continuum as a triplet of Riemannian manifolds: a material manifold, the ambient space manifold of material particles and a director field manifold. Green-Naghdi-Rivlin theorem and its extensions for structured continua are critically reviewed.

This paper studies the invariance of balance of
energy for a system of interacting particles under groups of
transformations. Balance of energy and its invariance is first
examined in Euclidean space. Unlike the case of continuous media,
it is shown that conservation and balance laws do not follow
from the assumption of invariance of balance of energy under
time-dependent isometries of the ambient space. However, the
postulate of invariance of balance of energy under arbitrary
diffeomorphisms of the ambient (Euclidean) space, does yield
the conservation laws. These ideas are then extended to the case
when the ambient space is a Riemannian manifold. Pairwise