Finite element formulation of hyperelastic plane frames subjected to nonconservative loads

Abstract

A finite element formulation for finite deformation static analysis of plane hyperelastic frames subjected to nonconservative loads is presented. A rubber-like material is considered for which the behaviour in tension and in compression differs substantially. A new proposal for the strain energy function of rubber at uniaxial stress state is given, convenient for the present deformation analysis. The finite element formulation is based on a new variational principle of the Hu-Washizu type where exact nonlinear kinematic equations of one-dimensional finite-strain beam theory are taken into account. The contribution of the shear deformations to the total potential energy and the initial curvature of the beam are neglected. The functional of the variational principle is expressed in terms of only one function, the rotation of the cross-section of the beam. Thus only one function needs to be approximated in the functional in the finite element implementation of the variational principle. The outstanding accuracy and high efficiency of the method are illustrated by numerical examples. The application of the present method for the analysis of hyperelastic frames subjected to static nonconservative forces is shown, and some results for critical loads for the dynamic instabilities in the form of flutter are given.