Inverse variational principle in finite elasticity

Abstract

Although there is a vast literature on the variational principles of the theory of nonlinear elasticity, some basic problems in this field are still open (cf. e.g. [1]). General speaking, almost all the variational principles were studied in the reference configuration ( i.e. undeformed configuration ). Since the loaded boundary is unknown, the well-known dead loading condition should be assumed in this kind of variational principle. Otherwise, the configuration-dependent loading should be considered [2]. In my opinion, the correct description of variational boundary value problem for finite deformation theory is a problem with a movable boundary under the true loading systems [3]. The complete solutions of this problem should include both field functions and deformed configuration. To solve this kind of problem, this paper establishes an Euler-type variational principle in the current configuration by introducing a new variational definition. The critical condition of this principle yields a complete solution of the boundary value problem and the optimal shape condition of the movable boundary. This principle can be applied to problems of optimal shape design.