**Geometrically-based Consequences of Internal Constraints**

Donald E. Carlson

Department of Theoretical and Applied Mechanics

University of Illinois at Urbana-Champaign

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, 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.

This reports on joint work with Eliot Fried and Daniel
A. Tortorelli, which may be found in full in the *Journal of Elasticity*
**70**: 101-109, 2003.