An energy–momentum integration scheme and enhanced strain finite elements for the non-linear dynamics of shells

Abstract

The paper is concerned with a dynamic formulation of shells and the development of a corresponding robust energy–momentum integration scheme within the framework of enhanced finite elements. Energy–momentum schemes preserve, by design, specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the system and the applied forces allow to. In a previous work, an energy–momentum scheme was developed by the authors which enjoys the feature of being applicable to any shell theory whatever the non-linearity in the strain–displacement relations may be. The method goes beyond a formulation by Simo and Tarnow which applies only when the non-linearity is of quadratic nature. In this paper, we build up on previous work and extend the formulation to encompass enhanced strain finite elements frequently used in structural analysis. The shell formulation used is characterized by seven degrees of freedom and the non-linearity in the strain–displacement relations is of cubic nature. Various examples of non-linear shell dynamics including free large overall motion and non-linear vibrations in conjunction with cases of dynamic stability are considered.