Balance Laws in Continua with Microstructure

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. We show that when the ambient space is Euclidean, postulating a single balance of energy and thinking of microstructure manifold as the tangent space of the ambient space manifold, postulating energy balance invariance under time-dependent isometries of the Euclidean ambient space, one obtains conservation of mass, balances of linear and angular momenta but not a separate balance of linear momentum. This will be associated with the rigid structure of Euclidean space. We develop a covariant elasticity theory for structured continua by postulating that energy balance is invariant under time-dependent spatial diffeomorphisms of the ambient space, which in this case is the product of two Riemannian manifolds. We then introduce two types of constrained continua in which microstructure manifold is linked to the reference and ambient space manifolds. We show that when at every material point the microstructure manifold is the tangent space of the ambient space manifold at the image of the material point, covariance gives us balances of linear and angular momenta with contributions from both forces and micro-forces and two Doyle-Ericksen formulas. We show that a generalized covariance can lead to two balances of linear momentum and a single coupled balance of angular momentum. We then covariantly obtain the balance laws for two specific examples, namely elastic solids with distributed voids and mixtures. Lagrangian field theory of structured elasticity is revisited and a connection is made between covariance and Noether's theorem.