Variational Characterization of a Quasi-rigid Body

Abstract

The rigidity of a body usually is characterized by the kinematical assumption that the mutual distance between any two of its particles remains unaltered in any possible deformation. However, from this alone nothing can be said about the internal contact forces exerted between adjacent sub-bodies. Therefore, the determination and form of an internal state of stress for a rigid body is problematical. Here, we will show that by considering such a kinematical characterization as an internal constraint for an elastic body, the constrained body inherits the mechanical structure of the elastic parent theory, i.e., the internal constraint generates an associated set of Lagrange multiplier fields which can be interpreted as an internal constraint reaction pseudo-stress field with the same structure as the state of stress in the parent elastic body. Thus, although the final deformation is the same for both the rigid body and the rigidly constrained elastic body, the latter corresponds to a richer model and, to emphasize this distinction, we refer to it as a quasi-rigid body. While in equilibrium the pseudo-stress field of a quasi-rigid body will satisfy equations identical to the equilibrium equations for the stress field in the elastic parent theory, such equations are not, in general, sufficient to assure uniqueness. In order to overcome this indeterminacy, we consider the quasi-rigid body as the limit of a sequence of deformable bodies, where each member of the sequence is identified by a material parameter such that, as this parameter tends to infinity, the body to which it refers is rigidified. Our approach is variational, i.e., we consider a sequence of minimization problems for hyperelastic bodies whose elastic strain energy is multiplied by a penalty term, say 1/ε . As ε→ 0, body distortions are more and more penalized so that the sequence of the displacement fields tends to a rigid displacement field, whereas the sequence of the associated stress fields tends to a definite non-zero limit. It will be shown that among all pseudo-stress fields that satisfy the equilibrium equations for the quasi-rigid body, the unique limit of the sequence as ε→0 minimizes a functional analogous to the complementary energy functional in classical linearized elasticity. This result permits its unique determination without having to consider the whole sequence of penalty problems.