Buckling of residually stressed cylindrical tubes under compression

We evaluate the loss of stability of axially compressed slender and thick-walled tubes subject to a residual stress distribution. The nonlinear theory of elasticity, when used to analyze the underlying deformation, shows that the residual stress induces preferred directions in the reference configuration. The incremental theory, given in Stroh form, is used to derive an exact bifurcation condition. The critical stretch and the associated critical buckling mode are identified for axisymmetric and asymmetric increments in the deformation. Mode transitions are illustrated as the tube slenderness varies. For slender tubes, Euler buckling is energetically favorable, and the effect of residual stress is negligible. However, for short and thick-walled tubes where barreling mode is dominant, the residual stress significantly affects the buckling behavior and may eliminate barreling instability. We show that, depending on its magnitude and direction, residual stress can either accelerate or delay instability. Phase diagrams for various modes are obtained and provide insight into pattern selection across different tube geometries.