Bridges, Ronald 1980- (2011-08). Implicit Rate-Type Models for Elastic Bodies: Development, Integration, Linearization and Application. Doctoral Dissertation.
In this dissertation, we use the second law of thermodynamics to find restrictions on the Gibbs potential so that there can be no dissipation associated with the deformation, in any process. We find that the use of the Gibbs potential leads to a much richer class of models than traditional elasticity wherein the Helmholtz potential is utilized, and the relationship between the stress and stretch is governed by a tensor-valued rate equation. For the special case of an isotropic body, we obtain a solution to this rate equation and after a linearization process, show that such models offer the possibility of the stress "blowing up," while the strain remains finite which is entirely consistent with the theory; such is not possible in traditional elasticity. This possibility has a wide array of applications including fracture in metals and delamination of composite bodies. A boundary value problem is proposed and solved numerically, for a particular Gibbs potential, in order to illustrate the efficacy of the framework. More applications such as boundary value problems within the context of finite elasticity and dissipative mechanisms are also discussed.