On the design of energy–momentum integration schemes for arbitrary continuum formulations. Applications to classical and chaotic motion of shells

Abstract

AbstractThe construction of energy–momentum methods depends heavily on three kinds of non‐linearities: (1) the geometric (non‐linearity of the strain–displacement relation), (2) the material (non‐linearity of the elastic constitutive law), and (3) the one exhibited in displacement‐dependent loading. In previous works, the authors have developed a general method which is valid for any kind of geometric non‐linearity. In this paper, we extend the method and combine it with a treatment of material non‐linearity as well as that exhibited in force terms.In addition, the dynamical formulation is presented in a general finite element framework where enhanced strains are incorporated as well. The non‐linearity of the constitutive law necessitates a new treatment of the enhanced strains in order to retain the energy conservation property.Use is made of the logarithmic strain tensor which allows for a highly non‐linear material law, while preserving the advantage of considering non‐linear vibrations of classical metallic structures. Various examples and applications to classical and non‐classical vibrations and non‐linear motion of shells are presented, including (1) chaotic motion of arches, cylinders and caps using a linear constitutive law and (2) large overall motion and non‐linear vibration of shells using non‐linear constitutive law. Copyright © 2004 John Wiley & Sons, Ltd.