On the bifurcation and global energy approach in propagating plasticity

A bifurcation method is proposed to solve a class of steady-state propagation problems in plastic shells. The method is explained on a simple one-dimensional example of a collapsing underwater pipeline. The pipe is modeled as a rigid–plastic beam/string resting on a plastic strain-softening foundation and is subjected to uniform pressure loading. The unknowns in the problem are the critical pressure for the buckle to propagate and the length and shape of the so-called transition zone. Using the method of local equilibrium closed-form solutions are derived for three different structural models: beam, string, and beam/string. The global formulation is then presented in which the rate of energy associated with the change in the length of the transition zone is included in the balance equations. The solution is obtained as an intersection of the equilibrium paths corresponding to a constant and variable length of the deformation zone. An interpretation of the “Calladine paradox” is offered so that the global energy approach to propagating plasticity remains unchallenged.