On the search for functionals of variational principles with applications in elasticity

Abstract

This thesis involves the calculus of variations and explores a method of construction for functionals of variational principles, particularly in the area of static elasticity. The background material in small displacement theory of static elasticity is briefly presented. The general idea and historical development of the calculus of variations are then briefly reviewed. A new precise definition of variations is proposed in connection with the ∂-operator which is introduced. The consequences of this definition and the properties of variations are discussed in detail. These matters are often not carefully examined by other authors. A more powerful basic lemma is developed so that the choice of the test functions becomes more flexible. The calculus of variations is applied to problems in static elasticity, which are classified according to different variational constraints. The variational process with zero complement is recommended for some particular cases. A method for finding functional of variational principles is developed. Fifty one functionals for distinct principles are then derived. These include the functionals of the Hellinger-Reissner principle, the Hu-Washizu principle, and all other important functionals of variational principles in static elasticity.