Geometrically-based consequences of internal constraints

When a body is subject to simple internal constraints, the deformation gradient must belong to a certain manifold. This is in contrast to the situation in the unconstrained case, where the deformation gradient is an element of the open subset of second-order tensors with positive determinant. Commonly, following Truesdell and Noll [1], modern treatments of constrained theories start with an a priori additive decomposition of the stress into reactive and active components with the reactive component assumed to be powerless in all motions that satisfy the constraints and the active component given by a constitutive equation. Here, we obtain this same decomposition automatically by making a purely geometrical and general direct sum decomposition of the space of all second-order tensors in terms of the normal and tangent spaces of the constraint manifold. As an example, our approach is used to recover the familiar theory of constrained hyperelasticity.