A new arc-length control method based on the rates of the internal and the dissipated energy

The purpose of this paper is to introduce a new arc-length control method for physically non-linear problems based on the rates of the internal and the dissipated energy.

In this paper, the authors derive from the second law of thermodynamics the arc-length method based on the rate of the dissipated energy and from the time derivative of the energy density the arc-length method based on the rate of the internal energy.

The method requires only two parameters and can automatically trace equilibrium paths which display multiple snap-through and/or snap-back phenomena.

A fully energy-based control procedure is developed, which facilitates switching between dissipative and non-dissipative arc-length control equations in a natural way. The method is applied to a plate with an eccentric hole using the phase field model for brittle fracture and to a perforated beam using interface elements with decohesion.