Saw-tooth softening/stiffening model

A sequentially linear saw-tooth continuum model which captures the nonlinear response via a series of linear steps is presented. In the model, the softening stress-strain curve with negative slope is replaced by a saw-tooth diagram of positive slopes, while the incremental-iterative procedure is replaced by a scaled sequentially linear procedure. Mesh-size objectivity is achieved by adjusting both the peaks and the ultimate strain of the saw-tooth diagram to the size of the finite elements, keeping the fracture energy invariant. First, considering large-scale dog-bone specimens in direct tension, it will be demonstrated that the model is capable of automatically providing the snap-back response. Furthermore, the bifurcation problem is circumvented as the scaling process triggers the lowest non-symmetric equilibrium path. Secondly, the model is extended from an isotropic to an orthotropic format, taking into account the direction of cracking and the anisotropy of the induced damage. In this way, the model can compare with classical fixed smeared crack models. This improvement allows for studying reinforced concrete structures, in which compressive struts develop parallel to the crack directions. Finally, the effectiveness of the enhanced model is proven considering a long-embedment tension-pull specimen and a slender reinforced beam, showing a good ability to capture both the load-displacement curve and the crack pattern.