Helical buckling and postbuckling of pre-stressed cylindrical tubes under finite torsion

Abstract

In this paper we analyze bifurcation and postbifurcation of a finite deformation boundary-value problem for a residually-stressed elastic body using the finite element method. In particular, the torsion of a circular cylindrical tube subject to radial and circumferential residual stresses is studied. The material model is given by a residual-stress dependent nonlinear elastic constitutive law in terms of invariants. Firstly, the numerical methodology able to capture the elastic stability of tubes under torsion is developed without including residual stress. The bifurcation point is initially estimated reducing the original problem to an algebraic eigenvalue problem. A nonlinear analysis is performed to capture the postcritical behavior. Secondly, prestress is introduced. The residual-stress dependent nonlinear elastic constitutive law is implemented in a finite element code. The dependence of bifurcation and postbifurcation behavior of tubes under torsion on residual stresses is illustrated and compared with results when there is no residual stress. Analytical solutions involve a complicated mathematical machinery that deals with specific material models and loads as well as, usually, a perfect geometry. Here, we provide a highly flexible approach that is implemented in a commercial finite element code, which allows this methodology to be easily used on other problems under different loading conditions and geometries. The numerical results obtained for the (thin) tubes are compared with the (analytical) solution for the torsion of an infinitely long circular cylindrical membrane to validate the methodology.