A sequentially linear saw-tooth model for interface elements

摘要

The sequentially linear approach in modeling strain softening materials and structures has recently been demonstrated to be effective especially when the ratio between the elastic energy stored and the energy that can be dissipated by fracture is very large, i.e. in the case of real scale structures (Rots & Invernizzi, 2004ab). The model replaces the downward stress-strain curve by a saw-tooth curve, either saw-tooth tension-softening for unreinforced material or saw-tooth tension stiffening for reinforced material. A linear analysis is performed, the most critical element is traced, the stiffness and strength of that element are reduced according to the saw-tooth curve, and the process is repeated. The sequence of scaled critical steps provides the global load-displacement response. In the present paper, the approach is implemented for a particular class of finite elements useful in describing discrete rather that smeared fracture, namely the so-called interface elements. These elements do not show the typical mesh dependency that affects continuous smeared crack approaches, because the softening tail is expressed in terms of stress versus displacement rather than stress versus strain; nevertheless, regularization is necessary since the softening curve is discretized with a finite number of teeth. The regularization criterion is the same followed in the continuous case in order to keep the energy dissipated in the softening regime independent from the number of teeth used in the discretization. Analogously to the continuous case, the best approach is to uplift the softening tail updating both the ultimate strength and strain. Some different choices for the saw-tooth discretization rule are described, and closed form solutions are provided, when possible, in the case of linear softening. A group of beams with different mesh size is analyzed to prove the independence of the results from the mesh-size and number of teeth. Moreover, a dog bone specimen is considered to emphasize the ability of the method to capture snap-back and to overcome bifurcation points.