Generalization of load–unload and force-release sequentially linear methods

Two different non-iterative solving approaches for mechanical systems composed of elements with saw-tooth constitutive laws have been described in the literature. The first algorithm called the load–unload ( L–U) method removes the whole external load after every rupture event and linearly increases it again from zero until the next rupture event occurs. The second algorithm, called the force-release ( F-R) method, keeps the load unchanged after a rupture and redistributes the stresses from damaged elements until an equilibrium state has been reached. Then, the load may increase again. It is known that these two algorithms yield different results with respect to crack paths and force–deflection curves, even for proportional loading paths. It will be shown that these two algorithms can be taken as extreme cases of a general approach that is developed in this paper. The general method is based on the simultaneous redistribution of stresses from damaged elements together with the scaling of external load. Both the L–U and F-R algorithms can be reproduced by the general method and thus compared within one framework. The general method also allows the tracing of snap-backs and enables the simulation to be controlled indirectly, e.g. via crack mouth opening displacement. It is applicable to both proportional and non-proportional load paths.