Localisation of initially diffused cracking in softening materials is a numerically challenging task, which is difficult to accomplish with conventional iterative methods. The sequentially linear approach has been used to overcome convergence problems. However, the use of such a total approach raises some difficulties, namely can not be treated in a straightforward way and significant load decays may occur. In this manuscript, new non-iterative numerical methods are proposed to overcome the difficulties mentioned above: (i) an improved total non-proportional analysis and (ii) a purely incremental approach. Similar to the Sequentially Linear Approach, the evolution of the material response is based on the control of the damage state. A mixed-mode discrete damage model is used to exemplify the implementation of these methods. Finally, examples are presented to illustrate the improvements obtained with these new techniques.

In this manuscript, mixed-mode fracture is studied. Conversely to previous plasticity based formulations, the purpose of this work is to derive discrete damage models to simulate the evolution of fracture under both normal and shear tractions. First, an energy based model is used. Next, deformation-based models are adopted, both with isotropic and non-isotropic damage evolution laws. Damage is usually considered as a deformation driven process. However, fracture criteria, such as crack initiation and crack evolution, are typically defined in the stress or traction space. This is why a new, more refined model is also introduced, in which damage evolution is traction-based. Several special cases are studied, such as: homotetic damage evolution, isotropic damage evolution and a general mixed-mode evolution law. Compressive tractions are also dealt with, namely under Mode-II fracture. In all cases, as a direct consequence of the damage approach, both the total/secant constitutive relation and the corresponding incremental/tangent stiffness are derived. Some elementary numerical results are obtained and discussed.